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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors prove the Cohen-Macaulayness of multicones over Schubert varieties.
Let G be a reductive algebraic group, B a Borel subgroup and X a Schubert variety in G/B. Let \(L_ i\), \(1\leq i\leq n\) be the line bundles corresponding to the fundamental weights (here, \(n=rank(G))\). Let \(L=\otimes^{n}_{i=1}L_ i^{a_ i},\quad a_ i\in {\mathbb{Z}}^+\), \(1\leq i\leq n\). Let \(R=\oplus_{L}H^ 0(X,L)\quad\) and \(C=Spec(R)\). The authors prove that the ring C is Cohen-Macaulay.
In the course of proving this, they also give a criterion for C to have rational singularities, if X does (similar to Serre's criterion for arithmetic Cohen-Macaulayness). The proof of the main theorem uses Frobenius splitting of Schubert varieties. The Frobenius splitting of Schubert varieties was first proved by Mehta and Ramanathan. Subsequently, Ramanathan has proved geometric properties for Schubert varieties like arithmetic normality, arithmetic Cohen-Macaulayness etc., using Frobenius splitting. The result about the Cohen-Macaulayness of multicones completes the picture regarding Cohen-Macaulay properties that arise in the context of Schubert varieties. The paper is a nice contribution to the geometric study of Schubert varieties. Cohen-Macaulayness of multicones over Schubert varieties; rational singularities; Frobenius splitting G. Kempf, A. Ramanathan, Multicones over Schubert varieties, Invent. Math. 87 (1987), 353--363. Grassmannians, Schubert varieties, flag manifolds Multi-cones over Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(z=\{z_{ij}\}_{1\leq i,j\leq n}\) be variables and \(v,w\in S_n\) permutations. Let \(Z^{(v)}\) be the specialized generic matrix, i.e. with entries \(z_{ij}\) but specialized to \(z_{n-v(i)+1,i}=1\), \(z_{n-v(i)+1,a}=0\;,\;z_ {b,i}=0\) if \(a>i, b>n-v(i)+1\). Let \(z^{(v)}\subset z\) be the remaining variables. Let \(r^w_{i,j}=\#\{k\;|\;w(k)\geq n-i+1, k\leq j\}\) and denote by \(Z^{(v)}_{ab}\) the southwest \(a\times b\) submatrix of \(Z^{(v)}\). The Kazhdan--Lusztig ideal \(I_{v,w}\subseteq\mathbb{C}[z^{(v)}]\) is the ideal generated by all minors of size \(1+r^w_{ij}\) of \(Z^{(v)}_{ij}\) for all \(i,j\) (the so--called defining minors). Consider the following monomial ordering:
\[
z_{ij}<z_{kl}, \text{ if } j<l\;\text{ or if } j=l\text{ and } i<k\;.
\]
It is proved that the defining minors are a Gröbner basis of \(I_{v,w}\).
Kazhdan--Lusztig ideals provide an explicite choice of coordinates and equations encoding a neighborhood of a torus--fixed point of a Schubert variety on a type \(A\) flag variety.
The Gröbner basis turns out to be a tool to explain several combinatorical formulas. Gröbner basis; Kazhdan-Lusztig ideal; Schubert variety Woo, Alexander; Yong, Alexander, A Gröbner basis for Kazhdan-Lusztig ideals, Amer. J. Math., 134, 4, 1089-1137, (2012) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Determinantal varieties A Gröbner basis for Kazhdan-Lusztig ideals | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(S= k[x_0,\ldots,x_n]\) be the polynomial ring in \(n+1\) variables over a field of characteristic \(p.\) For a finitely generated graded \(S\)-module \(M\) the Hilbert-Kunz function is defined on powers of the characteristic, \(q = p^n\), \(n \in \mathbb N,\) by \(HK_M(q)= \dim_k M/m^{[q]}M,\) where \(m^{[q]} = (x_0^q, \ldots, x_n^q)\) denotes the \(q\)-th Frobenius power of the maximal ideal \(m = (x_0, \ldots, x_n).\) It follows from a paper by \textit{P. Monsky} [Math. Ann. 263, 43-49 (1983; Zbl 0509.13023)] that \(HK_M(q) = cq^m +O(q^{m-1})\) with \(c\geq 1\) some real number and \(m = \dim M,\) the Krull dimension of \(M.\) The number \(c\) is called the Hilbert-Kunz multiplicity of \(M.\) In general, it seems very difficult to determine these functions explicitly and a conceptual interpretation of \(c\) is missing.
In the paper under review the authors study the case of \(n = 2\) and \(M = S/fS,\) where \(f\) corresponds to an irreducible cubic curve \(C\) in \(\mathbb P^2.\) They exhibit the Hilbert-Kunz function of plane elliptic curves of odd characteristic and of plane nodal cubics. In fact, they confirm the formulas conjectured by \textit{K. Pardue} [``Nonstandard Borel-fixed ideals'', Doctoral thesis (Brandeis University 1994)], where he also determined the Hilbert-Kunz function of reducible cubics. The remaining case of an elliptic curve \(C\) and \(p=2\) was solved by \textit{P. Monsky} [see J. Algebra 197, No. 1, 268-277 (1997; see the following review)]. In the case of a plane nodal cubic the authors use techniques from minimal free resolutions and Veronese embeddings. For \(C\) a plane elliptic curve in odd characteristic their proof is based on an unexpected use of a formula of Geronimus for the determinant of a certain matrix whose entries are Legendre polynomials. As a corollary of their work the authors show that for any \(d\geq 2\) and any field \(k\) of prime characteristic there exists a plane curve of degree \(d\) in \(\mathbb P^2\) whose Hilbert-Kunz multiplicity is \(3d/4\) -- and this is the minimal possible value for such curves. In particular, the minimal Hilbert-Kunz multiplicity is rational and independent of the characteristic.
Then the authors determine explicitly the Hilbert-Kunz function of Cayley's cubic surface in \(\mathbb P^3.\) There are similar conclusions about minimal Hilbert-Kunz multiplicities of surfaces of degree \(d\) in \(\mathbb P^4\) as shown in the case of curves. In higher dimensions it follows from the work by \textit{C. Han} and \textit{P. Monsky} [in: Math. Z. 214, No. 1, 119-135 (1993; Zbl 0788.13008)] that the minimal Hilbert-Kunz multiplicity depends upon the characteristic. For further results see also \textit{L. Chiang} and \textit{Y.-C. Hung} [J. Algebra 199, No. 2, 499-527 (1998; see the review after next)]. Hilbert-Kunz function; Hilbert-Kunz multiplicity; cubic curve; cubic surface; Frobenius power; characteristic \(p\) \beginbarticle \bauthor\binitsR.-O. \bsnmBuchweitz and \bauthor\binitsQ. \bsnmChen, \batitleHilbert-Kunz functions of cubic curves and surfaces, \bjtitleJ. Algebra \bvolume197 (\byear1997), no. \bissue1, page 246-\blpage267. \endbarticle \endbibitem Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Multiplicity theory and related topics, Special algebraic curves and curves of low genus, Special surfaces Hilbert-Kunz functions of cubic curves and surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Author's abstract: ``It is well-known that the coset spaces \(G(k((z)))/G(k[[z]])\), for a reductive group \(G\) over a field \(k\), carry the geometric structure of an inductive limit of projective \(k\)-schemes. This \(k\)-ind-scheme is known as the affine Grassmannian for \(G\). From the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form \(\mathcal {G}(\mathbf {W}(k)[1/p])/\mathcal {G}(\mathbf {W}(k))\), where \(p\) is a rational prime, \(\mathbf {W}\) denotes the ring scheme of \(p\)-typical Witt vectors, \(k\) is a perfect field of characteristic \(p\) and \(\mathcal {G}\) is a reductive group scheme over \(\mathbf {W}(k)\). The present paper is an attempt to describe which constructions carry over from the function field case to the \(p\)-adic case, more precisely to the situation of the \(p\)-adic affine Grassmannian for the special linear group \(\mathcal {G}=\mathbf {SL}_{n}\). We start with a description of the \(R\)-valued points of the \(p\)-adic affine Grassmannian for \(\mathbf {SL}_{n}\) in terms of lattices over \(\mathbf {W}(R)\), where \(R\) is a perfect \(k\)-algebra. In order to obtain a link with geometry we further construct projective \(k\)-subvarieties of the multigraded Hilbert scheme which map equivariantly to the \(p\)-adic affine Grassmannian. The images of these morphisms play the role of Schubert varieties in the \(p\)-adic setting. Further, for any reduced \(k\)-algebra \(R\) these morphisms induce bijective maps between the sets of \(R\)-valued points of the respective open orbits in the multigraded Hilbert scheme and the corresponding Schubert cells of the \(p\)-adic affine Grassmannian for \(\mathbf {SL}_{n}\).''
The main theorem of the paper is Theorem 6 which is concerned with the algebraic structure of the \(p\)-adic affine Grassmannian for for \(\mathbf {SL}_{n}\). This theorem relies on the Greenberg realizations which are explained in Section 3 of the paper. \(p\)-adic; special linear group; group actions; affine Grassmannian; Hilbert scheme; Greenberg realization Kreidl, M, On \(p\)-adic lattices and Grassmannians, Math. Z., 276, 859-888, (2014) Actions of groups on commutative rings; invariant theory, Positive characteristic ground fields in algebraic geometry, Group schemes, Group actions on varieties or schemes (quotients), Categories admitting limits (complete categories), functors preserving limits, completions On \(p\)-adic lattices and Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial \(K\)-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We show that the Newton polytopes of these Grothendieck polynomials and their homogeneous components have SNP. Moreover, the Newton polytope of each homogeneous component is a permutahedron. This addresses recent conjectures of C. Monical-N. Tokcan-A. Yong and of A. Fink-K. Mészáros-A. St. Dizier in this special case. symmetric Grothendieck polynomials; Newton polytopes Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Symmetric functions and generalizations, Toric varieties, Newton polyhedra, Okounkov bodies Newton polytopes and symmetric Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(O\) be a nilpotent orbit of a complex semisimple Lie algebra \(\mathfrak{g}\) and let \(\pi: X \to \bar{O}\) be the finite covering associated with the universal covering of \(O\). In [\textit{Y. Namikawa}, in: Handbook of moduli. Volume III. Somerville, MA: International Press; Beijing: Higher Education Press. 1--38 (2013; Zbl 1322.14013)], he has explicitly constructed a \(\mathbb{Q}\)-factorial terminalization \(\widetilde{X}\) of \(X\) when \(\mathfrak{g}\) is classical.
