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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities From the viewpoint of integrable systems on algebraic curves, we discuss linearization of birational maps arising from the seed mutations of types $A_1^{(1)}$ and $A_2^{(2)}$, which enables us to construct the set of all cluster variables generating the corresponding cluster algebras. These birational maps induce discrete integrable systems on algebraic curves referred to as the types of the seed mutations from which they are arising. The invariant curve of type $A_1^{(1)}$ is a conic, while the one of type $A_2^{(2)}$ is a singular quartic curve. By applying the blowing-up of the singular quartic curve, the discrete integrable system of type $A_2^{(2)}$ on the singular curve is transformed into the one on the conic, the invariant curve of type $A_1^{(1)}$. We show that both the discrete integrable systems of types $A_1^{(1)}$ and $A_2^{(2)}$ commute with each other on the conic, the common invariant curve. We moreover show that these integrable systems are simultaneously linearized by means of the conserved quantities and their general solutions are obtained. By using the general solutions, we construct the sets of all cluster variables generating the cluster algebras of types $A_1^{(1)}$ and $A_2^{(2)}$.
{\par\copyright 2019 American Institute of Physics} Cluster algebras, Relationships between algebraic curves and integrable systems, Rational and birational maps Generators of rank 2 cluster algebras of affine types via linearization of seed mutations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(N\) be a normal subgroup of a finite group \(G\) and \(V\) a fixed finite-dimensional \(G\)-module. The Poincaré series for the multiplicities of induced modules and restriction modules in the tensor algebra \(T(V) = \bigoplus_{k \geq 0} V^{\otimes k}\) are studied in connection with the McKay-Slodowy correspondence. In particular, it is shown that the closed formulas for the Poincaré series associated with the distinguished pairs of subgroups of \(\text{SU}_2\) give rise to the exponents of all untwisted and twisted affine Lie algebras except \(\mathrm{A}_{2n}^{(1)}\). Poincaré series; tensor algebras; invariants; McKay-Slodowy correspondence McKay correspondence, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Group rings of finite groups and their modules (group-theoretic aspects) Poincaré series, exponents of affine Lie algebras, and McKay-Slodowy correspondence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R_{3,n}\) denote the ring of ternary forms of degree \(n\) over the complex field \(\mathbb{C}\). Consider the action of \(\operatorname{SL} _3(\mathbb{C})\) on the ring \(R_{3,n}\). Explicit generators are known for \(n \leq 4\). While the cases \(n \leq 3\) are classically known, the case of \(n = 4\) was shown by \textit{J. Dixmier} [Adv. Math. 64, 279--304 (1987; Zbl 0668.14006)] and by Ohno (unpublished, see also \textit{A.-S. Elsenhans} [J. Symb. Comput. 68, Part 2, 109--115 (2015; Zbl 1360.13017)] and \textit{M. Girard} and \textit{D. R. Kohel} [Lect. Notes Comput. Sci. 4076, 346--360 (2006; Zbl 1143.14304)]). By the work of these authors it follows that the ring \(\mathbb{C}[R_{3,4}]^{\operatorname{SL} _3(\mathbb{C})}\) is generated by 13 elements, the so-called Dixmier-Ohno invariants of ternary quartics. The main result of the present paper is an explicit method that, given a generic tuple of Dixmier-Ohno invariants, reconstructs a corresponding plane quartic curve. The main technical tool is a method of \textit{J.-F. Mestre} [Prog. Math. 94, 313--334 (1991; Zbl 0752.14027)], see also the authors in [Open Book Ser. 1, 463--486 (2013; Zbl 1344.11049)]. A Magma package of the authors for reconstructing plane quartics from Dixmier-Ohno invariants is available under \url{https://github.com/JRSijsling/quartic\_reconstruction/}. plane quartic curves; invariant theory; Dixmier-Ohno invariants; moduli spaces; reconstruction Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Families, moduli of curves (algebraic), Arithmetic ground fields for curves Reconstructing plane quartics from their invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \textit{R} be a regular local ring. Let \textbf{G} be a reductive group scheme over \textit{R}. A well-known conjecture due to Grothendieck and Serre assertes that a principal \textbf{G}-bundle over \textit{R} is trivial, if it is trivial over the fraction field of \textit{R}. In other words, if \textit{K} is the fraction field of \textit{R}, then the map of non-abelian cohomology pointed sets
\[
H_{ét}^1(R,\mathbf{G})\rightarrow H_{ét}^1(K,\mathbf{G}),
\]
induced by the inclusion of \textit{R} into \textit{K}, has a trivial kernel. \textit{The conjecture is solved in positive for all regular local rings contaning a field}. More precisely, if the ring \textit{R} contains an infinite field, then this conjecture is proved in a joint paper due to \textit{R. Fedorov} and \textit{I. Panin} [Publ. Math., Inst. Hautes Étud. Sci. 122, 169--193 (2015; Zbl 1330.14077)]. If the ring R contains a finite field, then this conjecture is proved in 2015 in a preprint due to \textit{I. Panin} [``Proof of Grothendieck-Serre conjecture on principal bundles over regular local rings containing a finite field'', Preprint, \url{https://www.math.uni-bielefeld.de/LAG/man/559.pdf}] which can be found on preprint server Linear Algebraic Groups and Related Structures. A more structured exposition can be found in \textit{I. Panin}'s preprint of the year 2017 [``Proof of Grothendieck-Serre conjecture on principal bundles over regular local rings containing a finite field'', Preprint, \url{arXiv:1707.01767}].
This and other results concerning the conjecture are discussed in the present paper. We illustrate the exposition by many interesting examples. We begin with couple results for complex algebraic varieties and develop the exposition step by step to its full generality. linear algebraic groups; principal bundles; affine algebraic varieties Linear algebraic groups over arbitrary fields, Linear algebraic groups over adèles and other rings and schemes, Cohomology theory for linear algebraic groups, Group schemes On Grothendieck-Serre conjecture concerning principal bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Authors' abstract: Let \(G\) be a complex connected reductive group which is defined over \(\mathbb{R}\), let \(\mathfrak{G}\) be its Lie algebra, and let \(\mathcal{T}\) be the variety of maximal tori of \(G\). For \(\xi \in \mathfrak{G} (\mathbb{R})\), let \(\mathcal{T}_\xi\) be the variety of tori in \(\mathcal{T}\) whose Lie algebra is orthogonal to \(\xi\) with respect to the Killing form. We show, using the Fourier-Sato transform of conical sheaves on real vector bundles, that the weighted Euler characteristic of \(\mathcal{T}_\xi (\mathbb{R})\) is zero unless \(\xi\) is nilpotent, in which case it equals \((-1)^{\frac{\dim\mathcal{T}}{2}}\). Here `weighted Euler characteristic' means the sum of the Euler characteristics of the connected components, each weighted by a sign \(\pm 1\) which depends on the real structure of the tori in the relevant component. This is a real analogue of a result over finite fields which is connected with the Steinberg representation of a reductive group. maximal torus; weighted Euler characteristic Topology of real algebraic varieties, General properties and structure of real Lie groups, Homology and cohomology of homogeneous spaces of Lie groups, Linear algebraic groups over finite fields Euler characteristics of the real points of certain varieties of algebraic tori | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a compact Riemann surface of genus \(g\), and denote by \({\mathcal U}_ X (d)\) the moduli space of semistable rank-2 vector bundles of degree \(d\). On \({\mathcal U}_ X (d)\) there is a natural polarizing line bundle \(\theta_ X\) which generalizes the line bundle associated with the Riemann theta divisor on the Jacobian of \(X\). For any positive integer \(k\), the space \(H^ 0({\mathcal U}_ X (d), \theta_ X^{\otimes k})\) is called the space of generalized theta functions of order \(k\) on \({\mathcal U}_ X (d)\). These spaces correspond to certain objects arising in conformal quantum field theory, namely to the so-called conformal blocks of level \(k\), and the arguments of physicists suggest that the spaces \(H^ 0({\mathcal U}_ X (d), \theta_ X^{ \otimes k})\) should be related to the spaces of generalized order-\(k\) theta functions for suitable curves \(Y\) of (the lower) genus \(g - 1\). In physics, such a relationship is known under the name ``factorization rule'' (or ``glueing axiom'', respectively). The aim of the present paper is to establish such a relationship by rigorous algebro-geometric methods, i.e., to give an explicit geometric counterpart of that factorization principle in conformal quantum field theory. To this end, the authors consider degenerations of the given curve \(X\) into irreducible curves \(Y\) which are smooth except for a single node. Then the normalization \(\widetilde Y\) of \(Y\) is smooth of genus \(g - 1\). The authors' main theorem states that, for such a curve \(Y\), the inequality \(g \geq 4\) implies \(H^ 1 ({\mathcal U}_ Y(d), \theta_ Y^{\otimes k}) = H^ 1 ({\mathcal U}_ X (d), \theta_ X^{\otimes k}) = 0\) which, in turn, means that the spaces of generalized theta functions on \({\mathcal U}_ X (d)\) and \({\mathcal U}_ Y (d)\) of order \(k\) are isomorphic.
The second part of the main theorem establishes a ``factorization rule'' by constructing an isomorphism \(H^ 0 ({\mathcal U}_ Y (d), \theta_ Y^{\otimes k}) \cong \bigoplus_ j H^ 0({\mathcal U}_{\widetilde Y}^ j (d), \theta_ j)\), where \(j\) runs through a certain index domain depending on \(k\), \({\mathcal U}^ j_{\widetilde Y} (d)\) denotes the moduli space of semistable rank-2 vector bundles of degree \(d\) on \(\widetilde Y\) with a certain parabolic structure [in the sense of \textit{V. B. Mehta} and \textit{C. S. Seshadri}, Math. Ann. 248, 205-239 (1980; Zbl 0454.14006)], also depending on \(k\), and \(\theta_ j\) is the natural polarizing line bundle (generalized theta bundle) on \({\mathcal U}^ j_{\widetilde Y} (d)\).
The proof of this important relation confirming the physicists' arguments is rather involved and delicate, since the authors are dealing with nodal curves, for which the moduli spaces of parabolic vector bundles have not been constructed yet. Therefore a great part of the paper is devoted to establishing the existence of moduli spaces of semistable torsion-free sheaves of rank 2 on nodal curves and their properties, including the analysis of their generalized theta bundles. The authors announce that, in a subsequent work, they will remove the restrictional condition \(g \geq 4\) for the vanishing of \(H^ 1({\mathcal U}_ Y (d), \theta_ Y^{\otimes k})\). Then their results can be used to verify the Verlinde formula for the dimension of the spaces of generalized order-\(k\) theta functions on \({\mathcal U}_ X (d)\), which would give another approach in this case, different from the one made by \textit{A. Bertram} [Invent. Math. 113, No. 2, 351-372 (1993)]. factorization rule; glueing axiom; Riemann surface; moduli space; theta divisor on the Jacobian; conformal blocks; nodal curves; theta bundles; Verlinde formula Narasimhan M.S., Ramadas T.R.: Factorization of generalized theta functions. I. Invent. Math. 114, 565--623 (1993) Theta functions and curves; Schottky problem, Quantum field theory on curved space or space-time backgrounds, Vector bundles on curves and their moduli, Fine and coarse moduli spaces, Families, moduli of curves (algebraic), Theta functions and abelian varieties Factorisation of generalised theta functions. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Tate conjecture claims that if two isomorphic two-dimensional Galois representations occur in the étale cohomology of two varieties defined over \(\mathbb Q\), then there should be an algebraic correspondence defined over \(\mathbb Q\) between the two varieties which induces an isomorphism of the two Galois representations. An example confirming the Tate conjecture for rigid Calabi-Yau threefolds over \(\mathbb Q\) was first given in the paper of \textit{M.-H. Saito} and \textit{N. Yui} [J. Math. Kyoto Univ. 41, No. 2, 403--419 (2001; Zbl 1077.14546)].
The paper under review constructs several examples of rigid Calabi-Yau threefolds over \(\mathbb Q\) and prove the modularity of the two-dimensional Galois representations by means of establishing the Tate conjecture. These rigid Calabi-Yau threefolds are associated to the modular form of weight \(4\) and level \(8\), \(\eta(2\tau)^4\eta(4\tau)^4\). (Here \(\eta(\tau)\) is the Dedekind eta function.) Now twisting the modular form by Legendre symbols, one obtains modular forms of different levels, e.g., \(16, 64, 72, 144, 200, 392, 400\). Examples of rigid Calabi-Yau threefolds over \(\mathbb Q\) with \(h^{1,1}\in\{70, 50, 46, 44, 40, 36, 32, 28, 16\}\) are listed. All but three examples are associated to the self-fiber product of the modular curve for \(\Gamma_1(4)\cap\Gamma(2)\), and that for \(\Gamma_0(8)\cap\Gamma_1(4)\). The modularity is proved by exhibiting explicit algebraic correspondences thereby establishing the Tate conjecture for these examples. modular rigid Calabi-Yau threefolds; Tate's conjecture; modular forms of weight 4 and level 8 Cynk S., Meyer C.: Modular Calabi--Yau threefolds of level eight. Int. J. Math. 18(3), 331--347 (2007) Calabi-Yau manifolds (algebro-geometric aspects), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Modular Calabi-Yau threefolds of level eight | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present paper is a continuation of [the authors, Contemp. Math. 593, 241--261 (2013; Zbl 1297.14032)]. The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most \(n\) simple poles on genus \(g\) complex algebraic curves. This generalizes the previous results obtained by the authors on moduli spaces of holomorphic quadratic differentials. The paper is organized as follows : Section 1 is an introduction to the subject and statement of results. In Section 2 the authors introduce a twofold canonical cover corresponding to a pair consisting of a complex algebraic curve and a quadratic differential on it with \(n\) simple poles. The main objective of this section is to discuss the action of the covering involution on (co)homology of the cover and the associated matrix of \(b\)-periods. In Section 3 they define two tau functions corresponding to the eigenvalues \(\pm1\) of the covering map, discuss their basic properties and interpret them as holomorphic sections of line bundles on the moduli space of quadratic differentials with simple poles. In Section 4 they study the asymptotic behavior of the tau functions near a divisor and use it to express the Hodge and Prym classes via the classes of the boundary divisors and the tautological class. quadratic differentials; tau function; moduli spaces Families, moduli of curves (analytic), Relationships between algebraic curves and integrable systems, Analytic theory of abelian varieties; abelian integrals and differentials, Differentials on Riemann surfaces Tau function and moduli of meromorphic quadratic differentials | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article is the first one of a series of papers devoted to the foundations of algebraic geometry in homotopical and higher categorical contexts. The authors first define the notion of an \textit{S-topology} \(\tau\) on a simplicially enriched category (an \textit{S-category}) \(T\). They show how each S-topology \(\tau\) yields a model structure on the category of simplicial presheaves on \(T\); the resulting model category is denoted by \(\roman{SPr}_\tau(T)\). This model category is the homotopical equivalent to the category of ordinary sheaves over an ordinary site.
The first main result of the article gives then an analogue of the usual adjoint pair of sheafification and the forgetful functor between the category of presheaves and sheaves on a site to this homotopical context: the authors construct an adjoint pair between the homotopy category of simplicial presheaves and the homotopy category of \(\roman{SPr}_\tau\), the homotopy category of \textit{stacks over \((T, \tau)\)}. The image of the forgetful functor is described by a hyperdescent condition. The next main result shows which model categories are Quillen equivalent to model categories of the form \(\roman{SPr}_\tau(T)\) for fixed \(T\). It follows that \(\roman{SPr}_\tau(T)\) defines the S-topology \(\tau\) on \(T\) uniquely. This is a direct generalisation of the classical result about the reconstruction of a topology from a Grothendieck topos of sheaves on a category. Instead of starting with an S-category \(T\), one could have also started with a model category \(M\). The authors define the notion of a \textit{model pre-topology \(\tau\) on \(M\)}. Using this they define a model category \(M^{\sim, \tau}\) of \textit{stacks over \((M, \tau)\)}, which is the analogue of \(\roman{SPr}_\tau(T)\) defined earlier. They prove that the simplicial localisation \(LM\) of \(M\) carries a natural \(S\)-topology \(\tau\) and that the categories \(M^{\sim, \tau}\) and \(\roman{SPr}_\tau(LM)\) are naturally Quillen equivalent. This means that the two approaches --- stacks over a model site and stacks over an S-site --- are equivalent in a certain sense. For the approach with \textit{model sites \((M, \tau)\)}, the authors prove a homotopical analogue of Giraud's theorem thereby completely identifying the combinatorial model categories that are of the form \(M^{\sim, \tau}\).
Finally, the authors apply their ideas to give a solution of the problem of defining a notion of an étale \(K\)-theory of a commutative \(\mathbb S\)-algebra, i.e.\ of a commutative monoid in Elmendorf-Kriz-Mandell-May's category of \(\mathbb S\)-modules. Further applications of the general theory to algebraic geometry (e.g.\ derived moduli spaces) are not covered in this article. The \(\infty\)-topoi of \textit{J.~Lurie} [\texttt{math.CT/0306109}] may be seen complementary to this article's approach to model topoi. stacks; topoi; higher categories; simplicial categories; model categories; étale \(K\)-theory B.~Toën, G.~Vezzosi, Homotopical algebraic geometry. I. Topos theory, Adv. Math. 193 (2005), no. 2, 257--372. DOI 10.1016/j.aim.2004.05.004; zbl 1120.14012; MR2137288; arxiv math/0207028 Étale and other Grothendieck topologies and (co)homologies, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Nonabelian homotopical algebra, Generalizations (algebraic spaces, stacks), Grothendieck topologies and Grothendieck topoi, Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.), Topological categories, foundations of homotopy theory Homotopical algebraic geometry. I: Topos theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the first of two papers in which the authors intend to give concrete interpretations of the building of a reductive group G over a local field K and of the group scheme \(G_ x\) attached to a point x of that building [cf. Publ. Math., Inst. Hautes Étud. Sci. 41, 5-251 (1972; Zbl 0254.14017) and 60, 5-184 (1984)] in the special case where G is a classical group. Here, ''concrete'' means expressed in terms of the natural representation. The present paper deals with the (general and special) linear groups over a division ring D endowed with a discrete valuation and for finite rank over its center K. Unitary groups will be considered in a forthcoming part 2.
Section 1 presents more or less classical results on ''splittable'' norms (''normes scindables'', i.e., in the classical terminology, norms admitting an orthogonal basis) on \(D^ n\), and on the order \(M_{\alpha}\) in \(M_ n(D)\) associated to such a norm \(\alpha\) ; here, D is not assumed to be complete. - Section 2 shows that the ''enlarged building'' (''immeuble élargi'') \({\mathcal I}\) of \(GL_ n(D)\) can be canonically identified with the set \({\mathcal N}\) of splittable norms and that this provides an isomorphism of the abstract simplicial complex of all facets of \({\mathcal I}\) with the set of all hereditary orders in \(M_ n(D)\), made into a simplicial complex by the inclusion relation. - To every \(x\in {\mathcal I}={\mathcal N}\), section 3 associates a smooth and connected group scheme \({\mathfrak G}_ x\) with generic fiber G, namely the multiplicative group scheme of \(M_ x\); it has a ''big cell'', and its group of integral points is the stabilizer of x in \(GL_ n(D)\). By section 4 all that behaves well under étale Galois extension. It follows that if K is Henselian and D is split by an étale extension of K, then the schemes \({\mathfrak G}_ x\) are those of the general theory of the cited papers. This provides an easy proof of a theorem of \textit{V. P. Platonov} and \textit{V. I. Jančevskij} [Soviet Math., Doklady 16, 424-427 (1975); translation from Dokl. Akad. Nak SSSR 221, 784-787 (1975; Zbl 0333.20035)]. - Section 5 deals with \(SL_ n(D)\); the results are similar except that the schemes are no longer always smooth if D does not split over an etable extension of K. Most results of sections 1 and 2 are extended to non- discrete valuations in an appendix. classical groups; building of a reductive group over a local field; group scheme F. \textsc{Bruhat} and J. \textsc{Tits}, Schémas en groupes et immeubles des groupes classiques sur un corps local, \textit{Bull. Soc. Math. Fr.}, \textbf{112} (1984), 259-301. Group schemes, Classical groups (algebro-geometric aspects), Linear algebraic groups over local fields and their integers, Division rings and semisimple Artin rings Schémas en groupes et immeubles des groupes classiques sur un corps local | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The resolution of singularities of algebraic varieties defined over a field of characteristic zero by a sequence of blow-ups is a famous result proved by \textit{H. Hironaka} [Ann. Math. (2) 79, 109-203, 205-326 (1964; Zbl 0122.38603)], which has a lot of applications. Probably, relatively few mathematicians read the 200 page proof. This can change now. The article is arranged in a similar way to a talk in a colloquium (25\% should be understood by everyone, 25\% is for people who are interested, the next 25\% is for the specialists and the rest only the speaker will understand) with one difference: that also the rest is understandable. It starts with an overview explaining the result and giving rough ideas of the proof. The author suggests that very busy people should only read this.
The next chapter gives an introduction to the main problems (choice of the centre of the blow-up, equiconstant points, improvement of singularities under blow-up including many examples). This is for the next 25.
The next chapter (constructions and proofs) gives the technical details. It contains also several examples and a section on problems in positive characteristic. In an appendix, necessary basic facts from commutative algebra and the theory of blow-ups used in the previous chapters are collected.
It is advisable for everybody interested in resolution of singularities to read this article. Hironaka resolution of singularities; blowing up H.HAUSER,\textit{The Hironaka theorem on resolution of singularities (or: A proof we always wanted to} \textit{understand)}, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 323--403.http://dx.doi.org/ 10.1090/S0273-0979-03-00982-0.MR1978567 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Local complex singularities, Invariants of analytic local rings, Modifications; resolution of singularities (complex-analytic aspects) The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The goal of this paper is to study minimal model of dlt pairs of numerical Kodaira dimension zero. The main theorem is the following: let \((X,\Delta)\) be a projective \(\mathbb{Q}\)-factorial dlt pair such that \(\nu(K_{X}+\Delta)=0\), then there exists a minimal model of \((X,\Delta)\).
This result has been proved for klt pairs by \textit{S. Druel} [Math. Z. 267, No. 1, 413--423 (2011; Zbl 1216.14007)] using similar methods. The proof is based on results in [\textit{C. Birkar, P. Cascini, C. D. Hacon} and \textit{J. McKernan}, J. Am. Math. Soc. 23, No. 2, 405--468 (2010; Zbl 1210.14019)], especially on the termination of flips with scaling and the divisorial Zariski decomposition.
The generalization to dlt pairs is important since it allows the author to prove a particular case of the abundance conjecture. Let \(X\) be a normal projective variety and let \(\Delta\) be an effective \(\mathbb{Q}\)-divisor. Suppose that \((X,\Delta)\) is a log canonical pair such that \(\nu(K_{X}+\Delta)=0\), then \(\kappa(K_{X}+\Delta)=0\). Particular cases of this theorem were known before, for example Nakayama proved the case \((X,\Delta)\) is a klt pair, see [\textit{N. Nakayama}, Zariski-decomposition and abundance. MSJ Memoirs 14. Tokyo: Mathematical Society of Japan (2004; Zbl 1061.14018)]. See also [\textit{Y. Kawamata}, ``On the abundance theorem in the case \(v=0\)'', \url{arXiv:1002.2682}].
Remarkably the previous theorem is independent of Simpson's results. minimal model; abundance conjecture; numerical Kodaira dimension Gongyo, Y., On the minimal model theory for DLT pairs of numerical log Kodaira dimension zero, Math. Res. Lett., 18, 991-1000, (2011) Minimal model program (Mori theory, extremal rays), Divisors, linear systems, invertible sheaves On the minimal model theory for DLT pairs of numerical log Kodaira dimension zero | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a reductive complex Lie group and \(\theta\) an involutive automorphism. Then the pair \((G,\theta)\) is called a ``reductive symmetric pair''. The eigenspaces of the induced involution \(\theta'\) of the Lie algebra yield the ``Cartan decomposition'' \(g=k\oplus p\). An ``anisotropic torus'' is an abelian Lie subalgebra whose elements are semisimple and contained in \(p\). A ``generic reduction'' is a maximal anisotropic torus. The generic reductions may be regarded as points in the respective Grassmann variety. The ``variety of reduction'' is the closure of the set of these points in the Grassmann variety.
This article is devoted to a systematic study of such ``varieties of reduction'' and related questions in representation theory. A useful result obtained in this way is the rigidity of semisimple elements in deformations of algebraic subalgebras of Lie algebras.
A key motivation for this study is the quest for Fano varieties (projective varieties with ample anticanonical bundle). For example, a smooth projective compactification of \({\mathbb C}^n\) with \(b_2=1\) is necessarily Fano. Varieties of reduction often provide interesting examples of Fano varieties. This generalizes previous work of \textit{A. Iliev} and \textit{L. Manivel} [J. Reine Angew. Math. 585, 93--139 (2005; Zbl 1083.14060)]. variety of reduction; Fano variety; Cartan decomposition; symmetric space Barbier Grünewald, M, The variety of reductions for a reductive symmetric pair, Transform. Groups, 16, 1-26, (2011) Grassmannians, Schubert varieties, flag manifolds, Fano varieties The variety of reductions for a reductive symmetric pair | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0745.00066.]
Let \(k\) be a field of characteristic zero and \({\mathcal O} = k[[x]]\) the ring of formal power series in one variable. This paper is concerned with simple modules over the ring of differential operators \({\mathcal D} = {\mathcal D}({\mathcal O}) = k[[x]][\partial]\). In this one-dimensional case the simple modules are holonomic. Those with regular singularities are \(\mathcal O\), \(K/ {\mathcal O}\), and \(Kx^ \alpha\), for \(\alpha \in k \setminus \mathbb{Z}\) (where \(K\) denotes the field of fractions of \(\mathcal O\)). The main effort of this paper is to describe the simple \(\mathcal D\)-modules with irregular singularities. The authors achieve this by a rather elegant descent procedure. These modules are finite-dimensional \(K\)-vector spaces. A result of the second author states that \(\partial\) has an eigenvector over a finite field extension of \(K\). The authors use this to give an extremely explicit description of the simple \(\mathcal D\)-modules. For each finite-dimensional \(K\)-vector space \(M\) they describe all the actions of \(\partial\) on \(M\) that make \(M\) into a simple \(\mathcal D\)-module with irregular singularities. They also extend this classification to \({\mathcal D}(A)\), the ring of differential operators on a subalgebra \(A\) of \(\mathcal O\) with finite codimension, using Smith and Stafford's Morita equivalence between \({\mathcal D}(A)\) and \(\mathcal D\). holonomic modules; ring of formal power series; simple modules; ring of differential operators; regular singularities; irregular singularities; eigenvector; finite field extension; actions; Morita equivalence Den Essen, A. Van; Levelt, A.: An explicit description of all simple K[[X]][\partial]-modules. (1992) Rings of differential operators (associative algebraic aspects), Simple and semisimple modules, primitive rings and ideals in associative algebras, Commutative rings of differential operators and their modules, Valuations, completions, formal power series and related constructions (associative rings and algebras), Singularities of curves, local rings An explicit description of all simple \(k[[x]][\partial]\)-modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Consider a finite-dimensional associative \(K\)-algebra with unity, where \(K\) is a field of characteristic \(\neq 2\). If \(i\) is an involution of \(A\) over \(K\), then \(\{a\in A^*:a^ia=1\}\) is an algebraic group. If \(G\) denotes its connected component at the identity, then its Lie algebra is seen to be
\[
\text{Lie}(G)=\{a\in A:a^i+a=0\}.
\]
As \(\text{char\,}K\neq 2\), the map \(a\mapsto (1-a)(1+a)^{-1}\) defines a \(G\)-equivariant birational map \(\lambda\) from \(G\) to \(\text{Lie}(G)\). Its inverse from \(\text{Lie}(G)\) to \(G\) is also \(b\mapsto (1-b)(1+b)^{-1}\). Therefore, we have a \(K\)-birational isomorphism between \(G\) and \(\text{Lie}(G)\); such a map is said to be a Cayley \(K\)-map. Indeed, the first example of this is the `Cayley transform' due to Cayley in 1846; this is the case \(A=M_n(K)\) and \(i\colon M\mapsto M^t\).
In general, a linear algebraic group \(G\) over a field \(K\) is called a Cayley \(K\)-group if it has a Cayley \(K\)-map; viz., a \(K\)-birational \(G\)-equivariant (under the conjugation and adjoint actions) isomorphism between \(G\) and \(\text{Lie}(G)\). In an important paper [``Cayley groups'', J. Am. Math. Soc. 19, No. 4, 921-967 (2006; Zbl 1103.14026)] \textit{N. Lemire, V. L. Popov} and \textit{Z. Reichstein} studied this notion for algebraic groups over algebraically closed fields \(K\) of characteristic \(0\), raised many questions and answered some of them. In particular, they showed all connected solvable groups are Cayley. Also, over an algebraically closed field of characteristic \(0\), they proved that the groups \(\text{SL}_2\) and \(\text{SL}_3\) are Cayley (the latter was contrary to expectations) while \(\text{SL}_n\) for \(n\geq 4\) is not Cayley.
A weaker notion is that of being stably Cayley. If \(G\times_K\mathbb G_{m,K}^r\) is a Cayley \(K\)-group (where \(G_{m,K}^r\) is a \(K\)-split torus), one calls \(G\) a stably Cayley \(K\)-group. Observe that Cayley groups are automatically stably Cayley but, the converse is a difficult problem which is not completely settled yet. In their paper cited above, Lemire, Popov \& Reichstein classified Cayley and stably Cayley groups which are simple over an algebraically closed field \(K\) of characteristic \(0\). By the works of Borovoi, Iskovskikh, Kunyavskii, Lemire and Reichstein, as of now, a complete classification of stably Cayley \(K\)-groups which are semisimple, is known where \(K\) is any field of characteristic \(0\). Further, it is known that all reductive \(K\)-groups of absolute rank \(\leq 2\) (where \(\text{char\,}K=0\)) are stably Cayley.
In this paper, the case \(K=\mathbb R\) is considered, and the results show which -- among these (stably Cayley) reductive groups of rank \(\leq 2\) -- are Cayley and which are not.
The main result is: Let \(G\) be a connected, reductive \(\mathbb R\)-group of absolute rank \(\leq 2\). If \(G\) is simple of type \(G_2\), or is isomorphic to \(\text{SL}_3\), or is \(\text{PGU}_3\), or \(\text{PGU}(2,1)\), then \(G\) is NOT Cayley. In the rest of the cases, \(G\) is Cayley.
The proof is done case by case. The cases where \(G\) is Cayley (other than the case of \(\text{SU}_3\)) are proved in the main body of the paper. In appendix A, I. Dolgachev proves the Cayley-ness of \(\text{SU}_3\) and the non-Cayley-ness in the cases other than \(G_2\) which is already known from Lemire, Popov and Reichstein's work. In appendix B, some interesting remarks by a referee are given which deal with \(K\) of positive characterictic and \(G=\text{PGL}_1(A)\) where \(A\) is a central simple algebra over \(K\) whose degree is a multiple of the characteristic.
In the paper, the following remarkable result refining stable Cayley-ness is proved as well: Let \(G\) be a connected, reductive \(K\)-group of absolute rank \(\leq 2\) where \(\text{char\,}K=0\). If \(G\) has rank \(1\), then it is Cayley. If \(G\) has rank \(2\), then \(G\times_K\mathbb G^2_{m,K}\) is Cayley. linear algebraic groups; Cayley groups; Cayley maps; reductive algebraic groups; algebraic surfaces; equivariant birational isomorphisms Linear algebraic groups over the reals, the complexes, the quaternions, Linear algebraic groups over arbitrary fields, Rational and birational maps, Lie algebras of linear algebraic groups Real reductive Cayley groups of rank 1 and 2. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The treatise under review provides a fairly comprehensive account of the classical theory of hyperelliptic Kleinian functions, together with some new applications towards the description of hyperelliptic Jacobians and their Kummer varieties as well as to families of matrix differential operators and their associated soliton equations. The text consists of six chapters, one appendix, and a rich bibliography referring to the classical and to the contemporary literature on abelian functions and their use in solving various classes of nonlinear differential equations of KdV-type.
Chapter I presents a constructive approach to F. Klein's hyperelliptic sigma functions [cf. \textit{F. Klein}, Math. Ann. 32, 351--380 (1888; JFM 20.0491.01)], which is based on the modern concept of the moduli space of \(n\)-dimensional principally polarized abelian varieties and the universal bundle over it. In this framework, Klein's classical sigma functions appear as automorphic functions on the moduli subspace of Jacobians of plane hyperelliptic curves, and their properties are discussed, in greater detail, in chapter II. Chapter III compiles some background material for the theory of Kleinian functions, most of which is thoroughly treated in \textit{H. F. Baker}'s brilliant classic ``Abel's theorem and the allied theory of theta functions'' (Cambridge Univ. Press 1897; reprint 1995; Zbl 0848.14012)].
Chapter IV is devoted to the basic relations between the various Kleinian functions and their derivatives. These relations are then applied to the modern theory of integrable Hamiltonian systems, in particular to the construction of explicit solutions of the Korteweg--de Vries system (KdV) in terms of Kleinian functions on moduli spaces of hyperelliptic Jacobians, and to the problem of constructing families of matrix differential operators satisfying certain curvature conditions. Chapter V gives a more subtle analysis of the fundamental cubic and quartic relations for Kleinian functions derived in chapter IV. These deeper results are shown to lead to explicit matrix realizations of hyperelliptic Jacobians and their associated Kummer varieties, on the one hand, and to the description of a certain dynamical system defined on the universal space of Jacobians of hyperelliptic curves of given genus \(g\). Also, this construction allows to establish systems of linear differential operators for which the hyperelliptic base curve is their common spectral variety. -- In chapter VI, the authors study abelian Bloch functions, as solutions of Schrödinger equations, which are expressible in terms of Kleinian functions of genus 2. This investigation culminates in a new addition theorem for the Akhiezer-Baker function on the Jacobian of a genus-2 curve. In addition, spectral problems for singular Kummer surfaces and Humbert surfaces are discussed in this context.