In this paper, the author counts different \(\mathbb{Q}\)-factorial terminalizations of \(X\). He also constructs the universal Poisson deformation of \(\widetilde{X}\) over \(H^2(\widetilde{X}, \mathbb{C})\) and looks at the action of the Weyl group \(W(X)\) on \(H^2(\widetilde{X}, \mathbb{C})\), leading toward an explicit geometric description of \(W(X)\). birational geometry; nilpotent orbit of a complex Lie algebra; Poisson deformation Global theory and resolution of singularities (algebro-geometric aspects), Coadjoint orbits; nilpotent varieties Birational geometry for the covering of a nilpotent orbit closure. II. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An invertible polynomial in \(n\) variables is a quasi-homogeneous polynomial \(\sum _{i=1}^na_i\prod _{j=1}^nx_j^{E_{ij}}\) with \(a_i\in \mathbb{C}^*\), \(E_{ij}\in \mathbb{N}\cup 0\), \(\det E\neq 0\). Thus the weights of the variables and the quasi-homogeneous degree are well defined. Berglund, Hübsch and Henningson [\textit{P. Berglund} and \textit{T. Hübsch}, Nucl. Phys., B 393, No. 1--2, 377--391 (1993; Zbl 1245.14039); \textit{P. Berglund} and \textit{M. Henningson}, Nucl. Phys., B 433, No. 2, 311--332 (1995; Zbl 0899.58068)] have considered a pair \((f,G)\) consisting of an invertible polynomial \(f\) and an abelian group \(G\) of its symmetries. This has been done within the framework of the construction of mirror symmetric orbifold Landau-Ginzburg models. The authors of the present paper have studied reduced orbifold zeta functions for such pairs and their dual pairs and have shown that they coincide in the case of even \(n\) or are inverse to each other in the case of odd \(n\). invertible polynomial; group action; monodromy; orbifold zeta function Ebeling, W.; Gusein-Zade, S. M., Orbifold zeta functions for dual invertible polynomials, Proc. Edinb. Math. Soc., 60, 1, 99-106, (2017) Mirror symmetry (algebro-geometric aspects), Group actions on affine varieties, Topology and geometry of orbifolds, Monodromy on manifolds, Structure of families (Picard-Lefschetz, monodromy, etc.) Orbifold zeta functions for dual invertible polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathbb K\) be a field of characteristic 0 and let \({\mathbb K}[x_1,\ldots,x_n]\), \(W_n({\mathbb K})\), and \(P_n({\mathbb K})={\mathbb K}[x_1,\ldots,x_n,p_1,\ldots,p_n]\) be, respectively, the polynomial algebra in \(n\) variables, the \(n\)-th Weyl algebra, and the \(n\)-th Poisson algebra equipped with the standard Poisson bracket where the only nonzero brackets are \(\{p_i,x_i\}=1\). In the paper under review, the authors consider the problem for approximation of automorphisms of these three algebras by tame automorphisms. A classical theorem in [\textit{D. J. Anick}, J. Algebra 82, 459--468 (1983; Zbl 0535.13014)] gives that every endomorphism of \({\mathbb K}[x_1,\ldots,x_n]\) with invertible Jacobian is a limit of a sequence of tame automorphisms in the formal power series topology. The first main result of the paper is a slightly modified proof of this theorem with an automorphism with Jacobian equal to 1 as a limit. The proof is adapted to the problem of approximation of the automorphisms of the Poisson algebra \(P_n({\mathbb K})\). In the paper the automorphisms of \(P_n({\mathbb K})\) are called symplectomorphisms because they can be identified with the invertible polynomial mappings \({\mathbb A}_{\mathbb K}^{2n}\to {\mathbb A}_{\mathbb K}^{2n}\) of the affine space \({\mathbb A}_{\mathbb K}^{2n}\) which preserve the symplectic form \(\sum dp_i\wedge dx_i\). The second main result is that every symplectomorphism is a limit of a sequence of tame symplectomorphisms in the formal power series topology. By a theorem in [\textit{A. Belov-Kanel} and \textit{M. Kontsevich}, Lett. Math. Phys. 74, No. 2, 181--199 (2005; Zbl 1081.16031)] and [\textit{A. Belov-Kanel} and \textit{M. Kontsevich}, Mosc. Math. J. 7, No. 2, 209--218 (2007; Zbl 1128.16014)] the groups of tame automorphisms of \(W_n({\mathbb C})\) and \(P_n({\mathbb C})\) are isomorphic. The third main result of the paper is that if \(\sigma\) is a symplectomorphism of \(P_n({\mathbb C})\), then there exists a sequence of tame automorphisms of \(W_n({\mathbb C})\) such that their images in \(\text{Aut}(P_n({\mathbb C}))\) converge to \(\sigma\). In other words, the authors prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory. Finally, let \(\varphi\) be a polynomial automorphism of \({\mathbb C}[x_1,\ldots,x_n]\) and let \({\mathcal O}_{\varphi}\) be the local ring generated by the coefficients of \(\varphi\) and with maximal ideal \(\mathfrak m\). If the sequence of tame automorphisms \(\psi_1,\psi_2,\ldots\) converges to \(\varphi\) in the formal power series topology, then the coordinates of \(\psi_k\) converge to the coordinates of \(\varphi\) in the \(\mathfrak m\)-adic topology. A similar result is established for the symplectomorphisms of \(P_n({\mathbb C})\). Jacobian conjecture; polynomial automorphisms and symplectomorphisms; tame and wild automorphisms; quantization Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Polynomials over commutative rings, Jacobian problem Lifting of polynomial symplectomorphisms and deformation quantization | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a representation of a group \(G\) on the vector space \(V\), invariant varieties \(Y \subset V\) define equivariant cohomology classes \([Y] \in H^*_G(V)=H^*(BG)\). One key example is when \(V\) is the vector space of map germs \((C^n,0) \to (C^p,0)\) and \(Y\) is a certain collection of these germs -- called a singularity. In this context
\[
[Y]\in H^*\left(B(GL(n,C)\times GL(p,C)\right) = Z[a_1,\ldots,a_n,b_1,\ldots,b_p]
\]
is called the Thom polynomial of the singularity \(Y\). Thom polynomials govern the global behavior of singularities; namely they express cohomology classes represented by singularity submanifolds.
\textit{W. Fulton} and \textit{R. Lazarsfeld} [Positive polynomials for ample vector bundles, Ann. Math. (2) 118, 35--60 (1983; Zbl 0537.14009)] considered equivariant classes \([Y]\in H^*(BG)\) when \(Y\) is a cone, and showed certain positivity properties. The present paper applies these results to the Thom polynomial setting and obtains the following theorem. The Thom polynomial of a ``stable'' singularity, when expressed in Schur functions of the quotient variables \(c_i\) (\(\sum c_it^i = \sum b_i t^i / \sum a_i t^i\)), has nonnegative coefficients. Thom polynomial; Schur functions; numerical positivity P. Pragacz and A. Weber, ''Positivity of Schur function expansions of Thom polynomials,'' Fund. Math., vol. 195, iss. 1, pp. 85-95, 2007. Symmetric functions and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical problems, Schubert calculus, Homology of classifying spaces and characteristic classes in algebraic topology, Singularities of differentiable mappings in differential topology Positivity of Schur function expansions of Thom polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors use the theory of Thom polynomials of right-left singularities to prove enumerative results on the points of a variety $X$ in projective space that have prescribed contact with a line. \par Right-left singularities are obtained by considering map germs from $m$-dimension to $n$-dimension up to holomorphic re-parametrization of the source and the target. Consider a family of such germs. Thom polynomials are universal formulas for the number of (or cohomology class represented by) points in the family where the germ belongs to a given singularity. Thom polynomials are rather well known for so-called K-singularities, but not much is known for right-left singularities. Hence, the authors use the interpolation method to calculate the Thom polynomials of certain right-left singularities for maps between low dimensional spaces. \par The second part of the paper is devoted to geometric applications, as follows. Consider a variety $X^m$ in $\mathbb{P}^{n+1}$. The notion ``a line $l$ having prescribed local contact type with $X$'' is re-phrased as a certain germ from $\mathbb{C}^m$ to $\mathbb{C}^n$ having a particular right-left singularity. Then, the Thom polynomial formulas developed in the first half of the paper are used to calculate the number or rather the degree of such loci. For smooth surfaces in $\mathbb{P}^2$ classical formulas are rediscovered, and are generalized to singular surfaces. New formulas are proved for surfaces in $\mathbb{P}^3$ and 3-folds in $\mathbb{P}^4$, in similar spirit. Thom polynomial; right-left singularity; enumeration of contacts; singular projections Classical problems, Schubert calculus, Characteristic classes and numbers in differential topology, Singularities of differentiable mappings in differential topology, Global theory of complex singularities; cohomological properties Thom polynomials in \(\mathcal A\)-classification I: counting singular projections of a surface | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let Sym\((n,K)\) and Sk\((n, K)\) denote the sets of \(n\times n\) invertible symmetric and skew-symmetric matrices over a field \(K\) respectively. Let GL\((n, K)\) be the set of \(n\times n\) invertible matrices over \(K\) and define an action GL\((n, K)\times\text{Sym}(n, K)\to\text{Sym}(n, K)\) by \(g\cdot A= gA^t g\). An action of GL\((n, K)\) on Sk\((n, K)\) is defined in an analogous way. The first results presented in this paper are that the homotopy type of Sym\((n,\mathbb{R})\) can be determined by fibering the orbits over the Grassmann variety and then showing that the fibres are contractible and that the Betti numbers of the orbits of Sk\((n, \mathbb{R})\) can be determined by fibering them over the sphere and using the associated Wang exact sequence. We also show how these results can be derived using Iwasawa decompositon and theorems of Borel and Ehresmann.
Deligne showed that any complex algebraic variety \(X\) has a canonical mixed Hodge structure, i.e. a finite increasing filtration \(W\) on \(H^i (X; \mathbb{Q})\), called the weight filtration, and a finite decreasing filtration \(F\) on \(H^i (X; \mathbb{C})\), called the Hodge filtration such that the filtration induced by \(F\) on \(\text{Gr}^W_k H^i (X; \mathbb{C})\) is a Hodge structure of weight \(k\). Using the weight filtration an invariant for the variety can be defined known as the weight polynomial. The next result is that the weight polynomial of Sk\((n, \mathbb{C})\) is determined by applying a theorem of Dimca and Lehrer to the fibration
\[
\text{Sp}(n, \mathbb{C}) \hookrightarrow\text{GL}(n, \mathbb{C})\to\text{Sk}(n, \mathbb{C}).
\]
Let Sym\((i, n, K)\) denote the variety of \(n\times n\) symmetric matrices over \(K\) with rank \(i\) and \(G(k, n, K)\) the Grassmannian variety of \(k\)-dimensional subspaces of \(K^n\). An inductive formula for the weight polynomial of Sym\((n, \mathbb{C})\) is found by using the fibration
\[
\text{Sym}(i, \mathbb{C}) \hookrightarrow\text{Sym}(i, n, \mathbb{C})\to G(n-i, n, \mathbb{C})
\]
which is then solved to give an explicit result.
For general complex algebraic varieties a relationship between the number of \(\mathbb{F}_q\)-rational points of their reduction modulo \(q\) and the weight \(m\) Euler characteristic of their \(\mathbb{C}\)-rational points can be found using results of Deligne and the Grothendieck-Verdier fixed point theorem. We conclude by calculating the number of \(\mathbb{F}_q\)-rational points of the varieties considered in this article, thus giving an alternative method for computing their weight polynomials. We note that for these varieties there is also an interesting relationship between the compact Euler characteristics of the \(\mathbb{R}\)-rational points and the weight polynomials of the \(\mathbb{C}\)-rational points, namely
\[
\chi_c (X (\mathbb{R}))= W_c (X (\mathbb{C}), i).
\]
At the moment, this relationship is not well understood. number of rational points; homotopy type; Grassmann variety; Betti numbers; complex algebraic variety; mixed Hodge structure; weight filtration; Hodge filtration; invariant; weight polynomial; Euler characteristic Homotopy theory and fundamental groups in algebraic geometry, Rational points, Variation of Hodge structures (algebro-geometric aspects), Arithmetic ground fields (finite, local, global) and families or fibrations, Topological properties in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Arithmetic varieties and schemes; Arakelov theory; heights, Transcendental methods, Hodge theory (algebro-geometric aspects) On the geometry of varieties of invertible symmetric and skew-symmetric matrices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Stable Khovanov-Rozansky polynomials of algebraic knots are expected to coincide with certain generating functions, superpolynomials, of nested Hilbert schemes and flagged Jacobian factors of the corresponding plane curve singularities. Also, these 3 families conjecturally match the DAHA superpolynomials. These superpolynomials can be considered as singular counterparts and generalizations of the Hasse-Weil zeta-functions. We conjecture that all \(a\)-coefficients of the DAHA superpolynomials upon the substitution \(q \mapsto qt\) satisfy the Riemann Hypothesis for sufficiently small \(q\) for uncolored algebraic knots, presumably for \(q \leq 1/2\) as \(a = 0\). This can be partially extended to algebraic links at least for \(a = 0\). Colored links are also considered, though mostly for rectangle Young diagrams. Connections with Kapranov's motivic zeta and the Galkin-Stöhr zeta-functions are discussed. double affine Hecke algebras; Jones polynomials; HOMFLY-PT polynomials; plane curve singularities; compactified Jacobians; Hilbert scheme; Khovanov-Rozansky homology; iterated torus links; Macdonald polynomial; Hasse-Weil zeta-function; Riemann hypothesis Knots and links in the 3-sphere, Plane and space curves, Hecke algebras and their representations, Braid groups; Artin groups, Root systems, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Singular homology and cohomology theory Riemann hypothesis for DAHA superpolynomials and plane curve singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider a Riemann surface \(X\) defined by a polynomial \(f(x,y)\) of degree \(d\), whose coefficients are chosen randomly. Hence, we can suppose that \(X\) is smooth, that the discriminant \(\delta (x)\) of \(f\) has \(d(d - 1)\) simple roots, \(\Delta \), and that \(\delta (0)\neq 0\), i.e. the corresponding fiber has \(d\) distinct points \(\{y_{1}, \dots ,y_d\}\). When we lift a loop \(0 \in \gamma \subset \mathbb C - \Delta \) by a continuation method, we get \(d\) paths in \(X\) connecting \(\{y_{1},\dots ,y_d\}\), hence defining a permutation of that set. This is called monodromy.
Here we present experimentations in \texttt{Maple} to get statistics on the distribution of transpositions corresponding to loops around each point of \(\Delta \). Multiplying families of ``neighbor'' transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups.
Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions. bivariate polynomial; plane curve; random Riemann surface; absolute factorization; algebraic geometry; continuation methods; monodromy; symmetric group; algorithms; Maple code Galligo, A.; Poteaux, A., Computing monodromy via continuation methods on random Riemann surfaces, Theoret. Comput. Sci., 412, 1492-1507, (2011) Computational aspects of algebraic curves, Symbolic computation and algebraic computation, Polynomials, factorization in commutative rings, Numerical problems in dynamical systems Computing monodromy via continuation methods on random Riemann surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Homogeneous varieties are ubiquitous in mathematics. Especially the Grassmannians and flag varieties associated to the classical Lie groups play a central role in geometry, representation theory and combinatorics. In this paper, we survey recent developments in two important problems, the restriction and rigidity problems, in the geometry and cohomology of homogeneous varieties following [the author, J. Differ. Geom. 87, No. 3, 493--514 (2011; Zbl 1232.14032); Adv. Math. 228, No. 4, 2441--2502 (2011; Zbl 1262.14059); Clay Math. Proc. 18, 205--239 (2013; Zbl 1317.14107); ibid. 18, 205--239 (2013; Zbl 1317.14107); Isr. J. Math. 200, 85--126 (2014; Zbl 1354.14073)]. These notes grew out of lectures I gave at IMPAN in December 2013. The lectures were organized around the following three themes:
(1) Develop a concrete geometric theory of isotopic flag varieties in the spirit of the classical theory of Grassmannians, reducing the theory to a few simple principles of quadric geometry.