The appendix to the paper gives a brief survey on some very recently discovered addition theorems for hyperelliptic Kleinian functions, which are due to the authors and will be discussed, in greater detail, in a forthcoming paper. Altogether, the comprehensive article under review is based on recent results of the authors, which were partially announced in several short notes, contributions to conference proceedings, and preprints published during the past four years, and which are presented here in a thorough, detailed and coherent fashion. The paper contains a wealth of classical and new aspects of the theory of abelian functions and their applications to differential equations of mathematical physics. hyperelliptic Kleinian functions; Kummer varieties; soliton equations; Klein's hyperelliptic sigma functions; JFM 20.0491.01; matrix realizations of hyperelliptic Jacobians Buchstaber, VM; Enolskiĭ, VZ; Leĭkin, DV, Kleinian functions, hyperelliptic Jacobians and applications, Rev. Math. Math. Phys., 10, 1-103, (1997) Analytic theory of abelian varieties; abelian integrals and differentials, Theta functions and curves; Schottky problem, KdV equations (Korteweg-de Vries equations), Jacobians, Prym varieties, Theta functions and abelian varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Kleinian functions, hyperelliptic Jacobians and applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove that the space of coinvariants of functions on an affine variety by a Lie algebra of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem include Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves under Hamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities under the flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preserving groups under the incompressible vector fields, and arbitrary varieties with isolated singularities under the flow of all vector fields. We compute this quotient explicitly in many of these cases. The proofs involve constructing a natural \(\mathcal{D}\)-module representing the invariants under the flow of the vector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicity we study in more detail). We give many counter-examples to naive generalizations of our results. These examples have been a source of motivation for us. vector fields; Lie algebras; \(\mathcal{D}\)-modules; Poisson homology; Poisson varieties; Calabi-Yau varieties; Jacobi varieties Etingof, P., Schedler, T.: Coinvariants of Lie algebras of vector fields on algebraic varieties (2012). arXiv:1211.1883\textbf{(to appear in Asian J.~Math.)} Lie algebras of vector fields and related (super) algebras, Poisson algebras, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Calabi-Yau manifolds (algebro-geometric aspects), Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Co-invariants of Lie algebras of vector fields on algebraic varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a complex Lie group with a closed subgroup \(H\). A Cartan geometry of type \(G/H\) on a compact Kähler manifold \(M\) is a holomorphic \(H\)-bundle over \(M\) with a holomorphic \(\mathfrak{g}\)-valued \(1\)-form \(\theta\) on it satisfying certain conditions. The notion is modeled on the quotient bundle \(G\to G/H\) with \(\theta=g^{-1}dg\) known as the tautological Cartan geometry.
McKay conjectured recently that the only Calabi-Yau manifolds admitting a Cartan geometry are those étale covered by a complex torus. The main result of this note is a proof of this conjecture via the Bogomolov inequality for semistable sheaves. In addition, the authors show that a Cartan geometry on a projective and rationally connected \(M\) is holomorphically isomorphic to the tautological one. Cartan geometry; Calabi-Yau manifold; Bogomolov inequality; semistable sheaves; rationally connected Biswas, I; McKay, B, Holomorphic Cartan geometries, Calabi-Yau manifolds and rational curves, Diff. Geom. Appl., 28, 102-106, (2010) General geometric structures on manifolds (almost complex, almost product structures, etc.), Homogeneous spaces and generalizations, Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Calabi-Yau theory (complex-analytic aspects) Holomorphic Cartan geometries, Calabi-Yau manifolds and rational curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities If \(S\) is a smooth (polarized) projective variety over the complex numbers then the nonabelian Hodge theory as developed for example by Hitchin, Corlette and Simpson gives a one-to-one correspondence between the set of stable Higgs bundles with vanishing Chern classes and structure group \(G\) and the set of all irreducible representations of the fundamental group of \(S\) in \(G\). There exists a result of \textit{C. T. Simpson} [in Complex geometry and Lie theory, Proc. Sympos., Sundance 1989, Proc. Sympos. Pure Math. 53, 329-348 (1991; Zbl 0770.14014)] saying that every nonrigid Zariski dense representation of the fundamental group of such an \(S\) in the group \(\text{SL} (2, \mathbb{C})\) factors through the fundamental group of an orbicurve.
In the article under review the authors investigate the situation for other simple Lie groups \(G\). Their main theorem is that a Zariski dense representation of the fundamental group whose associated Higgs bundle is regular and semi-simple factors geometrically through an orbicurve if and only if a certain generalized Prym variety of the smooth spectral cover \(\widetilde S\) (constructed with the help of the Higgs bundle) over \(S\) does not vanish. In particular, the representation factors through a representation of a Fuchsian group generated by \(2g\) hyperbolic elements and a finite number of elliptic elements. -- To construct the orbicurve the Albanese of the spectral cover is used.
Some more tractable criteria are given. nonabelian Hodge theory; stable Higgs bundles; irreducible representations of the fundamental group [KP] Katzarkov, L., Pantev, T.: Representations of fundamental groups whose Higgs bundles are pullbacks. Preprint Homotopy theory and fundamental groups in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Transcendental methods, Hodge theory (algebro-geometric aspects), Variation of Hodge structures (algebro-geometric aspects) Representations of fundamental groups whose Higgs bundles are pullbacks | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the author connects the theory of equisingular deformations of plane curve singularities and sandwiched surface singularities. From a simultaneous embedded deformation of an equisingular family of plane curve singularities he identifies a simultaneous resolution of a family of sandwiched surface singularities. The main results are:
Theorem 1. Let \(C\) be a plane curve singularity and let \(a\in N^k\) where \(k\) is the number of analytic branches of \(C\). Let \(X=X_{(C,a)}\) and \(Y=Y_{(C,a)}\). Then there is a smooth map \(ES_C\to ES_X\) of deformation functors and
an \(a^*\in \mathbb{N}^k\) which depends only on the topological type of \(C\), such that if \(a\geq a^*\) then the map is an isomorphism.
Theorem 2. Assume \(C\) is a plane curve singularity with topological type \(\Phi\) and let \(\Gamma\) be the dual graph of \(X=X_{(C,a)}\). The there exists an \(a^*\), depending only on \(\Phi\) such that if \(a\geq a^*\) the isomorphism classes of plane curve singularities with topological type \(\Phi\) are in one to one correspondence with the isomorphism classes of (the complete local ring of) normal surface singularities with dual graph \(\Gamma\) (by the construction in Section 1). equisingularity; plane curve singularity Deformations of singularities, Singularities in algebraic geometry, Formal methods and deformations in algebraic geometry, Equisingularity (topological and analytic) Equisingular deformations of sandwiched singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main theorem of this paper supplies a very uniform presentation of the quantum cohomology ring of (co)minuscule homogeneous spaces, ranging from the usual and lagrangian grassmannians, quadrics, spinor varieties up to exceptional hermitian spaces like the Freudenthal variety or the Cayley plane. The methods are based upon the combinatorics of certain quivers. Pieri's and Giambelli's type formulas for all these kind of varieties are also known but, as the the authors remark, the techniques used in this paper do not apply, yet, to deduce them in a uniform way, all at once. This paper has a sequel, concerned with \textsl{hidden symmetries}, especially for grassmannians [Int. Math. Res. Not. 2007, No. 22, Article ID rnm107 (2007; Zbl 1142.14033)].
In spite of dealing with highly non trivial mathematics, the paper is written in a quite friendly way: for instance, the section that follows the introduction explains the basic terminology regarding minuscule and cominuscule homogeneous spaces. To help the potential reader to know in advance what is this paper about, recall that any parabolic subgroup contains a Borel subgroup \(B\) which contains a maximal torus \(T\) of \(G\). The pair \((G,T)\) defines \textsl{weights} (which are characters of \(T\) satisfying certain non triviality conditions) and a \textsl{root system}. A fundamental weight \(\omega\) is said to be \textsl{minuscule} if and only if \(|<\omega, \alpha>|\leq 0\), for each positive root \(\alpha\), where \(<,>\) is the pairing induced by the natural duality between the characters and the co-characters of \(T\). Furthermore \(\omega\) is said to be \textsl{co-minuscule} if and only if \(<\omega, \alpha^\vee>=1\), where \(\alpha\) denotes the highest root. To each such weight a parabolic subgroup \(P_\omega\) of \(G\) can be associated, and the corresponding quotient \(G/P_\omega\) is said to be a (co)minuscule homogeneous space.
One of the key remarks of the paper is based on the observation that the Gromow-Witten invariant of degree \(d\) of \(G/P\) can be seen as classical intersection numbers on certain auxiliary \(G\)-homogeneous varieties. This is well explained with details in the third section of the paper, suggestively entitled ``From classical to quantum invariants''. Section 4 is devoted to the quantum Chevalley formula (concerning the intersection of any Schubert cycle with a codimension \(1\) Schubert cycle) and higher Poincaré duality. In this section the problem, already studied by \textit{W. Fulton} and \textit{C. Woodward} for grassmannians [J. Algebr. Geom. 13, No. 4, 641--661 (2004; Zbl 1081.14076)], of the minimum power of the quantum parameter \(q\) occurring in the product of two Schubert classes, is also analyzed. The final section is devoted to the study of the quantum cohomology of two exceptional hermitian spaces, namely the Cayley plane and the Freudenthal variety. quantum cohomology; minuscule homogeneous spaces; quivers; Schubert calculus P. E. Chaput, L. Manivel, and N. Perrin, ''Quantum cohomology of minuscule homogeneous varieties,'' ccsd-0086927, 28 Sep 2006, 1--34. Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology of minuscule homogeneous spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The interplay between topology and algebra is a central theme in singularity theory. For example, the Milnor number of an isolated hypersurface singularity can be expressed in terms of the length of its local algebra. The present article establishes a similar result. It connects the purely algebraic object, the Hochster's Theta pairing associated with two modules on the ring of an isolated hypersurface singularity with the linking number of two associated cycles. This generalizes results of Hochster and proves a conjecture of Steenbrink (relating Hochster's Theta pairing to the variation mapping in the cohomology of the Milnor fiber). As a corollary the authors prove the vanishing of the Theta pairing for isolated hypersurfaces in an odd number of variables (as was conjectured by H. Dao). The proof involves higher algebraic \(K\)-theory, topological \(K\)-theory and the Chern character. matrix factorisation; hypersurface singularities; maximal Cohen-Macaulay modules; intersection form; linking number; variation map; K-theory Buchweitz, R.-O., van Straten, D.: An index theorem for modules on a hypersurface singularity. Mosc. Math. J. \textbf{12}, 237-259, 459 (2012) Complex surface and hypersurface singularities, Milnor fibration; relations with knot theory, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Algebraic \(K\)-theory of spaces, Riemann-Roch theorems, Chern characters, Differential topology An index theorem for modules on a hypersurface singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this very clear paper, the authors propose some classification results regarding the subcategory of projective objects of an exact category \(A\). By a result of Verdier, these subcategories correspond to thick subcategories of the bounded derived category \(D^b(A)\) containing all the projective objects. Let \(A\) be an exact category having enough projective objects. The main result (Theorem 1 in Section 2) is devoted to describe a bijection between these subcategories of \(D^b(A)\) and thick subcategories of \(A\) containing all the projective objects. The maps that realize the bijection are the one sending a subcategory \(D\) to \(D \cap A\) and the inverse sending a thick subcategory \(C\) of \(A\) to the replete closure of \(D^b(C)\).
The applications of this result are described in the next sections. Let \(A\) be a strict local complete intersection ring; there is a regular local ring \(B\) surjecting on \(A\) with kernel generated by the regular sequence \(\{b_1, \cdots, b_c\}\). Let \(Y\) be the hypersurface defined by the section \(\sum_{i=1}^c b_ix_i\) of \(\mathcal{O}_{\mathbb{P}^{c-1}_B}(1)\). There is an order-preserving bijection between thick subcategories of \(\operatorname {mod} A\) and specialization closed subsets of the singular locus of \(Y\).
Moreover, if \(G\) is a finite group, denote by \(H^*(G,k)\) the cohomology of \(G\) with coefficient in \(k\). There is an order-preserving bijection between the specialization closed subsets of \(\operatorname {Proj}H^*(G,k)\) and the thick tensor ideals of the category \(\operatorname {mod}kG\) containing \(kG\).
Eventually let \(X\) a separated Noetherian scheme. The authors prove a characterization of the category of coherent \(O_X\)-modules as the smallest thick subcategory of \(\operatorname {Coh}(X)\) containing a particular set of coherent \(\mathcal{O}_X\)-modules \(S\) described by an ample family of line bundles on \(X\). thick subcategories; stable derived categories Krause, H; Stevenson, G, A note on thick subcategories of stable derived categories, Nagoya Math. J., 212, 87-96, (2013) Derived categories, triangulated categories, Derived categories and commutative rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] A note on thick subcategories of stable derived categories | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A particular case in the superstring theory where a finite group \(G\) acts upon the target Calabi-Yau manifold \(M\) in the theory seems to attract both physicists' and mathematician's attention. Define the ``orbifold Euler characteristic'': \(\chi (M,G)= {1\over |G |} \sum_{gh=hg} \chi (M^{\langle g, h\rangle})\), where the summation runs over all the pairs \(g,h\) of commuting elements of \(G\), and \(M^{\langle g,h \rangle}\) denotes the subset of \(M\) of all the points fixed by both of \(g\) and \(h\).
Vafa's formula-conjecture. If a complex manifold \(M\) has trivial canonical bundle and if \(M/G\) has a (nonsingular) resolution of singularities \(\widetilde {M/G}\) with trivial canonical bundle, then we have \(\chi (\widetilde {M/G} ) = \chi (M,G)\).
In the special case where \(M= \mathbb{A}^n\) an \(n\)-dimensional affine space, \(\chi (M,G)\) turns out to be the number of conjugacy classes, or equivalently the number of equivalence classes of irreducible \(G\)-modules. If \(n=2\), then the formula is therefore a corollary to the classical McKay correspondence.
Let \(G\) be a finite subgroup of \(SL(2, \mathbb{C})\) and \(\text{Irr} (G)\) the set of all equivalence classes of nontrivial irreducible \(G\)-modules. Let \(X=X_G: =\text{Hilb}^G (\mathbb{A}^2)\), \(S=S_G: =\mathbb{A}^2/G\), \({\mathfrak m}\) (resp. \({\mathfrak m}_S)\) the maximal ideal of \(X\) (resp. \(S)\) at the origin and \({\mathfrak n}: ={\mathfrak m}_S {\mathcal O}_{\mathbb{A}^2}\). Let \(\pi: X\to S\) be the natural morphism and \(E\) the exceptional set of \(\pi\). Let \(\text{Irr} (E)\) be the set of irreducible components of \(E\). Any \(I\in X\) contained in \(E\) is a \(G\)-invariant ideal of \({\mathcal O}_{\mathbb{A}^2}\) which contains \({\mathfrak n}\).
Definition: \(V(I): =I/({\mathfrak m} I+{\mathfrak n})\).
For any \(\rho\), \(\rho'\), and \(\rho''\in \text{Irr} (G)\) define \(E(\rho): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho)\}\)
\(P(\rho, \rho'): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho) \oplus V(\rho')\}\)
\(Q(\rho, \rho', \rho''): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho) \oplus V(\rho') \oplus V(\rho'')\}\).
Main theorem: (1) The map \(\rho \mapsto E(\rho)\) is a bijective correspondence between \(\text{Irr} (G)\) and \(\text{Irr} (E)\).
(2) \(E(\rho)\) is a smooth rational curve for any \(\rho\in \text{Irr} (G)\).
(3) \(P(\rho, \rho)= Q(\rho, \rho',\rho'') = \emptyset\) for any \(\rho,\rho', \rho''\in \text{Irr} (G)\). Hilbert schemes; orbifold Euler characteristics; irreducible components of exceptional set; superstring theory; McKay correspondence Ito, Y., Nakamura, I.: McKay correspondence and Hilbert schemes. Proc. Japan Acad. Ser. A Math. Sci., 72, 135--138 (1996) Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Global theory and resolution of singularities (algebro-geometric aspects) McKay correspondence and Hilbert schemes | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review studies variations of moduli spaces of representations of preprojective \(K\)-algebras by applying tilting theory, to deal with minimal resolutions of Kleinian singularities.
To a finite subgroup \(G\subset SL(2,K)\) a McKay quiver \(Q\) is associated, and a \(K\)-algebra \(\Lambda\) associated to \(Q\), called preprojective algebra. Different resolutions of quotient singularities \(\mathbb{A}^{n}/G\) are encoded by different moduli spaces \(\mathcal{M} _{\theta,d}({\Lambda})\) of \(\theta\)-semistable \(\Lambda\)-modules of dimension vector \(d\), in the sense of \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)]. Hence, by studying variations of moduli spaces for different stability parameters \(\theta\) we can study quotient singularities.
In section 2, tilting modules \(I_{w}\) (generalization in homological algebra of being torsion or torsion free for a module) are constructed over preprojective algebras. If we denote by \(\mathcal{S}_{\theta}(\Lambda)\subset Mod \Lambda\) the full subcategory of \(\theta\)-semistable \(\Lambda\)-modules, in section 3 relations between the moduli spaces are given, by showing that the functors \(Hom_{\Lambda}(I_{w},-)\) and \(-\otimes_{\Lambda}I_{w}\) induce equivalences between the categories \(\mathcal{S}_{\theta}(\Lambda)\) and \(\mathcal{S}_{w\theta}(\Lambda)\) (c.f. Theorem 3.13) which preserve \(S\)-equivalence classes, where \(w\) are elements of the Coxeter group. This induces a bijection between the sets of closed points in the moduli spaces, and in section 4 the equivalence can be extended to the respective derived categories to show that the bijection can be extended to an isomorphism of \(K\)-schemes (c.f. Theorem 4.20), by using the functors of points.
Section 5 is devoted to use the previous results in the framework of Kleinian singularities to generalize some results of \textit{W. Crawley-Boevey} (see, e.g., [Am. J. Math. 122, No. 5, 1027--1037 (2000; Zbl 1001.14001)]). In section 6 a full example is provided.
Note that the proofs work even in the case where \(Q\) is a non-Dynkin quiver with no loops. Combined with the homological nature of the proofs, this is why the authors expect to use the results in higher dimensions. moduli spaces; preprojective algebras; tilting theory; McKay correspondence; Kleinian singularities Sekiya, Y.; Yamaura, K., \textit{tilting theoretical approach to moduli spaces over preprojective algebras}, Algebr. Represent. Theory, 16, 1733-1786, (2013) Fine and coarse moduli spaces, Representations of quivers and partially ordered sets, McKay correspondence, Homological functors on modules (Tor, Ext, etc.) in associative algebras, Families, moduli, classification: algebraic theory Tilting theoretical approach to moduli spaces over preprojective algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review provides one of the first examples of invariant Hilbert schemes with multiplicities introduced in [\textit{V. Alexeev} and \textit{M. Brion}, J. Algebr. Geom. 14, No. 1, 83--117 (2005; Zbl 1081.14005)]. \(\text{SL}_2\) acts on \((\mathbb{C}^2)^6\) and the moment map \(\mu\) defines the symplectic reduction \(\mu^{-1}(0)// \text{SL}_2\). The paper under review describes explicitly the invariant Hilbert scheme \(\text{Hilb}^{\text{SL}_2}_h(\mu^{-1}(0))\) for the Hilbert function
\[
h:\mathbb{N}_0\to \mathbb{N},\quad d\mapsto d+1,
\]
and proves it is connected and smooth. The Hilbert-Chow morphism
\[
\text{Hilb}^{\text{SL}_2}_h(\mu^{-1}(0))\to \mu^{-1}(0)//\text{SL}_2
\]
factors through the well known symplectic resolutions of \(\mu^{-1}(0)//\text{SL}_2\) given by the cotangent bundles of \(\mathbb{P}^3\) and its dual. The method of the paper provides a general procedure of these calculations that can be applied to similar examples. invariant Hilbert schemes T. Becker, \textit{An example of an} SL\_{}\{2\}-\textit{Hilbert scheme with multiplicities}, Transform. Groups \textbf{16} (2011), no. 4, 915-938. Formal groups, \(p\)-divisible groups, Birational geometry, Special varieties An example of an SL\(_{2}\)-Hilbert scheme with multiplicities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(W\) be a finite dimensional linear representation of a reductive algebraic group \(G\) over an algebraically closed field \(k\) of characteristic 0. Let \(\mathcal{H}\) denote the invariant Hilbert scheme \(\mathrm{Hilb}^G_{h_W}(W)\) parametrizing \(G\)-stable closed subschemes \(Z\) of \(W\) with \(h_W\) being the Hilbert function of the general fiber of the (categorical) quotient morphism \(\nu : W \rightarrow W/\!\!/G = \mathrm{Spec}(k[W]^G)\). The article under review addresses the following question: in which cases is the Hilbert-Chow morphism from \(\gamma : \mathcal{H} \rightarrow W/\!\!/G\), possibly restricted to the main component, a desingularization of \(W/\!\!/G\)?
This question was studied before only for finite groups \(G\), in which case the \(G\)-Hilbert scheme of \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)] coincides with the main component of~\(\mathcal{H}\). They gave a positive answer for finite groups of \(\mathrm{SL}(2)\), then \textit{T. Bridgeland} et al. [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] for finite subgroups of \(\mathrm{SL}(3)\), and \textit{M. Lehn} and \textit{C. Sorger} [in: Geometric methods in representation theory. II. Selected papers based on the presentations at the summer school, Grenoble, France, June 16 -- July 4, 2008. Paris: Société Mathématique de France. 429--435 (2012; Zbl 1312.14007)] for a single 4-dimensional symplectic group.
The article under review gives first results in the case of infinite group~\(G\). The author considers four classical groups (SL, O, Sp, GL) with chosen series of natural representations. The main theorem states that in cases which are small enough (that is, they satisfy certain bounds on parameters of chosen representations), \(\mathcal{H}\) is a desingularization of \(W/\!\!/G\). The proof is based on a reduction principle, which allows to obtain information on all cases from the description of certain small ones. Two of four series of representations are analysed in the article, the details for remaining two can be found in the author's PhD thesis. algebraic group; quotient; desingularization; Hilbert scheme R. Terpereau, \textit{Invariant Hilbert schemes and desingularizations of quotients by classical groups}, Transform. Groups \textbf{19} (2014), no. 1, 247-281. Algebraic cycles, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Invariant Hilbert schemes and desingularizations of quotients by classical groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main subject is the Hilbert scheme \(X^{[n]}\) of \(n\) points on a smooth quasi-projective algebraic surface \(X\). The tautological sheaves on the Hilbert scheme are defined by means of the Fourier--Mukai functor. The Bridgeland--King--Reid transform of a tautological sheaf is compute as well as the same for the tensor product of tautological sheaves.
Also Brion--Danila's formulas for the derived direct images of a tensor product of tautological sheaves are proven for the Hilbert-to-Chow morphism. The author obtains general formulas for the derived direct image of a tautological sheaf or for a tensor product of two of them.
As an application, author gives the explicit formulas for cohomology of \(X^{[n]}\) with value in any tautological sheaf or in the tensor product of torsion-free tautological sheaves with disjoint singular loci.
Relevant papers: \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)]; \textit{G. Danila} [J. Algebr. Geom. 10, No. 2, 247--280 (2001; Zbl 0988.14011)]; \textit{M. Haiman} [J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001)]; \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)]. Hilbert scheme; tautological sheaves; cohomology; smooth quasiprojective surface Scala, L, Some remarks on tautological sheaves on Hilbert schemes of points on a surface, Geom. Dedicata, 139, 313-329, (2009) Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Some remarks on tautological sheaves on Hilbert schemes of points on a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an expository paper which has two parts. In the first part, we study quiver varieties of affine \(A\)-type from a combinatorial point of view. We present a combinatorial method for obtaining a closed formula for the generating function of Poincaré polynomials of quiver varieties in rank 1 cases. Our main tools are cores and quotients of Young diagrams. In the second part, we give a brief survey of instanton counting in physics, where quiver varieties appear as moduli spaces of instantons, focusing on its combinatorial aspects. Young diagram; core; quotient; quiver variety; instanton Fujii, Sh., Minabe, S.: A combinatorial study on quiver varieties. SIGMA Symmetry Integrability Geom. Methods Appl. \textbf{13}, Art. No. 052 (2017) Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Parametrization (Chow and Hilbert schemes), Combinatorial identities, bijective combinatorics, Combinatorial aspects of representation theory, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series A combinatorial study on quiver varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G< \mathrm{GL}_2(\mathbb{C})\) be finite small subgroup. By a result of the first author [J. Reine Angew. Math. 549, 221--233 (2002; Zbl 1057.14057)], the \(G\)-Hilbert scheme \(Y\) gives the minimal resolution of the quotient singularity \(\mathbb{A}^2/G\). If \(G<\mathrm{SL}_2(\mathbb{C})\) it is known the resolution is crepant and it induces the equivalence of derived categories (McKay Correspondence):
\[
\Phi:D^b(\text{coh } Y)\to D^b(\text{coh }[\mathbb{A}^2/G]).
\]
If \(G\not <\mathrm{SL}_2(\mathbb{C})\), then \(Y\) is no longer a crepant resolution and \(\Phi\) is no longer an equivalence but is a full and faithful embedding with admissible essential image, which is generated by \(\{\mathcal{O}_{\mathbb{A}^2}\otimes\rho\}\) where \(\rho\) runs over the special irreducible representations of \(G\).
The main result of the paper under review proves that for a suitable fully faithful functor
\[
\Phi':D^b(\text{coh } Y)\to D^b(\text{coh }[\mathbb{A}^2/G])
\]
there is an exceptional collection \(E_1,\dots, E_n \in D^b(\text{coh }[\mathbb{A}^2/G]) \) and a semi-orthogonal decomposition
\[
D^b(\text{coh }[\mathbb{A}^2/G])=\langle E_1,\dots,E_n, \Phi'(D^b(\text{coh } Y))\rangle,
\]
where \(n\) is the number of non-special irreducible representations of \(G\). If \(G\) is cyclic then one can take \(\Phi'=\Phi\). This result can be viewed as a continuation of the result of Craw-Wemyss that describes \(D^b(\text{coh } Y)\) as the derived category of modules over the path algebra of the special McKay quiver. From this, the paper under review deduces the global version in which \(\mathcal{X}\) is the canonical stack associated to a surface \(X\) with at worst quotient singularities. The proof of the main result consists of several steps. It is first proven for cyclic subgroups. Next, for \(G_0=G\cap \mathrm{SL}_2(\mathbb{C})\) which is a normal subgroup of \(G\), and \(A=G/G_0\) one obtains an equivalence
\[
\Phi_0:D^b(\text{coh } [Y_0/A])\to D^b(\text{coh }[\mathbb{A}^2/G])
\]
where \(Y_0\) is the \(G_0\)-Hilbert scheme. The stack \([Y_0/A]\) is obtained by the iterated root constructions from a canonical stack whose coarse moduli space has a minimal resolution isomorphic to a (non-minimal resolution) of \(\mathbb{A}^2/G\). The theorem is proven by finding the orthogonal decompositions the derived categories of these various intermediate stacks and resolutions. McKay correspondence; exceptional collections Ishii, A., Ueda, K.: The special McKay correspondence and exceptional collections. Tohoku Math. J. (2) \textbf{67}(4), 585-609 (2015) McKay correspondence, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The special McKay correspondence and exceptional collections | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(H_n=\text{Hilb}^n(\mathbb{C}^2)\) be the Hilbert scheme which parametrizes the subschemes \(S\) of length \(n\) of \(\mathbb{C}^2\). To each such subscheme \(S\) corresponds a unordered \(n\)-tuple with possible repetitions \(\sigma(S)=[[P_1,...,P_n]]\) of points of \(\mathbb{C}^2\). There exists an algebraic variety \(X_n\) (called the isospectral Hilbert scheme) which is finite over \(H_n\) and which consists of all ordered \(n\)-tuples \((P_1,...,P_n)\in(\mathbb{C}^2)^n\) whose underlying unordered \(n\)-tuple is \(\sigma(S)\). The main aim of the paper under review is to study the geometry of \(X_n\), which is more complicated than the geometry of \(H_n\). For instance, a classical result of J. Fogarty asserts that \(H_n\) is irreducible and non-singular. The main result of the paper under review asserts that \(X_n\) is normal and Gorenstein (in particular, Cohen-Macaulay). Earlier work of the author indicated that there is a far-reaching correspondence between the geometry and sheaf cohomology of \(H_n\) and \(X_n\) on the one hand, and the theory of Macdonald polynomials on the other hand. The link between Macdonald polynomials and Hilbert schemes comes from some recent work [see \textit{A. M. Garsia} and \textit{M. Haiman}, Proc. Nat. Acad. Sci. USA 90, No. 8, 3607-3610 (1993; Zbl 0831.05062)].
The main result proved in this paper is expected to be an important step toward the proof of the so-called \(n!\)-conjecture and Macdonald positivity conjecture. The main theorem is based on a technical result (theorem 4.1) which asserts that the coordinate ring of a certain type of subspace arrangement is a free module over the polynomial ring generated by some of the coordinates. Macdonald polynomials; isospectral Hilbert schemes; \(n!\)-conjecture; Macdonald positivity conjecture; subspace arrangement K.B. Alkalaev and V.A. Belavin, \textit{Conformal blocks of}\( {\mathcal{W}}_n \)\textit{Minimal Models and AGT correspondence}, arXiv:1404.7094 [INSPIRE]. Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Hilbert schemes, polygraphs and the Macdonald positivity conjecture | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Consider a reductive group \(G\) over an algebraically closed field \(k\) of characteristic zero. For an affine \(G\)-scheme \(W\) of finite type over \(k\) and a given function \(h: \mathrm{Irr}(G) \to \mathbb{Z}_{\geq 0}\), \textit{V.~Alexeev} and \textit{M.~Brion} [J.~Algebraic~Geom. 14, No. 1, 83--117 (2005; Zbl 1081.14005)] studied the invariant Hilbert scheme \(\mathcal{H} = \mathrm{Hilb}^G_h(W)\); morally, it classifies the closed \(G\)-subschemes (or more generally, flat families of such objects over a base scheme) \(\mathcal{Z}\) of \(W\) such that the coordinate algebra of \(\mathcal{Z}\) decomposes under \(G\) according to \(h\). In particular, there is a universal family \(\mathcal{U} \subset W \times \mathcal{H}\). The Hilbert-Chow morphism \(\gamma: \mathcal{H} \to W /\!/ G\) is a canonically defined morphism covered by \(\mathrm{pr}_1: \mathcal{U} \to W\). It is known to be a projective morphism; furthermore, there is a largest open subset \(U \subset W /\!/ G\) over which the GIT quotient morphism from \(W\) is flat. The main component \(\mathcal{H}_{\mathrm{main}}\) is defined to be the closure of \(\gamma^{-1}(U)\).
Many geometric properties of the invariant Hilbert scheme remain inaccessible, such as the singularity, connectedness, etc. The main purpose of this paper is to give an algorithm for calculating the universal deformation at a point \([X]\) of \(\mathcal{H}\). An effective theory of invariant deformations is carefully set up under suitable hypotheses. As applications, the authors considered three cases: (1) \(\mathrm{GL}_3\) acts on \((k^3)^{\oplus n_1} \oplus (k^{3*})^{\oplus n_2}\), (2) \(\mathrm{SO}_3\) acts on \((k^3)^{\oplus 3}\), and (3) \(\mathrm{O}_3\) acts on \((k^3)^{\oplus n}\). In each case there is a \(\mathbb{G}_m \subset \mathrm{Aut}^G(W)\) with strictly positive weights on \(k[W]\) and on \((T_{[X]} \mathcal{H})^\vee\). Let \(h\) be the Hilbert function of the general fibers of \(W \to W/\!/G\).
In the cases (1) and (2), it turns out that \(\mathcal{H}_{\mathrm{main}}\) is smooth, there is a decomposition \(\mathcal{H} = \mathcal{H}_{\mathrm{main}} \cup \mathcal{H}'\) into irreducible components, such that \(\mathcal{H}'\) is smooth (resp. singular) in case (1) (resp. case (2)). In the case (3), \(\mathcal{H}\) turns out to be connected and has at least two irreducible components.
Grosso modo, the technique is to find a suitable algebraic subgroup \(H \subset \mathrm{Aut}^G(W)\), a Borel subgroup \(B \subset H\) and try to ``localize'' the geometric properties of \(\mathcal{H}\) at some \(B\)-fixed point; one looks then for \(\mathbb{G}_m \subset B\) satisfying the positivity property alluded to above. invariant Hilbert scheme; invariant deformation Lehn, C; Terpereau, R, Invariant deformation theory of affine schemes with reductive group action, J. Pure Appl. Algebra, 219, 4168-4202, (2015) Actions of groups on commutative rings; invariant theory, Representation theory for linear algebraic groups, Algebraic moduli of abelian varieties, classification, Group actions on varieties or schemes (quotients), Computational aspects in algebraic geometry, Local deformation theory, Artin approximation, etc. Invariant deformation theory of affine schemes with reductive group action | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subset \text{SL}(2,{\mathbb C})\) be a finite subgroup and \(Y\) the Hilbert scheme of \(G\)-orbits (\(G\)-Hilb) introduced by Nakamura. \textit{Y. Ito} and \textit{I. Nakamura} [in: New trends in algebraic geometry. Proc. Warwick 1996, Lond. Math. Soc. Lect. Note Ser. 264, 151--233 (1999; Zbl 0954.14001)] studied \(I/mI\) as representations of \(G\) for ideals \(I\) in the exceptional locus of \(Y\). They proved that
1) \(Y\) is the minimal resolution of \({\mathbb C}^2/G\) and
2) there is a bijection of irreducible exceptional curves \(\{E_1,\ldots,E_n\}\) with non-trivial irreducible representations \(\{\rho_1,\ldots,\rho_n\}\) such that \(I_y/(mI_y+a)\) is isomorphic either with \(\rho_i\oplus\rho_j\) if \(y\in E_i\cap E_j\), or with \(\rho_i\) if \(y\in E_i\) and \(y\not\in E_j\) for \(j\not =i\), where \(a=(m^G){\mathcal O}_{{\mathbb C}^2}\) (this bijection coincides with the original McKay's correspondence).
The aim of this paper is to show that the above result holds if \(G\) is a finite subgroup of GL\((2,{\mathbb C})\) choosing some \textit{special} representations defined by Wunram. This was conjectured by \textit{O. Riemenschneider} [On the two-dimensional McKay correspondence, Hamb. Beitr. Math. 94 (2000), \texttt{http://www.math.uni-hamburg.de/meta/preprints/ms/ms2000094.html]}. Hilbert scheme of \(G\)-orbits; exceptional curves; irreducible representations Ishii, A., On the mckay correspondence for a finite small subgroup of \(\operatorname{GL}(2, \mathbb{C})\), J. Reine Angew. Math., 549, 221-233, (2002), MR 1916656 Group actions on varieties or schemes (quotients), Geometric invariant theory, Global theory and resolution of singularities (algebro-geometric aspects), Representation theory for linear algebraic groups, Complex surface and hypersurface singularities, Global theory of complex singularities; cohomological properties On the McKay correspondence for a finite small subgroup of \(\text{GL}(2,\mathbb{C})\). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies tautological vector bundles and their tenor products over the Hilbert scheme of points on a smooth quasi-projective algebraic surface, with a particular interest in their cohomology. In section one, standard notations and preliminary results are recalled. For a smooth quasi-projective algebraic surface \(X\), let \(X^{[n]}\) be the Hilbert scheme parametrizing length-\(n\) \(0\)-dimensional closed subschemes of \(X\). It is well-known that there exists a derived equivalence
\[
\Phi: \text{D}^b(X^{[n]}) \to \text{D}_{S_n}^b(X^n)
\]
where \(\text{D}^b(X^{[n]})\) is the derived category of bounded complexes of coherent sheaves on \(X^{[n]}\), and \(S_n\) acts on \(X^n\) by permutation. In section two, let \(L^{[n]}\) be the tautological rank-\(n\) bundle on \(X^{[n]}\) corresponding to a line bundle \(L\) on \(X\). The author identifies the image \(\Phi(L^{[n]})\) in terms of a complex \(\mathcal C_L^\bullet\) of \(S_n\)-equivariant sheaves on \(X^n\), and characterize the image \(\Phi(L^{[n]} \otimes \cdots \otimes L^{[n]})\) in terms of the hyperderived spectral sequence associated to the derived tensor products of the complex \(\mathcal C_L^\bullet\). In sections three and four, the derived direct images of \(L^{[n]} \otimes L^{[n]}\) and \(\Lambda^k L^{[n]}\) under the Hilbert-Chow morphism are obtained. In section five, the author computed the cohomology of \(X^{[n]}\) with values in \(L^{[n]} \otimes L^{[n]}\) and \(\Lambda^k L^{[n]}\). Hilbert schemes of points; tautological bundles; cohomology Scala L: Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles. Duke Math. J 2009,150(2):211--267. 10.1215/00127094-2009-050 Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Representations of finite symmetric groups Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The finite subgroups of the multiplicative group of nonzero elements of the quaternions were classified by \textit{C. Jordan} [J. Reine Angew. Math. 84, 89--215 (1877; JFM 09.0234.01)] up to the action of \(\text{SO}_3(\mathbb R)\). \textit{J. McKay} [in: Finite groups, Santa Cruz Conf. 1970, Proc. Symp. Pure Math. 37, 183--186 (1980; Zbl 0451.05026)] associated to each of these subgroups a simply-laced Dynkin diagram, now called its McKay label. The present paper covers similar ground for octonions. In particular, any finite subloop is either associative, a double of a noncommutative associative subloop or the group of units in the octavian integers. Conjugacy under \(G_2\) depends on the McKay label. The authors remark that somewhat similar work was done in an unpublished manuscript of Robert Curtis. Although there are overlapping results, the methods are different and each work has its own special contributions. octonions; quaternions; loop; root system P. Boddington and D. Rumynin, ''On Curtis' theorem about finite octonionic loops,'' Proc. Amer. Math. Soc. 135 (2007), 1651--1657. Alternative rings, Simple, semisimple, reductive (super)algebras On Curtis' theorem about finite octonionic loops | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a given finite small binary dihedral group \(G\subset\mathrm{GL}(2,\mathbb{C})\) we provide an explicit description of the minimal resolution \(Y\) of the singularity \(\mathbb{C}^{2}/G\). The minimal resolution \(Y\) is known to be either the moduli space of \(G\)-clusters \(G\)-Hilb\((\mathbb{C}^{2})\), or the equivalent \(\mathcal{M}_{\theta}(Q,R)\), the moduli space of \(\theta\)-stable quiver representations of the McKay quiver. We use both moduli approaches to give an explicit open cover of \(Y\), by assigning to every distinguished \(G\)-graph \(\Gamma\) an open set \(U_{\Gamma}\subset\mathcal{M}_{\theta}(Q,R)\), and calculating the explicit equation of \(U_{\Gamma}\) using the McKay quiver with relations \((Q,R)\). mckay correspondence; \(G\)-Hilbert scheme; quiver representations Á. Nolla de Celis, Dihedral \({G}\)-Hilb via representations of the McKay quiver , Proc. Japan Acad. Ser. A 88 (2012), 78-83. McKay correspondence, Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets, Global theory and resolution of singularities (algebro-geometric aspects) Dihedral \(G\)-Hilb via representations of the McKay quiver | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an affine scheme of finite type over \(\mathbb{C}\), and let \(G\) be an infinite reductive group acting on \(X\). Denote by \(\mathrm {Irr } G\) the set of classes of irreducible representations \(\rho: G\to \mathrm {GL}(V_{\rho})\). For a Hilbert function \(h:\mathrm {Irr } G \to \mathbb{N}_0\) we refer a \(G\)-equivariant coherent \({\mathcal O}_X\) module \({\mathcal F}\) such that \(H^0({\mathcal F})\cong \bigoplus_{\rho\in\mathrm {Irr } G} \mathbb{C}^{ h(\rho)}\otimes_{\mathbb{C}}V_{\rho}\) as \(G\)-modules as a \((G,h)\)-constellation. While the set of all \((G,h)\)-constellations is too large to afford a parameterization, in the work under review the authors describe the moduli space \(M_{\theta}(X)\) of \((G,h)\)-constellations which are stabilized by a certain \(\theta:\bigoplus_{\rho\in\mathrm {Irr } G}\mathbb{N}\cdot\rho \to\mathbb{Q}\).
A significance of this construction is that it unifies the constructions found elsewhere. For example, if \(h(\rho_0)=1\), where \(\rho_0\) is the trivial representation, then for a suitable choice of \(\theta\) one recovers the invariant Hilbert scheme of Alexeev and Brion -- see, for example, [\textit{V. Alexeev} and \textit{M. Brion}, J. Algebr. Geom. 14, No. 1, 83--117 (2005; Zbl 1081.14005)]. Furthermore, this construction acts as an analogue to the case where \(G\) is finite as found in [\textit{A. Craw} and \textit{A. Ishii}, Duke Math. J. 124, No. 2, 259--307 (2004; Zbl 1082.14009)].
Additionally, the description of \(M_{\theta}(X)\) permits the authors to construct a morphism \(M_{\theta}(X)\to X/\!\!/G\), corresponding to the Hilbert-Chow morphism, which may allow for the resolution of the singularities of \(X/\!\!/G\). Hilbert scheme; irreducible representations; moduli spaces Becker, T.; Terpereau, R., Moduli spaces of \((G, h)\)-constellations, Transform. Groups, 20, 2, 335-366, (2015) Parametrization (Chow and Hilbert schemes), Group actions on affine varieties, Group actions on varieties or schemes (quotients), Geometric invariant theory, Representation theory for linear algebraic groups Moduli spaces of \((G, h)\)-constellations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite subgroup of \(\text{SL}(n,\mathbb{C})\). The generalized McKay correspondence aims to relate the geometry of crepant (i.e. with trivial canonical divisor) resolutions of singularities of the quotient \(\mathbb{C}^n/G\) to the representations of the group \(G\). This paper deals with the natural candidate given by the Hilbert scheme of \(G\)-regular orbits introduced by \textit{I. Nakamura} [J. Algebr. Geom. 10, No.~4, 757--779 (2001; Zbl 1104.14003)], parametrizing generalized \(G\)-orbits on \(\mathbb{C}^n\), denoted by \(G\text{-Hilb}(\mathbb{C}^n)=:Y\). By a theorem of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)], for \(n=3\) this provides the required resolution of singularities. The McKay correspondence is realized as follows: there exists a natural integral basis of the Grothendieck group \(K(Y)\) given by natural bundles \(\mathcal{R}_k\) indexed by the irreducible representations of \(G\). This provides, through Chern character, a rational basis of the cohomology \(H^*(Y,\mathbb{Q})\) in one-to-one correspondence with the irreducible representations of \(G\). It is still an open problem to get a similar correspondence for the integral cohomology \(H^*(Y,\mathbb{Z})\). Reid conjectured that some ``cookery'' with the Chern classes of these bundles \(\mathcal{R}_k\) should provide an integral basis. This paper establishes explicitly this integral McKay correspondence for all abelian subgroups \(A\) in \(\text{SL}(3,\mathbb{C})\) (Theorem 1.1). The method follows the recipe introduced by Reid, uses previous work of \textit{Y. Ito, H. Nakajima} [Topology 39, No.~6, 1155--1191 (2000; Zbl 0995.14001)] and an explicit algorithm of computation of \(A\text{-Hilb}(\mathbb{C}^3)\) already described by \textit{A. Craw} and \textit{M. Reid} [in: Geometry of toric varieties. Lect. summer school. Grenoble. 2000, Sémin. Congr. 6, 129--154 (2002; Zbl 1080.14502)] and extending the initial work of Nakamura [loc.cit.], based upon a decoration of the toric fan of \(A\text{-Hilb}(\mathbb{C}^3)\) with the characters of the group \(A\). The integral basis of \(H^2(Y,\mathbb{Z})\) is then given by the first Chern classes of some \(\mathcal{R}_k\)'s indexed by specific non-trivial characters (Proposition 7.1). In order to base \(H^4(Y,\mathbb{Z})\), the author computes all relations between the line bundles (since the group is abelian) \(\mathcal{R}_k\) in \(\text{Pic}(Y)\), and introduces a family of virtual bundles \(\mathcal{V}_m\) indexed by the remaining non-trivial irreducible representations, whose second Chern classes will give the expected integral basis (Proposition 7.3). Hilbert scheme of orbits; toric geometry Craw, A., An explicit construction of the McKay correspondence for \(A\)-Hilb \({\mathbb{C}^3}\), J. Algebra, 285, 682-705, (2005) Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(3\)-folds, Toric varieties, Newton polyhedra, Okounkov bodies, Classical real and complex (co)homology in algebraic geometry, Ordinary representations and characters An explicit construction of the McKay correspondence for \(A\)-Hilb \(\mathbb C^3\) | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A common way of studying algebraic varieties is to study their embeddings in projective space by linear series of base-point free line bundles. In this article, this technique is extended to multigraded linear series of a collection of globally generated vector bundles on a scheme, unifying several constructions in the study of algebraic varieties. The primary goal is to describe schemes as a moduli space to achieve geometric properties. The article gives several explicit families of examples.
Throughout the article, \(\Bbbk\) denotes an algebraically closed field of characteristic \(0\). Let \(Y\) be a projective scheme. Given a collection \(E_1,\dots,E_n\) of effective vector bundles on \(Y\), let \(E=\bigoplus_{0\leq i\leq n}E_i\) where \(E_0\simeq\mathcal O_Y\). Let \(A=\text{End}_Y(E)\) be the endomorphism algebra of \(E\) and let \(\mathbf{v}=(v_i)\) with \(v_i=\text{rk}(E_i)\) be the dimension vector. The \textit{multigraded linear series} of \(E\) is the fine moduli space \(\mathcal M(E)\) of \(0\)-generated \(A\)-modules with dimension vector \(\mathbf{v}\). The universal family of \(\mathcal M(E)\) is a vector bundle \(T=\bigoplus_{0\leq i\leq n}T_i\) together with a \(\Bbbk\)-algebra homomorphism \(A\rightarrow\text{End}(T)\) where each \(T_i\) is a tautological vector bundle of rank \(v_i\) and \(T_0\) is the trivial line bundle.
The first main result of the article generalizes the classical morphism \(\phi_{|L|}:Y\rightarrow |L|\) to the linear series of a single base-point free line bundle \(L\) on \(Y\), i.e. the morphism to a Grassmannian defined by a globally generated vector bundle on a projective variety. Verbatim:
Theorem 1.1. If the vector bundles \(E_1,\dots,E_n\) are globally generated, then there is a morphism \(f:Y\rightarrow\mathcal M(E)\) satisfying \(E_i=f^\ast(T_i)\) for \(0\leq i\leq n\) whose image is isomorphic to the image of the morphism \(\phi_{|L|}:Y\rightarrow|L|\) to the linear series of \(L:=\bigotimes_{1\leq i\leq n}\det(E_i)^{\otimes j}\) for some \(j>0.\)
If the line bundle \(\bigotimes_{1\leq i\leq n}\det(E_i)\) is ample, after taking a multiple of a linearisation if necessary, the resulting universal morphism \(f:Y\rightarrow\mathcal M(E)\) is a closed immersion. The next question is then if \(f\) is surjective. If that is the case, then \(f\) represents \(Y\) as the fine moduli space \(\mathcal M(E)\). It turns out that even when \(Y\) is isomorphic to \(\mathcal M(E),\) more insight can be gained by deleting some summands of \(E.\) When \(0\in C\subseteq\{0,1,\dots,n\},\) the subbundle \(E_C=\sum_{i\in C}E_i\) has the trivial subbundle \(E_0\) as a summand, and Theorem 1.1 proves the universal morphism \(g_C:\mathcal M(E)\rightarrow\mathcal M(E_C)\) between multigraded linear series. This can lead to a more geometric significant moduli space description of \(Y\).
Of such geometric significance is a moduli construction determined by a tilting bundle. \textit{A. Bergman} and \textit{N. J. Proudfoot} [Pac. J. Math. 237, No. 2, 201--221 (2008; Zbl 1151.18011)] proved that for a smooth variety \(Y\) with a tilting bundle \(E\), \(f\) is an isomorphism onto a connected component of \(\mathcal M(E)\). A main goal of this article is to establish several cases where \(f\) is an isomorphism onto \(\mathcal M(E)\) itself, implying a description of \(Y\) as a moduli space. Also, the authors manage to give results in situations where \(Y\) is singular.
A second main goal is to apply the theory to the \textit{special McKay correspondence}. Let \(G\subset\text{GL}(2,\Bbbk)\) be a finite subgroup without pseudo-reflections, let \(\text{Irr}(G)\) be the set of isomorphism classes of irreducible representations of \(G\), and let \(Y\) denote the minimal resolution of \(\mathbb A^2_{\Bbbk}/G\). \textit{R. Kidoh} [Hokkaido Math. J. 30, No. 1, 91--103 (2001; Zbl 1015.14004)] and \textit{A. Ishii} [J. Reine Angew. Math. 549, 221--233 (2002; Zbl 1057.14057)] proved that \(Y\) is isomorphic to the fine moduli space of \(G\)-equivariant coherent sheaves of the form \(\mathcal O_Z\), for subschemes \(Z\subset\mathbb A^2_{\Bbbk}\) such that \(\Gamma(\mathcal O_Z)\) is isomorphic to the regular representation of \(G\) (\(G\)-Hilbert scheme). Writing the tautological bundle on the \(G\)-Hilbert space \(T=\underset{\rho\in\text{Irr}(G)}{\sum} T_\rho^{\bigoplus\dim(\rho)}\) and noticing that \(\text{End}_Y(T)\) is isomorphic to the skew group algebra, it follows that the minimal resolution \(Y\cong G-\text{Hilb}\) is isomorphic to the multigraded linear series \(\mathcal M(T)\). When \(G\) is a finite subgroup of \(\text{SL}(2,\Bbbk)\), \(T\) is a tilting bundle on \(Y\) so that \(Y\) is derived equivalent to the category of modules over \(\text{End}_Y(T),\) but the authors prove that this is false in general.
A natural moduli description of \(Y\) is given by the special McKay correspondence of the finite subgroup \(G\subset\text{GL}(2,\Bbbk)\). For the set \(\text{Sp}(G)=\{\rho\in\text{Irr}(G)|H^1(T^\vee_\rho)=0\}\) of special representations, Van den Bergh proved that the \textit{reconstruction bundle} \(E:=\underset{\rho\in\text{Sp}(G)}{\bigoplus} T_\rho\) is a tilting bundle on \(Y\) so that \(Y\) is derived equivalent to the category of modules over the reconstruction algebra studied by \textit{M. Wemyss} [Math. Ann. 350, No. 3, 631--659 (2011; Zbl 1233.14012)]. The second main result of the article proves that \(E\) contains enough information to reconstruct \(Y\), and provides a moduli space description that trumps the \(G\)-Hilbert scheme in general. Verbatim:
Theorem 1.2. Let \(G\subset\text{GL}(2,\Bbbk)\) be a finite subgroup without pseudo-reflections. Then: (i) the minimal resolution \(Y\) of \(\mathbb A^2_{\Bbbk}/G\) is isomorphic to the multigraded linear series \(\mathcal M(E)\) of the reconstruction bundle; and (ii) for any partial resolution \(Y^\prime\) such that the minimal resolution \(Y\rightarrow\mathbb A^2_\Bbbk/G\) factors via \(Y^\prime,\) there is a summand \(E_C\subseteq E\) such that \(Y^\prime\) is isomorphic to \(\mathcal M(E_C).\)
The authors correctly remark that the approach in this article is closer in spirit to the geometric construction of the special McKay correspondence for cyclic subgroups of \(\text{GL}(2,\Bbbk)\) given by \textit{A. Craw} [Q. J. Math. 62, No. 3, 573--591 (2011; Zbl 1231.14010)] in his earlier work.
The main tools for proving the theorems is a homological criterion to decide when the morphism \(g_C\) is surjective. Any subset \(C\subseteq\{0,\dots,n\}\) containing \(0\) determines a subbundle \(E_C\) of \(E\), and the module categories of the algebras \(A=\text{End}_Y(E)\) and \(A_C=\text{End}_Y(E_C)\) are linked by \textit{recollement}. The authors prove that the morphism \(g_C:\mathcal M(E)\rightarrow\mathcal M(E_C)\) is surjective iff for each \(c\in\mathcal M(E_C)\), the \(A\)-module \(j_!(N_x)\) admits a surjective map onto an \(A\)-module of dimension vector \(\mathbf(v)\).
As a second main ingredient, the authors use the fact that a derived equivalence \(\Psi(-)=E^\vee\otimes_A-:D^b(A)\rightarrow D^b(Y)\) induces an isomorphism between the lattice of dimension vectors for \(A\) and the numerical Grothendieck group for compact support \(K_c^{\text{num}}(Y)\) introduced by Bayer-Craw-Zhang.
The article concludes with examples from NCCRs in dimension three. In very short terms, a resolution \(Y\rightarrow\mathbb A/G\) is given. Examples are given where one can reconstruct \(Y\) using only a proper summand of a tilting bundle \(T\).
The article is very important, contains very nice results with clear proofs, and show important applications of tilting- and moduli theory, and their connection. linear series; base-point free line bundles; effective vector bundles; multigraded linear series; summands; special McKay correspondance; pseudo-reflections; irreducible group-representations; G-Hilbert space; skew group algebra; derived equivalence; reconstruction bundle; reconstruction algebra; numerical Grothendieck group; NCCRs Craw, A., Ito, Y., Karmazyn, J.: Multigraded linear series and recollement (2017). arXiv:1701.01679 (to appear in Math. Z.) Noncommutative algebraic geometry, McKay correspondence, Representations of quivers and partially ordered sets, Grothendieck groups (category-theoretic aspects) Multigraded linear series and recollement | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subset \text{GL}(n,\mathbb C)\) be a finite group and consider the corresponding quotient singularity \(X=\mathbb A^n/G\). Inspired by the classical McKay correspondence, many interesting connections between resolutions \(Y\to X\) of the quotient singularity and the representation theory of \(G\) have been discovered. In particular, for \(G\subset \text{SL}(3,\mathbb C)\) abelian, every projective crepant resolution is given by a moduli space of \(G\)-constellations or, equivalently, a moduli space of representations of the McKay quiver of \(G\) of dimension vector \((1,\dots,1)\); see \textit{A. Craw} and \textit{A. Ishii} [Duke Math.\ J. 124, No. 2, 259--307 (2004; Zbl 1082.14009)]. Given a resolution \(Y\to \mathbb A^3/G\) with \(G\not\subset \text{SL}(3,\mathbb C)\) one may ask whether \(Y\) can be identified with a moduli space of representations of the McKay quiver too.
The author studies this question for the Danilov resolution (also known as the economic resolution) of the terminal quotient singularity of type \(\frac 1r(1,a,r-a)\) for coprime numbers \(a\), \(r\). This means that \(G\subset \text{GL}(3,\mathbb C)\) is the cyclic group of order \(r\) acting diagonally with eigenvalues \(\zeta\), \(\zeta^a\), and \(\zeta^{r-a}\) where \(\zeta\) is a primitive \(r\)-th root of unity. The Danilov resolution \(Y\to X=\mathbb A^3/G\) is a toric resolution given by a series of weighted blow-ups.
For \(G\subset \text{GL}(n,\mathbb C)\), a \(G\)-constellation is defined as a \(G\)-equivariant sheaf \(F\) on \(\mathbb A^n\) whose global sections \(\Gamma(F)\) are given by the regular representation \(R\) of \(G\). A stability parameter for \(G\)-constellations is given by a \(\mathbb Z\)-linear map \(\theta: \text R(G)\to \mathbb Q\) from the representation ring to the rational numbers such that \(\theta(R)=0\). A \(G\)-constellation \(F\) is called \(\theta\)-stable if for every non-trivial \(G\)-subsheaf \(E\subset F\) we have \(\theta(\Gamma(F))>0\). Due to more general results of \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)], there is a fine moduli space \(\mathcal M_\theta\) of \(\theta\)-stable \(G\)-constellations. As structure sheaves of free \(G\)-orbits do not have any non-trivial \(G\)-subsheaves, they are \(\theta\)-stable for every parameter \(\theta\). The irreducible component \(Y_\theta\) of \(\mathcal M_\theta\) containing the structure sheaves of free orbits is called the coherent component. Let \(f: Y\to X\) be a resolution of the singularities. A family \(\mathcal F\) of \(G\)-clusters over \(Y\) is called a gnat-family (short for \(G\)-natural) if \(\mathcal F_y\) is supported on the \(G\)-orbit \(f(y)\) for every \(y\in Y\). If \(\mathcal F\) is in addition \(\theta\)-stable, this means that the classifying morphism \(Y\to\mathcal M_\theta\) factorises over the coherent component and commutes with \(f\) and the canonical morphism \(Y_\theta\to X=\mathbb A^n/G\) given by sending a \(G\)-constellation to the \(G\)-orbit supporting it. For \(G\) abelian, \textit{T. Logvinenko} [Doc. Math., J. DMV 13, 803--823 (2008; Zbl 1160.14025)] gave a characterization of all gnat-families on \(Y\) in terms of \(G\)-Weyl divisors.
In the paper under review, toric divisors on the Danilov resolution \(Y\) are used in order to construct a gnat-family \(\mathcal F\). Then the cone of stability parameters \(\theta\) for which \(\mathcal F\) is \(\theta\)-stable is computed. Furthermore, it is shown that the fibres of \(\mathcal F\) are pairwise non-isomorphic. It follows that the classifying morphism \(Y\to Y_\theta\) is bijective and consequently that \(Y\) is the normalisation of \(Y_\theta\). It is conjectured that \(Y_\theta\) is normal so that \(Y\cong Y_\theta\). McKay correspondence; resolutions of terminal quotient singularities; Danilov resolution; moduli of quiver representations Kȩdzierski, O.: Danilov's resolution and representations of the mckay quiver. Tohoku math. J. (2) 66, No. 3, 355-375 (2014) McKay correspondence, Representations of quivers and partially ordered sets, Geometric invariant theory Danilov's resolution and representations of the McKay quiver | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite abelian group acting linearly on \(k^n\), where \(k\) is an algebraically closed field, and the order of \(G\) is invertible in \(k\) (so the action can be diagonalised). The present paper is a sequel to [Proc. Lond. Math. Soc. (3) 95, No. 1, 179--198 (2007; Zbl 1140.14046)], where the authors constructed expilitly an irreducible component \(Y_{\theta}\) (called the coherent component) of the moduli space of \(\theta\)-stable representations of the McKay quiver of \(G\).
Passing from a \(\theta\)-stable quiver representation to a \(G\)-constellation (i.e. a \(G\)-equivariant \(S\)-module isomorphic to the regular \(G\)-module \(kG\), where \(S\) is the coordinate ring of the \(G\)-module \(k^n\)), the authors can make use of Gröbner basis theory. They determine whether a given \(\theta\)-stable \(G\)-constellation corresponds to a point on the coherent component \(Y_{\theta}\). In the case when \(Y_{\theta}\) equals Nakamura's \(G\)-Hilbert scheme, they present explicit equations for a cover by local coordinate charts. The computational techniques introduced here are applied to construct a subgroup of \(GL(6,k)\) for which the \(G\)-Hilbert scheme is not normal. McKay quiver; Gröbner bases; \(G\)-Hilbert scheme Craw, A.; Maclagan, D.; Thomas, R.R., Moduli of mckay quiver representations II: Gröbner basis techniques, J. algebra, 316, 2, 514-535, (2007) Toric varieties, Newton polyhedra, Okounkov bodies, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Representations of quivers and partially ordered sets Moduli of McKay quiver representations. II: Gröbner basis techniques | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Fix an algebraically closed characteristic field \(k.\) Let \(Q=\left( Q_{0},Q_{1}\right) \) be a quiver whose arrows are labelled by a map \(l:\mathbb{Z}^{Q_{1}}\rightarrow\mathbb{Z}^{d},\) and let \(\mathbb{Z}^{Q_{0}}\) be the free abelian group generated by the vertices. Let \(\theta\) be a weight, that is, \(\theta=\sum_{i\in Q_{0}}\theta_{i}\mathbf{e}_{i}\), where \(\sum\theta_{i}=0\) and \(\left\{ \mathbf{e}_{i}\right\} _{i\in Q_{0}}\) is the standard basis for \(\mathbb{Z}^{Q_{0}}.\) Let \(R\) be the weights which arise from \(\ker l\) through the map \(\mathbf{e}_{a}\mapsto\mathbf{e}_{h\left( a\right) }-\mathbf{e}_{t\left( a\right) }\) (where \(a\) is an arrow with head \(h\left( a\right) \) and tail \(t\left( a\right) \)). Fix a basis \(\mathcal{B}\) for \(R.\) A refined representation of \(\left( Q,l\right) \) is a representation \(\overline{W}:=\left( W_{i},w_{a}\right) \) along with isomorphisms \(f_{b}:k\rightarrow\bigotimes_{i\in Q_{0}}\left( \det W\right) ^{\otimes b_{i}},\) where \(b=\sum_{i\in Q_{0}}b_{i}\mathbf{e}_{i}\in \mathcal{B}.\) This is independent of the choice of \(\mathcal{B}.\) If \(\alpha\) is the dimension vector for \(\overline{W}\) then \(\theta\) gives rise to a character \(\chi_{\theta}\;\)of \(\text{GL}\left( \alpha\right) =\prod_{i\in Q_{0}}\text{GL}\left( W_{i}\right) \) given by \(\chi_{\theta}\left( g\right) =\prod_{i\in Q_{0}}\det\left( g_{i}\right) ^{\theta_{i}}.\) Let \(\mathcal{R}\left( Q,l,\alpha\right) \) be the moduli space of isomorphism classes of refined representations of \(\text{GL} \left( \alpha\right) \) -- an explicit description is given in the paper.
We say the refined quiver representation \(\overline{W}:=\left( W_{i} ,w_{a}\right) \) is \(\theta\)-semistable if \(\sum\theta_{i}\dim\left( W_{i}^{\prime}\right) \geq0\) for all subrepresentations \(\overline{W^{\prime }}\) of \(\overline{W}\): if the equality is strict we say \(\overline{W}\) is \(\theta\)-stable. Then it is shown that \(\overline{W}\) \ is \(\theta\)-semistable if and only if the corresponding point in \(\mathcal{R}\left( Q,l,\alpha \right) \) is \(\chi_{\theta}\)-semistable. The result is still true if ``semistable'' is replaced by ``stable''. This allows for a construction of moduli stacks \(\mathcal{M}_{\theta}\left( Q,l,\alpha\right) \) of refined representations.
Let \(\mathcal{X}\) be a projective toric orbifold, and let \(\mathcal{L}\) \(=\left( L_{0},\dots,L_{r}\right) \) be a collection of line bundles on \(\mathcal{X}.\) Then one can define a labelled quiver \(\left( Q,\text{div} \right) \) with vertices \(\left\{ 0,\dots,r\right\} \) and arrows \(i\rightarrow j\) corresponding to the \(T_{\mathcal{X}}\)-invariant sections in \(\Gamma\left( \mathcal{X},L_{j}\otimes L_{i}^{\vee}\right) \) . The labelling div is induced by sending an arrow \(a\) to its corresponding divisor, and for any weight \(\theta\) there is a rational map \(\psi_{\theta}:\mathcal{X} \dashrightarrow\mathcal{M}_{\theta}\left( Q,\text{div},\left( 1,1,\dots ,1\right) \right) .\) If we let \(\mathcal{L}_{\text{bpf}}\) be the set of base-point free line bundles \(L_{i}^{\vee}\otimes L_{j}\) where \(L_{i} ,L_{j}\in\mathcal{L},\) then if rank\(_{\mathbb{Z}}\mathcal{L=}\) rank\(_{\mathbb{Z}}\mathcal{L}_{\text{bpf}}\) there is a \(\theta\) such that \(\psi_{\theta}\) is a morphism \(\mathcal{X}\rightarrow\mathcal{M}_{\theta }\left( Q,\text{div},\left( 1,1,\dots,1\right) \right) .\) Furthermore, this morphism is representable if any only if \(\bigoplus_{L_{j}\in\mathcal{L} }L_{j}\) is \(\pi\)-ample, that is, if for every \(k\)-rational point of \(\mathcal{X}\) the representation of the stabilizer group at that point is faithful.
Finally, if \(G\leq\text{GL}\left( n,k\right) \) is finite and abelian, then the authors show how to recover the stack \(\left[ \mathbb{A}^{n}/G\right] \) as well as Hilb\(^{G}\left( \mathbb{A}^{n}\right) .\) quivers; quiver representations; projective stacks; toric stacks DOI: 10.1016/j.jalgebra.2011.08.033 Stacks and moduli problems, Representations of quivers and partially ordered sets Quivers of sections on toric orbifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The relation between Hilbert schemes of finite-length subschemes and crepant resolution of singularities, via the McKay correspondence, is fairly well understood now after the work of Ito and many others in the case of surfaces in characteristic zero. For higher-dimensional varieties there is progress but still much more to be done. As far as the reviewer can tell, this is the first serious attempt to understand the situation in positive characteristic. Many of the constructions are valid for subschemes of arbitrary length on arbitrary smooth varieties, but the detailed results are limited to \({\text{Hilb}}^{2}(S)\) for \(S\) a smooth surface, which is the most accessible case but quite possibly also the most interesting one.
Another, more specific use of \({\text{Hilb}}^{n}(S)\) is in Beauville's generalised Kummer construction. In this case \(S=A\) is an abelian surface: the Hilbert-Chow morphism \({\text{Hilb}}^{n}(A)\to{\text{Sym}}^{n}(A)\), composed with addition in \(A\), gives a map \({\text{Hilb}}^{n}(A)\to A\) and the fibre over \(0\) is called the generalised Kummer variety \({\text{Km}}^{n}(A)\). In characteristic zero, both \({\text{Hilb}}^{n}(A)\) and \({\text{Km}}^{n}(A)\) are smooth (and have trivial dualising sheaf), but in positive characteristic this fails for \({\text{Km}}^{n}(A)\). This phenomenon, too, is explored here in characteristic~\(2\) for \(n=2\). It turns out that \({\text{Hilb}}^{2}(A)\to A\) is a quasifibration, i.e.\ all the fibres are nonsmooth. The third strand of this paper is to study the singularities of the fibres, especially \({\text{Km}}^{2}(A)\), in this case. They depend strongly on \(A\), via the singularities of the usual Kummer variety \(A/\pm 1\), which were studied by Shioda and by Katsura in the 1970s.
Thus the author is led to revisit a large part of classical surface singularity theory, but in characteristic~\(2\). A key role is played by Artin's wild involutions on surfaces. These, and the consequences of blowing up the associated surface singularities, are studied in Section~1 of the paper. The blow-up does not resolve the singularity (it is not even normal in general) but does behave cohomologically like a resolution, and this provides a counterpart in the cases under discussion to the McKay correspondence. This relation to the Hilbert scheme is discussed in Sections~2 and~3. A byproduct is examples of nonrational canonical singularities (another positive characteristic phenomenon).
Section~4 deals with the special case of rational double points (quotient singularities need not be rational in positive characteristic), listing the cases that may arise on \(G\)-Hilbert schemes.
The next part of the paper, Sections~5--7, is concerned with the \(S=A\) and \(G=\{\pm 1\}\), the Kummer surface case. The most difficult case by far is when \(A\) is supersingular: to understand it involves an excursion into Laufer's theory of minimal elliptic singularities. Within this, the superspecial case \(A=E\times E\), where \(E=(y^2=x^3+x)\), requires its own treatment.
Section~8 is a technical one about local algebraic conditions on symmetric products of surfaces. This, and the study of symmetric products of abelian surfaces that follows in Section~9, allows us to identify the Kummer variety \(A/\pm 1\) with the closed fibre of the addition map \({\text{Sym}}^{2}(A)\to A\). Now everything fits together and the relation between \({\text{Km}}^{2}(A)\) and \(A/\pm 1\) and its singularities is fully elucidated in the final Section~10. Hilbert-Chow morphism; wild involutions; McKay correspondence doi:10.1007/s11512-007-0065-6 Singularities of surfaces or higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), McKay correspondence The Hilbert scheme of points for supersingular abelian surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author presents (in expository form) results and recent questions on the relation between simple singularities and the corresponding finite groups. The questions treated here arose from conjectures of Grothendieck, solved by Brieskorn and published first time with complete proofs and additional contributions by the author in his book ``Simple singularities and simple algebraic groups'', Lect. Notes Math. 815 (1980; Zbl 0441.14002). Let \(X=\mathbb{C}^ 2/F\), \(F\subset SU(2,\mathbb{C})\) a finite subgroup, then \(X\) is a surface with an isolated singularity, and the dual graph of its minimal resolution is a Dynkin diagram of type \(A_ n\), \(D_ n\), \(E_ 6\), \(E_ 7\), \(E_ 8\) respectively, corresponding to the cyclic group of order \(n\), the binary dihedral group of order \(4n\), the binary tetrahedral, octahedral, icosahedral group, respectively. The 2-dimensional homology of the minimal resolution can be obtained from the corresponding root system \(A_ n\), \(D_ n\), \(E_ n\), its intersection form is given by the Cartan matrix \(C\). Conversely, if \(C\) is given corresponding to some Dynkin diagram, the group \(F\) can be expressed in terms of generators and relations by the elements of \(C\).
On the other hand, let \(G\) be a simply connected simple Lie group of type \(A_ n\), \(D_ n\), \(E_ n\), \(T\subset G\) a maximal torus, \(W=N_ G(T)/T\) the corresponding Weyl group, \({\mathcal X}:G\to T/W\) the quotient by the adjoint action of \(G\) on itself. Then the theorem of Brieskorn asserts: If \(S\) is a section transversal to the subregular unipotent orbit, then \(\mathcal X\) restricted to \(S\) is a versal deformation of the singularity of the same type. [Note that a different construction was recently found by \textit{F. Knop}, Invent. Math. 90, 579-604 (1987; Zbl 0648.14002); it gives some surprising deformations of the singularities in characteristic 2 and 3.]