(2) Construct explicit rational equivalences between subvarieties of homogeneous varieties and unions of Schubert varieties.
(3) Use explicit rational equivalences and intersection theory to study rigidity of Schubert classes. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Restriction varieties and the rigidity problem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{M. Kontsevich} and \textit{Y. Soibelman} [Commun. Number Theory Phys. 5, No. 2, 231--352 (2011; Zbl 1248.14060)] introduced cohomological aspects of Donaldson-Thomas theory for Jacobi algebras and the associated cohomological Hall algebra. \textit{B. Davison} and \textit{S. Meinhardt} [``Donaldson-Thomas theory for categories of homological dimension one with potential'', Preprint, \url{arXiv:1512.08898}] proved the Hodge-theoretic categorification of the integrality conjecture and the wall crossing formula (Theorem A, page 783, and Theorem B, page 787). Furthermore, the authors realize the isomorphism in Theorem A and Theorem B as Poincaré-Birkhoff-Witt isomorphisms for the associated cohomological Hall algebra (Theorem C, page 791).
This leads them to construct a degeneration of the cohomological Hall algebra, for generic stability condition and fixed slope, to a free supercommutative algebra generated by a mixed Hodge structure categorifying the BPS invariants (Theorem D, page 792). As a corollary of this construction, Davison and Meinhardt obtain a Lie algebra structure on this mixed Hodge structure, the Lie algebra of BPS invariants (Section 1.7, page 792), for which the entire cohomological Hall algebra can be seen as the positive part of a Yangian-type quantum group. Donaldson-Thomas theory; quiver with potential; moduli stack; quiver representations; BPS Lie algebra; monodromic mixed Hodge modules; quantum enveloping algebras Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations, Representations of quivers and partially ordered sets Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author gives an interpretation for the multiplicities of weights in a finite dimensional representations of a simple complex Lie algebra \({\mathfrak g}\) in terms of intersection cohomologies of Schubert varieties of the corresponding adjoint Lie group \(G\). The method, used in the paper, is the study of the Hecke algebra of the corresponding (``affine'') Coxeter group. Weyl's character formula; multiplicities of weights; simple complex Lie algebra; intersection cohomologies of Schubert varieties George Lusztig, Singularities, character formulas, and a \(q\)-analog of weight multiplicities, Astérisque101-102 (1983), p. 208-229 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Simple, semisimple, reductive (super)algebras, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Hecke algebras and their representations, Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Singularities, character formulas, and a \(q\)-analog of weight multiplicities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of \textit{M.-P. Schützenberger} [Lect. Notes Math. 579, 59--113 (1977; Zbl 0398.05011)] for standard Young tableaux. We apply this to give a new combinatorial rule for the \(K\)-theory Schubert calculus of Grassmannians via \(K\)-theoretic jeu de taquin, providing an alternative to the rules of Buch and others. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety \(G/P\), extending recent work of Thomas and Yong. We also present analogues of results of Fomin, Haiman, Schensted and Schützenberger. Schubert calculus; \(K\)-theory; jeu de taquin Hugh Thomas & Alexander Yong, ``A jeu de taquin theory for increasing tableaux, with applications to \(K\)-theoretic Schubert calculus'', Algebra Number Theory3 (2009) no. 2, p. 121-148 Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds A jeu de taquin theory for increasing tableaux, with application to \(K\)-theoretic Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The bisymplectic Grassmannian I\(_2\)Gr\((k,V)\) parametrizes \(k\)-dimensional subspaces of a vector space \(V\) which are isotropic with respect to two general skew-symmetric forms; it is a Fano projective variety which admits an action of a torus with a finite number of fixed points. In this work, we study its equivariant cohomology with complex coefficients when \(k=2\); the central result of the paper is an equivariant Chevalley formula for the multiplication of the hyperplane class by any Schubert class. Moreover, we study in detail the case of I\(_2\)Gr\((2,\mathbb{C}^6)\), which is a quasi-homogeneous variety, we analyse its deformations, and we give a presentation of its cohomology. Bisymplectic Grassmannians of planes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We extend the idea of interval pattern avoidance defined by \textit{A. Yong} and the author [J. Algebra 320, No. 2, 495--520 (2008; Zbl 1152.14046)] for \(S_n\) to arbitrary Weyl groups using the definition of pattern avoidance due to \textit{S. C.
Billey} and \textit{T. Braden} [Transform. Groups 8, No. 4, 321--332 (2003; Zbl 1063.20044)], and \textit{S. C.
Billey} and \textit{A. Postnikov} [Adv. Appl. Math. 34, No. 3, 447--466 (2005; Zbl 1072.14065)]. We show that, as previously shown by Yong and the author for GL\(_n\), interval pattern avoidance is a universal tool for characterizing which Schubert varieties have certain local properties, and where these local properties hold. 10.4153/CMB-2010-080-2 Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Interval pattern avoidance for arbitrary root systems | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\widetilde G\) be a semisimple group scheme of type \(G_2\), \(F_4\) or \(E_8\). The author points out interesting coincidences among two sets of polynomials associated with \(\widetilde G\), one set coming from the representation theory of the finite groups \(\widetilde G(\mathbb{F}_q)\), and the other coming from certain prehomogeneous vector spaces associated with nilpotent orbits in the Lie algebra \(\widetilde{\mathfrak G}\) of \(\widetilde G(\mathbb{C})\). More specifically, these polynomials arise as follows.
For \(q\) sufficiently large, the nilpotent orbits of \(\widetilde{\mathfrak G}(\mathbb{F}_q)\) have the same description as those of \(\widetilde{\mathfrak G}(\mathbb{C})\). In each case above there is a unique nilpotent orbit, \((N)\) say, such that the component group \(A(N)\) of the centraliser \(Z_{\widetilde G}(N)\) is a symmetric group \(S_k\), where \(k=3,4\) and \(5\) in the respective cases above. On the one hand, there is associated with each element \(h\in S_k\) an irreducible (unipotent) character \(R(h,1)\) of the finite group \(\widetilde G(\mathbb{F}_q)\). The ``generic degree'' of this character has denominator a polynomial \(d_{h,1}(t)\).
On the other hand, associated with the nilpotent orbit \((N)\) in \(\widetilde{\mathfrak G}(\mathbb{C})\), there is a prehomogeneous vector space \((G,V)\), which has ``singular contractions'' \((G^{((h))}, V^{((h))})\) for each \(h\in S_k\). These latter prehomogeneous vector spaces each have an associated ``exponential \(b\)-function'' \(b^{\exp}_{f^{((h))}}(t)\), which arises from functional equations for relative invariants of closed orbits in \(V^{((h))}\). The author observes by direct computation, that in the cases above, \(b^{\exp}_{f^{((h))}}(t)=d_{h,1}(t)\). Moreover, he gives an example from the group of type \(E_7\) which shows the result does not always hold. unipotent representations; unipotent characters; semisimple group schemes; prehomogeneous vector spaces; nilpotent orbits; relative invariants Linear algebraic groups over finite fields, Representation theory for linear algebraic groups, Homogeneous spaces and generalizations, Prehomogeneous vector spaces Certain unipotent representations of finite Chevalley groups and Picard-Lefschetz monodromy | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review is the authors' ongoing efforts to establish the \(K\)-equivalence conjecture [\textit{C.-L. Wang}, J. Algebr. Geom. 12, No. 2, 285--306 (2003; Zbl 1080.14510)] for general ordinary flops, which is an important case of the crepant transformation conjecture.
A flop is a birational surgery on a smooth algebraic variety which modifies a small part of the variety called the exceptional locus. The exceptional locus of an ordinary flop is in general a projectivised non-split vector bundles over an arbitrary base. For a simple flop where the base is a point, the \(K\)-equivalence conjecture has been proved in [\textit{Y.-P. Lee} et al., Ann. Math. (2) 172, No. 1, 243--290 (2010; Zbl 1272.14040)] in the genus zero case, and in [\textit{Y. Iwao} et al., J. Reine Angew. Math. 663, 67--90 (2012; Zbl 1260.14068)] for all genera.
The background and some recent development on the \(K\)-equivalence relation among birational manifolds has been surveyed in [\textit{C.-L. Wang}, in: Second international congress of Chinese mathematicians. Proceedings of the congress (ICCM2001), Taipei, Taiwan, December 17--22, 2001. Somerville: International Press. 199--216 (2004; Zbl 1328.14022)].
The current paper is a continuation of \textit{Y.-P. Lee} et al. [``Invariance of quantum rings under ordinary flops. I: Quantum corrections and reduction to local models'', Preprint, \url{arXiv:1109.5540}; ``Invariance of quantum rings under ordinary flops. II: A quantum Leray-Hirsch theorem'', Preprint, \url{arXiv:1311.5725}] where the authors proved that a general ordinary flop over an smooth base induces an isomorphism of big quantum rings in the genus zero case. In the first two papers, the authors showed that the general case can be reduced to the case of the standard local models for such flops; they then verified the case of split vector bundles for the local models. The major effort of the third paper is to further reduce the non-split case of local models to the split case.
To prove the reduction, the authors observed that after a sequence of blow-ups of the base of a given local model, the vector bundle can be deformed to a split one. Since Gromov-Witten invariants are deformation invariant, the challenge is to analyze the invariants under blow-ups of the base. To relate these invariants of the blow-ups, the authors employed similar ideas as in [\textit{D. Maulik} and \textit{R. Pandharipande}, Topology 45, No. 5, 887--918 (2006; Zbl 1112.14065)], and applied the degeneration formula for Gromov-Witten invariants [\textit{J. Li}, J. Differ. Geom. 60, No. 2, 199--293 (2002; Zbl 1063.14069)] and [\textit{A.-M. Li} and \textit{Y. Ruan}, Invent. Math. 145, No. 1, 151--218 (2001; Zbl 1062.53073)]. Lee, Yuan-Pin; Lin, Hui-Wen; Qu, Feng; Wang, Chin-Lung, Invariance of quantum rings under ordinary flops III: A quantum splitting principle, Camb. J. Math., 4, 3, 333-401, (2016) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Invariance of quantum rings under ordinary flops. III: A quantum splitting principle | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author shows that there is an affine Schubert variety in the infinite-dimensional partial Flag variety (associated to the two-step parabolic subgroup of the Kac-Moody group \(\widehat{\mathrm{SL}}_n\)) which is a natural compactification of the cotangent bundle to the Grassmann variety.
The motivation for this article is the following one:
Let \(\mathbb C\) be the field of complex numbers. Consider a cyclic quiver with 2 vertices and dimension vector \(d = (d_1,d_2)\). Denote \(V_i =\mathbb C^i\). Let \(Z =\mathrm{Hom}(V_1, V_2)\times\mathrm{Hom}(V_2,V_1)\); \(\mathrm{GL}_d =\mathrm{GL}(V_1)\times\mathrm{GL}(V_2)\). There is a natural action of \(\mathrm{GL}_d\) in \(Z\). Let \(N= \{(f_1,f_2)\in Z\mid f_1\circ f_2\text{ is nilpotent}\}=\{(f_1,f_2)\in Z\mid f_1\circ f_2\text{ and }f_2\circ f_1\text{ are nilpotent}\}\)
Clearly \(N\) is \(\mathrm{GL}_d\)-stable. \textit{G. Lusztig} [J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008)] has shown that an orbit closure in \(N\) is canonically isomorphic to an open subset of a Schubert variety in \(\widehat{\mathrm{SL}}_n/Q\), where \(n =d_1+d_2\) and \(Q\) is the parabolic subgroup of \(\widehat{\mathrm{SL}}_n\) corresponding to omitting \(\alpha_0,\alpha_{d_1}\) (Luztig consider the case of cyclic quiver with \(h\geq 2\) vertices).
Now let \(Z_0 =\{(f_1,f_2)\in Z | f_1\circ f_2=0\) and \(f_2\circ f_1=0\}\). \textit{E. Strickland} [J. Algebra 75, 523--537 (1982; Zbl 0493.14030)] has shown that each irreducible component of \(Z_0\) is the conormal variety to a determinantal variety in \(M_{d_1,d_2}\). A determinantal variety in \(M_{d_1,d_2}\) being canonically isomorphic to an open subset in a certain Schubert variety in the Grassmannian variety of \(d_2\)-dimensional subspaces of \(\mathbb C^{d1+d2}\) (see [\textit{V. Lakshmibai} and \textit{C. S. Seshadri}, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1--54 (1978; Zbl 0447.14011)]), the above two results of Lusztig and Strickland suggest a connection between conormal varieties to Schubert varieties in the (finite-dimensional) flag variety and the affine Schubert varieties. Schubert varieties; affine flag varieties; determinantal varieties Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Homogeneous spaces and generalizations Cotangent bundle to the Grassmann variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper studies the classical and quantum cohomology of a class of quasi-homogeneous, but not homogeneous, spaces known as Mihai's odd symplectic Grassmannian of lines. The classical cohomology ring can be described in terms of Pieri and Giambelli-type formulas by exploiting relations to the even symplectic and the ordinary Grassmannians. The quantum deformation is obtained by a careful study of enumerativity of Gromov-Witten invariants and a transversality lemma.
The quantum cohomology ring being semi-simple motivates the check of a conjecture of Dubrovin's, according to which semi-simplicity should be equivalent to the existence of a full exceptional collection. Pech constructs such an exceptional collection for the odd symplectic Grassmannian by modifying Kuznetsov's exceptional collection for the even symplectic Grassmannian.
The paper is a pleasant read and can be understood with familiarity with cohomology of homogeneous spaces and some background knowledge of Gromov-Witten theory. quantum cohomology; quasi-homogeneous spaces; Grassmannians; Pieri and Giambelli formulas; exceptional collections in derived categories Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum cohomology of the odd symplectic Grassmannian of lines | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We suggest a new combinatorial construction for the cohomology ring of the flag manifold. The degree 2 commutation relations satisfied by the divided difference operators corresponding to positive roots define a quadratic associative algebra. In this algebra, the formal analogues of Dunkl operators generate a commutative subring, which is shown to be canonically isomorphic to the cohomology of the flag manifold. This leads to yet another combinatorial version of the corresponding Schubert calculus. The paper contains numerous conjectures and open problems. We also discuss a generalization of the main construction to quantum cohomology. representation of the symmetric group; Pieri rule; Gromov-Witten invariants; Schubert polynomials; cohomology ring of the flag manifold; divided difference operators; quadratic associative algebra; Dunkl operators; Schubert calculus; quantum cohomology Fomin, Sergey; Kirillov, Anatol N., Quadratic algebras, Dunkl elements, and Schubert calculus. Advances in geometry, Progr. Math. 172, 147-182, (1999), Birkhäuser Boston, Boston, MA Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Quadratic algebras, Dunkl elements, and Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The primary aim of this monograph is to achieve part of Beilinson's program on mixed motives using Voevodsky's theories of \(A^1\)-homotopy and motivic complexes.
Historically, this book is the first to give a complete construction of a triangulated category of mixed motives with rational coefficients satisfying the full Grothendieck six functors formalism as well as fulfilling Beilinson's program, in particular the interpretation of rational higher Chow groups as extension groups. Apart from Voevodsky's entire work and Grothendieck's SGA4, our main sources are Gabber's work on étale cohomology and Ayoub's solution to Voevodsky's cross functors theory. We also thoroughly develop the theory of motivic complexes with integral coefficients over general bases, along the lines of Suslin and Voevodsky.
Besides this achievement, this volume provides a complete toolkit for the study of systems of coefficients satisfying Grothendieck' six functors formalism, including Grothendieck-Verdier duality. It gives a systematic account of cohomological descent theory with an emphasis on h-descent. It formalizes morphisms of coefficient systems with a view towards realization functors and comparison results. The latter allows to understand the polymorphic nature of rational mixed motives. They can be characterized by one of the following properties: existence of transfers, universality of rational algebraic K-theory, h-descent, étale descent, orientation theory.
This monograph is a longstanding research work of the two authors. The first three parts are written in a self-contained manner and could be accessible to graduate students with a background in algebraic geometry and homotopy theory. It is designed to be a reference work and could also be useful outside motivic homotopy theory. The last part, containing the most innovative results, assumes some knowledge of motivic homotopy theory, although precise statements and references are given. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to category theory, Cycles and subschemes, Derived categories, triangulated categories Triangulated categories of mixed motives | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove some general results concerning the cohomology of the Milnor fibre of a hyperplane arrangement, and apply them to the case when the arrangement has some symmetry properties, particularly the case of the set of reflecting hyperplanes of a unitary reflection group. We relate the isotypic components of the monodromy action on the cohomology to the cohomology degree and to the mixed Hodge structure of the cohomology. We also use monodromy eigenspaces to determine the spectrum in some cases, which in turn throws further light on the equivariant Hodge structure of the cohomology and on the determination of the equivariant Hodge-Deligne polynomials. When the arrangement is the set of reflecting hyperplanes of a unitary reflection group, then using eigenspace theory for reflection groups, we prove sum formulae for additive functions such as the equivariant weight polynomial and certain polynomials related to the Euler characteristic, such as the Hodge-Deligne polynomials. This leads to a case-free determination of the Euler characteristic in this case, answering a question of Denham-Lemire. We also give an alternative formula for the spectrum of an arrangement which permits its computation in low dimensions, and we provide several examples of such computations. equivariant Hodge-Deligne polynomial; hyperplane arrangement; Milnor fibre; monodromy 10.1007/978-3-319-31580-5_10 Configurations and arrangements of linear subspaces, Structure of families (Picard-Lefschetz, monodromy, etc.), Relations with arrangements of hyperplanes, Mixed Hodge theory of singular varieties (complex-analytic aspects), Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Cohomology of the Milnor fibre of a hyperplane arrangement with symmetry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The study of Schubert varieties in \(G/B\) has led to numerous advances in algebraic combinatorics and algebraic geometry. These varieties are indexed by elements of the corresponding Weyl group, an affine Weyl group, or one of their parabolic quotients. Often times, the goal is to determine which of the algebraic and topological properties of the Schubert variety can be described in terms of the combinatorics of its corresponding Weyl group element. A celebrated example of this occurs when \(G/B\) is of type \(A\), due to Lakshmibai and Sandhya. They showed that the smooth Schubert varieties are precisely those indexed by permutations that avoid the patterns 3412 and 4231. Our main result is a characterization of the rationally smooth Schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations. pattern avoidance; affine permutations; Schubert varieties Billey, S.; Crites, A., Rational smoothness and affine Schubert varieties of type \(A\), 171-181, (2011), Nancy Grassmannians, Schubert varieties, flag manifolds Rational smoothness and affine Schubert varieties of type \(A\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A symmetric polynomial \(q(x)\) in non-commutative variables \(x=\{x_{1},\dots,x_{n}\}\) is called matrix-positive if whenever the variables \(\{x_{1},\dots,x_{n}\}\) are replaced by matrices of any size, the resulting polynomial is positive semidefinite. In [\textit{J. Helton}, Ann. Math. (2) 156, 675--694 (2002; Zbl 1033.12001)] it was proved that a symmetric non-commutative polynomial is matrix-positive if and only if it is a sum of squares. Roughly speaking, the paper under review provides a weighted sum square representation for any non-commutative polynomial \(q(x)\) that is ``strictly positive'' on a bounded ``semi-algebraic'' set \(D_{\mathcal{P}}\) defined by a collection \(\mathcal{P}\) of such polynomials.
More precisely, given a collection \(\mathcal{P}\) of symmetric polynomials in non-commutative variables \(x=\{x_{1},\dots,x_{n}\}\) and a real Hilbert space \(\mathcal{H}\), denote by \(D_{\mathcal{P}}(\mathcal{H)}\) the set of tuples \(X=(X_{1},\dots,X_{n})\) where each \(X_{j}\) is an operator on \(\mathcal{H}\) and \(p(X_{1},\dots,X_{n})\succeq0\). Then we call the domain of positivity associated to \(\mathcal{P}\) the collection \(D_{\mathcal{P}}\) of all tuples \(X\in D_{\mathcal{P}}(\mathcal{H)}\) and all Hilbert spaces \(\mathcal{H}\). The domain of positivity \(D_{\mathcal{P}}\) is bounded, if there exists a constant \(c>0\) such that for any Hilbert space \(\mathcal{H}\), any \(X=(X_{1},\dots,X_{n})\in D_{\mathcal{P}}(\mathcal{H)}\) and any \(j\in\{1,\dots,n\}\), we have \(\| X_{j}\|\leq c\). The authors prove the following ``Positivstellensatz'' result: let \(\mathcal{P}\) be a collection of symmetric polynomials with a bounded domain of positivity \(D_{\mathcal{P}}\). If a polynomial \(q\) is strictly positive on \(D_{\mathcal{P}}\), it can be represented as a weighted sum of squares:
\[
q=\sum_{j=1}^{N}s_{j}^{T}p_{j}s_{j}+\sum_{k=1}^{M}r_{k}^{T}r_{k}+\sum_{m,\ell }t_{m,\ell}^{T}(C^{2}-x_{m}^{2})t_{m,\ell}
\]
for a finite number of polynomials \(p_{j}\in P\) and polynomials \(s_{j} ,r_{k},t_{m,l}\). The proof uses a Hahn-Banach separation argument. When the domain of positivity \(D_{\mathcal{P}}\) is ``convex'' (meaning that for every \(\mathcal{H}\), the set \(D_{\mathcal{P}}(\mathcal{H)}\) is convex), the Hilbert spaces (respectively, the operators) involved above can be taken to be finite-dimensional (respectively, matrices). Versions of the Positivstellensatz are presented for three classes of matrix-valued non-commutative polynomials, since the authors anticipate potential applications.
The paper finishes with a discussion on the real Nullstellensatz, and shows by means of an example that a non-commutative version is not possible along certain lines.
The paper is written in a clear and instructive way; in particular, it contains several examples and useful explanations which make it accessible to non-specialists. non-commutative polynomial; Positivstellensatz; operator positivity; semi-algebraic set J. W. Helton and S. McCullough, \textit{A Positivstellensatz for non-commutative polynomials}, Trans. Amer. Math. Soc., 356 (2004), pp. 3721--3737. Several-variable operator theory (spectral, Fredholm, etc.), Semialgebraic sets and related spaces A positivstellensatz for non-commutative polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The text is suitable for graduate students and researchers in algebraic geometry. It begins by introducing hyperbolicity cones of homogeneous polynomials, definite determinantal representations, the notion of spectrahedra and the generalized Lax conjecture. The second section defines generalized Clifford algebras of hyperbolic homogenous polynomials. The main result is a theorem relating that \(-1\) is not a sum of squares in the generalized Clifford algebra and the spectrahedral property. This is followed by a theorem relating a positive trace functional with the spectrahedral property. Traces are in turn related to determinantal representations. Finally, several computational aspects are explained and the paper concludes with a list of open questions surrounding the generalized Lax conjecture. homogenous polynomial; hyperbolicity; generalized Clifford algebra, spectrahedra; semidefinite programming; Lax conjecture; positive trace functional; determinantal representations Netzer, T; Thom, A, Hyperbolic polynomials and generalized Clifford algebras, Disc. Comput. Geom., 51, 802-814, (2014) Clifford algebras, spinors, Solving polynomial systems; resultants, Semidefinite programming, Determinantal varieties Hyperbolic polynomials and generalized Clifford algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors explain the relationship between the trace map that occurs in Grothendieck duality from \textit{J. Lipman's} version [Dualizing sheaves, differential and residues on algebraic varieties, Astérisque 117 (1984; Zbl 0562.14003)] and its counterpart (the integral) in the de Rham theory.
Theorem. On a smooth \(n\)-dimensional complete variety \(X\) over \(\mathbb{C}\) the trace map \(\widetilde\theta_X: H^n(X,\Omega^n_X)\to \mathbb{C}\) arising from Lipman's version of Grothendieck duality agrees with
\[
(2\pi i)^{-n}(- 1)^{n(n-1)/2} \int_X: H^{2n}_{\text{DR}}(X,\mathbb{C})\to \mathbb{C}
\]
under the Dolbeault isomorphism. Grothendieck duality; de Rham integral; trace map Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Integral representations; canonical kernels (Szegő, Bergman, etc.), Schemes and morphisms, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories The Grothendieck trace and the de Rham integral | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper works on a family of FJRW potentials which is viewed as the counterpart of a nonconvex Gromov-Witten potential via the Landau-Ginzburg/Calabi-Yau correspondence. The author provides an explicit formula for Polishchuk and Vaintrob's virtual cycle in genus 0, constructed algebraically by using matrix factorizations [\textit{A. Polishchuk} and \textit{A. Vaintrob}, J. Reine Angew. Math. 714, 1--122 (2016; Zbl 1357.14024)]. In the nonconcave case of the so-called chain invertible polynomials \(W=x_1^{a_1}x_2+\cdots+x_{N-1}^{a_{N-1}}x_N+x_N^{a_N+1}\), it yields a compatibility theorem between Polishchuk and Vaintrob's class and FJRW virtual class, and also a proof of mirror symmetry for FJRW theory when the chain polynomial is of Calabi-Yau type, where the B-side concerns the local system given by the primitive cohomology of the fibration \([\{W^\vee-t\prod x_j=0\}/\underline{SL}(W^\vee)]_t\to \Delta ^*\) over a punctured disk. Here \(W^\vee=y_1^{a_1}+y_1y_2^{a_2}+\cdots+y_{N-1}y_N^{a_N+1}\) and \(\underline{SL}(W^\vee)\) is a group containing automorphisms of \(W^\vee\) of determinant 1. mirror symmetry; FJRW theory; nonconcavity; virtual cycle; matrix factorization; LG model; spin curves; recursive complex; chain invertible polynomial; Givental's formalism; J-function Guéré, Jérémy: A Landau-Ginzburg mirror theorem without concavity. (2013) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Families, moduli of curves (algebraic), Mirror symmetry (algebro-geometric aspects) A Landauc-Ginzburg mirror theorem without concavity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The book contains an introduction to some applications of the theory of formal functions on subvarieties \(Y\) of projective varieties \(X\). Formal functions and formal schemes were introduced by Zariski and Grothendieck as the algebraic counterpart of the analytic concept of functions over tubular neighbourhoods of \(Y\) inside \(X\). Technically, formal schemes are defined by taking the algebraic completion of the structure sheaf of \(X\) modulo the ideal sheaf defining \(Y\), i.e. the direct limit of successive infinitesimal neighbourhoods of \(Y\) in \(X\). Formal functions are thus a tool towards the study of the extensions of functions on \(Y\) to functions defined in the ambient variety \(X\).