There arises the question of a more direct connection between the Lie group we started with and the finite group \(F\). A remark of \textit{J. McKay} [cf. Proc. Am. Math. Soc. 81, 153-154 (1981; Zbl 0477.20006)] relates \(F\) with the Dynkin diagram by means of representation theory: If \(N\) is the 2-dimensional representation \(F\subset SU(2,\mathbb{C})\), \(R_ 0,\ldots,R_ r\) (representants of) the irreducible representations of \(F\) and \(N\otimes R_ i=\oplus_ ja_{ij}R_ j\), then \(2E_{r+1}- (a_{ij})\) is the Cartan matrix of the extended Dynkin diagram associated to the group \(F\) [later efforts to understand McKay's correspondence are related with the socalled Auslander-Reiten theory, cf. \textit{M. Auslander} and \textit{I. Reiten}, Trans. Am. Math. Soc. 302, 87-97 (1987; Zbl 0617.13018); results of Artin, Verdier, Esnault, Knörrer, Buchweitz, Greuel, Schreyer and others concern the maximal Cohen-Macaulay modules over simple singularities, cf. e.g. \textit{H. Knörrer} in Representations of algebras, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 116, 147-164 (1986; Zbl 0613.14004)]. Let \({\mathcal X}_ i\) denote the character of the representation \(R_ i\), \(d_ i={\mathcal X}_ i(1)\), then \(d=(d_ 0,\ldots,d_ r)\) generates the kernel of \(2E_{r+1}-(a_{ij})\), \(d_ i\) are the coefficients of the maximal root in the corresponding root system and give the fundamental cycle of the minimal resolution of \(\mathbb{C}^ 2/F\). Further, \(\sum d_ i\) is the Coxeter number of the root system. Until now, the Dynkin diagrams of type \(B\), \(C\), \(F\) and \(G\) are missing; this is explained in the following way: They can be obtained by factorizing some of the homogeneous diagrams by a group \(\Gamma\) of symmetries. A simple singularity of type \(B_ n\), \(C_ n\), \(F_ 4\), \(G_ 2\), respectively is defined to be a couple \((X,\Gamma)\), where \(X\) is one of \(A_{2n-1}\), \(D_{n+1}\), \(E_ 6\), \(D_ 4\), respectively, such that \(\Gamma\) acts as a group of automorphisms. The author's results generalize the Brieskorn theorem to singularities of type \(B\), \(C\), \(F\), \(G\): If \(G\) is a simple Lie group of one of the above types, \(T\subset G\) a maximal torus, \(W\) the corresponding Weyl group, \({\mathcal X}:G\to T/W\) the adjoint quotient, a versal deformation of the singularity of the same type is obtained as follows: Take a transversal section \(S\) to the orbit of a unipotent subregular element \(x\in S\) such that \(S\) is stabilized by a reductive subset of \(Z_ G(x)\). Then \(\mathcal X\) restricted to \(S\) induces an equivariant versal deformation. \(S\cap\text{Uni}(G)\) is a singularity of type \(A_{2n-1}\), \(D_{n+1}\), \(E_ 6\), \(D_ 4\), respectively, and the action of \(\Gamma\) is given by a subgroup of \(Z_ G(x)\). A refined result is obtained considering the mentioned groups together with their automorphisms. -- Again, the preceding diagrams \(B\), \(C\), \(F\), \(G\) are considered from the viewpoint of representation theory of finite groups, this time in the relative case.
The author formulates as a dominating question the explanation of the relation between the finite group \(F\) and the corresponding Lie group, to show why the simple singularity \(\mathbb{C}^ 2/F\) appears on the unipotent variety of \(G\). finite subgroup of \(Sl(2,\mathbb{C})\); simple singularities; Dynkin diagram; homology of minimal resolution; root system; Cartan matrix; simple Lie group; McKay correspondence; Coxeter number Singularities in algebraic geometry, Representations of finite symmetric groups, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, General properties and structure of complex Lie groups On finite groups associated to simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let V be a finite-dimensional vector space over an algebraically closed field K of characteristic zero, G a subgroup of GL(V). The class of G is, by definition, the minimum of the numbers rk(h-1), taken over h running through G, \(h\neq 1\); it is denoted by cl(G). A finite group is called known if the simple noncyclic quotients of its composition series are either alternative groups, or groups of Lie type or simple groups taken from some finite list (if one assumes the classification of finite simple groups to be finished then any finite group is known).
The following theorems are proved in the paper under review: Theorem. Let \(G\subseteq GL(V)\) be a known finite group. Assume that G is irreducible, quasiprimitive and \(cl(K^*G)\ll n\), where \(n=\dim V\). Then either \(cl(K^*G)=cl(G)\) or \(cl(K^*G)=(1/2)cl(G)\) and \(A_ m\otimes G_ 1\triangleleft G\triangleleft S_ m\otimes G_ 2\), where \(G_ 1\leq G_ 2\leq GL_ s(K)\), \((m-1)s=n\), \(s=cl(G)\) or \(s=(1/2)cl(G)\), \([G_ 2:G_ 1]\leq 2.\)
Let \(R=K[V]^ G\) be the algebra of invariant polynomial functions on V. Theorem. Let G be a known finite group. Assume that G is irreducible, quasiprimitive and codim \(R\ll n=\dim V\). Then \(G=A_{n+1}\) or \(S_{n+1}\). Theorem. Let G be a semisimple algebraic group. Then:
(1) codim \(R\geq (cl G/d(G))-2\dim G-1;\)
(2) codim \(R\geq (\sqrt{\dim V}/12[G:G^ 0]| Z(G^ 0)| \cdot rk G\cdot d(G))-2\dim G-1\), if \(V^ G=0;\)
(3) codim \(R\geq (\dim V/12[G:G^ 0]\cdot rk G\cdot d(G))-2\dim G-1\) and
codim \(R\geq (\dim V/34[G:G^ 0]\cdot d(G))-2\dim G-1\)
if V is an irreducible G-module, where d(G) is the order of a certain element of the Weyl group of G. finite-dimensional vector space; class; finite group; composition series; alternative groups; groups of Lie type; simple groups; algebra of invariant polynomial functions; semisimple algebraic group; Weyl group N. L. Gordeev, Coranks of elements of linear groups and the complexity of algebras of invariants, Algebra i Analiz 2 (1990), no. 2, 39 -- 64 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 2, 245 -- 267. Linear algebraic groups over arbitrary fields, Classical groups (algebro-geometric aspects), Vector and tensor algebra, theory of invariants, Simple groups Coranks of elements of linear groups and complexity of algebras of invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the review in Zbl 0703.20038. finite-dimensional vector space; class; finite group; composition series; alternative groups; groups of Lie type; simple groups; algebra of invariant polynomial functions; semisimple algebraic group; Weyl group Gordeev, N.: Coranks of elements of linear groups and the complexity of algebras of invariants. Leningrad math. J. 2, 245-267 (1991) Linear algebraic groups over arbitrary fields, Classical groups (algebro-geometric aspects), Vector and tensor algebra, theory of invariants, Simple groups, Finite simple groups and their classification Coranks of elements of linear groups, and complexity of algebras of invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0597.00009.]
In this elegant survey paper, the authors survey some recent results of V. Kac's, that describe the dimension types of all indecomposables of arbitrary quivers. The starting points of the paper are the well-known facts due to P. Gabriel resp. P. Donovan - M. R. Freislich and L. A. Nazarova, that a connected quiver is of finite representation type resp. tame if and only if its associated graph is a Dynkin diagram resp. an extended Dynkin diagram. In the first case, the dimension type \underbar{dim} induces a bijection between (isomorphism classes of) indecomposable representations of the quiver Q and positive roots of Q, i.e. the vectors \(\alpha \in {\mathbb{N}}^ n\) with \(q(\alpha)=1\), where q is the (positive definite!) Tits form of Q and \(n=| Q|\). In the tame case a similar relation may be given as well. Of course, all other quivers are wild, showing the depth of Kac's surprising results.
Although most results in the paper are interesting in their own right, let us concentrate on its main result: Kac's theorem. From a geometric point of view, one considers the \(\alpha\)-representation space of Q, i.e. the affine varieties R(Q,\(\alpha)\) of all representations of Q of dimension type \(\alpha =(\alpha (1),...,\alpha (n))\in {\mathbb{N}}^ n\). The group \(G\ell (\alpha)=\prod G\ell (\alpha (i))\) acts linearly on R(Q,\(\alpha)\), and one wants to study the orbits of \(G\ell (\alpha)\) in R(Q,\(\alpha)\). One of the tools to study these, is by considering their number of parameters. In general, if G is an algebraic group acting on some variety V and if \(X\subset V\) is a G-stable subset, we let \(X_{(s)}\) consist of all \(x\in X\) with dim \(O_ x=s\) and we then define the number of parameters of X to be \(\mu (X)=\max (\dim X_{(s)}- s)\). In particular, we may thus consider the number of parameters \(\mu (R(Q,\alpha)_{ind})\) of the indecomposables in R(Q,\(\alpha)\). Let us call a vector \(\alpha \in {\mathbb{N}}^ n\) a root of Q if R(Q,\(\alpha)\) contains an indecomposable representation. If \(\mu (R(Q,\alpha)_{ind})\geq 1\), then we call \(\alpha\) an imaginary root, otherwise, we speak of a real root (i.e. \(R(Q,\alpha)_{ind}\) contains a finite number of orbits). So, the set of all roots \(\Delta\) (Q) is partitioned into \(\Delta (Q)=\Delta (Q)_{re}\cup \Delta (Q)_{im}.\)
On the other hand, a simple root is a standard basic vector \(e_ i\) of \({\mathbb{Q}}^ n\), where i is a vertex to which no loop is attached. We denote by \(\pi_ Q=\Delta (Q)_{re}\) the set of all simple roots. Clearly, \(e_ i\) is a simple root if and only if \(e_ i\) does not belong to \(F_ Q\), the fundamental set of Q, which consists of all non- zero \(\alpha \in {\mathbb{N}}^ n\) with connected support, such that \((\alpha,e_ i)\leq 0\). Finally, denote by \(W_ Q\subset Gl({\mathbb{Z}}^ n)\) the Weyl group of Q, generated by the reflections \(r_ i: {\mathbb{Z}}^ n\to {\mathbb{Z}}^ n:\alpha \to \alpha -(\alpha,e_ i)e_ i\), associated to the simple roots \(e_ i\). Of course, if Q is a Dynkin quiver, then these definitions are just the classical ones, so in particular \(W_ Q\pi_ Q=\Delta (Q)\cup -\Delta (Q)\), is the corresponding root system.
Kac's theorem now states that \(\Delta (Q)_{re}=W_ Q\pi_ Q\cap {\mathbb{N}}^ n\) resp. \(\Delta (Q)_{im}=W_ QF_ Q\). Moreover, if \(\alpha \in \Delta (Q)_{re}\), then \(R(Q,\alpha)_{ind}\) is one orbit, whereas for \(\alpha \in \Delta (\Omega)_{im}\), we have \(\mu (R(Q,\alpha)_{ind})=1-q(\alpha)\), where, as before, q is the Tits form associated to Q. The proof of this result heavily relies upon the fact that for any \(\alpha \in {\mathbb{N}}^ n\) the number of isomorphism classes of indecomposables V with \underbar{dim} V\(=\alpha\), as well as \(\mu (R(Q,\alpha)_{ind})\) only depend upon the underlying graph of Q. dimension types; indecomposables of arbitrary quivers; finite representation type; extended Dynkin diagram; indecomposable representations; positive roots; Tits form; tame; Kac's theorem; affine varieties; orbits; simple roots; reflections; Dynkin quiver; root system H. Kraft and Ch. Riedtmann, Geometry of representations of quivers, Representations of algebras (Durham, 1985) London Math. Soc. Lecture Note Ser., vol. 116, Cambridge Univ. Press, Cambridge, 1986, pp. 109 -- 145. Representation theory of associative rings and algebras, Finite rings and finite-dimensional associative algebras, Simple, semisimple, reductive (super)algebras, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Group actions on varieties or schemes (quotients), Vector and tensor algebra, theory of invariants Geometry of representations of quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field of finite characteristic \(p\), and let \(G\) be a finite group scheme whose order is a multiple of \(p\). Let \(kG\) be the dual of the coordinate algebra \(k[G]\). In the study of \(kG\)-modules one can consider \(\pi\)-points, flat \(K\)-algebra maps \(\alpha_K\colon K[t]/(t^p)\to KG\) for \(K\) an extension of \(k\). Two \(\pi\)-points \(\alpha_K,\beta_L\) are equivalent if for any finite dimensional \(kG\)-module \(M\) the restriction of \(M_K\) along \(\alpha_K\) is a free \(KG\)-module if and only if the restriction of \(M_L\) along \(\beta_L\) is free. If the rank of the linear operator \(\alpha_K(t)\) on \(M_K\) is independent of the choice of \(\alpha_K\) we say \(M\) has constant rank; moreover if the Jordan type of \(\alpha_K(t)\) is independent of the choice of \(\pi\)-point we say \(M\) is of constant Jordan type. This work is a study of \(k(\mathbb Z/p\times\mathbb Z/p)\)-modules of constant Jordan type, paying particular attention to a subcategory of such modules, \(W\)-modules, which are introduced here.
For \(G=(\mathbb Z/p)^r\), \(kG=k[t_1,\dots,t_r]/(t_1^p,\dots,t_r^p)\), a \(\pi\)-point \(\alpha_K\) is equivalent to the image of \(t\in K[t]/(t^p)\) -- denote this point in \(KG\) by \(\ell_{\alpha_K}\), or \(\ell_\alpha\) for short. A finite dimensional \(kG\)-module \(M\) has the equal images property if for any two \(\pi\)-points \(\alpha_K\) and \(\beta_L\) the images of \(\ell_\alpha(M_K)\) and \(\ell_\beta(M_L)\) agree after base change to a field extension \(\Omega\) of both \(K\) and \(L\). There are a number of equivalent formulations of the equal images property, such as for all \(\ell\in\text{Rad}(KG)\setminus\text{Rad}(KG)^2\) we have \(\ell(M_K)=\text{Rad}(M_K)\). As an example, if \(I=\text{Rad}(kG)\) then \(I^j\), a module of constant Jordan type, has the equal images property if and only if \((r-1)(p-1)\leq j\). More generally, if \(M\) has the equal images property, then \(M\) has constant Jordan type. Furthermore, for \(L\subset M\) a \(kG\)-submodule and \(M\) possesses the equal images property, then so does the quotient \(M/L\).
After the formulation of the above property, attention is focused on the case \(G=\mathbb Z/p\times\mathbb Z/p\). Then we may write \(kG=k[t_1,t_2]/(t_1^p,t_2^p)\). Let \(x\) and \(y\) denote the classes of \(t_1\) and \(t_2\) in \(\text{Rad}(kG)\): \(x\) and \(y\) clearly generate this radical. For \(1\leq d\leq n\), \(d\leq p\), let \(W_{n,d}\) be the \(kG\)-module generated by \(\{v_1,\dots,v_n\}\) subject to the relations \(xv_1=yv_n=x^dv_n=0\) and \(x^dv_i=yv_i-xv_{i+1}=0\) for \(1\leq i<n-1\). For \(d>n\) let \(W_{n,d}=W_{n,n}\). Collectively, these are called \(W\)-modules. It is shown that \(W\)-modules have the equal images property and a formula for the (constant) Jordan type is given. For \(n\leq p\) the classes \([W_{n,n}]\) form a minimal generating set of the Grothendieck group \(K_0(\mathcal CW)\) of the category of \(kG\)-modules of the form \(W_{n,d}\); furthermore \(K_0(\mathcal CW)\cong\mathbb Z^p\), the isomorphism arising from composing the map \(K_0(\mathcal CW)\to K_0(\mathcal C)\), where \(\mathcal C\) is the category of modules of constant Jordan type, with the Jordan type mapping.
One reason for focusing on the \(W\)-modules is their prevalence in \(\mathcal C\). For example, if \(\text{Rad}^2(M)\) is trivial, then \(M\) decomposes into a direct sum of \(W\)-modules; furthermore the decomposition is into \(W\)-modules with \(n=1\) or \(2\). Also, any \(kG\) module with the equal images property can be realized as a quotient of \(W_{n,d}\) for some \(n\), where \(\text{Rad}^d(M)\) is trivial.
Suppose for this paragraph that \(k\) is infinite. To any finite dimensional \(kG\)-module \(M\) and any point \(\langle a,b\rangle\in\mathbb P^1(k)\) we can define \(_{\langle a,b\rangle}M=\text{Ker}\{ax+by\colon M\to M\}\); more generally for \(S\subset\mathbb P^1(k)\) we let \(_SM\) be the sum of \(_{\langle a,b\rangle}M\) over all \(\langle a,b\rangle\in\mathbb P^1(k)\). The generic kernel \(\mathfrak R(M)\) is then the intersection of all \(_SM\) with \(S\subset\mathbb P^1(k)\) cofinite. The generic kernel is compatible with extending the field \(k\) in a natural way, a fact needed to prove that \(\mathfrak R(M)\) also has the equal images property. In fact, \(\mathfrak R(M)\) is the maximal submodule of \(M\) with the equal images property, and \(\mathfrak R(M)\) is the maximal submodule of \(M\) arising as the quotient of a \(W\)-module.
Let \(W\) be the generic kernel of \(M\), a module of constant rank. Then an increasing filtration of \(M\) is given, namely by submodules of the form \(x^iW\) for \(1-p\leq i\leq p-1\). In this filtration, \(x^iW\) has the equal images property and \(M/x^iW\) has the equal kernels property.
Finally, it is shown that any cyclic \(kG\)-module of constant Jordan type is a quotient of \(kG\) by a power of the augmentation ideal. This holds regardless of the characteristic of \(k\). finite group schemes; coordinate algebras; modules of constant Jordan type; \(W\)-modules; equal images property; modular representations; Grothendieck groups Carlson, J. F.; Friedlander, E. M.; Suslin, A. A., Modules for \(\mathbb{Z} / p \times \mathbb{Z} / p\), Comment. Math. Helv., 86, 609-657, (2011) Representation theory for linear algebraic groups, Group schemes, Cohomology theory for linear algebraic groups, Modular representations and characters, Representations of associative Artinian rings Modules for \(\mathbb Z/p\times\mathbb Z/p\). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper deals with the following problem: Given a finitely generated associative algebra over an algebraically closed field, the algebraic variety of \(d\)-dimensional modules is endowed with a natural \(\text{Gl}_d\)-action. Various problems associated with this action are studied in the paper. Some strong results are obtained for algebras of finite representation type.
The author considers pointed varieties of the form \((\overline {O(m)},n)\) where \(\overline {O(m)}\) is the closure of the orbit of a module \(M\) and \(n\) is a minimal degeneration of \(M\) (see the paper for definitions). He calls the pointed varieties of the form \((\overline {O(m)}, n)\) minimal singularities.
The main theorem asserts that all minimal singularities occurring in representations of Dynkin quivers are very smoothly equivalent to \((D(p,q),0)\) where \(D(p,q)\) is the set of \(p \times q\) matrices with rank \(\leq 1\). The theorem at the end of section 1 relates degenerations of two distinct finite dimensional modules. It is a fundamental result used in the rest of the paper.
With this result the author derives the famous result of \textit{H. Kraft} and \textit{C. Procesi} on minimal singularities of conjugacy classes of nilpotent matrices [which appeared in Invent. Math. 53, 227-247 (1979; Zbl 0434.14026)] and states that in this setting any minimal singularity is equivalent to the subregular singularity inside the set of nilpotent matrices of some smaller size or to the singularity at 0 inside the set of all nilpotent matrices of rank at most one.
Section 4 deals with tilting modules, in particular corollary 1 shows a very close relation between the \(\text{Gl}_d\)-stable subsets of \(Y(\underline {d})\) and \(\text{Gl}_e\)-stable subsets of \(Y(\underline {e})\). Here a tilting module \(Y(\underline {d})\) consists of the category of torsion \(A\)-modules of vector dimension \(\underline {d}\). \(Y\) is the subcategory of \(B\)-mod corresponding to \(\tau\) (the torsion free part), and \(Y(\underline {e})\) is the full subcategory of \(Y\) whose objects have vector dimension \(\underline {e}\).
If \([M] = \underline {d}\) then \(\underline {e} = [\text{Hom} (T,M)] - [\text{Ext}^1 (T,M)]\). Theorem 3 and its corollary are very beautiful applications to tilting theory. Section 5 studies possible reduction of the underlying Gabriel quiver. Under reduction of the Gabriel quiver with some technical hypotheses the author gets associated pointed varieties which are very smoothly equivalent. Section 6 studies the minimal degeneration in the cases where the partial orders \(\leq\) and \(\leq_{\text{Ext}}\) are equivalent. This equivalence of the partial orders \(\leq_{\text{Ext}}\) and \(\leq\) is valid for preprojective modules; the author uses this fact and gets that any minimal degeneration of representations of a Dynkin quiver is of codimension one and then gets the main result which is Theorem 6. The proof of this result is very technical and complex.
The entire paper uses techniques of algebraic geometry applied to the representation theory of algebras. This paper is certainly a very nice one, although it requires from the reader a good knowledge of representation theory and algebraic geometry. finitely generated algebras; algebraic variety; \(\text{Gl}_ d\)-actions; algebras of finite representation type; pointed varieties; minimal degenerations; minimal singularities; representations of Dynkin quivers; nilpotent matrices; tilting modules; tilting theories; Gabriel quivers; preprojective modules Bongartz, K.: Minimal singularities for representations of Dynkin quivers. Comment. Math. Helv. 69(4), 575--611 (1994) Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Group actions on varieties or schemes (quotients), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities in algebraic geometry Minimal singularities for representations of Dynkin quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Dynkin diagram; simple Lie groups; simple singularities; representations of Weyl groups; monodromy transformations Slodowy, P., Four lectures on simple groups and singularities, \textit{Communications of the Math. Institute}, (1980) Semisimple Lie groups and their representations, Research exposition (monographs, survey articles) pertaining to topological groups, Singularities in algebraic geometry, Local complex singularities, Modifications; resolution of singularities (complex-analytic aspects) Four lectures on simple groups and singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In his previous work [\textit{W. Wang}, Duke Math. J. 103, 1--23 (2000; Zbl 0947.19004)], the author has indicated the deep analogy and connections between (A) the theory of the Hilbert scheme \(X^{[n]}\) of \(n\) points on a (quasi-) projective surface \(X\) [cf. e.g. \textit{H. Nakajima}, Lectures on Hilbert schemes of points on surfaces (University Lecture Series 18, Providence, RI: AMS)(1999; Zbl 0949.14001)], and (B) the theory of wreath products \(\Gamma_{[n]} =\Gamma^n\rtimes S_n\) between a power \(\Gamma^n\) of a finite group \(\Gamma\) and the symmetric group \(S_n\) [cf. e.g. \textit{A. Zelevinsky}, Representations of finite classical groups. A Hopf algebra approach (Lecture Notes in Mathematics 869, Berlin: Springer) (1981; Zbl 0465.20009)]. The reviewed article is a survey that is related to many questions and works of this topic: first of all, the construction of the Heisenberg and Virasoro algebras in the framework of (A), with description of the cohomology ring of \(X^{[x]}\) and in the framework of (B), with description of vertex representations of affine and toroidal Lie algebras, whose Dynkin diagrams correspond to finite subgroups \(\Gamma\) of \(\text{SL}_2(\mathbb{C})\), a.o.
If \(Y\) is a quasi-projective surface acted by a finite group \(\Gamma\), and a resolution of singularities \(X\to Y/\Gamma\) (1) is given, then, according to the author [loc.cit.], one obtains the resolution of singularities \(X^{[n]} \to Y^n/\Gamma_{[n]}\) (2) that can preserve many properties of the resolution (1), in particular, some of them that are concerned with the Euler and Hodge numbers of the corresponding orbifolds.
Under some additional conditions, equivalence between the bounded derived categories \(D_{\Gamma_{[n]}}(Y^n)\) and \(D(X^{[n]})\) of \(\Gamma_{[n]}\)-equivariant coherent sheaves on \(Y^n\) and coherent sheaves on \(X^{[n]}\) respectively is established. The case \(Y= \mathbb{C}^2\) is considered in more detail, in particular, the question of replacement of \(X^{[n]}\) (in case of a certain known minimal \(X\) in (1)) by a special subvariety \(Y_{\Gamma,n}\) of \((\mathbb{C}^2)^{[nN]}\) (where \(N\) is the order of \(\Gamma\)) that admits a quiver description.
Finally, a short dictionary for comparison of analogous notions in the theories (A) and (B) is presented. algebraic surfaces; action of finite groups; resolution of singularities; Heisenberg algebra; Virasoro algebra; representation of Lie algebras; vertex algebras; equivariant \(K\)-theory of schemes W. Wang, Algebraic structures behind Hilbert schemes and wreath products, in: S. Berman et al. (Eds.), Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory, Charlottesville, VA, 2000; Contemp. Math. 297 (2002) 271-295. Parametrization (Chow and Hilbert schemes), Virasoro and related algebras, Vertex operators; vertex operator algebras and related structures, Extensions, wreath products, and other compositions of groups Algebraic structures behind Hilbert schemes and wreath products. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities To any finite group \(\Gamma\subset\text{Sp}(V)\) of automorphisms of a symplectic vector space \(V\) we associate a new multi-parameter deformation, \(H_\kappa\) of the algebra \(\mathbb{C}[V]\#\Gamma\), smash product of \(\Gamma\) with the polynomial algebra on \(V\). The parameter \(\kappa\) runs over points of \(\mathbb{P}^r\), where \(r=\) number of conjugacy classes of symplectic reflections in \(\Gamma\). The algebra \(H_\kappa\), called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity \(V/\Gamma\). This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If \(\Gamma\) is the Weyl group of a root system in a vector space \(\mathfrak h\) and \(V={\mathfrak h}\oplus{\mathfrak h}^*\), then the algebras \(H_\kappa\) are certain `rational' degenerations of the double affine Hecke algebra introduced earlier by Cherednik.
Let \(\Gamma=S_n\), the Weyl group of \({\mathfrak g}=\mathfrak{gl}_n\). We construct a 1-parameter deformation of the Harish-Chandra homomorphism from \({\mathcal D}({\mathfrak g})^{\mathfrak g}\), the algebra of invariant polynomial differential operators on \(\mathfrak{gl}_n\), to the algebra of \(S_n\)-invariant differential operators with rational coefficients on the space \(\mathbb{C}^n\) of diagonal matrices. The second order Laplacian on \(\mathfrak g\) goes, under the deformed homomorphism, to the Calogero-Moser differential operator on \(\mathbb{C}^n\), with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: \({\mathcal D}({\mathfrak g})^{\mathfrak g}\twoheadrightarrow\) spherical subalgebra in \(H_\kappa\), where \(H_\kappa\) is the symplectic reflection algebra associated to the group \(\Gamma=S_n\). This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of `quantum' Hamiltonian reduction.
In the `classical' limit \(\kappa\to\infty\), our construction gives an isomorphism between the spherical subalgebra in \(H_\infty\) and the coordinate ring of the Calogero-Moser space. We prove that all simple \(H_\infty\)-modules have dimension \(n!\), and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of \(S_n\). Moreover, we prove that the algebra \(H_\infty\) is isomorphic to the endomorphism algebra of that vector bundle. finite groups of automorphisms; symplectic vector spaces; multi-parameter deformations; symplectic reflection algebras; coordinate rings; McKay correspondence; Weyl groups; double affine Hecke algebras; Harish-Chandra homomorphisms; invariant polynomial differential operators; Calogero-Moser differential operators P. Etingof and V. Ginzburg, Symplectic reflection algebras, Calogero--Moser space, and deformed Harish-Chandra homomorphism, \textit{Invent. Math.}, 147 (2002), no. 2, 243--348. Zbl 1061.16032 MR 1881922 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Group actions on varieties or schemes (quotients), Lie algebras of vector fields and related (super) algebras, Applications of Lie algebras and superalgebras to integrable systems, Hecke algebras and their representations, Rings arising from noncommutative algebraic geometry, Deformations of associative rings, Noncommutative algebraic geometry Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic \(p>0\). Let \(\mathcal G\) be a finite algebraic group, and let \(k\mathcal G\) be its algebra of measures. Then \(k\mathcal G\) is a finite dimensional cocommutative \(k\)-Hopf algebra, and there is a unique block -- the ``principal block'' -- denoted \(\mathcal B_0(\mathcal G)\) of \(k\mathcal G\) such that \(\varepsilon(\mathcal B_0(\mathcal G))\neq 0\), where \(\varepsilon\colon k\mathcal G\to k\) is the counit. For \(\Lambda\) an associative \(k\)-algebra, the group \(\text{GL}_d(k)\) acts on the variety of \(d\)-dimensional \(\Lambda\)-modules, and we let \(C_d\) be a closed subset of minimal dimension such that \(\text{GL}_d(k)\cdot C_d\) contains the set of indecomposable \(\Lambda\)-modules of dimension \(d\). If \(C_d=0\) for all \(d\) then \(\Lambda\) is said to be representation finite; if this is not the case and \(C_d\) has dimension at most \(1\) for all \(d\) then \(\Lambda\) is said to be tame; otherwise, \(\Lambda\) is said to be wild. Information about the principal block \(\mathcal B_0(\mathcal G)\) can give results about the representation type of \(\mathcal G\): for example, \(\mathcal B_0(\mathcal G)\) is simple if and only if \(k\mathcal G\) is semisimple, in which case \(\mathcal G\) is representation-finite.
As the classification of the indecomposable modules in the wild case is (from the paper) ``deemed hopeless'', in this survey article (based on a series of lectures at the Advances in Group Theory and its Applications conference in 2011), the author investigates conditions for when \(\mathcal B_0(\mathcal G)\) is representation-finite or tame. The objective of this work is to give an overview of these conditions, consequently most of the results are provided without proof (although a citation is provided for each unproven result). representation types; Dynkin diagrams; quivers; representations of finite group schemes; cohomological support varieties; indecomposable modules; cocommutative Hopf algebras; module categories; finite representation type Representation type (finite, tame, wild, etc.) of associative algebras, Representations of quivers and partially ordered sets, Group schemes, Hopf algebras and their applications, Modular Lie (super)algebras Dynkin diagrams, support spaces and representation type. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities As the title indicates, the book under review is concerned with the conjugacy classes and the representation theory of finite groups of Lie type. Let G be a connected reductive algebraic group defined over a finite field, F a Frobenius morphism of G, and \(G^ F\) the finite group of F-fixed points. The finite groups obtained in this way are the groups of Lie type. These groups inherit some of the structure of the algebraic group G, and in particular they have a split BN-pair with Weyl group W, where B is an F-fixed Borel subgroup containing an F-fixed maximal torus T (such a torus is said to be maximally split), \(N=N(T)\) and \(W=N/T\). The book therefore begins in Chapter 1 with an exposition (without proofs) of the theory of algebraic groups over an algebraically closed field. The BN-pair axioms are given and some consequences are derived in Chapter 2, including the pattern of intersections of parabolic subgroups. Chapter 3 continues the basic material by describing the classification of \(G^ F\)-conjugacy classes of maximal tori in G; these are in bijection with the F-conjugacy classes of W.
The representation theory of finite groups of Lie type entered a new phase in the early seventies with the development of what is now known as Harish-Chandra theory by Harish-Chandra and \textit{T. A. Springer} [see Cusp forms in finite groups, in Lect. Notes Math. 131, C1-C24 (1970; Zbl 0263.20024)]. An even more exciting breakthrough was achieved by the paper of \textit{P. Deligne} and \textit{G. Lusztig} in 1976 [Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)]. Let T be any F-fixed maximal torus of the algebraic group G as above. Then for any character \(\theta\) of \(T^ F\) (i.e. a homomorphism into \(\bar Q^*_{\ell}\), for a prime \(\ell\) not equal to the characteristic of the finite field) Deligne and Lusztig (loc. cit.) defined a virtual representation, or equivalently, a generalized character \(R^ G_ T(\theta)\) of \(G^ F\) over \(\bar Q_{\ell}\) by using the \(\ell\)-adic cohomology of certain subvarieties of the flag variety. These characters have the expected properties of orthogonality, and indeed \(R^ G_ T(\theta)\) and \(R^ G_{T'}(\theta ')\) have no irreducible constituents in common unless the pairs (T,\(\theta)\), (T',\(\theta\) ') are related by ''geometric conjugacy''. If \(G^*\) is a group in duality with G, the geometric conjugacy classes of pairs (T,\(\theta)\) are in bijection with the F-stable semisimple conjugacy classes of \(G^{*F}\). The concepts of geometric conjugacy and duality are explained in Chapter 4. If \(\theta\) is in ''general position'', \(R^ G_ T(\theta)\) is irreducible up to sign. Moreover, if \(x=su\) is the Jordan decomposition of \(x\in G^ F\), the value of \(R^ G_ T(\theta)\) at x can be described in terms of \(\theta\) and of the values of the analogous generalized characters for \(C^ F_ G(s)\) at unipotent elements. Assuming the basic properties of \(\ell\)-adic cohomology (which are stated in an appendix) the author derives these properties of the Deligne-Lusztig characters in Chapter 7. In Chapter 8 he describes (when the center of G is connected) certain rational linear combinations of the \(R^ G_ T(\theta)\) which turn out to be irreducible characters and which he calls semisimple and regular characters. There is precisely one semisimple and one regular character in each geometric conjugacy class, and the regular characters are those appearing in the Gelfand-Graev representation, which is also described in this chapter.
The Harish-Chandra theory states that the irreducible representations of G fall into families in the following way: if \(\rho\) is an irreducible character of \(G^ F\) then \(\rho\) is a constituent of an induced character \(Ind^{G^ F}_{P^ F}(\tau)\) where \(\tau\) is the pullback of a parabolic subgroup \(P^ F\) of a ''cuspidal'' character of a Levi factor \(L^ F\) of \(P^ F\), and the pair (L,\(\tau)\) is unique up to \(G^ F\)-conjugacy. A connection of this theory with the Deligne-Lusztig theory is given by the fact that if \(\theta\) is a character of \(T^ F\) in general position, then the irreducible (up to sign) character \(R^ G_ T(\theta)\) is cuspidal if and only if T is anisotropic, i.e. not contained in any proper parabolic subgroup of G. These results are described in Chapter 9. Thus the Harish-Chandra theory leads to the problems of classifying the cuspidal characters and of decomposing the characters induced from cuspidal characters of parabolic subgroups. \textit{R. B. Howlett} and \textit{G. I. Lehrer} [Invent. Math. 58, 37-64 (1980; Zbl 0435.20023)] described the structure of the endomorphism algebras of such induced representations and this theory is described in Chapter 10.
The Deligne-Lusztig theory, on the other hand, leads naturally to the problems of decomposing the \(R^ G_ T(\theta)\) when \(\theta\) is not in general position, and of classifying the cuspidal constituents. As a first step one can take \(\theta =1\), and then the constituents of the \(R^ G_ T(1)\) are the unipotent characters. In a series of papers starting from 1977 Lusztig, working case by case, classified the unipotent characters, identified the cuspidal ones, and decomposed the \(R^ G_ T(1)\) (in some cases, for large q). For the classical groups this was completed in 1982. Finally in a brilliant tour de force [Characters of reductive groups over a finite field. Ann. Math. Stud. 107 (1984; Zbl 0556.20033)] he gave a decomposition of all the \(R^ G_ T(\theta)\) when G has a connected center. For this work he used the intersection cohomology theory defined by Goresky and MacPherson. A consequence of this work is that the irreducible characters of \(G^ F\) are in bijection with \(G^{*F}\)-conjugacy classes of pairs (s,\(\phi)\) where s is a semisimple element in \(G^{*F}\) and \(\phi\) is a unipotent character of \(C_{G^*}(s)\), thus yielding a ''Jordan decomposition'' of characters.