From this point of view, formal functions have a natural application to the study of subvarieties of \(X\) which contain \(Y\) (an approach which can be used in any characteristic). The author points out mainly the relations of formal geometry with the problem of extending a variety \(Y\subset \mathbb P^n\) as a hyperplane section of a subvariety \(Y'\subset \mathbb P^{n+1}\), extending bundles from \(Y\) to \(X\) and Hartshorne's conjecture for varieties of low codimension. A first motivation for this analysis is the remark that the first proof of the celebrated Zak extension principle was obtained by the machinery of formal extensions. A simplified proof, by means of the Fulton--Hansen connectedness principle, was introduced later.
The author exploits carefully the interplay between formal geometry and the connectedness principle. For instance, \(Y\) is defined of type G3 in \(X\) if the field \(K(X_Y)\) of meromorphic formal functions of \(Y\) in \(X\) is isomorphic to \(K(X)\). Then it is a general fact that subvarieties of low codimension are likely to be of type G3. The author shows that (in any characteristic), for any closed irreducible subvariety \(X\subset \mathbb P^n\times\mathbb P^n\) of dimension \(>n\), the intersection of \(X\) with the diagonal is G3 in \(X\). This result, an improvement for the connectedness principle, can be used to determine the geometry of varieties of low codimension and their deformations. For instance, it is applied in the book to prove restrictions on subvarieties of \(X\) which are \(0\)-loci of sections of vector bundles. Further applications to generating subvarieties of homogeneous spaces and the geometry of quasi--lines (rational curves with many infinitesimal deformations) are discussed. Bădescu, L., IMPAN Monogr. Mat. (N. S.), 65, (2004), Birkhäuser Verlag: Birkhäuser Verlag, Basel Research exposition (monographs, survey articles) pertaining to algebraic geometry, Formal neighborhoods in algebraic geometry, Low codimension problems in algebraic geometry Projective geometry and formal geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\widetilde G_{n,k}\) be the Grassmann manifold of oriented \(k\)-dimensional subspaces in \(\mathbb R^n\) (\(1\leq k\leq n-k\)). It is well known that \(\widetilde G_{n,k}\) is a closed orientable \(k(n-k)\)-dimensional manifold. It was shown in [\textit{V. Ramani} and \textit{P. Sankaran}, Proc. Indian Acad. Sci., Math. Sci. 107, No. 1, 13--19 (1997; Zbl 0884.55002)] that if \((n,k)\neq(m,l)\), \(l\geq2\) and \(k(n-k)=l(m-l)\), then every map \(f:\widetilde G_{n,k}\rightarrow\widetilde G_{m,l}\) has degree zero.
In the paper under review, the authors consider the Grassmann manifold \(\widetilde I_{2n,k}\) (where \(1\leq k\leq n\)) of oriented isotropic \(k\)-dimensional subspaces of \(\mathbb R^{2n}\) (which is equipped with the standard symplectic form) and obtain an analogous result: if \(\widetilde I_{2n,k}\) and \(\widetilde I_{2m,l}\) are two distinct ``oriented isotropic Grassmannians'' of the same dimension, and if \(k,l\geq2\), then \(\deg f=0\) for all maps \(f:\widetilde I_{2n,k}\rightarrow\widetilde I_{2m,l}\). Moreover, they establish that the same conclusion holds for all maps of the form \(\widetilde I_{2n,k}\rightarrow\widetilde G_{m,l}\) and \(\widetilde G_{m,l}\rightarrow\widetilde I_{2n,k}\), provided that \(\dim\widetilde I_{2n,k}=\dim\widetilde G_{m,l}\) and \(k,l\geq2\). The authors actually prove that there is no ring monomorphism between rational cohomology rings of the manifolds in question and use the following fact (easily obtained from Poincaré duality): if \(f:M\rightarrow N\) is a nonzero degree map between oriented closed connected manifolds of the same dimension, then \(f^*:H^*(N;\mathbb Q)\rightarrow H^*(M;\mathbb Q)\) is a monomorphism. isotropic Grassmann manifolds; Brouwer degree; characteristic classes Degree, winding number, Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Homology and cohomology of homogeneous spaces of Lie groups, Homology of classifying spaces and characteristic classes in algebraic topology Degrees of maps between isotropic Grassmann manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials One of the main reasons for studying character varieties is their prominent role in non-abelian Hodge theory. It is important to point out that the framework developed in this paper might be useful for understanding the mirror symmetry conjectures related to character varieties.
The authors in the present paper construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group \(G\). They also extend the result to the parabolic case with any number of punctures and arbitrary monodromies. As a byproduct, the authors obtain formulas for these polynomials in terms of homomorphisms between the space of mixed Hodge modules on \(G\). The construction is developed in a categorical-theoretic framework that can be applied to other situations.
This research proposes a general categorical framework for studying the recursive pattern of \(E\)-polynomials of representation varieties found by the geometric method. This framework is valid for any complex algebraic group \(G\), any manifold and any parabolic configuration.
The organization of the article is the following: Section 2 is devoted to reviewing the fundamentals of Hodge theory and Saito's mixed Hodge modules as a way of tracing variations of Hodge structures with nice functorial properties [\textit{M. Saito}, Publ. Res. Inst. Math. Sci. 26, No. 2, 221--333 (1990; Zbl 0727.14004)]. In Section 3 one finds an introduction to the main categorical tool that will allow one to formalize the recursion required.
In Section 4 the authors apply these ideas to the computation of Deligne-Hodge polynomials of representation varieties by defining the geometrisation in terms of the fundamental groupoid of the underlying manifolds and using the previously developed theory of mixed Hodge modules for an algebraisation. Under this formulation the authors obtain an explicit formula for these polynomials as a main result. TQFT; moduli spaces; \(E\)-polynomial; representation varieties Topological quantum field theories (aspects of differential topology), Transcendental methods, Hodge theory (algebro-geometric aspects), Variation of Hodge structures (algebro-geometric aspects), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) A lax monoidal topological quantum field theory for representation varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A quantum Grassmannian \(\mathcal O_q(G(m,n))\) is the subalgebra of coordinate algebra \(\mathcal O_q(M(m,n))\) of \(m\times n\)-matrices, generated by all \(m\times m\)-minors. It is shown that the long cycle \((12\dots n)\) does not preserve \(\mathcal O_q(G(m,n))\). Let \(M_\alpha\) be a minor associated with a choice of column \(\{\alpha,\alpha+1,\dots,\alpha+m-1\}\). Then one can consider localization of \(\mathcal O_q(G(m,n))\) by \(M_\alpha\). There is found is presentation of the localized algebra as a skew polynomial extension of a matrix coordinate algebra.
The previous cycling can be obtained by means of a cocycle twist. There are also found \(H\)-prime ideals in \(\mathcal O_q(G(m,n))\), where \(H\) is the torus acting on quantum Grassmannian. quantum Grassmannians; quantum matrices; prime spectra; localizations; cocycle twists DOI: 10.1090/S0002-9939-2010-10478-1 Ring-theoretic aspects of quantum groups, Noetherian rings and modules (associative rings and algebras), Rings arising from noncommutative algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations, Ideals in associative algebras, Group actions on varieties or schemes (quotients) Twisting the quantum Grassmannian. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathfrak g\) be a semisimple Lie algebra whose Dynkin diagram is simply-laced. The positive part \(U_{\nu}(\mathfrak g)^+\) of the quantized enveloping algebra of \(\mathfrak g\) has a PBW-basis and also admits a bar involution. These are two basic ingredients for Lusztig's construction of the canonical basis [\textit{G. Lusztig}, J. Am. Math. Soc. 3, 447--498 (1990; Zbl 0703.17008)]. The purpose of the present paper is to give two geometric interpretations of the coefficients of the bar automorphism in the PBW-basis. The first is in the framework of the Hall algebra approach to quantum groups; the coefficients are computed in terms of rational points over finite fields of quiver analogues of orbital varieties. The second interpretation, a consequence of the first, is in terms of a duality of constructible functions in preprojective varieties of quivers, see [\textit{G. Lusztig}, ``Canonical bases arising from quantized enveloping algebras. II'', Prog. Theor. Phys. 102, Suppl., 175--201 (1990; Zbl 0776.17012)]. canonical basis Quantum groups (quantized enveloping algebras) and related deformations, Group actions on varieties or schemes (quotients) The bar automorphism in quantum groups and geometry of quiver representations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Two compact Kähler manifolds \(X\) and \(Y\) of the same dimension are called geometric mirrors of each other if their tables of Hodge numbers are mirror-symmetric. The phenomenon of the existence of mirror symmetry in complex geometry, first predicted and discovered by physicists in the context of conformal quantum field theory (string models), has recently become a topic of great interest and significance in both complex algebraic geometry and mathematical physics.
The present article, written about three years ago, provides a concise survey on the state of knowledge of mirror symmetry (up to 1994) and its relations to various topics in algebraic geometry and topological conformal quantum field theory. In the meantime, most of the material sketched here can be found, in a systematic, detailed, self-contained and up-to-date form, in the recent monograph ``Symétrie miroir'' by \textit{C. Voisin} (Panoramas et Synthèses, Vol. 2, (1996; Zbl 0849.14001), where also the author's own contributions to the subject [see the author, Sel. Math., New Ser. 1, No. 2, 325-345 (1995) and the author and \textit{B. Kim}, Commun. Math. Phys. 168, No. 3, 609-641 (1995; Zbl 0828.55004)] are thoroughly discussed within the general, brand-new theory of mirror manifolds. toric varieties; quantum cohomology; Gromov-Witten invariants; Floer cohomology; homological geometry; string models; mirror symmetry A. Givental, Homological geometry and mirror symmetry, In \textit{Proceedings of the Inter-} \textit{national Congress of Mathematicians}, Vol. 1, 2 (Z''urich, 1994), 472-480, Birkh''auser, Basel, 1995. \(3\)-folds, Quantization in field theory; cohomological methods, Applications of global analysis to the sciences, Toric varieties, Newton polyhedra, Okounkov bodies, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Homological geometry and mirror symmetry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a compact connected Lie group and \(H\) be the centralizer of a one-parameter subgroup in \(G\). The problem posed by the author in section 1 is to determine numbers \( a_{u,v}^w \) appearing in the classical cases such as the Littlewood-Richardson rule, the Chevalley formula and the classical Pieri formula in terms of certain Cartan numbers of \(G\). His main result is stated as a theorem in subsection 2.3 providing a formula to compute Schubert classes.
The rest of the paper is organized as follows. In subsection 2.4 the author introduces an algorithm to compute the numbers \(a_{u,v}^w\). In section 3 he develops preliminary results from algebraic topology in which he defines in subsection 3.3 a special class of spaces known as oriented twisted product of 2-spheres of rank \(k\). In section 4, 2-spherical involutions are studied (subsection 4.3) which allow to construct the so-called Bott-Samelson cycles associated to a sequence of \(k\) roots (definition 6 of section 4.4). In section I, the image of a Schubert class as an element of the cohomology ring of the associated cycle in lemma 5.1 is given and proved in section 5.1. In section 5.2 the author defines a Bott-Samelson resolution of \(X_w\) for \( w \in W \), the Weyl group of \(G\), completing the proof of lemma 5.1 at the end of subsection 5.4. Section 6 is devoted to the proof of the main theorem of the paper. For that he uses lemma 5.1 and the description of the additive map given in section 3.4 lemma 3.4 in terms of the operator \(T_A\) defined in subsection 2.2. Section 7 recalls developments of a more historical nature used in this paper. In subsection 7.1 the author states as proposition 2 the generalization of \(K\)-cycles of \textit{R. Bott, H. Samelson} [Am. J. Math. 80, 964--1029 (1958; Zbl 0101.39702)] to Bott-Samelson cycles as here defined. In subsection 7.3 he recalls the Bruhat-Chevalley decomposition [\textit{C. Chevalley}, in: Algebraic groups and their generalization, Proc. Symp. Pure Math. 56, 1--26 (1994; Zbl 0824.14042)] of the quotient \(K/B\) where \(K\) is a linear algebraic group and \(B \subset K \) is a Borel subgroup in terms of the open cells of Schubert varieties. In subsection 7.4, definition 7.4 he gives the version of the Schubert variety in \(G / T\) used in this paper. In subsection 7.5 a linkage between the operator introduced in definition 3 of subsection 3.2 and the divided difference operator given by \textit{I. N. Bernstein} et al. [Russ. Math. Surv. 28, 1--26 (1973; Zbl 0289.57024)] and \textit{M. Demazure} [Invent. Math. 21, 287--301 (1973; Zbl 0269.22010)] is given in proposition 6. The author concludes the paper in subsection 7.6 commenting that his formula falls short of positivity using the additive basis introduced in lemma 3.3. classical problems; Schubert calculus; homogeneous spaces and generalizations H. Duan, Multiplicative rule of Schubert classes, Invent. Math. 159, 407--436 (2005). Homogeneous spaces and generalizations, Classical problems, Schubert calculus Multiplicative rule of Schubert classes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected complex reductive group and let \(\widetilde G\) be a connected reductive subgroup of \(G\). To any one parameter subgroup \(\lambda: \mathbb{C}^*\to\widetilde G\), there are corresponding parabolic subgroups \(\widetilde P=\widetilde P(\lambda)\subset\widetilde G\) and \(P= P(\lambda)\subset G\), and a \(\widetilde G\)-equivariant map of flag varieties \(\phi_\lambda:\widetilde G/\widetilde P\to G/P\). In the paper under review, the authors construct a map \(\phi^\odot_\lambda: H^*(G/P)\to H^*(\widetilde G/\widetilde P)\) generalizing the deformed cup product given by \textit{P. Belkale} and \textit{S. Kumar} (denoted \(\odot_0\)) in [Invent. Math. 166, No. 1, 185--228 (2006; Zbl 1106.14037)]. It is a generalization in the sense that under the diagonal embedding \(\widetilde G/\widetilde P\subset\widetilde G/\widetilde P\times\widetilde G/\widetilde P\), if \([\Lambda_i]\) denotes Schubert classes in \(H^*(\widetilde C/\widetilde P)\), then \(\phi^\odot_\lambda([\Lambda_1\times \Lambda_2])= [\Lambda_1]\oplus_0[\Lambda_2]\).