Lusztig's classification shows that the unipotent characters fall into families in a remarkable way. When G is F-split and \(\phi\) is an irreducible character of W let \(R_ w\) be \(R^ G_{T_ w}(1)\), where \(T_ w\) is a torus obtained from a maximally split torus by twisting by \(w\in W\), and let \(R_{\phi}=(1/| w|)\sum_{w}\phi (w)R_ w\). (There is a similar definition in the twisted case.) Then it is sufficient to decompose the \(R_{\phi}\), and Lusztig's results show that the constituents of the \(R_{\phi}\) are all in the same family. The decomposition is described by a ''Fourier transform'' matrix. These results are described without proofs in Chapters 12 and 13. Other topics described in these two chapters include cells of Weyl groups, the Springer correspondence between unipotent classes and Weyl group representations, and the Jordan decomposition of irreducible characters due to Lusztig mentioned earlier. Chapter 11 discusses representations of Coxeter groups as a preparation for Chapter 12.
As for conjugacy classes, a standard argument reduces the problem of classifying the conjugacy classes to that of classifying the semisimple and the unipotent classes. In good characteristics, Springer defined a G- equivariant bijection between the nilpotent variety in the Lie algebra of G and the unipotent variety in G. Thus we can look at the orbit of G on the nilpotent variety in the Lie algebra. Springer and Steinberg gave a classification of these G-orbits by weighted Dynkin diagrams by using the Jacobson-Morozov theorem when the characteristic is sufficiently large. The author and Bela gave another approach using results of Richardson. Both these theories are described in Chapter 5. The case of small primes is also described without proofs, and this is a useful addition to the literature. Finally the semisimple classes are also described here, using a geometric approach due to Deriziotis.
The author has given in Chapters 5-10 a careful, detailed treatment of the unipotent conjugacy classes and of the representation theory including the paper of Deligne-Lusztig and the results of Howlett-Lehrer. The tables and other information in Chapter 13 and the extensive bibliography will also be useful to workers in the field; for example all the ''generic degrees'' for exceptional groups are available for the first time in one place. In the later chapters 11-12 he has given an account of the deep results of Lusztig on unipotent characters and the decomposition of the \(R^ G_ T(1)\), but in the author's own words, the exposition here is very sketchy. In the reviewer's opinion what is missing are indications of the proofs and of the ideas involved in this deep work. It is also disappointing that no mention is made of recent work on the computation of Green functions which play an important role in the character tables of the groups. However, in sum the book is a most valuable source of information and is excellent preparation for the book of Lusztig (loc. cit.) in which the whole story is told. conjugacy classes; representation theory of finite groups of Lie type; connected reductive algebraic group; Frobenius morphism; finite group of F-fixed points; BN-pair; Weyl group; Borel subgroup; maximal torus; intersections of parabolic subgroups; virtual representation; generalized character; \(\ell \)-adic cohomology; flag variety; geometric conjugacy; Jordan decomposition; unipotent elements; Deligne-Lusztig characters; irreducible characters; regular characters; Gelfand-Graev representation; cuspidal characters; unipotent characters; intersection cohomology; Springer correspondence; representations of Coxeter groups; Dynkin diagrams; bibliography; exceptional groups; character tables R.\ W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, John Wiley, New York, 1985. Representation theory for linear algebraic groups, Research exposition (monographs, survey articles) pertaining to group theory, Cohomology theory for linear algebraic groups, Linear algebraic groups over finite fields, Classical groups (algebro-geometric aspects), Universal enveloping (super)algebras Finite groups of Lie type. Conjugacy classes and complex characters | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a simply connected semisimple algebraic group scheme over an algebraically closed field \(k\) of prime characteristic \(p\) with \(G\) being split over \(\mathbb F_p\). Associated to \(G\) are two infinite families of groups: the finite groups of Lie type \(G(\mathbb F_{p^r}):=G^{F^r}\) and the Frobenius kernels (infinitesimal group schemes) \(G_r:=\ker(F^r)\) where \(F\colon G\to G\) is the Frobenius morphism. The representation theories of \(G(\mathbb F_{p^r})\) (over \(k\)) and \(G_r\) are known to be closely related. Of particular interest here is a conjecture (and generalizations thereof) of B. Parshall that if a finite dimensional rational \(G\)-module \(M\) is projective over \(G_1\), then it is also projective over \(kG(\mathbb F_p)\).
Using the notion of complexity of a module and cohomological support varieties, \textit{Z. Lin} and \textit{D. K. Nakano} [Invent. Math. 138, No. 1, 85-101 (1999; Zbl 0937.17006)] verified Parshall's conjecture. Later, [in Bull. Lond. Math. Soc. 39, No. 6, 1019-1028 (2007; Zbl 1192.20030)], \textit{Z. Lin} and \textit{D. K. Nakano} claimed to prove a generalized version of the conjecture: for any \(r\geq 1\), if \(M\) (as above) is projective over \(G_r\), then it is necessarily projective over \(kG(\mathbb F_{p^r})\). In this work, the author observes that there is a flaw in the latter work of Lin and Nakano and then provides a new proof of the claimed generalization. The proof here is representation-theoretic in nature rather than geometric (with no use of support varieties) and makes use of the distribution algebra of a Frobenius kernel. The key step is proving the analogous conjecture with \(G\) replaced by a Borel subgroup \(B\) of \(G\). Let \(U\subset B\) be the unipotent radical of \(B\). Then key to the reduction and the result for \(B\) is the fact that \(U(\mathbb F_{p^r})\) is a \(p\)-Sylow subgroup of \(B(\mathbb F_{p^r})\) or \(G(\mathbb F_{p^r})\), and so projectivity over \(kB(\mathbb F_{p^r})\) or \(kG(\mathbb F_{p^r})\) is equivalent to that over \(kU(\mathbb F_{p^r})\). The author also shows by example that the conjecture can fail to hold if \(G\) is replaced by \(U\). That is, one can have a finite dimensional rational \(U\)-module \(M\) which is projective over \(U_r\) but is not projective over \(kU(\mathbb F_{p^r})\).
Let \(\mathfrak g\) denote the (restricted) Lie algebra of \(G\) over \(k\) and \(u(\mathfrak g)\) denote the associated restricted enveloping algebra. Note that the representation theory of \(u(\mathfrak g)\) is equivalent to that of \(G_1\), and so the original Parshall Conjecture could equivalently be stated in terms of projectivity over \(u(\mathfrak g)\). In the last section of the paper, the author considers an alternate generalization of the Parshall Conjecture given by \textit{E. M. Friedlander} [J. Reine Angew. Math. 648, 183-200 (2010; Zbl 1225.20007)]. Using a Chevalley basis for \(\mathfrak g\), one can construct a (restricted) Lie algebra \(\mathfrak g_{\mathbb F_p}\) over \(\mathbb F_p\). One can then extend scalars to obtain \(\mathfrak g_{\mathbb F_{p^r}}:=\mathfrak g_{\mathbb F_p}\otimes_{\mathbb F_p}\mathbb F_{p^r}\). By Weil restriction, the Lie algebra \(\mathfrak g_{\mathbb F_{p^r}}\) can be considered as a restricted Lie algebra over \(\mathbb F_p\), from which one can extend scalars back to \(k\): \(\mathfrak g_{\mathbb F_{p^r}}\otimes_{\mathbb F_p}k\) (which is isomorphic to \(\mathfrak g^{\oplus r}\)). Friedlander showed, for any \(r\geq 1\), that for a finite dimensional rational \(G\)-module \(M\), the complexity of the module over \(kG(\mathbb F_{p^r})\) is related to that over \(u(\mathfrak g_{\mathbb F_{p^r}}\otimes_{\mathbb F_p}k)\). From that relationship, it follows that if a rational \(G\)-module is projective over \(u(\mathfrak g_{\mathbb F_{p^r}}\otimes_{\mathbb F_p}k)\), then it is projective over \(kG(\mathbb F_{p^r})\), giving an alternate generalization of the Parshall Conjecture. However, in this work, the author shows that, if \(r\geq 2\), then there are in fact no finite dimensional rational \(G\)-modules which are projective over \(u(\mathfrak g_{\mathbb F_{p^r}}\otimes_{\mathbb F_p}k)\). finite Chevalley groups; finite groups of Lie type; Frobenius kernels; Parshall conjecture; projective modules; restricted Lie algebras; semisimple algebraic group schemes; restricted Lie algebras; restricted enveloping algebras; complexity of modules; Borel subgroups; unipotent radical Drupieski, CM, On projective modules for Frobenius kernels and finite Chevalley groups, Bull. Lond. Math. Soc., 45, 715-720, (2013) Representations of finite groups of Lie type, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Cohomology of Lie (super)algebras, Modular Lie (super)algebras, Universal enveloping (super)algebras, Group schemes, Linear algebraic groups over finite fields On projective modules for Frobenius kernels and finite Chevalley groups. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field and let \(A\) be a finitely generated algebra. Let \(M_d\) be the \(k\)-scheme such that \(M_d(R)\) is the \(k\)-algebra of \(d\times d\)-matrices with coefficients in \(R\), for any commutative \(k\)-algebra \(R\), and for \(d\geq 1\), a positive integer. Then the affine scheme \(\text{mod}_A^d\) is represented by the functor \(\Hom_{k\text{-alg}}(A,M_d(-))\). The general linear group \(\text{GL}_d\) acts on \(\text{mod}_A^d\) by conjugation so that for any \(d\)-dimensional left \(A\)-module \(M\) the corresponding orbit \(O_M\) in \(\text{mod}_A^d(k)\) is well-defined.
The paper under review is devoted to the study of types of singularities in the closures of orbits in module schemes \(\text{mod}_A^d\). The type of singularity \((X,x)\) is defined as the equivalence class with respect to the following relation: two pointed schemes \((X,x)\) and \((Y,y)\) are (smoothly) equivalent if there are smooth morphisms \(f\colon Z\to X\) and \(g\colon Z\to Y\) and a point \(z\in Z\) such that \(f(z)=x\) and \(g(z)=y\) [\textit{W. Hesselink}, Trans. Am. Math. Soc. 222, 1-32 (1976; Zbl 0332.14017)]. Among other things the author proves the following. Let \(0\to U\to M\to U\to 0\) be an exact sequence of finite dimensional left \(A\)-modules such that the codimension of the orbit \(O_{U\oplus U}\) in \(\overline O_M\) is equal to 2. Then \(\text{Sing}(M\oplus U^{p-1},U^{p+1})\) for any \(p\geq 1\) is nothing but the type of singularity of the nilpotent \((p+1)\times(p+1)\) matrices of rank at most 1 at the zero matrix. The author also gives the reference [\textit{G. Bobiński} and \textit{G. Zwara}, Manuscr. Math. 105, No. 1, 103-109 (2001; Zbl 1031.16012)] where his results are used in order to prove the normality of orbit closures for quivers of Dynkin type \(A_n\). module schemes; orbit closures; smooth morphisms; Grothendieck groups; Dynkin quivers; representations of finite-dimensional algebras Zwara, G.: Smooth morphisms of module schemes. Proc. Lond. Math. Soc. (3) 84(3), 539--558 (2002) Representations of quivers and partially ordered sets, Finite rings and finite-dimensional associative algebras, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry Smooth morphisms of module schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We construct an action of the Neron-Severi part of the Looijenga-Lunts-Verbitsky Lie algebra on the Chow ring of the Hilbert scheme of points on a \(K3\) surface. This yields a simplification of \textit{D. Maulik} and \textit{A. Neguţ}'s [``Lehn's formula in Chow and Conjectures of Beauville and Voisin'', Preprint, \url{arXiv:1904.05262}] proof that the cycle class map is injective on the subring generated by divisor classes as conjectured by Beauville (see [\textit{A. Beauville} and \textit{C. Voisin}, J. Algebr. Geom. 13, No. 3, 417--426 (2004; Zbl 1069.14006)]). The key step in the construction is an explicit formula for Lefschetz duals in terms of Nakajima operators. Our results also lead to a formula for the monodromy action on Hilbert schemes in terms of Nakajima operators. \(K3\) surfaces; Chow groups; Hilbert schemes of points; Lie algebras \(K3\) surfaces and Enriques surfaces, (Equivariant) Chow groups and rings; motives, Parametrization (Chow and Hilbert schemes) A Lie algebra action on the Chow ring of the Hilbert scheme of points of a \(K3\) surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper of exceptional importance, the authors have been able to ``explain'' many combinatorial phenomena that are associated to a Weyl group in different contexts. The key discovery is that of a set of certain polynomials \(P_{x,y}\) in one variable with integral coefficients associated to a pair \((x,y)\) of elements of a Coxeter group \(W\). These polynomials are used extensively to (1) construct certain representations of the Hecke algebra of \(W\) thereby obtaining important information on representations of \(W\), (2) give a formula (conjecturally) for the multiplicities in the Jordan-Hölder series of Verma modules or equivalently for the formal characters of irreducible highest weight modules (extending the celebrated Weyl character formula to ``nondominant'' weights), (3) give a complete description for the inclusion-relations between various primitive ideals in the enveloping algebra of a complex semisimple Lie algebra \(\mathfrak g\); in case \(\mathfrak g\) is of type \(A_n\), this is further tied up with the dimensions of certain representations of the corresponding Weyl group (via the ``Jantzen conjecture'' -- cf. a paper by \textit{A. Joseph} [Lect. Notes Math. 728, 116--135 (1979; Zbl 0422.17004)], (4) give a measure of the failure of local Poincaré duality in the geometry of Schubert cells in flag varieties. (In a later paper [\textit{D. Kazhdan} and \textit{G. Lusztig}, Proc. Symp. Pure Math. 36, 185--203 (1980; Zbl 0461.14015)] the authors give a more precise interpretation of the coefficients of \(P_{x,y}\) in terms of a certain cohomology theory called ``middle intersection cohomology'' associated with the geometry of Schubert cells.)
Considering the various topics involved and their importance, it seems worthwhile to give a detailed review in order to give some idea of the wealth of information contained in this paper.
We first describe the combinatorial setup involved. Let \((W,S)\) be a Coxeter group. Let \(\mathbb Z[q^{1/2},q^{-1/2}]\) be the ring of Laurent polynomials in the indeterminate \(q^{1/2}\) over \(\mathbb Z\). Let \(\mathcal H\) be the free \(\mathbb Z[q^{1/2},q^{-1/2}]\)-module with \(\{T_y\mid y\in W\}\) as a basis; the multiplication in \(\mathcal H\) is given by: for \(s\in S\), \(y\in W\), \(T_s\cdot T_y=T_{sy}\) if \(l(sy)\geq l(y)\) and \(T_s\cdot T_y=(q-1)T_y+q\cdot T_{sy}\) if \(l(sy)\leq l(y)\). (Classically, one considers \(\mathbb Z[q]\)-coefficients only; the algebra \(\mathcal H\) thus obtained, called the Hecke algebra of \(W\), is isomorphic to the space of intertwining operators on the ``\(1_B^G\)''-representation of a finite Chevalley group with \(W\) as the Weyl group.)
It can be seen that \(T_y\) is invertible in \(\mathcal H\) and so \(\mathcal H\) has an involution \({}^-\) under which \(T_y\) goes to \(T_{y^{-1}}^{-1}\) and \(q^{1/2}\) goes to \(q^{-1/2}\). The main discovery of the paper can now be stated as the theorem: For any \(y\in W\), there is a unique element \(C_y\in\mathcal H\) such that (i) \(\overline C_y=C_y\) and (ii) \(C_y=\sum_{x\in W}(-1)^{l(x)+l(y)}\cdot(q^{1/2})^{l(y)}\cdot q^{-l(x)}\overline P_{x,y}\cdot T_x\), where \(P_{x,y}\in\mathbb Z[q]\) with \(P_{y,y}=1\) and \(\deg P_{x,y}\leq(l(y)-l(x)-1)/2\) if \(x\mathop{<}\limits_{\neq}y\) (\(\leq\) is the Bruhat ordering on \(W\)) and \(P_{x,y}=0\) otherwise.
The authors give an inductive formula for \(P_{x,y}\); however, no closed formula is available as yet. It is conjectured that the coefficients of \(P_{x,y}\) are nonnegative; the authors have proved it in the case of Weyl groups and affine Weyl groups by showing them to be dimensions of certain cohomology groups [cf. the authors, op. cit.].
We now describe the various applications of the polynomials \(P_{x,y}\).
(1) Representations of \(\mathcal H\): In order to obtain certain representations of \(H\), the authors introduce the notion of a \(W\)-graph as follows: It is a graph \(\Gamma\) without loops such that to each vertex \(x\in\Gamma\) is associated a subset \(I_x\) of \(S\) (the set of simple reflections in \(W\)) and to each edge \((x,y)\) is associated a nonzero integer \(\mu(x,y)\) which is required to satisfy certain compatibility conditions. (These conditions ensure that one can define a representation of \(\mathcal H\).) Define a preorder \(x\leq_\Gamma y\) on vertices of \(\Gamma\) by: \(x\leq_\Gamma y\) if there exist \(x=x_0,x_1,\dots,x_n=y\in\Gamma\) such that for all \(i\), \((x_i,x_{i+1})\) is an edge of \(\Gamma\) with \(I_{x_i}\not\subset I_{x_{i+1}}\). Let \(\sim\) be the equivalence relation associated with \(\leq_\Gamma\) (i.e. \(x\sim y\) if \(x\leq_\Gamma y\) and \(y\leq_\Gamma x\)). Then each equivalence class considered as a full subgraph of \(\Gamma\) and the assignments ``\(I_x\) and \(\mu(x,y)\)'' coming from \(\Gamma\) is a \(W\)-graph itself and thus one gets many representations of \(\mathcal H\) (e.g., the ``reflection'' representation of \(\mathcal H\) can be obtained in this way). The authors construct a \(W\)-graph from the polynomials \(P_{x,y}\) in the following way: The set \(W\) is the set of vertices and \((x,y)\) is an edge if either \(x\mathop{<}\limits_{\neq}y\) with \(\deg P_{x,y}=(l(y)-l(x)-1)/2\) or \(y<x\) with \(\deg P_{y,x}=(l(x)-l(y)-1)/2\) (one denotes such pairs by \(x\prec y\) or \(y\prec x\) as the case may be). For \(x\in W\), assign \(I_x=L_x=\{s\in S\mid l(sx)\leq l(x)\}\) and for an edge \((x,y)\), assign \(\mu(x,y)=\) leading coefficient of \(P_{x,y}\) (or \(P_{y,x}\) as the case may be). The authors show that this gives a \(W\)-graph and the corresponding representation is in fact the left-regular representation of \(\mathcal H\). The equivalence classes of this \(W\)-graph are called left cells (``left'' because the set \(L_x\) involves multiplication on the left by the elements of \(S\)). One has a similar \(W\)-graph by considering multiplication on the right and a \(W\times W^0\)-graph (\(W^0\) is the opposite group) by considering the left and right multiplications simultaneously. The equivalence classes are called right cells and two-sided cells, respectively. It turns out that the configuration of these cells forms important combinatorial data from which much information can be obtained. In case \(W=S_n\), the representations of \(W\) obtained from left cells of \(W\) by specializing \(q=1\) cover all complex representations equipped with a distinguished basis.
(2) Characters of highest weight representations of a complex semisimple Lie algebra \(\mathfrak g\): Fix a Cartan subalgebra \(\mathfrak h\) and a set of simple roots \(\Pi\) for the root system of \((\mathfrak g,\mathfrak h)\). Let \((W,S)\) be the corresponding Coxeter system. For \(x\in W\), let \(M_x\) be the Verma module with highest weight \(x\rho-\rho\) (\(\rho\) is the half-sum of positive roots) and let \(L_x\) denote the (unique) irreducible quotient of \(M_x\). For \(x,y\in W\), let \(\text{mtp}(x,y)\) be the multiplicity with which \(L_y\) occurs in a Jordan-Hölder series of \(M_x\). It is then known that \(\text{mtp}(x,y)\neq 0\) if and only if \(x\leq y\) (\(\leq\) being the Bruhat ordering). The problem of determining \(\text{mtp}(x,y)\) has been considered by several people (e.g., [\textit{J. Lepowsky} and the reviewer, J. Algebra 49, 512--524 (1977; Zbl 0381.17004); \textit{J. C. Jantzen}, Moduln mit einem höchsten Gewicht. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0426.17001)]). The authors conjecture that \(\text{mtp}(x,y)=P_{x,y}(1)\). (This conjecture has been proved recently by Brylinski and Kashiwara and, independently, by Beilinson and Bernstein.) This has an equivalent formulation in terms of the formal character of \(L_x\). (Recall: If \(x=\text{id}\) then one has the Weyl character formula for a finite-dimensional representation of \(\mathfrak g\).) In his talk at the AMS Santa Cruz Conference on Finite Groups [\textit{G. Lusztig}, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 313--317 (1980; Zbl 0453.20005)] the second author proposed a modular analogue analogue the above conjecture from which the character formula of a rational irreducible representation of a Chevalley group \(G\) over an algebraically closed field of positive characteristic can be obtained.
(3) Theory of primitive ideals in the enveloping algebra \(U(\mathfrak g)\) of a complex semisimple Lie algebra \(\mathfrak g\): A two-sided ideal \(I\) in \(U(\mathfrak g)\) is said to be ``primitive'' if it is the annihilator of an irreducible \(\mathfrak g\)-module. One is then interested in the space \(X\) of primitive ideals (equipped with the Jacobson topology). As every irreducible \(\mathfrak g\)-module has a central character, one gets a map \(X\rightarrow Z(\hat{\mathfrak g})\) (\(=\text{Hom}_{\mathbb C-\text{alg}}(Z(\mathfrak g),{\mathbb C})\) where \(Z(\mathfrak g)\) is the centre of \(U(\mathfrak g)\)). By a well-known theorem of Harish-Chandra \(Z(\hat{\mathfrak g})\simeq\mathfrak h^\ast|_W\). Let \(\Lambda\) be an equivalence class. One is interested in the fibre \(X_\Lambda\) over \(\Lambda\). Let \(\lambda\in\Lambda\) and \(J(\lambda)=\text{Ann}\,L(\lambda)\), where \(L(\lambda)\) is the irreducible \(\mathfrak g\)-module with highest weight \(\lambda\). Then \(J(\lambda)\in X_\lambda\) and in fact a deep result of Duflo asserts that every element of \(X_\Lambda\) is of this form. There are various results known about the structure of fibres, ``similarity'' of two fibres, etc. The most important problem is to determine the inclusion relation between two elements of the same fibre. Known results by Borho, Duflo, Jantzen, Joseph, Vogan, etc. give sufficient conditions for it. However, since the conjecture ``\(\text{mtp}(x,y)=P_{x,y}(1)\)'' is proved to be true, the left cells of \(W\) determine the inclusion completely. To be more precise, let \(\lambda\) be an antidominant integral element. Let \(\Lambda\) be the equivalence class to which it belongs. Then every element of \(X_\Lambda\) is of the form \(J(x\cdot\lambda)\) where \(x\cdot\lambda=x(\lambda+\rho)-\rho\). Then \(J(x\cdot\lambda)\subseteq J(y\cdot\lambda)\) if and only if \(y\leq_Lx\). (The inclusion relations in other fibres can be written down with the help of a ``suitable'' subgroup \(W\) which corresponds to a sub-root-system.) It may be recalled that the primitive spectrum \(X\) is related to nilpotent orbits in \(\mathfrak g\).
(4) Geometry of Schubert cells in flag varieties: Let \(G\) be a complex semisimple algebraic group, \(B\) a Borel subgroup and \(G/B\) be the corresponding flag variety. \(B\) acts on \(G/B\) and one has the Bruhat decomposition \(G/B=\bigcup_{x\in W}BxB\). For \(y\in W\), let \(X(y)=\overline{ByB}=\bigcup_{x\leq y}BxB\) (\(\leq\) is the Bruhat ordering). Let \(e_x\) be the point \(xB\in G/B\). Then one is interested in determining the nature of the singularity at \(e_x\) when considered as a point of \(X(y)\;(x\leq y)\). The authors show that the condition ``\(P_{x',y}\equiv 1\) for all \(x\leq x'\leq y\)'' is related to the singularity of \(e_x\) in \(X(y)\). A closer tie-up is given in a later paper [the authors, loc. cit.].
Considering the central role played by the polynomials \(P_{x,y}\) in above-mentioned contexts, it is desirable to have an explicit knowledge of their coefficients in terms of certain combinatorial data. The relation \(x\prec y\) is another key notion which should be investigated further. Weyl groups; Coxeter groups; representations of Hecke algebras; Jordan-Hölder series of Verma modules; irreducible highest weight modules; Weyl character formula; primitive ideals in enveloping algebras; complex semisimple Lie algebras; local Poincaré duality; geometry of Schubert cells; flag varieties; intersection cohomology; Laurent polynomials; intertwining operators; finite Chevalley groups; affine Weyl groups; cohomology groups; simple reflections; highest weight representations; Cartan subalgebras D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, \textit{Invent.} \textit{Math.}, 53 (1979), no. 2, 165--184.Zbl 0499.20035 MR 560412 Reflection and Coxeter groups (group-theoretic aspects), Representation theory for linear algebraic groups, Hecke algebras and their representations, Universal enveloping (super)algebras, Linear algebraic groups over arbitrary fields, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Group actions on varieties or schemes (quotients) Representations of Coxeter groups and Hecke algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author's stated objective is to provide a framework for the study of representations of restricted Lie algebras and certain quantum groups. Let \(H\) be a given finite-dimensional Hopf algebra over a field. An \(H\)-Galois extension \(U\) of a subalgebra \(O\), which is contained in the center of \(U\), is regarded as a sheaf which extends the affine scheme of \(O\). The author states that he does not know whether an \(H\)-Galois extension of a local ring is cleft or, in other words, has a normal basis. The question has been answered, in the affirmative, for commutative or cocommutative Hopf algebras by the reviewer and \textit{P. M. Cook} [J. Algebra 43, 115-121 (1976; Zbl 0343.16025)]. The author suggests applications of the Miyashita-Ulbrich action of \(H\) on \(U\) and equivariant splittings of a cleft \(H\)-Galois extension for cocommutative \(H\) to the determination of the dimension function for the center of \(U\) over \(O\). Also considered in this paper are a bilinear form on \(U\) over \(O\) determined by the action of a nonzero left integral of \(H^*\) on \(U\) and various types of connections, which are actions on \(U\) by the Lie algebra of derivations of \(O\). Finally, an \(H\)-torsor on a scheme is defined to be a sheaf \(\mathcal U\) of \(H\)-comodule algebras for which the scheme is the subsheaf \(\mathcal O\) of \(H\)-coinvariants and which is locally \(H\)-Galois. The author uses techniques of faithfully flat descent to show that \({\mathcal U}(V)\) is an \(H\)-Galois extension of \({\mathcal O}(V)\) for ``good'' open subschemes \(V\). torsors; Galois extensions; representations of restricted Lie algebras; quantum groups; finite-dimensional Hopf algebras; affine schemes; cocommutative Hopf algebras; Miyashita-Ulbrich actions; equivariant splittings; sheaves; faithfully flat descent [Ru1] D. Rumynin,Hopf-Galois extensions with central invariants and their geometric properties, Alg. Rep. Th.,1 (1998), 353--381. Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Rings arising from noncommutative algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations Hopf-Galois extensions with central invariants and their geometric properties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the second edition of a work that first appeared in 1987. The first printing was reviewed extensively by \textit{V. L. Popov}, see Zbl 0654.20039. Let me just say that Jantzen's book has been an indispensable reference for anyone involved with representations of algebraic groups, in particular of reductive groups in positive characteristic.
The second edition differs substantially from the first. Again it is a book one must have. About one third is new. The old part has been sprinkled with new comments, but has been left intact as much as is feasible. New chapters at the end, identified with capital letters, take care of several developments in the intervening years.
The following topics are now also covered in the familiar clear manner. Chapter A: Truncated categories and Schur algebras. Here the truncations are those of Donkin, associated to a finite `saturated' set of dominant weights. Chapter B: Results over the integers. This globalizes several cohomological results. Chapter C: Lusztig's Conjecture and some consequences. Chapter D: Radical filtrations and Kazhdan-Lusztig polynomials. These two chapters belong together and discuss various aspects of this central conjecture concerning characters of certain simple modules. Chapter E: Tilting modules. Chapter F: Frobenius splitting. Chapter G: Frobenius splitting and good filtrations. This chapter gives the proof by Mathieu of the theorem about tensor products of modules with good filtration. Logically chapters F and G should be read early. Chapter H: Representations of quantum groups. This chapter is in a different style. It is just a survey.
It is a pity that the author has still not adopted certain widely used terminology. For instance, the usual form of `sum formula' is `Jantzen sum formula'. representation theory; reductive algebraic groups; simple modules; highest weights; character formulas; Weyl's character formula; affine group schemes; injective modules; injective resolutions; derived functors; Hochschild cohomology groups; hyperalgebra; split reductive group schemes; Steinberg's tensor product theorem; irreducible representations; Kempf's vanishing theorem; Borel-Bott-Weil theorem; characters; linkage principle; dominant weights; filtrations; Steinberg modules; cohomology rings; rings of regular functions; Schubert schemes; line bundles; Schur algebras; quantum groups; Kazhdan-Lusztig polynomials J. C. Jantzen, \textit{Representations of Algebraic Groups. Second edition}, Amer. Math. Soc., Providence (2003). Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Group schemes, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Affine algebraic groups, hyperalgebra constructions, Linear algebraic groups over arbitrary fields Representations of algebraic groups. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove a recent conjecture by \textit{ Gyenge} et al. [Int. Math. Res. Not. 2017, No. 13, 4152--4159 (2017; Zbl 1405.14010); Eur. J. Math. 4, No. 2, 439--524 (2018; Zbl 1445.14007)] giving a formula of the generating function of Euler numbers of Hilbert schemes of points \(\operatorname{Hilb}^N(\mathbb{C}^2/ \Gamma )\) on a simple singularity \({\mathbb{C}^2}/ \Gamma \), where \(\Gamma\) is a finite subgroup of \(\operatorname{SL}(2)\). We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with \(\Gamma\) at \(\zeta =\exp (\frac{2\pi \sqrt{-1}}{2({h^{\vee }}+1)})\) are always 1, which is a special case of an earlier conjecture by Kuniba. Here \({h^{\vee }}\) is the dual Coxeter number. We also prove the claim, which was not known for \({E_7}\), \({E_8}\) before. Hilbert schemes of points; quantum affine algebras; quantum dimensions; simple surface singularities Parametrization (Chow and Hilbert schemes), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Quantum groups (quantized enveloping algebras) and related deformations Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper concerns the Bruhat-Renner decomposition of reductive monoids and is a survey on the recent developments on the associated Bruhat-Chevalley orders, Stanley-Reisner rings and Hecke algebras. The author also introduces the concept of a triangular Hecke algebra. Quite a few relevant examples are presented. The author is a main and original contributor to the development. In the paper, he also poses two conjectures about the associated Hecke algebras. The interested reader is also advised to read \textit{L. E. Renner}'s complementary paper in these Proceedings [ibid. 125-143 (2004; Zbl 1061.20059)]. Bruhat-Chevalley orders; Bruhat-Renner decompositions; Cohen-Macaulay rings; conjugacy classes; finite groups of Lie type; finite monoids of Lie type; Gorenstein rings; Hecke algebras; Kazhdan-Lusztig polynomials; linear algebraic monoids; reductive groups; reductive monoids; Putcha lattices of cross-sections; Renner monoids; representation theory; shellability; Stanley-Reisner rings; Weyl groups Semigroups of transformations, relations, partitions, etc., Hecke algebras and their representations, Linear algebraic groups over arbitrary fields, Combinatorics of partially ordered sets, Group actions on varieties or schemes (quotients), Representation of semigroups; actions of semigroups on sets, Representations of finite groups of Lie type Bruhat-Renner decomposition and Hecke algebras of reductive monoids. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C\) be a smooth curve with distinguished point \(0 \in C\). A \textit{simple degeneration} is a flat morphism \(f:X \to C\) with \(X\) a smooth algebraic space such that \(f\) is smooth outside \(X_0 = f^{-1} (0)\), \(X_0\) has normal crossing singularities, and the singular locus \(D \subset X_0\) is smooth. It is \textit{strict} if the irreducible components of \(X_0\) are also smooth. Given an orientation on the dual graph \(\Gamma (X_0)\) of \(X_0\) for a strict simple degeneration, \textit{J. Li} [J. Differ. Geom. 57, No. 3, 509--578 (2001; Zbl 1076.14540)] constructed an \textit{expanded degeneration} \(X[n] \to C[n]\), a \(G[n]\)-equivariant morphism in which \(C[n]/G[n] \cong C\) (\(G[n]\) is the \(n\)-torus \((\mathbb G_m)^n\)). \textit{M. G. Gulbrandsen} et al. used expanded degenerations to study degenerations of Hilbert schemes of \(n\) points on varying fibers \(X_t\), constructing a degeneration \(I^n_{X/C} \to C\) which coincides with the relative Hilbert scheme \(\text{Hilb}^n (X/C) \to C\) over \(C - \{0\}\), but differing at the central fiber. \(I^n_{X/C}\) is constructed as a GIT quotient of \(\text{Hilb}^n (X[n]/C[n])\) by the \(G[n]\)-action [Doc. Math. 24, 421--472 (2019; Zbl 1423.14068)]. If \(X\) is a scheme, then \(X[n]\) is a scheme if and only if \(\Gamma (X_0)\) has no loops and \(X \to C\) is projective if and only if \(\Gamma (X_0)\) has no directed cycles.
The authors study \(I^n_{X/C} \to C\) when \(\dim X_t \leq 2\) and \(\Gamma (X_0)\) is bipartite. They prove that \(I^n_{X/C}\) is normal with finite quotient singularities and \(\mathbb Q\)-factorial, the special fiber \((I^n_{X/C})_0\) is reduced, and the pair \((I^n_{X/C}, (I^n_{X/C})_0)\) is divisorial log terminal in the sense of the minimal model program. The crux of the proof is to show that the divisorial log terminal property passes from \(\text{Hilb}^n (X[n]/C[n])^{ss}\) to \(I^n_{X/C}\) under the GIT quotient. They also prove that if \(X \to C\) is a good minimal divisorial log terminal model, then so is \(I^n_{X/C} \to C\). With the same hypothesis, the authors prove that the dual complex for \(X \to C\) is a graph \(\Gamma\) and the dual complex \(\mathcal D ((I^n_{X/C})_0)\) is isomorphic to the \(n\)th symmetric product \(\text{Sym}^n (\Gamma)\) as a \(\Delta\)-complex. In particular, some results of \textit{M. V. Brown} and \textit{E. Mazzon} [Compos. Math. 155, No. 7, 1259--1300 (2019; Zbl 1440.14131)] on the essential skeleton of \(\text{Hilb}^n X\) in terms of the essential skeleton of \(X\) for K3 surfaces are recovered. In the motivating case where \(X \to C\) is a projective type II degeneration of K3 surfaces, they prove that the stack \(I^n_{X/C}\) is proper and semi-stable over \(C\); moreover, if \(K_{X/C}\) is trivial, then \(I^n_{X/C}\) carries an everywhere non-degenerate relative logarithmic \(2\)-form.
In the closing section, the authors compare their results with earlier work of \textit{Y. Nagai} [Math. Z. 258, 407--426 (2008; Zbl 1140.14008)] in the special case \(n=2\). There \textit{Y. Nagai} constructs a different degeneration \(H^2_{X/C} \to C\), even if \(\Gamma (X_0)\) is not bipartite; in case \(\Gamma (X_0)\) is bipartite, the authors relate \(H^2_{X/C} \to C\) to \(I^2_{X/C}\) by explicit birational maps. Recent work of \textit{Y. Nagai} constructs the Hilbert scheme degeneration \(I^n_{X/C} \to C\) using toric methods, describing the local structure of the singularities in \(\text{Sym}^n (X/C)\). He also gives an explicit \(\mathbb Q\)-factorial terminalization \(Y^{(n)} \to \text{Sym}^n (X/C)\) and isomorphism \(I^n_{X/C} \to \text{Sym}^n (X/C)\) [Math. Z. 289, 1143--1168 (2018; Zbl 1423.14069)]. strict simple degenerations; geometric Invariant theory; Hilbert schemes; good minimal dlt models; type II degenerations of \(K3\) surfaces Fibrations, degenerations in algebraic geometry, Parametrization (Chow and Hilbert schemes), \(K3\) surfaces and Enriques surfaces, Geometric invariant theory The geometry of degenerations of Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a simple algebraic group over a field \(k\), defined and split over the finite field \(\mathbb F_p\) for a prime \(p\). The Lie algebra \(\mathfrak g\) of \(G\) admits the structure of a \(p\)-restricted Lie algebra with restricted enveloping algebra \(u(\mathfrak g)\). Let \(G(\mathbb F_q)\) denote the finite Chevalley group of \(\mathbb F_q\)-rational points of \(G\) (for \(q=p^d\)). Given a rational \(G\)-module \(M\), by restriction, \(M\) can be considered as either a \(kG(\mathbb F_q)\)-module or a \(u(\mathfrak g)\)-module.
The first main result, which extends work of \textit{J. F. Carlson, Z. Lin} and \textit{D. K. Nakano} [Trans. Am. Math. Soc. 360, No. 4, 1879-1906 (2008; Zbl 1182.20039)] for \(q=p\) to an arbitrary \(q=p^d\), is a relationship between the (projectivized) cohomological support variety of \(G(\mathbb F_q)\) and that of \(u(\mathfrak g_{\mathbb F_q}^{\oplus d})\) as long as \(p\) is at least the Coxeter number of \(G\). A crucial tool in proving this relationship is use of the Weil restriction functor. For a Galois extension of fields \(E/F\), the Weil restriction functor is a functor from affine \(E\)-schemes to affine \(F\)-schemes that is right adjoint to base change from \(F\) to \(E\). The Weil restriction functor is applied for example to \(G_{\mathbb F_q}\) relative to the extension \(\mathbb F_q/\mathbb F_p\).
A second main result is that the complexity of a rational \(G\)-module \(M\) when considered as a \(kG(\mathbb F_q)\)-module is bounded by one-half its complexity when considered as a \(u(\mathfrak g_{\mathbb F_q}\otimes_{\mathbb F_p}k)\)-module. From this it follows that if \(M\) is projective upon restriction to \(u(\mathfrak g_{\mathbb F_q}\otimes_{\mathbb F_p}k)\), then it is necessarily projective upon restriction to \(kG(\mathbb F_q)\). These results extend to arbitrary \(q\) results for \(q=p\) of \textit{Z. Lin} and \textit{D. K. Nakano} [Invent. Math. 138, No. 1, 85-101 (1999; Zbl 0937.17006)].
Lastly, the author obtains a comparison theorem on non-maximal support varieties for a rational \(G\)-module with ``small'' high weights; comparing the support over \(kG(\mathbb F_p)\) with that over \(u(\mathfrak g)\). It follows that if such a module has constant Jordan type as a \(u(\mathfrak g)\)-module, then it has constant Jordan type as a \(kG(\mathbb F_p)\)-module. group schemes; cohomological support varieties; restricted Lie algebras; complexity; \(\pi\)-points; Weil restriction; projectivity; finite Chevalley groups; non-maximal support varieties; constant Jordan type Eric M. Friedlander, Weil restriction and support varieties, J. Reine Angew. Math. 648 (2010), 183 -- 200. Representations of finite groups of Lie type, Modular Lie (super)algebras, Cohomology of Lie (super)algebras, Modular representations and characters, Cohomology of groups, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Group schemes Weil restriction and support varieties. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Based in part on the authors' preface: The Kleinian singularities \({\mathbb C}^2/G\) associated to finite subgroups \(G \subset SL_2({\mathbb C})\) are important in algebraic geometry, singularity theory and other branches of mathematics. New remarkable properties continue to be discovered, one such being the McKay correspondence and its interpretation by Gonzalez-Springberg and Verdier in terms of the minimal resolution \({\mathbb C}^2//G\). Their results give identifications
\[
K_0({\mathbb C}^2//G) \simeq Rep(G) \simeq \widehat{\mathbf h}_Z
\]
where \(K_0\) is the Grothendieck group, Rep is the representation ring and \(\widehat{\mathbf h}_Z\) is the root lattice of the affine Lie algebra (of type A-D-E) associated to \(G\). The paper under review first extends these results by describing the derived category of coherent sheaves on \({\mathbb C}^2//G\), rather than just \(K_0\). The approach is a refinement of techniques by Kronheimer and Nakajima. Then the authors define an Euler-characteristic version \({\mathbb H}\) of the Hall algebra of the category of coherent sheaves on \({\mathbb C}^2//G\), and exhibit a subalgebra in \({\mathbb H}\) isomorphic to \(U(\text{g}_G^+)\), where \(\text{g}_G^+\) is the nilpotent part of the finite dimensional Lie-algabra (of type A-D-E) corresponding to G. As a consequence, taking the intersection graph of the \({\mathbb P}^1_i\) as a Dynkin graph, they get a possibly infinite-dimensional Kac-Moody Lie algebra, and prove that the positive part of this algebra acts in the space of functions on isomorphism classes of coherent sheaves in S. This partly extends results of Nakajima to a wider geometric context. Kleinian singularities; McKay correspondence; Lie algebras; derived category of coherent sheaves; Hall algebra; Kac-Moody Lie algebra M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and hall algebras, Math. Ann. 316 (2000), no. 3, 565-576. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Singularities of surfaces or higher-dimensional varieties, Derived categories, triangulated categories, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Kleinian singularities, derived categories and Hall algebras | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the Lie algebra of the prounipotent radical of the relative completion of the mapping class group of genus two. In particular, we partially determine a minimal presentation of the Lie algebra by determining the generators and bounding the degree of the relations of it. moduli space of curves; moduli space of principally polarized abelian varieties; mapping class groups; unipotent completion; Hodge Lie algebra; presentation of nilpotent Lie algebras; classical modular forms; special values of L-functions Families, moduli of curves (algebraic), Solvable, nilpotent (super)algebras, Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic moduli of abelian varieties, classification, Geometric group theory, Period matrices, variation of Hodge structure; degenerations, Homology of classifying spaces and characteristic classes in algebraic topology On the completion of the mapping class group of genus two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Die einfachen algebraischen Gruppen über einem algebraisch abgeschlossenen Körper werden bis auf Überlagerungen durch ihre Wurzelsysteme \(A_n\), \(B_n\), \(C_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\), \(F_4\) und \(G_2\) klassifiziert. In \(A_n\), \(D_n\) und \(E_n\) haben alle Wurzeln die gleiche Länge, man nennt diese Systeme homogen, die übrigen inhomogen.
Die einfachen Hyperflächensingularitäten lassen sich unter allen Singularitäten durch gewisse Schnittdiagramme charakterisieren, die den Coxeter-Dynkin-Witt-Diagrammen der homogenen Wurzelsysteme entsprechen. Im Falle der Charakteristik 0 haben Grothendieck und Brieskorn eine Konstruktion angegeben, die aus einem homogenen Wurzelsystem eine einfache Singularität vom entsprechenden Typ liefert. 1987 fand Knop noch eine weitere Konstruktion (für beliebige Charakteristik).
In der vorliegenden Dissertation wird nun untersucht, welche Singularitäten sich bei dieser Konstruktion im inhomogenen Fall ergeben und deren Zusammenhang mit den in diesem Fall bei der Brieskorn- Grothendieck-Slodowy-Konstruktion auftretenden Singularitäten geklärt. Schließlich wird noch auf die Nahmsche Konstruktion modular invarianter Partitionsfunktionen aus Wurzelsystemen eingegangen, wobei diese im inhomogenen Falle so modifiziert wird, daß sich auch hier nicht-triviale Partitionsfunktionen ergeben. (Deren Klassifikation im homogenen Fall, die man in der konformen Quantenfeldtheorie benötigt, wurde von Cappelli, Itzykson und Zuber bzw. Kato 1987 gegeben). orbits; simple singularities; modular invariant partition functions; root systems; hypersurface singularities Lie algebras of Lie groups, Singularities in algebraic geometry, Simple, semisimple, reductive (super)algebras, Spinor and twistor methods applied to problems in quantum theory On nilpotent orbits, simple singularities and modular invariant partition functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper proposes to use minimal logarithmic signatures for finite groups of Lie type. The method presented in the paper is working for the families \(\text{PSL}_n(q)\) and \(\text{PSp}_{2n}(q)\) and uses Singer subgroups and the Levi decomposition of parabolic subgroups for these groups.
To study the minimal logarithmic signature (MLS) conjecture, the authors give some basic results about LS for an arbitrary subset \(A\) of a group \(G\). Their aim is to obtain general methods to construct MLS for (simple) groups. By using Remark 3.12, they describe parabolic subgroups, sharply transitive sets and standard Levi subgroups for the corresponding (simple) groups. Then, they use the remark as a tool to create MLSs and other interesting LSs for such groups. They also obtain MLSs for \(\text{GL}_n(q)\) and \(\text{PGL}_n(q)\) using a slightly different method than \textit{W. Lempken} and \textit{Tran van Trung} [Exp. Math. 14, No. 3, 257--269 (2005; Zbl 1081.94038)].
The authors construct MLSs for the groups \(\text{GL}_{n}(q)\), \(\text{PGL}_n(q)\), \(\text{SL}_n(q)\) and \(\text{Sp}_{2n}(q)\). The blocks of the LSs are obtained from Singer subgroups of the classical (sub)groups. This also produces a spread in the corresponding projective or polar space. Their methods are general and may prove sufficiently strong as tools for constructing MLSs for all finite simple groups. logarithmic signatures; finite groups of Lie type; simple groups; Singer groups Singhi, [Singhi et al. 10] N.; Singhi, N.; Magliveras., S., Minimal logarithmic signatures for finite groups of Lie type., \textit{Des. Codes Cryptogr.}, 55, 2-3, 243-260, (2010) Cryptography, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry, Simple groups: alternating groups and groups of Lie type, Linear algebraic groups over finite fields, Authentication, digital signatures and secret sharing Minimal logarithmic signatures for finite groups of Lie type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field of finite characteristic \(p\), and let \(G\) be a finite group scheme whose order is a multiple of \(p\). Let \(kG\) be the dual of the coordinate algebra \(k[G]\). In previous work the authors, along with others, have created \(\pi\)-points, flat \(K\)-algebra maps \(\alpha_K\colon K[t]/(t^p)\to KG\) for \(K\) an extension of \(k\), as an aide in studying support varieties of \(kG\)-modules. The scheme of equivalence classes of \(\pi\)-points is denoted \(\Pi(G)\). Each finite dimensional \(kG\)-module \(M\) gives rise to a closed subset \(\Pi(G)_M\) of \(\Pi(G)\), and every closed subset is of this form. However, this is not a one-to-one correspondence, i.e., it is possible for \(M_1\) and \(M_2\) to be nonisomorphic \(kG\)-modules which give the same closed subset of \(\Pi(G)\): examples can be constructed using the observation that \(\Pi(G)_M\) is empty whenever \(M\) is a projective \(kG\)-module. The non-maximal subvariety \(\Gamma(G)_M\) provides some refinement to the support variety \(\Pi(G)_M\).
In the work under review, the authors introduce a new family of invariants which they call ``generalized support varieties''. For a finite dimensional \(kG\)-module \(M\) and \(1\leq j<p\) they define \(\Gamma^j(G)\) to be the subsets of \(\Pi(G)\) which satisfy a certain non-maximal rank condition which depends on both \(j\) and \(M\). The \(\Gamma^j(G)\) are called non-maximal rank varieties. The collection \(\{\Gamma^j(G)_M\}\) is finer than \(\Pi(G)\), and each \(\Gamma^j(G)_M\) is a proper closed subset of \(\Pi(G)\). Other properties are proved, such as \(\Gamma^j(G)_M\) is empty if and only if \(M\) have constant \(j\)-rank, these varieties do not differentiate between stably isomorphic \(kG\)-modules nor modules in the same component of the stable Auslander-Reiten quiver, and the union of the \(\Gamma^j(G)_M\) is equal to \(\Gamma(G)_M\).
Another class of invariants is introduced when \(M\) is of constant rank, i.e., then the rank of the operator \(M_K\to M_K\) induced from a \(\pi\)-point is independent of the choice of \(\pi\)-point. A cohomology class \(\zeta\in H^1(G,M)\) gives rise to an extension \(E_\eta\) of \(k\) by \(M\). In an effort to generalize the zero locus in \(\text{Spec}(H^\bullet(G,k))\) a subset \(Z(\zeta)\) is constructed. Here \(Z(\zeta)\) is shown to be \(\Pi(G)\) if the extension with \(E_\zeta\) as above is locally split, otherwise \(Z(\zeta)=\Gamma^1(G)_{E_\zeta}\), establishing that \(Z(\zeta)\) is necessarily closed in \(\Pi(G)\). This construction is then generalized to extension classes \(\zeta\in\text{Ext}_G^n(M,N)\) where \(M\) and \(N\) are \(kG\) modules of constant Jordan type. support varieties; finite group schemes; \(\pi\)-points; coordinate algebras; modules of constant Jordan type; modular representations Eric M. Friedlander and Julia Pevtsova, Generalized support varieties for finite group schemes, Doc. Math. Extra vol.: Andrei A. Suslin sixtieth birthday (2010), 197 -- 222. Representation theory for linear algebraic groups, Group schemes, Modular representations and characters, Cohomology theory for linear algebraic groups, Representations of associative Artinian rings Generalized support varieties for finite group schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We compute the number of rational points of classifying stacks of Chevalley group schemes using the Lefschetz-Grothendieck trace formula of Behrend for \(\ell\)-adic cohomology of algebraic stacks. From this we also derive associated zeta functions for these classifying stacks. Chevalley group schemes; algebraic groups; finite groups of Lie type; classifying stacks; \(\ell\)-adic cohomology Group schemes, \(p\)-adic cohomology, crystalline cohomology On the number of rational points of classifying stacks for Chevalley group schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a simple algebraic group over an algebraically closed field. Assume the characteristic is zero or good. Let \(e\) be a subregular nilpotent element of the Lie algebra of a Borel group \(B\). Its Springer fibre is a union of projective lines, known as a Dynkin curve. There is an action of the centralizer \(C_G(e)\) on this Dynkin curve.
The authors determine in what cases \(C_G(e)\) acts with finitely many orbits, or when it acts trivially. The projective lines that make up a Dynkin curve each have a simple root as type and the answer is given type by type. By Spaltenstein these results imply answers to similar questions for the action of \(B\) on the orbital varieties associated to \(e\). subregular classes; Dynkin curves; orbital varieties; simple algebraic groups; Borel subgroups; adjoint actions; Springer fibers; Lie algebras; irreducible components; numbers of orbits Goodwin, SM; Hille, L; Röhrle, G, The orbit structure of Dynkin curves, Math. Z., 257, 439-451, (2007) Linear algebraic groups over arbitrary fields, Classical groups (algebro-geometric aspects), Lie algebras of linear algebraic groups The orbit structure of Dynkin curves. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite group \(G\subset \text{SO}(3, {\mathbb R})\) let \(\tilde{G}\) denote the inverse image via the double covering \(\text{SU}(2) \to \text{SO}(3,{\mathbb R})\). The moduli space of clusters \(\tilde{G}\text{-Hilb}({\mathbb C}^2)\) is a natural resolution of the quotient singularity \({\mathbb C}^2/\tilde{G}\), and where the exceptional curves correspond to irreducible representations of \(\tilde{G}\). Moreover, any irreducible representation of \(G\) is also an irreducible representation of \(\tilde{G}\). The authors construct a map between the moduli spaces \(\tilde{G}\text{-Hilb}({\mathbb C}^2) \to G\text{-Hilb}(\mathbb{C}^3)\), and they show that there is an induced map of exceptional divisors which contracts components that do not correspond to irreducible representations of \(G\). quotient singularities; McKay correspondence; Hilbert schemes; polyhedral groups Boissière, S.; Sarti, A., Contraction of excess fibres between the McKay correspondences in dimensions two and three, Ann. Inst. Fourier (Grenoble), 57, 1839-1861, (2007) Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Ordinary representations and characters Contraction of excess fibres between the McKay correspondences in dimensions two and three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author discusses how to generalize the classical invariant theory for Kleinian groups and the theory of simple elliptic singularities by his theory of the systems of regular weights (not the weight system in the theory of representations of semi-simple Lie algebras). Let \(G\) be the group generated by \(x, y, z\) with the fundamental relations \(x^p=y^q=z^r=xyz=1\) i.e. the polyhedral group of type \((p,q,r)\). \(G\) is a finite group if and only if \((p,q,r)=(2,2,n)\), \((2,3,3)\), \((2,3,4)\) or \((2,3,5)\). The cyclic group of degree \(n\) together with these groups are called Kleinian groups. To these groups correspond the extended Dynkin diagrams \(\tilde A_{n-1}\), \(\tilde D_{n+2}\), \(\tilde E_6\), \(\tilde E_7\) and \(\tilde E_8\). Conversely, from the exponents \(m_1,\ldots,m_{\mu}\) of the Coxeter transformation in the Weyl group of the diagram, the author finds that the function \(\chi (T)=\sum_{i}T^{m_i}\) is given by
\[
\chi (T)=T^{-h}((T^ h - T^a)(T^h - T^b)(T^h - T^c))/((T^a - 1) (T^b-1)(T^ c-1))
\]
for some natural numbers \(a, b, c\) and the order \(h\) of the Coxeter transformation. In general, a quadruple \((a,b,c,h)\) of natural numbers is defined to be a regular system of weights if \(h>\max (a,b,c)\) and \(\chi(T)\) defined as above has no pole except at \(T=0\). Then \(\chi(T)\) can be expressed as \(\sum_{i}a_{m_ i}T^{m_ i}\) where \(m_ i\) (1\(\leq i\leq \mu)\) are integers (not necessarily positive) which are called the exponent of the system. If the exponents are all positive, the regular systems of weights correspond bijectively to the Kleinian groups. In this case, \(G\) acts naturally on \(C[u,v]\) and the invariant subring \(C[u,v]^G\) is generated by three homogeneous polynomials \(x, y, z\) and it is isomorphic to \(C[X,Y,Z]/(f(X,Y,Z))\) for some polynomial \(f\in C[X,Y,Z]\). Let \(X_0\) be the hypersurface in \(C^3\) defined by \(f\), then \(G\) is given by \(\pi_1(X_0 - \{0\})\). In general, the author classifies some types of regular systems of weights and discusses the corresponding groups and singularities. When the smallest weight is zero or the only negative one, to the systems of the weights correspond certain Heisenberg groups (and the corresponding \(X_0\) are simple elliptic singularities) or some Fuchsian groups of the first kind [cf. the author's papers Invent. Math. 23, 289--325 (1974; Zbl 0296.14019); Publ. Res. Inst. Math. Sci. 21, 75--179 (1985; Zbl 0573.17012); ibid. 19, 1231--1264 (1983; Zbl 0539.58003); Adv. Stud. Pure Math. 8, 479--526 (1986; Zbl 0626.14028)].
This paper is continued in a second part, reviewed in Zbl 0629.17006. invariant theory for Kleinian groups; simple elliptic singularities; systems of regular weights; polyhedral group; extended Dynkin diagrams; Coxeter transformation; Weyl group; Heisenberg groups; Fuchsian groups Saito, K.: Around the theory of the generalized weight system: relations with singularity theory, the generalized Weyl group and its invariant theory, etc.. Sugaku 38, 97-115 (1986) Fuchsian groups and their generalizations (group-theoretic aspects), Simple, semisimple, reductive (super)algebras, Infinite-dimensional Lie groups and their Lie algebras: general properties, Singularities of surfaces or higher-dimensional varieties, Geometric invariant theory, Infinite-dimensional Lie (super)algebras, Applications of Lie groups to the sciences; explicit representations, Lie algebras of Lie groups Around the general theory of weight systems. I: Theory of singularities, the general Weyl group and its relation to the theory of invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Lambda\) be a self-injective tame algebra over an algebraically closed field \(k\). A result of \textit{J.\ Rickard} [Bull. Lond. Math. Soc. 22, No. 6, 540-546 (1990; Zbl 0742.16007)] states that if the complexity \(\text{cx}_\Lambda(M)\) of a finite dimensional \(\Lambda\)-module \(M\) is at least \(3\) then \(M\) is a wild algebra (or, equivalently, if \(M\) is tame then \(\text{cx}_\Lambda(M)\leq 2\)). However, the first lemma of that paper is a test for wildness which was shown to be invalid in 2004, and therefore the statement above remains unproven.
In this work, the author establishes this theorem in the case where \(\Lambda=\mathcal B\) is a block of a cocommutative Hopf algebra \(H\). This is accomplished using support varieties. Recall that, for \(\mathcal G\) a finite group scheme with algebra of measures \(H\), the cohomological support variety for a simple \(\mathcal B\)-module \(S\) is the radical of the kernel of the homomorphism \([f]\mapsto[f\otimes\text{id}_M]\colon H^\bullet(\mathcal G,k)\to\text{Ext}_{\mathcal G}^\bullet(S,S)\). Furthermore, the support variety \(\mathcal V_{\mathcal B}\) for the block \(\mathcal B\) is the union of the support varieties above for a collection \(\{S_i\}\) of simple \(\mathcal B\)-modules which form a complete set of such modules. The main theorem is that if \(\dim\mathcal V _{\mathcal B}\geq 3\) then \(\mathcal B\) is wild. The relationship of complexity to this dimension establishes the desired result.
There is an application of the above theorem to reduced enveloping algebras of finite-dimensional restricted Lie algebras. Given \((\mathfrak g,[p])\) a finite-dimensional restricted Lie algebra and \(\chi\in\mathfrak g^*\) we may form the \(\chi\)-reduced enveloping algebra \(U_\chi(\mathfrak g):=U(\mathfrak g)/I_\chi\), where \(U(\mathfrak g)\) is the universal enveloping algebra of \(\mathfrak g\) and \(I_\chi\) is an ideal generated by \(\{x^p-x^{[p]}-\chi(x)^p1\mid x\in\mathfrak g\}\). After defining the support variety for a block \(\mathcal B\subset U_\chi(\mathfrak g)\) the work above quickly establishes that if \(\mathcal B\) is tame then \(\text{cx}_{U_\chi(\mathfrak g)}(M)\leq 2\). tame algebras; support varieties; representations of Hopf algebras; tame representation type; complexity of finite-dimensional modules; finite group schemes; cocommutative Hopf algebras; reduced enveloping algebras; restricted Lie algebras Farnsteiner, Rolf, Tameness and complexity of finite group schemes, Bull. Lond. Math. Soc., 39, 1, 63-70, (2007) Representation type (finite, tame, wild, etc.) of associative algebras, Group schemes, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Modular Lie (super)algebras, Representation theory for linear algebraic groups Tameness and complexity of finite group schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a finite group $\text{G}$ and a field $K$, the \textit{faithful dimension} of $\text{G}$ over $K$ is defined to be the smallest integer $n$ such that $\text{G}$ embeds into $\mathrm{GL}_{n}(K)$. We address the problem of determining the faithful dimension of a $p$-group of the form $\mathscr{G}_{q}:=\exp (\mathfrak{g}\otimes_{\mathbb{Z}}\mathbb{F}_{q})$ associated to $\mathfrak{g}_{q}:=\mathfrak{g}\otimes_{\mathbb{Z}}\mathbb{F}_{q}$ in the Lazard correspondence, where $\mathfrak{g}$ is a nilpotent $\mathbb{Z}$-Lie algebra which is finitely generated as an abelian group. We show that in general the faithful dimension of $\mathscr{G}_{p}$ is a piecewise polynomial function of $p$ on a partition of primes into Frobenius sets. Furthermore, we prove that for $p$ sufficiently large, there exists a partition of $\mathbb{N}$ by sets from the Boolean algebra generated by arithmetic progressions, such that on each part the faithful dimension of $\mathscr{G}_{q}$ for $q:=p^{f}$ is equal to $fg(p^{f})$ for a polynomial $g(T)$. We show that for many naturally arising $p$-groups, including a vast class of groups defined by partial orders, the faithful dimension is given by a single formula of the latter form. The arguments rely on various tools from number theory, model theory, combinatorics and Lie theory. faithful dimension of finite groups; Kirillov's orbit method; Lazard correspondence; Frobenius sets; free nilpotent Lie algebras Representation theory for linear algebraic groups, Ordinary representations and characters, Finite nilpotent groups, \(p\)-groups, Rational points Kirillov's orbit method and polynomiality of the faithful dimension of $p$-groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We discuss the Lie-Poisson group structure associated to splittings of the loop group \(LGL(N,\mathbb{C})\), due to Sklyanin. Concentrating on the finite-dimensional leaves of the associated Poisson structure, we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a \(\mathbb{C}^*\)-invariant Poisson structure. In particular, Sklyanin's Lie-Poisson structure admits a suitable abelianisation once one passes to an appropriate spectral curve. The Sklyanin structure is then equivalent to one considered by Mukai, Tyurin and Bottacin on a moduli space of sheaves on the Poisson surface. The abelianization procedure gives rise to natural Darboux coordinates for these leaves, as well as separation of variables for the integrable Hamiltonian systems associated to invariant functions on the group. Hilbert schemes; spectral curves; integrable systems on loop algebras and loop groups; Lie-Poisson group structure; abelianisation; separation of variables; integrable Hamiltonian systems Hurtubise, J.; Markman, E., Surfaces and the Sklyanin bracket, Commun. Math. Phys., 230, 485-502, (2002) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Parametrization (Chow and Hilbert schemes), Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Surfaces and the Sklyanin bracket | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Étant donné un nombre premier \(p\), on appelle pro-\(p\)-groupe une limite projective de \(p\)-groupes finis, chacun étant muni de la topologie discrète; ce sont donc des groupes compacts totalement discontinus, dans lesquels la suite des \(p^n\)-èmes puissances d'un élément quelconque tend vers l'élément neutre \(e\) lorsque \(n\) tend vers \(+\infty\). D'autre part, soit \(\mathbb Q_p\) le corps \(p\)-adique; un groupe analytique \(p\)-adique est un ensemble \(G\) sur lequel sont définies: (1) une structure de variété analytique sur \(\mathbb Q_p\); (2) une application analytique \(G\times G\rightarrow G\) qui définit une loi de groupe sur \(G\). Il est facile de voir qu'un groupe analytique \(p\)-adique est un pro-\(p\)-groupe. L'A. se propose surtout d'étudier les groupes analytiques \(p\)-adiques compacts et de les caractériser parmi les pro-\(p\)-groupes.
A cet effet, il introduit et utilise la notion de groupe \(p\)-valué. C'est un groupe \(G\) muni d'une application \(\omega\) de \(G\) dans \(]0, +\infty]\) (dite \(p\)-valuation) vérifiant les 5 conditions: 1. \(\omega(x) < +\infty\) pour \(x\ne e\); 2. \(\omega(x)> 1/(p - 1)\) pour tout \(x\in G\); 3. \(\omega(xy^{-1}) \ge \inf (\omega(x), \omega(y))\); 4. \(\omega(x^{-1}y^{-1}xy) \ge \omega(x) + \omega(y)\); 5. \(\omega(x^p) = \omega(x) + 1\).
Les ensembles \(G_a\) définis par \(\omega(x) \ge a\) (pour \(a > 0)\) sont des sous-groupes distingués de \(G\) qui forment un système fondamental de voisinages de \(e\) pour une topologie de groupe sur \(G\); \(G\) est dit \(p\)-saturé s'il est complet pour cette topologie, et de plus si, pour \(\omega(x) > p/(p - 1)\) il existe \(y\in G\) tel que \(y^p = x\). Les sous-groupes \(G_a\) constituent ce que l'A. appelle une filtration sur \(G\) (en un sens généralisé du sens usuel où les indices sont entiers), et il lui associe une \(\mathbb Z\)-algèbre de Lie graduée (en un sens généralisé de la même façon) \(\operatorname{gr}(G)\), qui est naturellement munie d'une structure d'algèbre de Lie sur l'anneau \(\mathbb F_p[T]\) des polynômes sur le corps premier \(\mathbb F_p\).
L'étude de ces notions est faite de façon approfondie dans les chap. I et II, et leur application aux groupes analytiques \(p\)-adiques commence avec le chap. III. Pour un groupe \(p\)-valué \(G\), \(\operatorname{gr}(G)\) est un \(\mathbb F_p[T]\)-module libre; son rang est ce qu'on appelle le rang de \(G\). Si \(G\) est un groupe analytique \(p\)-adique de dimension \(r\) en tant que variété analytique sur \(\mathbb Q_p\), il contient un sous-groupe ouvert \(H\) dont la topologie peut être définie par une \(p\)-valuation à valeurs entières pour laquelle \(H\) est de rang \(r\), et est \(p\)-saturé. Le sous-groupe \(H_{n+1}\) si \(p > 2\) (resp. \(H_{n+2}\) si \(p = 2)\) est alors l'ensemble des \(p^n\)-èmes puissances des éléments de \(H\), autrement dit la \(p\)-valuation de \(H\) est entièrement déterminée par sa structure de groupe abstrait. On dit qu'un groupe abstrait \(G\) est \(p\)-valuable (resp. \(p\)-saturable) s'il existe une \(p\)-valuation de \(G\) pour laquelle \(G\) est \(p\)-valué complet (resp. \(p\)-saturé) et de rang fini. Tout groupe \(p\)-valuable possède une structure canonique de pro-\(p\)-groupe analytique; et inversement, un pro-\(p\)-groupe analytique \(G\) possède un sous-groupe \(p\)-valuable d'indice fini égal à une puissance de \(p\); la dimension de \(G\) est donnée par \(r = \displaystyle\lim_{n\to\infty} n^{-1} \log_p\left(G:G^{p^n}\right)\), où \(G^{p^n}\) désigne le sous-groupe engendré par les puissances \(p^n\)-èmes des éléments de \(G\).
L'A. donne alors de remarquables critères pour qu'un pro-\(p\)-groupe de type fini (c'est-à-dire engendré topologiquement par un ensemble fini) soit un pro-\(p\)-groupe analytique: par exemple, si \(p > 2\), il suffit que tout commutateur soit contenu dans le sous-groupe de \(G\) engendré par les \(p\)-èmes puissances; ou encore, pour tout \(p\), il suffit que chaque commutateur soit contenu dans le sous-groupe engendré par les \(p^2\)-èmes puissances. D'autres critères intéressants sont donnés dans l'Appendice.
Le chap. IV développe la correspondance biunivoque canonique entre groupes analytiques \(p\)-saturés et algèbres de Lie sur l'anneau \(\mathbb Z_p\) des entiers \(p\)-adiques, valuées et saturées (en des sens correspondant à ceux définis pour les groupes). Pour cela, on définit canoniquement, à partir d'un groupe \(p\)-saturé \(G\), une ``application diagonale'' \(\Delta: A\to A\otimes A\) sur la saturée \(A\) de l'algèbre de groupe \(\mathbb Z_p[G]\); \(G\) se reconstitue alors à partir de \(A\) comme l'ensemble des \(x\) tels que \(\Delta(x) = x \otimes x\) et tels que la valuation de \(x - 1\) soit \(> 1/(p - 1)\). On définit sur ce même ensemble une structure d'algèbre de Lie au moyen de la formule de Hausdorff, qui permet inversement de remonter de cette structure d'algèbre de Lie à la structure de groupe de \(G\).
Le dernier chapitre traite de la cohomologie des groupes analytiques \(p\)-adiques. Deux sortes de cohomologie sont introduites, la cohomologie ``continue'' et la cohomologie ``analytique'', à valeurs dans un \(\mathbb Z_p\)-module complet \(M\); on prouve qu'elles sont isomorphes lorsque \(q\) contient un sous-groupe \(p\)-valuable d'indice fini et que \(M\) est sans torsion et de rang fini sur \(\mathbb Z_p\). D'autre part, la cohomologie continue de \(G\) dans \(M\) s'identifie canoniquement à la cohomologie de son algèbre de Lie, moyennant des hypothèses convenables sur \(M\). On a enfin une ``dualité de Poincaré'' pour un groupe \(p\)-valué complet de rang fini \(G\) lorsque \(\operatorname{gr}(G)\) est un \(\mathbb Z_p[T]\)-module engendré par ses éléments d'un même degré (on dit alors que \(G\) est équi-\(p\)-valué; tout pro-\(p\)-groupe analytique contient un sous-groupe ouvert équi-\(p\)-valué); l'algèbre de cohomologie continue \(H_c^*(G; \mathbb F_p)\) est isomorphe à l'algèbre extérieure du \(\mathbb F_p\)-espace vectoriel \(H_c^1(G; \mathbb F_p)\). characterization of p-adic analytic groups; projective limit of finite p-groups; totally discontinuous compact groups; correspondence with Lie algebras; cohomological dimension; continuous cohomology; analytic cohomology; cohomology of Lie algebras M. Lazard, Groupes analytiques \textit{p}-adiques, Inst. Hautes Études Sci. Publ. Math. (1965), no. 26, 389-603. Algebraic groups, Representations of Lie and linear algebraic groups over global fields and adèle rings \(p\)-adic analytic groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a connected, simple algebraic group of adjoint type over \(\mathbb C\) with corresponding Lie algebra \(\mathfrak g\). The orbit \(\mathcal O\) of a nilpotent element in \(\mathfrak g\) under the adjoint action is called a nilpotent orbit, and its closure \(\overline{\mathcal O}\) is a union of finitely many nilpotent orbits. The closure inclusion is a partial order on the set of nilpotent orbits, and this article considers the generic singularities, the singularities of \(\overline{\mathcal O}\) at points of maximal orbits of its singular locus.