The primary result is that \(\phi^\oplus_\lambda\) is a graded ring homomorphism from \((H^*(G/P), \oplus_0)\) to \((H^*(\widetilde G/\widetilde P),\oplus_0)\), which is interesting since cohomology equipped with \(\oplus_0\) is not functorial in general. Furthermore, the authors show how to compute \(\phi\oplus_\lambda\) in terms of the usual comorphism \(\phi^*_\lambda\) on cohomology. Techniques in the construction of \(\phi^\oplus_\lambda\) are mostly geometric and similar to those employed by Belkale and Kumar in the aforementioned paper. The construction relies on a proposition given in the second author's thesis [J. Algebr. Comb. 30, No. 1, 1--17 (2009; Zbl 1239.14043)].
A number of illuminating examples are given where \(\phi^\oplus_\lambda\) is computed explicitly, including examples where \(\phi^\oplus_\lambda\) and \(\phi^*_\lambda\) are equal and not equal. Also, for the case when \(\widetilde G\subset\widetilde G\times\widetilde G\) is diagonally embedded, connections are drawn with the \textit{H. Azad}, \textit{M. Barry} and \textit{G. Seitz} theorem (see [Commun. Algebra 18, No. 2, 551--562 (1990; Zbl 0717.20029)]). Lastly, a result by the first author related to the generalized eigencone problem is restated in this context (see [Invent. Math. 180, No. 2, 389--441 (2010; Zbl 1197.14051)]). Schubert calculus; Belkale-Kumar product; eigencone; structure constants Ressayre, N; Richmond, E, Branching Schubert calculus and the belkale-kumar product on cohomology, Proc. Amer. Math. Soc., 139, 835-848, (2011) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Homology and cohomology of homogeneous spaces of Lie groups Branching Schubert calculus and the Belkale-Kumar product on cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The purpose of this paper is to find explicitly a polynomial equation \(f(z,w)=0\) such that (z,w)\(\to z\) extends to a map of the Riemann surface defined by this equation on \({\mathbb{P}}^ 1({\mathbb{C}})\) which has to be a covering of degree \(n,\) branched at three points, say \(a_ 1, a_ 2=0\) and \(a_ 3,\) and for which the monodromy permutations corresponding to these points are (2,1,...,1), (n), \((n_ 1)(n_ 2)\), respectively. The equation found has the form \(f(z,w)=w^ n+\alpha zw^{n-1}+\gamma zw^{n-2}+...+\mu z=0\) where the coefficients \(\alpha\),...,\(\mu\) are given explicitly in terms of the branch points \(a_ 1, a_ 2, a_ 3\) and a parameter d. covering of Riemann surface; monodromy permutations Coverings in algebraic geometry Constructin of an algebraic function field corresponding to an n-sheeted covering of a sphere | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0719.00022.]
A close connection between curve flattenings and Schubert cells of the Grassmann manifold is described. Unexpectedly, flattenings turn out to have a classification that is more delicate than that of Schubert cells - the author gives an example of two flattenings that correspond to the same Schubert cells although their bifurcation diagrams are not diffeomorphic. Next, he proves the duality theorem and gives a new construction of transformations of curves that do not change the flattenings. As an application, he describes disintegration of ramification and Weierstrass points. curve flattenings; Schubert cells of the Grassmann manifold; bifurcation; ramification; Weierstrass points Families, moduli of curves (analytic), Grassmannians, Schubert varieties, flag manifolds, Manifolds of solutions, Bifurcations in context of PDEs Bifurcation of flattenings and Schubert cells | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study a BGG-type category of infinite-dimensional representations of
\(\mathcal H[W]\), a semidirect product of the quantum torus with parameter \(q\), built on the root lattice of a semisimple group \(G\), and the Weyl group of \(G\). Irreducible objects of our category turn out to be parametrized by semistable \(G\)-bundles on the elliptic curve \(\mathbb C^*/q^\mathbb Z\).
We introduce a noncommutative deformation of the algebra of regular functions
on a torus. This deformation \(\mathcal H\), called quantum torus algebra, depends on a complex parameter \(q\in\mathbb C^*\). We further introduce a certain category \(\mathcal M(\mathcal H,\mathcal A)\) of representations of \(\mathcal H\) which are locally-finite with respect to a commutative subalgebra \(\mathcal A\subset\mathcal H\) whose ``size'' is one-half of that of \(\mathcal H\) (our definition is modeled on the definition of the category \(\mathcal O\) of Bernstein-Gelfand-Gelfand). We classify all simple objects of \(\mathcal M(\mathcal H,\mathcal A)\) and show that any object of \(\mathcal M(\mathcal H,\mathcal A)\) has finite length.
In \S3 we consider quantum tori arising from a pair of lattices coming from
a finite reduced root system. Let \(W\) be the Weyl group of this root system. We
classify all simple modules over the twisted group ring \(\mathcal H[W]\) which belong to \(\mathcal M(\mathcal H,\mathcal A)\) as \(\mathcal H\)-modules. In \S4 we show that the twisted group ring \(\mathcal H[W]\) is Morita equivalent to \(\mathcal HW\), the ring of \(W\)-invariants.
In \S5 we establish a bijection between the set of simple modules over the algebra \(\mathcal H[W]\) associated with a semisimple simply-connected group \(G\), and the set of pairs \((P, \alpha)\), where \(P\) is a semistable principal \(G\)-bundle on the elliptic curve \(E = \mathbb C^*/q^\mathbb Z\), and \(\alpha\) is a certain ``admissible representation'' (cf. Definition 5.4) of the finite group \(\Aut(P)/(\Aut P)^\circ\). Our bijection is constructed by combining the results of \S3 with a bijection between \(q\)-conjugacy classes in a loop group and \(G\)-bundles the elliptic curve \(E\), established earlier by some of us in [Baranovsky and Ginzburg, Int. Math. Res. Not. 1996, No. 15, 733--751 (1996; Zbl 0992.20034)]. Baranovsky, V.; Evens, S.; Ginzburg, V., Representations of quantum tori and \(G\)-bundles on elliptic curves, (The Orbit Method in Geometry and Physics (Marseille, 2000), Progr. Math., vol. 213, (2003), Birkhauser Boston Boston, MA), 29-48 Quantum groups (quantized enveloping algebras) and related deformations, Ring-theoretic aspects of quantum groups, Vector bundles on curves and their moduli Representations of quantum tori and \(G\)-bundles of elliptic curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a permutation pattern avoidance criteria for determining when the projection map from the flag variety to a Grassmannian induces a fiber bundle structure on a Schubert variety. In particular, we introduce the notion of a split pattern and show that a Schubert variety has such a fiber bundle structure if and only if the corresponding permutation avoids the split patterns \(3|12\) and \(23|1\). Continuing, we show that a Schubert variety is an iterated fiber bundle of Grassmannian Schubert varieties if and only if the corresponding permutation avoids (non-split) patterns 3412, 52341, and 635241. This extends a combined result of \textit{V. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45--52 (1990; Zbl 0714.14033)], \textit{K. M. Ryan} [Math. Ann. 276, 205--224 (1987; Zbl 0579.14045)], and \textit{J. S. Wolper} [Adv. Math. 76, No. 2, 184--193 (1989; Zbl 0705.14048)] who prove that Schubert varieties whose permutation avoids the ``smooth'' patterns 3412 and 4231 are iterated fiber bundles of smooth Grassmannian Schubert varieties. Schubert variety; flag variety; Grassmannian; permutation; pattern avoidance; fiber bundle Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry Pattern avoidance and fiber bundle structures on Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a \(d\)-dimensional Kähler manifold and write \({\mathcal J}^p = {\mathcal J}^p (X)\) for its \(p\)-th intermediate Jacobian. Via Poincaré duality one may identify \({\mathcal J}^q = {\mathcal J}^q (X)\), \(q = d + 1 - p\), with \({\mathcal J}^p (X)^\vee = \text{Pic}^0 ({\mathcal J}^p)\). Then on \({\mathcal J}^p\times{\mathcal J}^q\), \(p+q=d+1\), one has the Poincaré line bundle \(P\). Let \(\mathbb{P}:=P\backslash\) zero section. Then \(\mathbb{P}\) is called the Poincaré biextension associated to \({\mathcal J}^p\). The projection \(p : \mathbb{P} \to {\mathcal J}^p \times {\mathcal J}^q\) makes it a \(\mathbb{C}^\times\)-torsor over \({\mathcal J}^p \times {\mathcal J}^{p \vee}\). For any \(\lambda \in {\mathcal J}^{p \vee}\), \(\mathbb{P}_\lambda : = p^{-1} ({\mathcal J}^p \times \lambda)\) fits in an exact sequence \(0 \to \mathbb{C}^\times \to \mathbb{P}_\lambda \to {\mathcal J}^p \to 0\). In particular, any \(W \in\)CH\(^q_{\text{hom}} (X)\) defines via the Abel-Jacobi map \(\psi = \psi^q : \text{CH}^q_{\text{hom}} (X) \to {\mathcal J}^q\) an exact sequence \((*)\): \(0 \to \mathbb{C}^\times \to \mathbb{P}_W \to {\mathcal J}^p \to 0\), where \(\mathbb{P}_W : = \mathbb{P}_{\psi (W)}\). On the other hand, using his higher Chow groups and their relation to Deligne-Beilinson cohomology, \textit{S. Bloch} constructed for \(W \in \text{CH}^q_{\text{hom}} (X)\), extensions of the form \((**)\): \(0 \to \mathbb{C}^\times \to \mathbb{E} W \to \text{CH}^p_{\text{hom}} (X) \to 0\). These glue together to obtain a biextension \(\mathbb{E} \to \text{CH}^p_{\text{hom}}(X)\times\text{CH}^q_{\text{hom}}(X)\). The main result of the underlying paper relates \((*)\) and \((**)\), thus proving a conjecture due to Bloch: \(\mathbb{E}\) is the pull-back of \(\mathbb{P}\) as torsors under the Abel-Jacobi homomorphism.