\textit{H. Kraft} and \textit{C. Procesi} [Invent. Math. 62, 503--515 (1981; Zbl 0478.14040)] have determined the generic singularities for the classical types of Lie algebras, \textit{E. Brieskorn} [Actes Congr. internat. Math. 1970, 2, 279-284 (1971; Zbl 0223.22012)] and \textit{P. Slodowy} [``Simple singularities and simple algebraic groups''. Lect. Notes Math. 815 (1980; Zbl 0441.14002)] determined them for the whole nilpotent cones \(\mathcal N\) for \(\mathfrak g\) of any type. The goal of this paper is to determine the generic singularities of \(\overline{\mathcal O}\) when \(\mathfrak g\) is of exceptional type.
A basic result is that the singular locus of \(\overline{\mathcal O}\) coincides with the boundary of \(\mathcal O\) in \(\overline{\mathcal O}\). To study generic singularities of \(\overline{\mathcal O}\), it is sufficient to study each maximal orbit \(\mathcal O^\prime\) in the boundary \(\overline{\mathcal O}\) of \(\mathcal O\). Such an \(\mathcal O^\prime\) is called a \textit{minimal degeneration} of \(\mathcal O\).
The local geometry of \(\overline{\mathcal O}\) at \(e\in\mathcal O^\prime\) is determined by the intersection of \(\overline{\mathcal O}\) with a transverse slice to \(\mathcal O^\prime\) at \(e\) in \(\mathfrak g\). Such slices exists in all cases, given by the affine space \(\mathcal S_e=e+\mathfrak g^f\), called the \textit{Slodowy slice}, where \(e,f\) are the nilpotent parts of an \(\mathfrak{sl}_2 \)-triple, and \(\mathfrak g^f\) is the centralizer of \(f\) in \(\mathfrak g\). The local geometry is thus encoded in \(\mathcal S_{\mathcal O,e}=\overline{\mathcal O}\cap\mathcal S_e\) which is named a \textit{nilpotent Slodowy slice}. If \(\mathcal O^\prime\) is a minimal degeneration of \(\mathcal O\), \(\mathcal S_{\mathcal O,e}\) has an isolated singularity at \(e\), and the generic singularities of \(\overline{\mathcal O}\) can be determined by studying the various \(\mathcal S_{\mathcal O,e}\) as \(\mathcal O^\prime\) runs over all minimal degenerations and \(e\in\mathcal O^\prime\), the isomorphism type of \(\mathcal S_{\mathcal O,e}\) being independent of \(e\).
The main result of the article is a classification of \(\mathcal S_{\mathcal O,e}\) up to algebraic isomorphism for each minimal degeneration \(\mathcal O^\prime\) of \(\mathcal O\) in the exceptional types. In a few cases, the result is able to determine the normalization of \(\mathcal S_{\mathcal O,e}\), and in another few cases, \(\mathcal S_{\mathcal O,e}\) is only determined up to local analytic isomorphism.
The article uses the theory of symplectic varieties. By \textit{Y. Namikawa} [J. Reine Angew. Math. 539, 123--147 (2001; Zbl 0996.53050)], a normal variety is symplectic if and only if its singularities are rational, Gorenstein, and its smooth part carries a holomorphic symplectic form, and it is proved that the normalization of a nilpotent orbit \(\overline{\mathcal O}\) is a symplectic variety. Because the normalization \(\overline{\mathcal O}\) has rational Gorenstein singularities, so has the normalization \(\tilde{\mathcal S}_{\mathcal O,e}\), and it is a symplectic variety. A singularity of a symplectic variety is what is called a \textit{symplectic singularity}, and the authors claim that a better understanding of those is important for studying the conjecture that a Fano contact manifold is homogeneous. Thus it is important to find examples of symplectic singularities, and the study of the isolated symplectic singularity \(\tilde{\mathcal S}_{\mathcal O,e}\) contributes to this.
The results in the article is motivated by representation theory. The geometry of the nilpotent cone \(\mathcal N\) was important in Springer's construction of Weyl group representations and the resulting Springer correspondence. It is proved that modular representation theory of the Weyl group of \(\mathfrak g\) is encoded in the geometry of \(\mathcal N\). Its decomposition matrix is a part of the decomposition matrix for equivariant perverse sheaves on \(\mathcal N\). Also, the authors remark that the reappearance of certain singularities in different nilpotent cones leads to equalities between parts of decomposition matrices. In the \(\text{GL}_n\)-case, the row and column removal rule for nilpotent singularities gives a geometric explanation for a similar rule for decomposition matrices of symmetric groups.
The main results in the article concerns simple surface singularities and their symmetries. A finite subgroup \(\Gamma\subset\text{SL}_2(\mathbb C)\cong\text{Sp}_2\) acts on \(\mathbb C^2\) and the quotient variety is an affine symplectic variety with an isolated singularity at the image of \(0\), known as a simple surface singularity, a double point, a du Val singularity, or a Kleinian singularity. Up to conjugacy in \(\text{SL}_2(\mathbb C)\), such \(\Gamma\) are in one-to-one correspondence with the simply-laced, simple Lie-algebras over \(\mathbb C\). Thus the simple singularities can be denoted as \(A_k\), \(D_k\;(k\geq 4)\), \(E_6\), \(E_7\), \(E_8\), according to the associated simple Lie algebra. The article contains a proof of the fact that in dimension 2, an isolated symplectic singularity is equivalent to a simple surface singularity. Also, the more general case is considered.
An automorphism of \(X=\mathbb C^2/\Gamma\) gives rise to a graph automorphism of the dual graph \(\Delta\) of \(X\). The authors ask when the action of \(\text{Aut}(\Delta)\) on the dual graph comes from an algebraic action on \(X\), and proves for which types of \(X\) this is true.
The article contains the study of the regular nilpotent orbit, starting with the generic singularities of the nilpotent cone.
In proving a conjecture of Grothendieck, Brieskorn and Slodowy [loc. cit.] described the generic singularities of the nilpotent cone \(\mathcal N\) of \(\mathfrak g\). In this particular case, \(\mathcal O\) is the regular nilpotent orbit, and so \(\overline{\mathcal O}\) equals \(\mathcal N\) with only one degeneration at the subregular nilpotent orbit \(\mathcal O^\prime\). Slodowy concludes that when \(e\in\mathcal O^\prime\), the slice \(\mathcal S_{\mathcal O,e}\) is algebraically isomorphic to a simple surface singularity. Also, when the Dynkin diagram of \(\mathfrak g\) is simply-laced, the Lie algebra associated to the simple surface singularity is \(\mathfrak g\). When it is not simply-laced, the singularity \(\mathcal S_{\mathcal O,e}\) belongs to a list given explicitly in the article. The authors explain an intrinsic realization of the symmetry of \(\mathcal S_{\mathcal O,e}\) when \(\mathfrak g\) is not simply-laced.
Kraft and Procesi [loc. cit]. classified the generic singularities of nilpotent orbit closures for all the classical groups, up to smooth equivalence. The \textit{minimal singularities} are those corresponding to the equivalence classes of a particular class of singularities, explicitly defined by their orbit closures, and denoted \(a_k,b_k,c_k,d_k(k\geq 4),g_2,f_4,e_6,e_7,e_8\). The \textit{generic singularities} in the classical types are classified: An irreducible component of a generic singularity is either a simple surface singularity or a minimal singularity, up to smooth equivalence. When a generic singularity is not irreducible, then it is smoothly equivalent to a union of two simple surface singularities of type \(A_{2k-1}\) meeting transversally in the singular point, denoted \(2A_{2k-1}\).
The main goal of the article is to describe the classification of generic singularities in the exceptional Lie algebras. The symmetry of minimal singularities and the generalization to the general case is essential. The exceptional Lie algebras introduce additional singularities, and and the non-normal cases is treated thoroughly. The main theorem classify generic singularities of nilpotent orbit closures in a simple Lie algebra of exceptional type, and graphs at the end of the article list precise results.
The object of the article is very interesting, and its goal is justified. In addition to treating the subject in a very interesting way, the authors introduce new theories, and explain established ones in a good way, making this a good overview of the subject. nilpotent orbits; symplectic singularities; Slodowy slice; Dynkin diagrams; Dynkin classification; transverse slice; generic singularities; degenerate orbits; closure relation; simple singularities,isolated singularities; simple surface singularities; generic orbits; generic singularities; exceptional Lie algebras; Springer correspondence B. Fu, D. Juteau, P. Levy and E. Sommers, \textit{Generic singularities of nilpotent orbit closures}, [arXiv:1502.05770]. Singularities in algebraic geometry, Geometric invariant theory, Lie algebras of linear algebraic groups Generic singularities of nilpotent orbit closures | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be an algebraically closed field of arbitrary characteristic \(p\), and \(G\) be a finite group. Then the group algebra \(KG\) is a product of connected algebras \(B_i\), \(i=1,\ldots,m\), called the blocks of \(KG=B_1\times\cdots\times B_m\). Let \(B\) be a block of \(KG\). The authors prove that every degeneration of finite dimensional \(B\)-modules is given by short exact sequences if and only if the block \(B\) is of finite representation type. As a result, for such block \(B\) the partial orders \(\leq_{\text{deg}}\) and \(\leq_{\text{ext}}\) on the set of isomorphism classes of finite dimensional \(B\)-modules [see \textit{G. Zwara}, J. Algebra 198, No. 2, 563-581 (1997; Zbl 0902.16015)] coincide for all \(B\)-modules. group algebras of finite groups; finitely generated algebras; Morita equivalences; quivers; degenerations of modules; decomposable modules; biserial algebras; representation type Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Module categories in associative algebras, Group rings of finite groups and their modules (group-theoretic aspects), Group actions on varieties or schemes (quotients) Degenerations for modules over blocks of group algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We mainly deal with the generic behaviour of Cartan invariants for finite groups of Lie type in this paper. Following \textit{L. Chastkofsky}'s paper [J. Algebra 103, 466-478 (1986; Zbl 0605.20046)], we describe the generic Cartan invariants for finite groups of Lie type by using the representation theory of Frobenius kernels, and show their symmetry properties. Moreover, we specialize to the case \(n=1\), and clarify the transition from \(n=1\) to general n. Finally, we explain the connection between Cartan invariants and generic Cartan invariants.
An abridged version of this paper was included in the proceedings of a conference in honor of L. K. Hua [Contemp. Math. 82, 235-241 (1989; Zbl 0673.20005)]. \textit{J. E. Humphrey}'s paper [J. Algebra 122, 345-352 (1989; Zbl 0674.20023)] is closer in spirit to the author's paper than to Chastkofsky's paper, but avoids the complicated calculations. He offers a more conceptual treatment, emphasizing the much simpler picture for the group schemes \(G_ nT\) associated with the Frobenius kernels \(G_ n\) in the ambient algebraic group. finite groups of Lie type; generic Cartan invariants; Frobenius kernels; group schemes Representation theory for linear algebraic groups, Modular representations and characters, Linear algebraic groups over finite fields, Representations of finite symmetric groups, Simple groups: alternating groups and groups of Lie type, Group schemes Cartan invariants of finite groups of Lie type. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The relatively new Putcha-Renner theory of algebraic monoids (with ground field algebraically closed) represents a beautiful blend of ideas from abstract semigroup theory, algebraic geometry, and the theory of linear algebraic groups. The group of units \(G(M)\) of an algebraic monoid \(M\) is an (affine) algebraic group. Every algebraic monoid is isomorphic to a Zariski closed submonoid of some total matrix monoid \(M_n(K)\). If \(M\) is irreducible (as a variety), \(M\) is the Zariski closure of \(G(M)\). Any algebraic monoid \(M\) is group-bounded: for each \(a\in M\), a power of \(a\) lies in a subgroup of \(M\). Hence \(M\) carries plenty of idempotents, which are intimately connected to the group structure: each idempotent of \(M\) lies in the closure of a maximal torus of \(G(M)\). Assume that \(M\) is an irreducible monoid with zero: observe that every connected algebraic group with nontrivial characters occurs as the group of units of an irreducible monoid with zero.
Most surprisingly, there is a confluence of the mainstreams in algebraic groups and in abstract semigroups: \(M\) (with zero) is reductive (i.e., \(G(M)\) is so as an algebraic group) iff \(M\) is regular (as a semigroup). (If \(M\) has no zero, the reviewer [cf. Semigroup Forum 52, No. 3, 319-323 (1996; Zbl 0855.20052)] has recently proved that \(M\) is reductive iff \(M\) is regular with its kernel a reductive group.) So a reductive monoid \(M\) is pieced together from its group of units and idempotents. This confluence is the outset and the base of the subject of reductive monoids -- the heart of the Putcha-Renner theory. The theory of reductive monoids has highly structured problems which, like those in the theory of reductive groups, are rooted in the combinatorics of the Weyl groups. It is remarkable that the natural questions about idempotents lead to all the standard constructs in semisimple Lie theory: weights, roots, parabolic subgroups, Tits building, and so forth, in particular to the most important special constructs for reductive monoids: the Putcha cross section lattice and the Renner monoid.
The paper under review is a wonderful introduction to the theory of reductive monoids, for readers with general background and interest in algebra but with no special knowledge of semigroup theory or reductive algebraic groups. The prerequisites for the theory to those readers seem formidable. To make the introduction more elementary and suggestive, the author presents the relative combinatorial data in a few familiar classic groups (such as \(GL_n\), \(SL_n\), \(Sp_n\)), \(M_n\) and some reductive monoids constructed from \(SL_n\) and \(Sp_n\). This is a marvelous way to describe how the key ideas from reductive groups go to reductive monoids and how certain constructs in reductive monoids over \(\overline{\mathbb{F}_q}\) descent to finite reductive monoids. Indeed, elementary but interesting examples are more important than theorems. The paper also contains many helpful footnotes for providing some further references as well as bits of arguments and hints concerning the depth of unsupported statements. It allows the reader to enter the subject with minimal prerequisites and hence is highly recommended to every willing beginner.
For a systematic theoretic introduction to the theory, the reader may read \textit{M. S. Putcha}'s monograph ``Linear algebraic monoids'' [Cambridge Univ. Press (1988; Zbl 0647.20066)]. linear algebraic monoids; linear algebraic groups; groups of units; Zariski closed submonoids; total matrix monoids; idempotents; maximal tori; reductive monoids; Weyl groups; weights; roots; parabolic subgroups; Tits buildings; Putcha cross section lattices; Renner monoids; reductive algebraic groups; finite reductive monoids; \(BN\)-pairs; Borel subgroups; Bruhat decompositions; finite groups of Lie type; finite monoids of Lie type; finite reductive groups; Frobenius maps; Hecke algebras Louis Solomon, An introduction to reductive monoids, Semigroups, formal languages and groups (York, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 466, Kluwer Acad. Publ., Dordrecht, 1995, pp. 295 -- 352. Semigroups of transformations, relations, partitions, etc., Linear algebraic groups over arbitrary fields, Research exposition (monographs, survey articles) pertaining to group theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Groups with a \(BN\)-pair; buildings, Linear algebraic groups over finite fields, Other geometric groups, including crystallographic groups, Regular semigroups, Inverse semigroups An introduction to reductive monoids | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this survey, we describe and relate various occurrences of quivers of type \(A\) (both finite and affine) and their canonical bases in combinatorics, in algebraic geometry and in representation theory. The ubiquity of these quivers makes them especially important to study: they are pervasive in very classical
topics (such as the theory of symmetric functions) as well as in some of the most recent and exciting areas of representation theory (such as representation theory of quantum affine algebras, affine Hecke algebras or Cherednik algebras).
There is a vast literature on the subject and we have been forced to make a choice in the selection of the results presented here. We believe that one of the main reasons for the omnipresence of quivers of type \(A\) is the fact that they are related (in more than one way) to classical and fundamental objects in geometric representation theory of type \(A\) such as (partial) flag varieties and nilpotent orbits. This is the point we tried to emphasize in this survey. For that reason, many interesting and important results are only alluded to or sketched here and we apologize to all those whose work we did not mention by lack of space and competence. Also, we do not give complete proofs of all results presented here and refer to the original papers whenever possible. Rather, we have tried to convey the fundamental ideas in a nontechnical way as much as possible. quivers of type \(A\); representation spaces; Ringel-Hall algebras; canonical bases; categories of representations; symmetric functions; quantum groups; flag varieties; nilpotent orbits; Schubert varieties; Kazhdan-Lusztig polynomials; Schur-Weyl duality; affine Hecke algebras Schiffmann, O., Quivers of type \textit{A}, flag varieties and representation theory, Fields inst. commun., vol. 40, 453-479, (2004), Amer. Math. Soc. Providence, RI Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Hecke algebras and their representations, Symmetric functions and generalizations Quivers of type \(A\), flag varieties and representation theory. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities If \(w\) is a word in \(d>1\) letters and \(G\) is a finite group, evaluation of \(w\) on a uniformly randomly chosen \(d\)-tuple in \(G\) gives a random variable with values in \(G\), which may or may not be uniform. It is known that if \(G\) ranges over finite simple groups of given root system and characteristic, a positive proportion of words \(w\) give a distribution which approaches uniformity in the limit as \(|G|\to\infty\). In this paper, we show that the proportion is in fact \(1\). word maps; random walks on finite simple groups; groups of Lie type Probabilistic methods in group theory, Varieties over finite and local fields, Finite ground fields in algebraic geometry, Linear algebraic groups over finite fields Most words are geometrically almost uniform | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This amazing book provides, in only 229 pages, an introduction to a large part of modern number theory -- from A. Weil's proof of the Riemann hypothesis for zeta functions of curves over finite fields to the theory of modular forms -- both in the classical and adelic languages. It includes most of the necessary background needed to derive estimates of many sorts of exponential sums which arise in the spectra of explicit Ramanujan graphs constructed as Cayley graphs of various finite groups. In particular, there is a discussion of the finite upper half plane graphs considered by the reviewer and co-workers in a series of papers. See the reviewer [Survey of spectra of Laplacians on finite symmetric spaces, Exp. Math. 5, No. 1, 15-32 (1996)] for a survey. The author provides a different method from that of N. Katz to show that these finite upper half plane graphs are Ramanujan. No \(\ell\)-adic cohomology is required. However, she does provide an introduction to this subject in Chapter 2.
The book begins with a chapter on finite fields. Then Chapter 2 shows how Weil arrived at the Weil conjectures (proved by P. Deligne) on zeta functions for projective varieties over finite fields starting with the estimation of the number of solutions to polynomial equations over finite fields using Jacobi sums. Chapter 3 concerns valuations of function fields over finite fields, completions, the adeles and the ideles. Chapters 4 and 5 concern zeta and \(L\)-functions of idele class characters of function fields. Here one finds proofs of the Riemann-Roch theorem and the Riemann hypothesis for the zeta function of a nonsingular projective curve over a finite field. Chapter 6 uses the preceding results plus a bit of class field theory from \textit{A. Weil} [Basic number theory, Springer-Verlag, Berlin (1973; Zbl 0267.12001)] to estimate character sums.
Chapter 7 sketches the classical theory of modular forms for congruence subgroups of \(SL(2, \mathbb{Z})\), including the theory of Hecke operators and \(L\)-functions attached to modular forms. Some of the background for the work of A. Wiles and R. Taylor proving Fermat's Last Theorem can be found here.
Chapter 8 gives the adelic representation theoretic version of the theory of modular forms. Work of H. Jacquet, R. Langlands, S. Gelbart, \dots including converse theorems, strong multiplicity one theorems, and the correspondence between automorphic representations of quaternion groups and admissible cuspidal representations of the adelic \(GL(2)\) are discussed.
Chapter 9 applies the preceding work to a part of graph theory which is of interest in computer science. A connected \(k\)-regular graph \(X\) is Ramanujan if for any eigenvalue \(\lambda\) of the adjacency matrix with \(|\lambda |\neq k\), we have \(|\lambda |\leq 2\sqrt {k-1}\). This bound implies the graph is a best possible expander among \(k\)-regular graphs as the number of vertices goes to infinity by a theorem of Alon and Boppana for which the author gives two proofs. It also implies that the simple random walk on \(X\) converges extremely rapidly to uniform. Some references for Ramanujan graphs are \textit{A. Lubotzky} [Discrete groups, expanding graphs, and invariant measures, Birkhäuser, Prog. Math. 125 (1994; Zbl 0826.22012)] and \textit{P. Sarnak} [Some applications of modular forms, Cambridge Tracts Math. 99 (1990; Zbl 0721.11015)].
The author allows her graphs to be directed if the adjacency matrix is diagonalizable by unitary matrices. It had been known for a long time that Ramanujan graphs exist, but G. Margulis and, independently, A. Lubotzky, R. Phillips and P. Sarnak constructed the first explicit examples in 1988 using quaternion groups. The author considers these constructions as well as others constructed using finite abelian groups by Fan Chung and herself, independently. Her earlier character sum estimates are necessary for the estimates of the eigenvalues of adjacency operators. Similarly these estimates are needed to show that the finite upper half plane graphs mentioned above are Ramanujan.
Finally she gives a proof due to herself and K. Feng of a result of B. D. McKay. It supposes one is given a sequence \(X_m\) of connected \(k\)-regular graphs with \(|X_m|\) going to infinity with \(m\), such that the number of primitive cycles (no backtracking and containing no proper subcycle) in \(X_m\) has moderate growth. Then the distribution of eigenvalues approaches the Wigner semi-circle distribution (alias the Sato-Tate distribution) as \(m\to \infty\). This is then applied to obtain a result of Feng and the author estimating the growth of the dimension of cusp forms of weight 2 for \(\Gamma_0 (N)\) which are eigenfunctions of the Hecke operator \(T_p\) with integral eigenvalues.
It is very interesting that number theory leads to graph theory and then back to new results in number theory, showing how fruitful this approach has become. In short, this book is highly recommended to those with an interest in some of the deepest and most beautiful results in modern number theory as well as the applications to the spectral theory of graphs. It is not an easy book for a reader without any familiarity with the modern adelic point of view. However, the reader will certainly be rewarded by contemplating this introduction to some very powerful mathematics. finite fields; character sums; Weil conjectures; Riemann-Roch theorem; points on curves over finite fields; zeta-functions; \(L\)-functions; idele class characters; modular forms; automorphic representations; Ramanujan graphs; Alon-Boppana theorem; regular graphs; Riemann hypothesis for zeta functions of curves over finite fields; exponential sums; Cayley graphs; finite upper half plane graphs; valuations of function fields; projective curve; Hecke operators; automorphic representations of quaternion groups; expander; simple random walk; spectral theory of graphs Li, W. -C. Winnie: Number theory with applications. Series of university mathematics 7 (1996) Research exposition (monographs, survey articles) pertaining to number theory, Modular and automorphic functions, Graph theory, Arithmetic theory of algebraic function fields, Curves over finite and local fields, Representation-theoretic methods; automorphic representations over local and global fields, Holomorphic modular forms of integral weight, Estimates on exponential sums, Exponential sums, Adèle rings and groups, Representations of Lie and linear algebraic groups over global fields and adèle rings, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Number theory with applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the book under review, the authors develop the theory of real hyperplane arrangements. In order to be a little bit more precise, a hyperplane arrangement \(\mathcal{A}\) is a finite set of codimension \(1\) affine subspaces in a finite-dimensional real vector space \(V\). The theory of hyperplane arrangements is one of the most classical subjects of research, and it develops in different directions. One can, for instance, try to classify all simplicial hyperplane arrangements (this problem is widely open, even in the case when \(\mathcal{A}\) is central and contained in \(V\) of dimension \(3\)), or just use hyperplane arrangements in different areas of mathematics, for instance in topology, algebraic geometry, or combinatorics. The authors decided to work over the field of real numbers, which is in fact no restriction at all because the theory which is presented in book is extremely rich and instructive. Starting from the very beginning (i.e., after defining all basic notions and objects), the authors develop the theory using mostly algebraic language, which is the key asset of the book (at least according to my subjective point of view). Before I will present a sketch of the content of this book, mostly due to the fact that there are a lot of extremely interesting results that I cannot present in a concise way, I will elaborate a little bit about the structure and some other feature of the book. It is divided into two parts and it contains also five appendices devoted to basics on regular cell complexes, posets and their properties, incidence algebras of posets, algebras and modules, and at the end on bands and distance functions. I strongly encourage every reader to have a short look on the mentioned appendices, just to check the terminology or recall many notions. I am quite sure that the reader will benefit after this effort. The book is written with a high level of clarity, contains a lot of interesting examples, and it is almost self-contained. I believe that this is the second key asset of this book which makes it a graduate-student-friendly book. At the end of each chapter, the authors provide the so-called notes where we can find details about the history of some objects/results, references to the articles related to the content, or just explanations about possible discrepancies between the terminology which is used by the authors and the existing one in the literature. The book contains both classical results and newly develop subjects (like a general theory of lunes), so we can think about this book as a research monograph. According to my view, this monograph together with a very recent book by \textit{A. Dimca} [Hyperplane arrangements. An introduction. Cham: Springer (2017; Zbl 1362.14001)] can be viewed as a modern introduction to the theory of hyperplane arrangements of a great value.
Now I am going to present sketchly the content of the book. In the Part I, the authors recall basics on hyperplane arrangements, flats and faces, posets, separating hyperplanes, characteristic polynomials and Zaslavsky formulae, isomorphisms of arrangements (geometric and combinatorial). The key point of the first part is to define the notion of Birkhoff and Tits monoids. We can view the lattice of flats \(\Pi[\mathcal{A}]\) as a commutative monoid with product given by the join operation, which leads to the Birkhoff monoid. The poset of faces \(\Sigma[\mathcal{A}]\) is not a lattice, but it carries a (non-commutative) monoid structure, and this leads to the notion of the Tits monoid (this is in fact an example of a left regular band), and for faces \(F,G\) the Tits product is denoted by \(FG\). A bi-face is a pair \((F,F')\) of faces such that \(F\) and \(F'\) have the same support. Let \(J[\mathcal{A}]\) denote the set of bi-faces, and we define the operation
\[
(F,F')(G,G') = (FG,G'F').
\]
This in turns allow us to define the Janus monoid. After that the authors focus on cones of an arrangement \(\mathcal{A}\), i.e, these are the subsets of the ambient space which can be obtained by intersecting some subset of half-spaces in the arrangement. In the case of cones, the authors focus mostly on the so-called base and case maps, and gallery intervals for chambers determined by real hyperplane arrangements. In the last part of Chapter 2, the authors recall the notion of rank functions, semimodularity, and join-distributivity. In Chapter 3, the authors develop the theory of lunes. For a cone \(V\) (determined by an arrangement) we can define the base \(b(V)\) which is the largest flat contained in that cone. For a hyperplane \(H\), we denote by \(H^{+}\) and \(H^{-}\) its two associated half-spaces. A \textit{lune} is a cone \(V\) with the following property that if a hyperplane \(H\) contains \(b(V)\), then either \(H^{+}\) contains \(V\) or \(H^{-}\) contains \(V\). The key result about lunes tells us that any cone can be optimally cut up into lunes by hyperplanes containing the base of the cone. In Chapter 4, the authors develop \textit{the category of lunes} -- this is a category whose objects are flats and morphisms are lunes. In Chapter 5, the authors present a theory of reflection arrangements and Coxter groups. This is a classical subject of research. At the end of this chapter, the authors classify the so-called \textit{good reflection arrangements}. It is well-known that if \(\mathcal{A}\) is a reflection arrangement, then for any flat \(X\) the localization \(\mathcal{A}_{X}\) is also reflection arrangement, but the restriction \(\mathcal{A}^{X}\) might not be a reflection arrangement. A reflection arrangement \(\mathcal{A}\) is good if for each flat \(X\) the arrangement \(\mathcal{A}^{X}\) is combinatorially isomorphic to a reflection arrangement. The main result (Theorem 5.28) provides a complete classification of irreducible good reflection arrangements. The whole Chapter 6 is devoted to braid arrangements and arrangements of type \(B\) and \(D\). For instance, the authors present a detailed description of enumerative features of a given braid arrangement (like the characteristic polynomial, Möbius number, etc.), and at the end they discuss the case of graphic arrangements (which are defined with use of simple graphs). In Chapter 7, the authors consider the so-called descent equations for chambers, i.e., for fixed chambers \(C\) and \(D\) the descent equation is \(HC=D\). In other words, one needs to solve for faces \(H\) such that the Tits product of \(H\) and \(C\) is equal to \(D\). More generally, we can fix faces \(F\) and \(G\), and consider the equation \(HF = G\). Apart from finding the solutions, there is also interest in computing the sum \(\sum (-1)^{rk(H)}\) as \(H\) ranges over the solution set, with \(\mathrm{rk}(H)\) denoting the rank of \(H\). For this, one can attach to the solution set a relative pair \((X,A)\) of cell complexes whose Euler characteristic is the given sum. A similar considerations can be done for the lune equations \(HC=D\), where a face \(H\) and a chamber \(D\) are fixed. It turns out that the solution set is precisely the set of chambers in some top-lune (i.e., a lune which is a top-cone). In Chapter 8, the authors study distance functions and Varchenko matrices. We say that a hyperplane separates two chambers if they lie on its opposite sides, so the distance between two chambers is the number of hyperplanes which separate them. Fix a scalar \(q\), and define a bilinear form on the set of chambers \(\Gamma[\mathcal{A}]\) by
\[
\langle C,D \rangle := q^{\mathrm{dist}(C,D)},
\]
where \(\mathrm{dist}(C,D)\) denotes the distance between two chambers \(C\) and \(D\). The determinant of this matrix factorizes with the factors of the form \(1-q^{i}\), and the bilinear form is non-degenerate if \(q\) is not a root of unity. These considerations can be further generalized by assigning to each half-space a weight, and then define \(\langle C, D \rangle\) to be the product of the weights of all half-spaces which contain \(C\), but do not contain \(D\). This idea leads to Varchenko's result on a factorization of the determinant of the associated matrix.
In Part II, the authors begin with the notion of Birkhoff and Tits algebras (which are linearizations of the associated monoids over a field \(\mathbb{K}\)). These algebras are \(\mathbb{K}\)-finitely dimensional algebras. One can show, for instance, that the Birkhoff algebra is isomorphic to \(\mathbb{K}^{n}\), where \(n\) is the number of flats. Moreover, one can show that the radical (the largest nilpotent ideal) of the Birkhoff algebra is trivial. In contrast, the Tits algebra has many nilpotent elements. Using a linearization argument applied to the Janus monoid one obtains the Janus algebra. This algebra is elementary (like in the case of the Tits algebra). Since the Janus algebra admits a deformation by scalar \(q\), one can show that if \(q\) is not a root of unity, then \(q\)-Janus algebra is isomorphic to a product of matrix algebras over \(\mathbb{K}\). In Chapter 10, the author study Lie and Zie elements. Given an arrangement \(\mathcal{A}\), the Tits algebra acts on the left module of chambers, so the primitive part of this module is the space of Lie elements. For instance, if \(\mathcal{A}\) is a braid arrangement then the space of Lie elements is the multilinear part of the free Lie algebra (so the so-called classical Lie elements). An interesting result about Lie elements is about the dimension of this space, it turns out that it is equal to the absolute value of the Möbius numer of \(\mathcal{A}\). Now we can consider the left action of the Tits algebra on itself, and the primitive part is called the space of Zie elements. It can be shown, for instance, that the space of Zie elements is a right ideal of the Tits algebra. In Chapter 11, the authors study Eulerian idempotents. An Eulerian family \(E\) is a complete system of primitive orthogonal idempotents of the Tits algebra and an Eulerian family constitutes of the so-called Eulerian idempotents. One of the main results of this chapter is Saliola's construction which allows to construct an Eulerian family starting with a homogeneous section (such a section is equivalent to an assigment of a scalar \(u^{F}\) to each face \(F\) such that for any flat \(X\) the sum of \(u^{F}\) over all \(F\) with support \(X\) is 1). In Chapter 12, the authors consider diagonalizability. An element of an algebra is diagonalizable if it can be expressed as a linear combination of orthogonal idempotents. One can show that all elements of the Birkhoff algebra are diagonalizable. However, this is no longer true for the Tits algebra -- no nonzero elements of the radical of the Tits algebra is diagonalizable. The main aim of this chapter is to provide a characterization of diagonalizable elements and in order to do so the authors follow Saliola's method using the existence of eigensections. In addition, the authors consider diagonalizablity of specific elements such Takeuchi elements (12.3.1), Fulman elements (12.4.8), and Adams elements (12.5.3). In the last four chapters, rather technical, the authors study Loewy series and Peirce decompositions, Dynkin idempotents (this includes an interesting Joyal-Klyachko-Stanley theorem about a natural isomorphism between the Lie space of an arrangement \(\mathcal{A}\) and the order top-cohomology of the lattice of flats), incidence algebras, and invariant Birkhoff and Tits algebras for reflection arrangements.
Let me emphasize here that my outline presents only a permil of the whole bunch of results of different flavors. On the other side, as an interested reader, I would like to see even a small chapter about matroids and their relations with hyperplane arrangements, or some deeper results on simplicial arrangements (like Deligne's result about \(K(\pi,1)\)-spaces), but this is my very subjective view which might not find any justification.
Summing up, this is a great and significant book about an active research area. It is written in great detail, in a self-contained way, and might be also useful for the experts. It is highly recommended to those people that want to learn things about real hyperplane arrangements from scratch. Coxeter groups; reflection arrangements; cones; posets; characteristic polynomials; lattices; distance functions; freeness; real hyperplane arrangements; lunes; Braid arrangements; descent and lune equations; Varchenko matrices; Birkhoff algebras; Tits algebras; Lie and Zie elements; Eulerian idempotents; homologies and cohomologies; Loewy series; Dynkin idempotents; Incidence algebras; Invariant algebras; Moebius functions , Hyperplane Arrangements, Universitext, Springer, Cham, 2017. Configurations and arrangements of linear subspaces, Research exposition (monographs, survey articles) pertaining to combinatorics, Combinatorial aspects of representation theory, Research exposition (monographs, survey articles) pertaining to ordered structures, Combinatorics of partially ordered sets, Algebraic aspects of posets, Semimodular lattices, geometric lattices Topics in hyperplane arrangements | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite group scheme \(G\), we continue our investigation of those finite-dimensional \(kG\)-modules that are of constant Jordan type. We introduce a Quillen exact category structure \(\mathcal C(kG)\) on these modules and investigate \(K_0(\mathcal C(kG))\). We study which Jordan types can be realized as the Jordan types of (virtual) modules of constant Jordan type. We also briefly consider thickenings of \(\mathcal C(kG)\) inside the triangulated category \(\text{stmod}(kG)\).
Together with \textit{J. Pevtsova}, the authors introduced [in J. Reine Angew. Math. 614, 191-234 (2008; Zbl 1144.20025)] an intriguing class of modules for a finite group \(G\) (or, more generally, for an arbitrary finite group scheme), the \(kG\)-modules of constant Jordan type. This class includes projective modules and endotrivial modules. It is closed under taking direct sums, direct summands, \(k\)-linear duals, and tensor products. We have several methods for constructing modules of constant Jordan type, typically using cohomological techniques. In the very special case that \(G=\mathbb Z/p\mathbb Z\times\mathbb Z/p\mathbb Z\), the authors and \textit{A. Suslin} have recently introduced several interesting constructions that associate modules of constant Jordan type to an arbitrary finite-dimensional \(kG\)-module and have identified cyclic \(kG\)-modules of constant Jordan type [Comment. Math. Helv. 86, No. 3, 609-657 (2011; Zbl 1229.20039)].