A cycle \(W \in \text{CH}^q_{\text{hom}} (X)\) is called incidence equivalent to zero, written \(W \sim_{\text{inc}} 0\), if for all pairs \((T,B)\) with \(T\) smooth projective and \(B \in \text{CH}^{d + 1 - q} (T \times X)\), the divisor \((\text{pr}_T)_* (B \cap \text{pr}^*_X (W))\) is linear equivalent to zero. Griffiths conjectured: Let \(W \sim_{\text{alg}} 0\). If \(W \sim_{\text{inc}} 0\), then some multiple of \(W\) is Abel-Jacobi equivalent to zero. The truth of this conjecture would follow from Grothendieck's generalized Hodge conjecture. For \(W \in\)CH\(^q_{\text{alg}} (X)\) one has for the restriction \(\mathbb{E}_W^{\text{alg}}\) of \(\mathbb{E}\) to the fiber over \(W\) an extension: \(0 \to \mathbb{C}^\times \to \mathbb{E}_W^{\text{alg}} \to \text{CH}^p_{\text{alg}} (X) \to 0\). It is shown that this extension splits iff \(W\) is incidence equivalent to zero. For Griffiths' conjecture one shows that this implies: Let \(X\) be smooth and projective over \(\mathbb{C}\) and \(W \in \text{CH}^q_{\text{alg}} (X)\) such that, for some \(N \in \mathbb{N}\), \(\psi (N \cdot W)\) is contained in the dual of \({\mathcal J}^p_{\text{alg}} (X) = \psi (\text{CH}^p_{\text{alg}} (X))\). Then \(W \sim_{\text{inc}} 0\) implies that \(\psi (W)\) is torsion in \({\mathcal J}^q (X)\). This result, in turn, can be applied to prove Griffiths' conjecture in codimension two, a result actually due to \textit{J. Murre}. biextensions; higher Chow groups; algebraic cycles; incidence equivalence; Poincaré biextension; Abel-Jacobi map; Deligne-Beilinson cohomology; torsors; Griffiths' conjecture S. Müller-Stach, \(\mathbb{C}^{*}\)-extensions of tori, higher Chow groups and applications to incidence equivalence relations for algebraic cycles , \(K\)-Theory 9 (1995), 395-406. Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Algebraic cycles \(\mathbb{C}^*\)-extensions of tori, higher Chow groups and applications to incidence equivalence relations for algebraic cycles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article, the author presents a quantum cohomology analogue of skew Schur polynomials. These are certain symmetric polynomials labeled by shapes that are embedded in a torus. The author shows that the Gromov-Witten invariants are the expansion coefficients of these toric Schur polynomials in the basis of the ordinary Schur polynomials. The toric Schur polynomials are defined as sums over certain cylindrical semistandard tableaux. This paper makes an important contribution to the quantum cohomology theory of the Grassmannians. quantum cohomology; Schur polynomials; Gronov-Witten invariants A. Postnikov. ''Affine approach to quantum Schubert calculus''. Duke Math. J. 128 (2005), pp. 473--509.DOI. Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Affine approach to quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a vector bundle on an algebraic variety \(X\), \(L\) a line bundle on \(X\) and \(Q: V \otimes V \to L\) a nondegenerate skew-symmetric or symmetric bilinear form. Let \({\mathcal F}\) be the bundle of complete flags of isotropic subbundles of \(V\) with respect to \(Q\) and let \(F_\bullet\) be such a flag. For each permutation \(w\) in the corresponding Weyl group \(W\) one can define, using \(F_\bullet\), a Schubert variety \({\mathcal X}_w \subset {\mathcal F}\). In this paper, the author gives a formula for the class of \({\mathcal X}_w\) in the Chow ring of \({\mathcal F}\), thus completing his results from Duke Math. J. 65, No. 3, 381-420 (1992; Zbl 0788.14044) where he considered the case of the Schubert subvarieties of the bundle of complete flags of subbundles of \(V\). The formulas found by the author imply formulas for the corresponding degeneracy loci whenever one has a bundle with a symplectic or orthogonal form and two flags of isotropic subbundles. This was worked out in detail by the author [J. Differ. Geom. 43, No. 2, 276-290 (1996)]. The author uses some splitting principles to reduce the proof of the formulas to the case where \(V\) is a direct sum of lines bundles and the symplectic or orthogonal forms are diagonal. Then, he proves the formulas for \(w=\) identity and deduces the general case by relating one locus to the next by a \(\mathbb{P}^1\)-bundle (or conic bundle) correspondence. bundle of complete flags; Schubert variety; Chow ring; splitting principles William Fulton, Schubert varieties in flag bundles for the classical groups, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 241 -- 262. Grassmannians, Schubert varieties, flag manifolds, Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Classical groups (algebro-geometric aspects) Schubert varieties in flag bundles for the classical groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author investigates quadratic relations for Gromov-Witten invariants, arising from the associativity of the quantum product (i.e., the famous WDVV equations). He concentrates to the case of rational curves (i.e. invariants associated to the moduli space of stable maps of genus zero).
Here, the relations can be described by Feynman diagrams relating trees of rational curves.
In several examples, the associativity has been used to compute all Gromov-Witten numbers recursively from some starting data. Considered as a purely algebraic problem, one may ask which invariants are needed in general as starting data such that the WDVV equations can be solved uniquely and consistently. The main result obtained by the author is the following.
Strong reconstruction theorem. Let \(X\) be a complex projective manifold, and assume that \(H^{2*}(X,\mathbb{Q})\) is generated by divisors. Then it is sufficient to know all Gromov-Witten invariants \(N(\beta,d)\) with \(\sum_{i=1}^s d_s\leq 2\) (which have to fulfil a simple initial relation)
to compute uniquely and consistently all Gromov-Witten invariants of \(X\) by means of the WDVV equations.
The proof is constructive and gives in principle an algorithm (possibly quite complicated) for the computation. It is illustrated in detail with the examples of the product of a smooth quadric threefold with itself, \( \text{Sym}^2 {\mathbb{P}}^2 \), and the Grassmannians \(\text{Gr}(2,4)\) and \(\text{Gr}(2,5)\). Actually, only the first example satisfies the conditions of the reconstruction theorem, but similar techniques work well in the remaining cases. The author also points out that there are in general non-geometric solutions to a WDVV system, coming from other initial data than the geometric ones. WDVV equation; Feynman diagrams; genus zero Gromov-Witten invariants Kresch, A.: Associativity relations in quantum cohomology, Adv. math. 142, No. 1, 151-169 (1999) Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Associativity relations in quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In 1997 \textit{M. Kontsevich} [Lett. Math. Phys. 48, No. 1, 35--72 (1999; Zbl 0945.18008)] showed that every Poisson manifold can be quantized to obtain a noncommutative algebra. In his quantization formula a formal power series appears in the deformation parameter \(\hbar\). It is conjectured that the coefficients in this power series are expressible as \(\mathbb{Q}[1/(2\pi i)]\)-linear combinations of multiple zeta values (MZVs) which are infinite sums of the form
\[\zeta\left(n_{1}, \ldots,n_{d}\right)=\sum_{0<k_{1}<\cdots<k_{d}}\frac{1}{k_{1}^{n_{1}} \cdots k_{d}^{n_{d}}} \in \mathbb{R}\]
with positive integers \(n_1,\dots,n_d\) such that \(\sum_jn_j=n\), and \(n_d\ge2\).
The conjecture is not only proven in the paper, but the authors give bounds on the weights of the MZVs that appear at a given order of \(\hbar\). Moreover, it is also shown that the coefficients of the linear combinations are, in fact, integers.
The analysis of the paper is based on a systematic theory of integration on the moduli spaces of marked disks via suitable algebras of polylogarithms. This theory is developed in the paper, based on earlier works of Brown and Goncharov.
The authors' results are constructive in the sense that they present an algorithm to determine the power series coefficients in the quantization formula. multiple zeta values; deformation quantization of Poisson brackets; Kontsevich's integrals Multiple Dirichlet series and zeta functions and multizeta values, Families, moduli of curves (algebraic), Deformation quantization, star products, Feynman integrals and graphs; applications of algebraic topology and algebraic geometry Multiple zeta values in deformation quantization | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Recently, \textit{H.Naruse} [``Schubert calculus and hook formula'', talk held at 73rd Séminaire Lotharingien de Combinatoire, Strobl, Austria, 2014, \url{https://www.emis.de/journals/SLC/wpapers/s73vortrag/naruse.pdf}] discovered a hook length formula for the number of standard Young tableaux of a skew shape. \textit{A. H. Morales} et al. [J. Comb. Theory, Ser. A 154, 350--405 (2018; Zbl 1373.05026)] found two \(q\)-analogs of Naruse's hook length formula over semistandard Young tableaux (SSYTs) and reverse plane partitions (RPPs). As an application of their formula, they expressed certain \(q\)-Euler numbers, which are generating functions for SSYTs and RPPs of a zigzag border strip, in terms of weighted Dyck paths. They found a determinantal formula for the generating function for SSYTs of a skew staircase shape and proposed two conjectures related to RPPs of the same shape. One conjecture is a determinantal formula for the number of pleasant diagrams in terms of Schröder paths and the other conjecture is a determinantal formula for the generating function for RPPs of a skew staircase shape in terms of \(q\)-Euler numbers.
In this paper, we show that the results of A. H. Morales et al. [loc. cit.] on the \(q\)-Euler numbers can be derived from previously known results due to \textit{H. Prodinger} [Sémin. Lothar. Comb. 60, B60b, 3 p. (2008; Zbl 1179.33026)] by manipulating continued fractions. These \(q\)-Euler numbers are naturally expressed as generating functions for alternating permutations with certain statistics involving \textit{maj}. It has been proved by \textit{T. Huber} and \textit{A. J. Yee} [J. Comb. Theory, Ser. A 117, No. 4, 361--388 (2010; Zbl 1228.05015)] that these \(q\)-Euler numbers are generating functions for alternating permutations with certain statistics involving \textit{inv}. By modifying Foata's bijection we construct a bijection on alternating permutations which sends the statistics involving \textit{maj} to the statistic involving \textit{inv}. We also prove the aforementioned two conjectures of A. H. Morales et al. [loc. cit.]. reverse plane partition; Euler number; alternating permutation; lattice path; continued fraction Combinatorial aspects of representation theory, Classical problems, Schubert calculus, \(q\)-calculus and related topics, Bernoulli and Euler numbers and polynomials, Basic hypergeometric functions in one variable, \({}_r\phi_s\), Continued fractions Reverse plane partitions of skew staircase shapes and \(q\)-Euler numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The involution Stanley symmetric functions \( \hat{F}_y\) are the stable limits of the analogs of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for involution words and are indexed by the involutions in the symmetric group. By construction, each \( \hat{F}_y\) is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur \(P\)-positive. We give an algorithm to efficiently compute the decomposition of \( \hat{F}_y\) into Schur \(P\)-summands and prove that this decomposition is triangular with respect to the dominance order on partitions. As an application, we derive pattern avoidance conditions which characterize the involution Stanley symmetric functions which are equal to Schur \(P\)-functions. We deduce as a corollary that the involution Stanley symmetric function of the reverse permutation is a Schur \(P\)-function indexed by a shifted staircase shape. These results lead to alternate proofs of theorems of Ardila-Serrano and DeWitt on skew Schur functions which are Schur \(P\)-functions. We also prove new Pfaffian formulas for certain related involution Schubert polynomials. flag variety; Schur \(P\)-function Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Schur \(P\)-positivity and involution Stanley symmetric functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Algebraic combinatorics lies at the junction of not only algebra and combinatorics, as the name suggests, but also areas such as number theory, algebraic geometry and topology. In this paper, a leader in the field surveys some of the recent progress that has been made in relation to the latter three areas. It is a tale that will appeal to both the experts and the novices, with its clarity, illustrative examples and comprehensive bibliography.
First we are given an informal taste of cluster algebras and the work of Fomin and Zelevinsky proving the integrability of the sequences Somos-4 through -7. We also get to glimpse at the relation between Somos-4 and perfect matchings. Second, after connecting symmetric functions to Schubert calculus we are introduced to the recent work of Postnikov on toric Schur functions, whose expansion into Schur functions yield Gromov-Witten invariants as the coefficients of the expansion. Lastly we are taken on a tour of increasingly complex polytopes. Our first stop is simplicial polytopes whose h-vectors are completely characterized by the g-theorem. We then meet a combinatorial description of the toric h-vector for rational polytopes. Moving to non-rational polytopes we are introduced to the recent work of Karu on the hard Lefschetz theorem that a certain bijection exists. Finally, we are told of the half hard Lefschetz theorem that a certain injection exists for matroid complexes, recently proved by Swartz. cluster algebra; toric Schur function; hard Lefschetz theorem Stanley, R. P.: Recent developments in algebraic combinatorics. Israel J. Math. 143, 317-339 (2004) Symmetric functions and generalizations, Combinatorial aspects of representation theory, Group actions on posets and homology groups of posets [See also 06A09], Special sequences and polynomials, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Polyhedra and polytopes; regular figures, division of spaces Recent developments in algebraic combinatorics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials 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 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum cohomology ring of a projective manifold \(V\) can be regarded as a formal deformation of its cohomology ring. The structure constants are given by a formal series \(\Phi^V\) (called the potential), which is defined in terms of Gromov-Witten invariants. The Künneth formula, mentioned in the title, gives an explicit formula for the potential \(\Phi^{V\times W}\) of a product.
In the theory of quantum cohomology it is important to have a good knowledge of the intersection theory of the moduli space \(\overline{M}_{0,n}\) of stable genus-0 curves with \(n\) marked points. The cohomology ring of this smooth projective variety was determined by \textit{S. Keel} [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)]. He described generators of this ring corresponding to boundary divisors and a complete set of relations between them. In principle, this solves the problem of algorithmic calculations, however, in practice the most basic properties of this ring are not obvious. Further progress was made by \textit{M. Kontsevich} and \textit{Yu. Manin} [Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020) and Invent. Math. 124, No.1-3, 313-339 (1996; Zbl 0853.14021)], who studied the additive structure of this ring.
In the paper under review, the author proves a general formula for the intersection product of two arbitrary monomials in boundary divisors (or Keel's generators). Furthermore, a basis of the cohomology ring of \(\overline{M}_{0,n}\) is presented, using the language of trees. This presentation is inspired by the work of \textit{S. Yuzvinsky} [``Cohomology bases for the De Concini-Procesi models of hyperplane arrangements and sums over trees'', Invent. Math. 127, No. 2, 319-335 (1997)]. The author determines the Gram matrix and its inverse for this basis with the help of the above mentioned knowledge on the intersection form. This is enough to derive the Künneth formula for quantum cohomology. rational curves; quantum cohomology ring; operads; stable trees; Gromov-Witten invariants; Frobenius manifold; WDVV equation; Künneth formula; intersection theory of the moduli space; stable genus-0 curves with \(n\) marked points Kaufmann, R, The intersection form in \(H^### ({\overline{M}}_{0, n})\) and the explicit Künneth formula in quantum cohomology, Int. Math. Res. Not., 19, 929-952, (1996) Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Quantization in field theory; cohomological methods The intersection form in \(H^*(\overline{M}_{On})\) and the explicit Künneth formula in quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a projective algebraic variety of dimension \(n \geq 2\) over an algebraically closed field \(k\). Let \(Y\) be a proper nontrivial closed subscheme of \(X\) and \(\widehat{X} | Y\) the formal completion of \(X\) along \(Y\). Denote by \(M\) the sheaf of total quotient rings of \({\mathcal O}_{\widehat{X}}\). The ring (of sections) \(K(\widehat{X}) := H^ 0(\widehat{X},M)\) is called the ring of formal rational functions of \(X\) along \(Y\). If the canonical injection \(K(X) \to K(\widehat{X})\) is an isomorphism (resp. a finite field extension) then \(Y\) is said to be \(G_ 3\) (resp. \(G_ 2\)) in \(X\).