What strikes us as remarkable is how challenging the problem of classifying modules of constant Jordan type is even for relatively simple finite group schemes. In this paper we address two other aspects of the theory that also present formidable challenges. The first is the realization problem of determining which Jordan types can actually occur for modules of constant Jordan type. The second question concerns stratification of the entire module category by modules of constant Jordan type.
To consider realization, we give the class of \(kG\)-modules of constant Jordan type the structure of a Quillen exact category \(\mathcal C(kG)\) using ``locally split short exact sequences.'' This structure suggests itself naturally once \(kG\)-modules are treated from the point of view of \(\pi\)-points as in [\textit{E. M. Friedlander, J. Pevtsova}, Duke Math. J. 139, No. 2, 317-368 (2007; Zbl 1128.20031)], a point of view necessary to even define modules of constant Jordan type. With respect to this exact category structure, the Grothendieck group \(K_0(\mathcal C(kG))\) arises as a natural invariant. There are natural Jordan type functions JType, \(\overline{\text{JType}}\) defined on \(K_0(\mathcal C(kG))\) that are useful for formulating questions of realizability of (virtual) modules of constant Jordan type. The reader will find several results concerning the surjectivity of these functions.
A seemingly very difficult goal is the classification of \(kG\)-modules of constant Jordan type, or at least the determination of \(K_0(\mathcal C(kG))\). In this paper, we provide a calculation of \(K_0(\mathcal C(kG))\) for two very simple examples: the Klein four group and the first infinitesimal kernel of \(\mathrm{SL}_2\).
The category \(\mathcal C(kG)\) possesses many closure properties. However, the complexity of this category is reflected in the observation that an extension of modules of constant Jordan type need not be of constant Jordan type. We conclude this paper by a brief consideration of a stratification of the stable module category \(\text{stmod}(kG)\) by ``thickenings'' of \(\mathcal C(kG)\). finite group schemes; modules of constant Jordan type; exact categories; group algebras; Grothendieck groups; categories of finitely generated projective modules; thickenings; stable module categories; Frobenius kernels Jon F. Carlson and Eric M. Friedlander, Exact category of modules of constant Jordan type, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser Boston, Inc., Boston, MA, 2009, pp. 267 -- 290. Representation theory for linear algebraic groups, Group schemes, Modular representations and characters, Cohomology theory for linear algebraic groups, Group rings of finite groups and their modules (group-theoretic aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Cohomology of groups, Representations of associative Artinian rings, Module categories in associative algebras, \(K_0\) of group rings and orders Exact category of modules of constant Jordan type. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This work is devoted to explain the relationship between invariants appearing in different subjects in mathematics. The first field are the modular invariants of the partition function of a rational conformal field theory (RCFT) on a torus, specially in the case of \(su(3)\) models.
In the second section, the authors discuss modular invariance for \(su(3)\) theories. The partition function of an RCFT of height \(n\) on a torus \({\mathbb{C}}/({\mathbb{Z}}+\tau{\mathbb{Z}})\), \({\mathfrak I}\tau>0\), is determined by a matrix of non-negative integers \(N:=\{N_{p,p'}\}_{p,p'\in B_n}\), where
\[
B_n:=\{p:=(r,s,t)\in{\mathbb{Z}}^3\mid r,s,t\geq 1, r+s+t=n\}.
\]
The modular action of \(PSL_2({\mathbb{Z}})\) on \(\{{\mathfrak I}\tau>0\}\) induces two matrices \(S\), \(T\) which reflect the isomorphism of the corresponding elliptic curves. A function determined by a matrix \(N\) is a modular invariant if and only if \(N\) commutes with \(S\) and \(T\). The matrices \(S\) and \(T\) are rational combinations of \(3n\)-roots of unity and Galois theory is involved [see \textit{A. Coste} and \textit{T. Gannon}, Phys. Lett. B 323, 316-321 (1994)]. This theory produces a parity selection rule which makes the computation of modular invariants easier.
In the third section, the authors consider the Jacobian of a Fermat curve \(x^n+y^n=z^n\). The first coincidence with previous results is that a basis of holomorphic differentials is parametrized in a natural way by \(B_n\); in fact the Jacobian is isogenous to a product of Abelian varieties also indexed by \(B_n\). These factors possess complex multiplication (CM) and the general theory for simpleness of abelian varieties with CM may be applied, see the Shimura-Taniyama theorem [\textit{G. Shimura} and \textit{Y. Taniyama}, ``Complex multiplication of abelian varieties and its application to number theory'', Publ. Math. Soc. Jap. 6 (1961; Zbl 0112.03502)].
In the fourth section, combinatorial groups for triangulated surfaces are studied, following Grothendieck \textit{dessins d'enfants}. The starting point is the standard triangulation of the Riemann sphere with vertices \(0,1,\infty\); if \(h\) is a meromorphic map from \(\Sigma\) ramified at \(0,1,\infty\), the standard triangulation induces a special one of \(\Sigma\) which may be encoded by the cartographic group. The universal cartographic group is the modular group and any subgroup of finite index of the uniformizing group \(\Gamma_2\) determines up to isomorphism a pair \((\Sigma,h)\). The Kummer projection defines a cartographic group for Fermat curves which is also related with \(B_n\). A rational triangular billiard associated to a triangle of angles \(\pi/r\), \(\pi/s\), \(\pi/t\) with \(r+s+t=n\) is the classical phase space of a particle moving in the corresponding orbifold. The trajectories determine a closed Riemann surface \(C_{r,s,t}\) (called triangular curve) which projects onto the triangle and produce a cartographic group. It may happen that a Fermat curve projects onto a triangular curve.
In the fifth section, the authors study the Riemann surface of a RFCT on a torus. The goal is to show that a compact Riemann surface can be associated with any RCFT. The authors show for example that the curve associated with \(su(3)_1\) is a triangular curve admitting as covering the Fermat curve of degree \(12\).
The paper finishes with some conclusions and questions about the relationships stated in the paper and two appendices. affine Lie algebras; abelian varieties; modular invariant; partition function; rational conformal field theory; Jacobian of a Fermat curve; triangulated surfaces; Riemann surface M. Bauer, A. Coste, C. Itzykson and P. Ruelle, ''Comments on the links between SU(3) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards,'' J. Geom. Phys. 22 (1997), 134--189. Jacobians, Prym varieties, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Complex multiplication and abelian varieties Comments on the links between \(su(3)\) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field of characteristic \(p\). Any \(k[t]/(t^p)\)-module of dimension \(n\) is given by a partition of \(n\) into blocks of size no larger than \(p\). Thus, for a finite-dimensional \(k[t]/(t^p)\)-module we define its Jordan type to be \(a_p[p]+\cdots+a_1[1]\), where the number of blocks of size \(i\) in the partition of \(n\) is \(a_i\). Now let \(G\) be a \(k\)-group scheme with group algebra \(kG\) (dual to the coordinate algebra \(k[G]\)), and let \(K\) be an extension of \(k\). Let \(\alpha_K\) be a \(\pi\)-point, that is, a map \(K[t]/(t^p)\to KG\) which factors through the group algebra \(KC_K\subset KG_K\) of some unipotent Abelian subgroup scheme \(C_K\subset G_K\). (Throughout, the subscript \(K\) indicates base change.) For \(M\) a finite-dimensional \(kG\) module the Jordan type of \(\alpha_K\) on \(M\) is the Jordan type of the \(K[t]/(t^p)\)-module \(\alpha_K^*(M_K)\), the restriction of \(M_K\) to a \(K[t]/(t^p)\)-module along the map \(\alpha_K\). If the Jordan type of \(\alpha_K^*(M_K)\) is independent of the \(\pi\)-point chosen, then \(M\) is said to be of constant Jordan type. The collection of modules of constant Jordan type includes all finite-dimensional projective \(kG\)-modules, and is closed under direct sums, duality, Heller shifts, and tensor products.
The authors construct modules of constant Jordan type using several different techniques. A condition is given on when a module arising as an extension of two modules of constant Jordan type is also of constant Jordan type. Additionally, if a particular map \(\varphi\) represented by a matrix with coefficients in \(\widehat H^*(G,k)\) has maximal possible rank when restricted to any \(\pi\)-point then \(\ker\varphi\) has constant Jordan type. Examples are given in the case where \(G\) is an elementary Abelian \(p\)-group. These results are used to obtain a new proof of a special case of Macaulay's generalized principal ideal theorem.
Using the Auslander-Reiten theory of almost split sequences more results are obtained. Specifically, for \(M\) an indecomposable non-projective module of constant Jordan type, let \(\Theta\) be a component of the stable Auslander-Reiten quiver of \(kG\) containing \(M\). If \(k\) is perfect or all of the vertices of \(\Theta\) are absolutely indecomposable then any element in \(\Theta\) has constant Jordan type. Additionally, if \(G\) is a finite group of \(p\)-rank at least 2 whose Sylow \(p\)-subgroup is not dihedral or semidihedral, or the collection of \(\pi\)-points has dimension at least 2, then there exists an indecomposable module of stable constant Jordan type \(n[1]\) for all \(n\). Here two Jordan types \(a_p[p]+\cdots+a_1[1]\), \(b_p[p]+\cdots+b_1[1]\) are stable if \(a_i=b_i\) for all \(i\neq p\).
The paper concludes with many open questions and conjectures. For example, it is asked for which Jordan types do there exist \(kG\)-modules of constant type -- it is shown that no such module exists with constant Jordan type \([2]+[p]\) for \(p>3\) and \(G\) an elementary Abelian \(p\)-group of rank 2. It is conjectured that for \(p>3\) and \(G\) an elementary \(p\)-group of rank at least \(2\) there is no module with constant Jordan type \([2]\). finite group schemes; modules over finite group schemes; modules of constant Jordan type; group algebras; almost split sequences; Auslander-Reiten quivers Carlson J.F., Friedlander E.M., Pevtsova J.: Modules of constant Jordan type. J. Reine \& Angew. Math. 614, 191--234 (2008) Representation theory for linear algebraic groups, Group schemes, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Group rings of finite groups and their modules (group-theoretic aspects) Modules of constant Jordan type. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite regular monoid \(M\) with unit group \(G\) the algebra of invariants \(\mathbb{C}[M]^G\) of the complex semigroup algebra \(\mathbb{C}[M]\) is studied. So, \(\mathbb{C}[M]^G\) is the centralizer of \(G\) in \(\mathbb{C}[M]\). For every irreducible character \(\theta\) of a maximal subgroup \(H\) of \(M\), certain induced characters \(\theta^+\), \(\theta^-\) and \(\widetilde\theta\) of \(G\) were defined by the author [in Proc. Lond. Math. Soc., III. Ser. 73, No. 3, 623-641 (1996; Zbl 0860.20051)]. Then \(M\) is called balanced, if for every \(H\), \(\theta\), every component of \(\theta^+\cap\theta^-\) is a component of \(\widetilde\theta\). For such monoids it is shown that \(\mathbb{C}[M]^G\) is a quasi-hereditary algebra and the blocks of \(\mathbb{C}[M]^G\) are determined. Moreover, a theory of cuspidal characters is developed, extending the case of universal canonical monoids of Lie type, considered by the author and the reviewer [Int. J. Algebra Comput. 1, No. 1, 33-47 (1991; Zbl 0752.20034)]. So the irreducible characters of \(G\) are classified into series depending on which \(\mathcal J\)-class of \(M\) they come from. The examples of balanced monoids include the full transformation semigroup \({\mathcal T}^n\) of all self-maps on \(\{1,\dots,n\}\) and the semigroup \(T_n(\mathbb{F}_q)\) of upper triangular matrices over the field \(\mathbb{F}_q\). semigroups of upper triangular matrices; finite regular monoids; unit groups; algebras of invariants; complex semigroup algebras; irreducible characters; induced characters; quasi-hereditary algebras; cuspidal characters; canonical monoids of Lie type; balanced monoids; full transformation semigroups Putcha M., J. Algebra 163 pp 632-- Semigroup rings, multiplicative semigroups of rings, Representation of semigroups; actions of semigroups on sets, Semigroups of transformations, relations, partitions, etc., Other matrix groups over finite fields, Representations of associative Artinian rings, Ordinary and skew polynomial rings and semigroup rings, Automorphisms and endomorphisms, Other algebraic groups (geometric aspects), Group actions on varieties or schemes (quotients) Invariant algebra and cuspidal representations of finite monoids | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(W\) be a finite complex reflection group acting on the vector space \(V=\mathbb C^l\), \(X\) a subset of \(V\), \(N_X\) denote the setwise stabilizer of \(X\), \(Z_X\) its pointwise stabilizer, and let \(C_X=N_X/Z_X\). Then restriction defines a homomorphism \(\rho\) from the algebra of \(W\)-invariant polynomial functions on \(V\) to the algebra of \(C_X\)-invariant functions on \(X\).
The authors consider the special case when \(W\) is a Coxeter group, \(V\) is the complexified reflection representation of \(W\), and \(X\) is in the lattice of the arrangement \(\mathcal A\) of the reflection hyperplanes of \(W\). The main result of the paper is a combinatorial criterion for surjectivity of the map \(\rho\) in terms of the exponents of \(W\) and \(C_X\). In the course of the proof the authors obtain a complete list of finite irreducible Coxeter groups \(W\) for which the conditions of the criterion are satisfied.
As an application, the authors consider the case when \(W\) is the Weyl group of a complex semisimple Lie algebra \(\mathfrak g\). Using a theorem of \textit{R. W. Richardson} [Lect. Notes Math. 1271, 243-264 (1987; Zbl 0632.14011)] and the main result of the present paper they give a complete classification of the regular decomposition classes in \(\mathfrak g\) whose closure is a normal variety. arrangements; finite Coxeter groups; finite complex reflection groups; decomposition classes; invariants; invariant polynomial functions; semisimple Lie algebras Douglass, J. Matthew; Röhrle, Gerhard, Invariants of reflection groups, arrangements, and normality of decomposition classes in Lie algebras, Compos. Math., 148, 3, 921-930, (2012) Reflection and Coxeter groups (group-theoretic aspects), Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Simple, semisimple, reductive (super)algebras, Configurations and arrangements of linear subspaces Invariants of reflection groups, arrangements, and normality of decomposition classes in Lie algebras. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Using the equivariant compactification of symmetric varieties, the author gives a Langlands-type parametrization of character sheaves of finite groups of Lie type. equivariant compactification; symmetric varieties; character sheaves; finite groups of Lie type Semisimple Lie groups and their representations, Group varieties Compactifications of symmetric varieties and applications to representation theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove that the Deligne-Lusztig variety associated to minimal length elements in any \(\delta\)-conjugacy class of the Weyl group is affine, which was conjectured by \textit{S. Orlik} and \textit{M. Rapoport} [in J. Algebra 320, No. 3, 1220-1234 (2008; Zbl 1222.14102)].
Another proof of the same statement is given by \textit{C. Bonnafé} and \textit{R. Rouquier} [in J. Algebra 320, No. 3, 1200-1206 (2008; Zbl 1195.20048)]. Deligne-Lusztig varieties; finite groups of Lie type; Weyl groups; conjugacy classes; minimal length elements He, X., On the affineness of Deligne-Lusztig varieties, J. Algebra, 320, 3, 1207-1219, (2008) Representation theory for linear algebraic groups, Classical groups (algebro-geometric aspects), Linear algebraic groups over finite fields, Reflection and Coxeter groups (group-theoretic aspects) On the affineness of Deligne-Lusztig varieties. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper provides a classification of the simple integrable modules of double affine Hecke algebras via perverse sheaves. Let \(\underline G\) be a simple connected simply connected linear algebraic group. Let \(\underline{\text{Lie}}\,\underline G\) denote the Lie algebra of \(\underline G\), let \(\underline{\text{Lie}}\,\underline H\subset\underline{\text{Lie}}\,\underline G\) be a Cartan subalgebra and let \(\underline{\text{Lie}}\,\underline B\subset\underline{\text{Lie}}\,\underline G\) be a Borel subalgebra containing \(\underline{\text{Lie}}\,\underline H\). Let \(\underline\Phi\) be the root system of \(\underline G\) and let \(\Phi^\vee\) be the root system dual to \(\underline\Phi\). Let \(\{\alpha_i:i\in\underline I\}\), \(\{\alpha_i^\vee:i\in\underline I\}\) be the set of simple roots and of simple coroots, respectively. Let \(I:=\underline I\sqcup\{0\}\). Let \(\underline W\), \(W\) be the Weyl group and the affine Weyl group of \(\underline G\). We identify \(\underline I\) (resp. \(I\)) with the set of simple reflections in \(\underline W\) (resp. \(W\)). Let \(s_i\in W\) be the simple reflection corresponding to \(i\in I\). For all \(i,j\in I\), let \(m_{ij}\) denote the order of the element \(s_is_j\) in \(W\). Let \(\underline X\) be the weight lattice of \(\underline\Phi\) and let \(Y^\vee\) be the root lattice of \(\underline\Phi v\). Let \(\{\omega_i:i\in\underline I\}\) be the set of fundamental weights. Consider the lattices \(Y=\bigoplus_{i\in I}\mathbb{Z}\alpha_i\subset X=\mathbb{Z}\delta\oplus\bigoplus_{i\in I}\mathbb{Z}\omega_i\), \(Y^\vee=\bigoplus_{i\in I}\mathbb{Z}\alpha_i^\vee\), where \(\delta\) is a new variable. There is unique pairing \(X\times Y^\vee\to\mathbb{Z}\) such that \((\omega_i:\alpha_j^\vee)=\delta_{ij}\) and \((\delta:\alpha_j^\vee)=0\).
The double affine Hecke algebra \(\mathbf H\) is the unital associative \(\mathbb{C}[q,q^{-1},t,t^{-1}]\)-algebra generated by \(\{t_i,x_\lambda:i\in I\), \(\lambda\in X\}\) modulo the following defining relations:
\[
x_\delta=t,\quad x_\lambda x_\mu=x_{\lambda+\mu}(t_i-q)(t_i+1)=0,
\]
\[
t_it_jt_i\cdots=t_jt_it_j\cdots\text{ if }i\neq j\;(m_{ij}\text{ factors in both products),}
\]
\[
t_ix_\lambda-x_\lambda t_i=0\text{ if }(\lambda:\alpha_i^\vee)=0,\quad t_ix_\lambda-x_{s_i(\lambda)}t_i=(q-1)x_\lambda\text{ if }(\lambda:\alpha_i^\vee)=1,
\]
for all \(i,j\in I\), \(\lambda,\mu\in X\).
One important step of the proof is the construction of a ring homomorphism from \(\mathbf H\) to a ring defined via the equivariant \(K\)-theory of an affine analogue \(\mathcal Z\) of the Steinberg variety. \(\mathcal Z\) is an ind-scheme of ind-infinite type. It comes with a filtration by subsets \({\mathcal Z}_{\leq y}\) with \(y\) in the affine Weyl group \(W\). The subsets \({\mathcal Z}_{\leq y}\) are reduced separated schemes of infinite type, and the inclusions \({\mathcal Z}_{\leq y'}\subset{\mathcal Z}_{\leq y}\) with \(y'\leq y\) are closed immersions. The set \(\mathcal Z\) is endowed with an action of a torus \(A\) which preserves each term of the filtration. For a well-chosen element \(a\in A\), the fixed point set \({\mathcal Z}^a\subset{\mathcal Z}\) is a scheme locally of finite type. Hence there is a convolution ring \(\mathbf K^A({\mathcal Z}^a)\): it is the inductive limit of the system of \({\mathbf R}_A\)-modules \(\mathbf K^A(({\mathcal Z}_{\leq y})^a)\) with \(y\in W\). (Here \({\mathbf R}_A\) means \({\mathbf K}_A(\text{point})\).) The author defines a ring homomorphism \(\Psi_a\colon{\mathbf H}\to\mathbf K^A({\mathcal Z}^a)_a\), where the subscript \(a\) means specialization at the maximal ideal \(J_a\subset{\mathbf R}_A\) associated to \(a\). The map \(\Psi_a\) becomes surjective after a suitable completion of \(\mathbf H\). It is certainly not injective. Using \(\Psi_a\), a standard sheaf-theoretic construction, due to Ginzburg in the case of affine Hecke algebras, provides a collection of simple \(\mathbf H\)-modules. These are precisely the simple integrable modules. -- The paper also give some estimates for the Jordan-Hölder multiplicities of induced modules. simple integrable modules; double affine Hecke algebras; perverse sheaves; linear algebraic groups; Lie algebras; Cartan subalgebras; Borel subalgebras; root systems; affine Weyl groups; simple reflections; pairings; equivariant \(K\)-theory; Jordan-Hölder multiplicities of induced modules Vasserot, Eric, Induced and simple modules of double affine Hecke algebras, Duke Math. J., 126, 2, 251-323, (2005) Hecke algebras and their representations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Lie algebras of linear algebraic groups, Grothendieck groups, \(K\)-theory, etc., Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Induced and simple modules of double affine Hecke algebras. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We notice that one of the Diophantine equations, \(knm = 2kn + 2km + 2nm\), arising in the universality originated Diophantine classification of simple Lie algebras, has interesting interpretations for two different sets of signs of variables. In both cases it describes ``regular polyhedra'' with \(k\) edges in each vertex, \(n\) edges of each face, with total number of edges \(| m |\), and Euler characteristics \(\chi = \pm 2\). In the case of negative \(m\) this equation corresponds to \(\chi = 2\) and describes true regular polyhedra, Platonic solids. The case with positive \(m\) corresponds to Euler characteristic \(\chi = - 2\) and describes the so called equivelar maps (charts) on the surface of genus \(2\). In the former case there are two routes from Platonic solids to simple Lie algebras -- above mentioned Diophantine classification and McKay correspondence. We compare them for all solutions of this type, and find coincidence in the case of icosahedron (dodecahedron), corresponding to \(E_8\) algebra. In the case of positive \(k\), \(n\) and \(m\) we obtain in this way the interpretation of (some of) the mysterious solutions (\(Y\)-objects), appearing in the Diophantine classification and having some similarities with simple Lie algebras. simple Lie algebras; McKay correspondence; Vogel's universality; Diophantine equations; regular maps Khudaverdian, H.M.; Mkrtchyan, R.L., Diophantine equations, platonic solids, mckay correspondence, equivelar maps and Vogel's universality, J. geom. phys., 114, 85-90, (2017) Cubic and quartic Diophantine equations, Curves of arbitrary genus or genus \(\ne 1\) over global fields, McKay correspondence, Simple, semisimple, reductive (super)algebras, Three-dimensional polytopes Diophantine equations, platonic solids, McKay correspondence, equivelar maps and Vogel's universality | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article under review studies Hilbert schemes of points of the total spaces \(X_{n}\) of line bundles \(\mathcal{O}_{\mathbb{P}^{1}}(-n)\) in terms of ADHM data and realizes them as irreducible connected components of moduli spaces of quiver representations. The surfaces \(X_{n}\) are minimal resolutions of toric singularities of type \(\frac{1}{n}(1,1)\) having Hizebruch surfaces \(\Sigma_{n}\) as projective compactifications, connecting with string theory in physics.
In the paper, the authors construct ADHM data for the Hilbert schemes \(\mathrm{Hilb}^{c}(X_{n})\), by going through the description of moduli spaces of framed sheaves of Hirzebruch surfaces \(\Sigma_{n}\), \(\mathcal{M}^{n}(1,0,n)\simeq \mathrm{Hilb}^{c}(X_{n})\). This ADHM data turns out to provide a principal bundle over the Hilbert scheme (c.f. Theorem 3.1).
In section \(4\), it is shown how these Hilbert schemes are irreducible connected components of GIT quotients of representation spaces of certain quivers for a suitable choice of the stability parameter (c.f. Theorem 4.5), therefore they can be seen as embedded components into quiver varieties associated, in a natural way, with ADHM data of the Hilbert schemes.
It is worth noting that the quiver varieties of this article fall outside the notion of a Nakajima quiver variety carrying naturally a simplectic structure, while \(\mathrm{Hilb}^{c}(X_{n})\) encodes a Poisson structure in general. The authors propose to study further the wall-crossing of the stability parameters to shed light on questions in geometric representation theory, and also to study the Poisson structure of these spaces. Hilbert schemes of points; quiver varieties; HIrzebruch surfaces; ADHM data; monads; Nakajima quivers; McKay quivers; quiver varieties, moduli spaces of quiver representations Bartocci, C.; Bruzzo, U.; Lanza, V.; Rava, C.L.S., Hilbert schemes of points of \(\mathcal{O}_{\mathbb{P}^1}(- n)\) as quiver varieties, J. pure appl. algebra, 221, 2132-2155, (2017) Algebraic moduli problems, moduli of vector bundles, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Representations of quivers and partially ordered sets Hilbert schemes of points of \(\mathcal{O}_{\mathbb{P}^1}(- n)\) as quiver varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors study equidistribution of solutions of word equations of the form \(w(x,y)=g\) in the family of finite groups \(\mathrm{SL}(2,q)\). They provide criteria for equidistribution in terms of the trace polynomial of \(w\). This allows them to get an explicit description of certain classes of words possessing the equidistribution property and show that this property is generic within these classes. word maps; finite groups of Lie type; equidistribution; trace polynomials Bandman, T.; Kunyavskii, B., Criteria for equidistribution of solutions of word equations on \(S L(2)\), J. Algebra, 382, 282-302, (2013) Algebraic geometry over groups; equations over groups, Linear algebraic groups over finite fields, Simple groups: alternating groups and groups of Lie type, Probabilistic methods in group theory, Arithmetic and combinatorial problems involving abstract finite groups, Classical groups, Rational points, Finite ground fields in algebraic geometry Criteria for equidistribution of solutions of word equations on \(\mathrm{SL}(2)\). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We formulate a conjecture on the motivic McKay correspondence for the group scheme \(\alpha_p\) in characteristic \(p>0\) and give a few evidences. The conjecture especially claims that there would be a close relation between quotient varieties by \(\alpha_p\) and ones by the cyclic group of order \(p\). McKay correspondence; motivic integration; quotient singularities; finite group schemes McKay correspondence, Group schemes Notes on the motivic McKay correspondence for the group scheme \(\alpha_p\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors give a full account of important results announced some years earlier [Bull. Am. Math. Soc., New Ser. 3, 1057--1061 (1980; Zbl 0457.17007)]. Several appendices discuss further implications of the theory presented and announce further results. The paper is written with recognition of the broad audience its content will interest. It includes background on affine Kac-Moody Lie algebras, irreducible highest weight representations, and classical theta functions and modular forms, as well as an account of recently discovered connections among those areas. In broad terms, the paper's principal results fall into two main categories.
First, a theta-function interpretation of the Macdonald identities due to \textit{E. Looijenga} [Invent. Math. 38, 17--32 (1976; Zbl 0358.17016)] is exploited through an observation of the first author [Adv. Math. 35, 264--273 (1980; Zbl 0431.17009)] that most generating functions for multiplicities appearing in the representation theory of affine Lie algebras become q-series of modular forms when multiplied by a suitable power of q. The character of a highest weight representation of an affine Lie algebra is rewritten in terms of theta functions of the modular forms. Then the authors use classical functional equations for theta functions to deduce transformation properties of the modular forms. The ``very strange'' formula [see the first author's paper in Adv. Math. 30, 85--136 (1978; Zbl 0391.17010)] is then used to estimate the order of the poles at the cusps.
In short, the modular form theory makes it possible to compute with the forms, and a Tauberian theorem of Ingham is used to obtain the asymptotics of the multiplicities in question. If \(L\) is the affine algebra and \(\Lambda\) is the highest weight, then the key step in this part of the paper is establishing that \(mult_{\Lambda}(\lambda -n\delta)\) is an increasing function of \(n\). Here \(\lambda\) is in the dual space of the Cartan subalgebra \(H\) of the affine algebra \(L\) and \(\delta\) is the unique element of that space that annihilates \(\bar H \) (the Cartan subalgebra of the underlying classical simple finite dimensional algebra \(L)\) and \(c\) (where \(H=H+Cc+Cd\), \(d\) the derivation that acts on \(C[t,t^{- 1}]\otimes_ C \bar L\) as \(t(d/dt)\) and annihilates \(c)\) and maps \(d\) to 1. The approach taken by the authors involves use of a Heisenberg algebra. The results in this direction make it possible to explicitly determine the string functions in many cases. The multiplicities do not appear to be given by any simple combinatorial functions such as the classical partition function, but rather to depend on the fact that \(q^{1/24}(q,q)\) is a modular form.
The second main theme of the paper is the use of the second author's explicit formulas for Kostant's partition function. These make it possible to derive explicit formulas for generalized Kostant partition functions for certain affine algebras. That in turn affords a way of computing multiplicities directly for the algebra of type \(A_ 1^{(1)}\). The corresponding generating series are closely related to Hecke modular forms associated to real quadratic fields [see Math. Werke E. Hecke, 418--427 (1959; Zbl 0092.001)].
While the complexity of the formulas and identities obtained in the paper makes it impractical to be more specific here, it is to be noted that at the end of the paper, a collection of new (and old) identities for modular forms and elliptic theta functions is given. These formulas, which are natural consequences of the representation theory and its connections to modular forms in the simplest case \((A_ 1^{(1)})\), can be read independently of the rest of the paper. multiplicity; affine Kac-Moody Lie algebras; irreducible highest weight representations; theta functions; modular forms; Macdonald identities; string functions; generalized Kostant partition functions; identities for modular forms; elliptic theta functions Kač, VG; Peterson, DH, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math., 53, 125, (1984) Infinite-dimensional Lie (super)algebras, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Holomorphic modular forms of integral weight, Representation-theoretic methods; automorphic representations over local and global fields, Infinite-dimensional Lie groups and their Lie algebras: general properties, Theta functions and abelian varieties Infinite-dimensional Lie algebras, theta functions and modular forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study orbifolds of \(N=4\) U\((n)\) super-Yang-Mills theory given by discrete subgroups of SU(2) and SU(3). We have reached many interesting observations that have graph-theoretic interpretations. For the subgroups of SU(2), we have shown how the matter content agrees with current quiver theories and have offered a possible explanation. In the case of SU(3) we have constructed a catalogue of candidates for finite (chiral) \(N=1\) theories, giving the gauge group and matter content. Finally, we conjecture a McKay-type correspondence for Gorenstein singularities in dimension 3 with modular invariants of WZW conformal models. This implies a connection between a class of finite \(N=1\) supersymmetric gauge theories in four dimensions and the classification of affine SU(3) modular invariant partition. McKay-type correspondence; Gorenstein singularities; modular invariants; orbifolds; super Yang-Mills theory Hanany, A.; He, Y-H, Non-abelian finite gauge theories, JHEP, 02, 013, (1999) Supersymmetric field theories in quantum mechanics, Relationships between surfaces, higher-dimensional varieties, and physics, Yang-Mills and other gauge theories in quantum field theory, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Non-abelian finite gauge theories | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Consider the Hilbert scheme \(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}})\) of generalized \(n\)-tuples of points on the affine plane: \(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}})\) is a smooth quasi-projective manifold of complex dimension \(2n\) and is a natural resolution of the quotient \((\mathbb{A}^2_{\mathbb{C}})^n/S_n\). Its odd cohomology vanishes, and there is no torsion. In this paper, the authors describe a natural model for the cohomology ring \(H^*(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}});\mathbb{Z})\) in terms of a graded version of the ring \(\mathcal{C}(S_n)\) of integer-valued class functions on the symmetric group \(S_n\).
The model. Starting from the group ring \(\mathbb{Z}[S_n]\), define a degree by setting \(\text{deg}(\pi)=d\) if the permutation \(\pi\in S_n\) can be written as a minimal product of \(d\) transpositions. The product in \(\mathbb{Z}[S_n]\) is compatible with the increasing filtration associated to the degree, and induces a product on the graded space \(\text{gr}\mathbb{Z}[S_n]\). The subring \(\mathcal{C}(S_n)\subset \mathbb{Z}[S_n]\) inherits a structure of commutative graded ring.
The theorem. The graded rings \(H^*(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}});\mathbb{Z})\) and \(\mathcal{C}(S_n)\) are naturally isomorphic (Theorem 1.1).
The proof. First, the authors prove the isomorphism over \(\mathbb{Q}\) by considering all values of \(n\) together and by induction on both \(n\) and the cohomological degree (Proposition 5.3). This uses the identification of \(\mathcal{C}:=\bigoplus_n\mathcal{C}(S_n)\otimes \mathbb{Q}\) and \(\mathbb{H}=\bigoplus_n H^*(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}});\mathbb{Q})\) with the bosonic Fock space \(\mathcal{P}=\mathbb{Q}[p_1,p_2,\ldots]\). The vertex algebra isomorphisms \(\mathcal{C}\cong \mathcal{P}\) and \(\mathcal{P}\cong \mathbb{H}\) come respectively from \textit{I. B. Frenkel, W. Wang} [J. Algebra 242, No. 2, 656--671 (2001; Zbl 0981.17021)] and \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)]. Then the assertion results from the similarity of Goulden's operator on \(\mathcal{C}\) with Lehn's operator on \(\mathbb{H}\) (see Theorem 4.1 and \textit{M.Lehn} [Invent. Math. 136, No. 1, 157--207 (1999; Zbl 0919.14001)]). In a second part, the authors show that the ring isomorphism \(\mathcal{C}(S_n)\otimes \mathbb{Q}\cong H^*(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}});\mathbb{Q})\) also holds over the integers by exhibiting appropriate \(\mathbb{Z}\)-basis of both spaces (\S 6).
Remarks. 1) This result has been independently obtained by \textit{E. Vasserot} [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 1, 7--12 (2001; Zbl 0991.14001)] using other methods.
2) The isomorphism has been further interpreted in the context of the Chen-Ruan conjecture by \textit{B. Fantechi, L. Göttsche} [Duke Math. J. 117, No. 2, 197--227 (2003; Zbl 1086.14046)].
3) Generalizing this result, a similar description of the complex cohomology ring of any symplectic resolution of the quotient of a vector space by a finite group of symplectic automorphisms has been obtained later by \textit{V. Ginzburg} and \textit{D. Kaledin} [Adv. Math. 186, No. 1, 1--57 (2004; Zbl 1062.53074)]. Hilbert schemes of points; symmetric functions; vertex algebras Lehn, M., Sorger, C.: Symmetric groups and the cup product on the cohomology of Hilbert schemes. Duke Math. J. \textbf{110}(2), 345-357 (2001). math/0009131 Parametrization (Chow and Hilbert schemes), (Equivariant) Chow groups and rings; motives, Symmetric groups, Virasoro and related algebras, Vertex operators; vertex operator algebras and related structures, Group rings of finite groups and their modules (group-theoretic aspects) Symmetric groups and the cup product on the cohomology of Hilbert schemes. | 0 |
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