In the first part of the paper, following Hartshorne's ideas, the authors extend Hartshorne's results on formal duality for the pair \((X,Y)\), cohomological dimension of \(X - Y\) etc. to the case when \(X\) is singular and \(Y\) is a local complete intersection subvariety contained in the nonsingular part of \(X\). -- The second part of the paper deals with the situation when \(Y\) is the zero scheme of a global section of an ample vector bundle \(E\) of rank \(r\) on \(X\) with \(\dim Y = n - r > 0\). The main results of this paper are the following:
(1) Assume further that \(\text{char }k = 0\) and the intersection of the singular locus of \(X\) with \(Y\) is of codimension \(\geq 2\) in \(Y\), \(Y- W\) is \(G_ 3\) in \(X - W\) and \(Y\) meets every closed subvariety (in particular every divisor) of \(X\) of dimension \(\geq r\). (2) Assume that \(X\) is an irreducible subvariety of \(\mathbb{P}^ m\), \(Y = X \cap L\), \(L =\) a linear subspace of dimension \(m - r\). Then for every closed subscheme \(W\) of \(Y\) of codimension \(\geq 2\) in \(Y\), \(Y - W\) is \(G_ 3\) in \(X - W\).
The final part of the paper gives the following generalisation of the Fulton-Hansen connectedness theorem [cf. \textit{W. Fulton} and \textit{J. Hansen}, Ann. Math., II. Ser. 110, 159-166 (1979; Zbl 0389.14002)]. Let \(f : X \to \mathbb{P}^ n \times \mathbb{P}^ n\) be a proper morphism such that \(\dim f(X) \geq n + 1\) and the diagonal \(\Delta\) intersects \(f(X)\) transversally (e.g. \(f\) is surjective). Then \(f^{-1}(\Delta - W)\) is \(G_ 3\) in \(X - f^{-1}(X)\) for every closed subscheme \(W\) of \(f(X) \cap \Delta\) such that \(\dim W \leq \dim(f(X) \cap \Delta) - 2\). If in addition the fibres of \(f\) have all dimension \(\leq \dim X - n - 1\) then \(f^{-1}(\Delta)\) meets every divisor of \(X\). formal completion; ring of formal rational functions; formal duality; cohomological dimension; zero scheme of a global section of an ample vector bundle L. Badescu, E. Ballico: Formal rational functions along zero loci of sections of ample vector bundles. Rev. Roumaine Math. Pur. Appl.38 (1993), 609-630 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Formal neighborhoods in algebraic geometry, Complete rings, completion Formal rational functions along zero loci of sections of ample vector bundles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book may well be used as an introduction to quantum groups -- or some class of noncommutative, noncocommutative Hopf algebras. The starting point of the author is the Hopf algebra of ``noncommutative functions'' on a quantum group, which is in some sense predual to the Hopf algebras considered by V. Drinfel'd using deformations of enveloping algebras. The author generalizes the original concept of quantum group and its Hopf algebra of noncommutative functions to the concept of ``quadratic Hopf algebras'' defined by arbitrary quadratic relations in a finitely generated tensor algebra. The definition makes sense and seems to be natural in view of the fact, that the commutation relation between some elements of quantum matrix groups are more complicated, than the integrated Heisenberg commutation relation satisfied in linear quantum spaces. The considerations of the author are based on category theory arguments as well as on direct computations. There are several instructive examples included. The methods are purely algebraic in the spirit of algebraic geometry, no differential calculus on quantum groups or quantum spaces is discussed. The relation to the Yang-Baxter equation is pointed out and put in a category-theoretic framework.
The content is given as follows: Introduction, 1. The quantum group \(\mathrm{GL}(2)\), 2. Bialgebras and Hopf algebras, 3. Quadratic algebras as quantum linear spaces, 4. Quantum matrix spaces I. Categorical viewpoint, 5. Quantum matrix spaces II. Coordinate approach, 6. Adding missing relations, 7. From semigroups to groups, 8. Frobenius algebras and the quantum determinant, 9. Koszul complexes and growth rate of quadratic algebras, 10. Hopf *-algebras and compact matrix pseudogroups, 11. Yang-Baxter equations, 12. Algebras in tensor categories and Yang-Baxter functors, 13. Some open problems, Bibliography. quantum groups; quadratic Hopf algebras; compact matrix pseudogroups; Yang-Baxter equations; tensor categories Manin, Yu I., Quantum groups and non-commutative geometry, Publications du C.R.M., (1988), Université de Montreal Quantum groups (quantized enveloping algebras) and related deformations, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Hopf algebras and their applications, Bialgebras, Ring-theoretic aspects of quantum groups, Yang-Baxter equations, Noncommutative algebraic geometry, Quantum groups and related algebraic methods applied to problems in quantum theory Quantum groups and non-commutative geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Much has been written about Grothendieck duality. This survey will make the point that most of this literature is now obsolete: there is a brilliant article by \textit{J. L. Verdier} [in: Algebr. Geom., Bombay Colloq. 1968, 393--408 (1969; Zbl 0202.19902)] with the right idea on how to approach the subject. Verdier's article was largely forgotten for two decades until \textit{J. Lipman} resurrected it, reworked it and developed the ideas to obtain the right statements for what had before been a complicated theory [in: \textit{J. Lipman} and \textit{M. Hashimoto}, Foundations of Grothendieck duality for diagrams of schemes. Berlin: Springer, 1--259 (2009; Zbl 1467.14052)].
For the reader's benefit, Sections 1 through 5, which present the (large) portion of the theory that can nowadays be obtained from formal nonsense about rigidly compactly generated tensor triangulated categories, are all post-Verdier. The major landmarks in developing this approach were a 1996 article by me [J. Am. Math. Soc. 9, No. 1, 205--236 (1996; Zbl 0864.14008)] which was later generalized and improved on by \textit{P. Balmer} et al. [Compos. Math. 152, No. 8, 1740--1776 (2016; Zbl 1408.18026)], and a much more recent article of mine [Contemp. Math. 749, 279--325 (2020; Zbl 1442.14062)] about improvements to the Verdier base-change theorem and the functor \(f^!\). Section 6 is where Verdier's 1968 ideas still play a pivotal role [loc. cit.], but in the cleaned-up version due to Lipman and with new, short and direct proofs. Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Derived categories, triangulated categories New progress on Grothendieck duality, explained to those familiar with category theory and with algebraic geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper discusses a surprising relationship between the quantum cohomology of the variety of complete flags and the partially ordered set of Newton polygons associated to an element in the affine Weyl group. One primary key to establishing this connection is the fact that paths in the quantum Bruhat graph, which is a weighted directed graph with vertices indexed by elements in the finite Weyl group, encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. In addition, using some geometry associated to the poset of Newton polygons, one obtains independent proofs for several combinatorial statements about paths in the quantum Bruhat graph and its symmetries, which were originally proved by Postnikov using the tilted Bruhat order. An important geometric application
of this work is an inequality which provides a necessary condition for non-emptiness of certain affine Deligne-Lusztig varieties in the affine flag variety. quantum Bruhat graph; quantum Schubert calculus; affine Weyl group; Newton polygon; affine Deligne-Lusztig variety; Mazur's inequality Symmetric functions and generalizations, Combinatorial aspects of groups and algebras, Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Maximal Newton polygons via the quantum Bruhat graph | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review is a summary of Ohmoto's minicourse delivered at the School on Real and Complex Singularities in Sao Carlos 2012. It contains important new results on Thom polynomial theory and its applications.
For a map \(f\) between complex algebraic or analytic varieties and a singularity type \(\eta\) one considers the \(\eta\)-singularity subset \(\eta(f)\) to be the collection of points of the domain where the map \(f\) has singularity \(\eta\). The classical starting point of global singularity theory is the result claiming that the cohomological fundamental class of \(\overline{\eta(f)}\) can be expressed as a multivariate polynomial depending only on the singularity, if one plugs in the Chern classes of the source and target manifolds. The polynomial is called the Thom polynomial \(tp(\eta)\) of the singularity \(\eta\).
The main addition of the present paper to global singularity theory is the extension of this result from the \textit{fundamental class} represented by \(\overline{\eta(f)}\) to the \textit{Chern-Schwartz-MacPherson (CSM) class} of \(\eta(f)\). The CSM class of a variety is an inhomogeneous deformation of its fundamental class, it is an additive invariant, it is consistent with push-forward maps, and equals the total Chern class of the tangent bundle if the variety is smooth. A version of the CSM class, called Segre-Schwartz-MacPherson (SSM) class is consistent with pullback. In Section 3 Ohmoto reviews the CSM and equivariant CSM theory (invented by himself), and in Section 4 he lays dows the foundations of the theory of SSM Thom polynomials (\(tp^{SM}(\eta)\)). A calculating method (``restriction method'') is used to calculate some terms, and an excursion to multisingularities is included.
The author's first application is the proof of several universal weighted Euler characteristics formulas for singularity loci, some of them reprove and generalize such formulas of the 19th century geometers. These results now follow not by case-by-case arguments but from the general framework of SSM-Thom polynomials.
The last two sections are devoted to another spectacular application: the conceptual treatment of the vanishing topology of finitely determined weighted homogeneous map germs. In particular, the multiplicities of stable map germs within degenerate ones are calculated, as well as the image and discriminant Milnor numbers. These are important singularity theory notions which had been studied before without Thom polynomials. Ohmoto's calculations follow from the universal formulas of SSM-Thom polynomials.
The paper is a very important contribution to global singularity theory. Thom polynomial; Chern-Schwartz-MacPherson class; Segre class; vanishing cohomology; Milnor number ; Ohmoto, School on real and complex singularities. Adv. Stud. Pure Math., 68, 191, (2016) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Singularities of maps and characteristic classes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(f \in \mathbb C[x_{1},\dots, x_{m}]\) and \(g \in \mathbb C[x_{m+1}, \dots, x_{m+n}]\) be two weighted homogeneous polynomials that are neither constant nor power of some other polynomial; let \(p,q\) be two positive integers. In this short paper, the author shows that, under these hypotheses, the fundamental group of the complement of the affine hypersurface \(\{f^{p}+g^{q}=0\}\) in \(\mathbb C^{m+n}\) does not depend from \(f\) and \(g\), but only from \(p\) and \(q\). In particular it equals the well-known fundamental group of the complement of the affine plane curve \(\{x^{p}+y^{q}=0\}\).
The author expresses explicitly the fundamental group of the (affine) zero locus of a weighted homogeneous polynomial \(h(x_{1}, \dots, x_{l})\) as semidirect product of the fundamental group of the affine hypersurface \(\{h=1\}\) and \(\mathbb Z\), using the \(\mathbb C^{*}\) action on the polynomial induced by the weights.
Then he shows that in his hypotheses the locus \(\{f^{p}+g^{q}=1\}\) is homotopy equivalent to the join of \(p\) copies of \(\{f=1\}\) and q copies of \(\{g=1\}\) whose fundamental group depends only from \(p\) and \(q\), and finally he checks that also the automorphism defining the semidirect product depends only from \(p\) and \(q\). weighted homogeneous polynomials; fundamental group; automorphism Homotopy theory and fundamental groups in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects), Coverings of curves, fundamental group On the fundamental group of the complement of certain affine hypersurfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Based on the formula for multiplying Schubert classes obtained by the first author [Invent. Math. 159, No. 2, 407--436 (2005; Zbl 1077.14067)], we develop an algorithm computing the product of two arbitrary Schubert classes in a flag manifold \(G/H\), where \(G\) is a compact connected Lie group and \(H \subset G\) is the centralizer of a one-parameter subgroup in \(G\).
Since all Schubert classes on \(G/H\) constitute a basis for the integral cohomology \(H^* (G/H)\), the algorithm gives also a method to compute the integral cohomology ring \(H^*(G/H)\) independent of the classical spectral sequence method of Leray and Borel. flag manifolds; Schubert varieties; cohomology; Cartan matrix H. Duan, X. Zhao, Algorithm for multiplying Schubert classes, Int. J. Algebra Comput. 16, 1197--1210 (2006). Classical problems, Schubert calculus, Complete intersections, Intersection homology and cohomology in algebraic topology, Lie algebras of Lie groups Algorithm for multiplying Schubert classes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main goal of this review is to compare different approaches to constructing the geometry associated with a Hecke type braiding (in particular, with that related to the quantum group \(U_{q}(sl(n)))\). We place emphasis on the affine braided geometry related to the so-called reflection equation algebra (REA). All objects of such a type of geometry are defined in the spirit of affine algebraic geometry via polynomial relations on generators. We begin by comparing the Poisson counterparts of `quantum varieties' and describe different approaches to their quantization. Also, we exhibit two approaches to introducing \(q\)-analogs of vector bundles and defining the Chern-Connes index for them on quantum spheres. In accordance with the Serre-Swan approach, the \(q\)-vector bundles are treated as finitely generated projective modules over the corresponding quantum algebras. Besides, we describe the basic properties of the REA used in this construction and compare different ways of defining \(q\)-analogs of partial derivatives and differentials on the REA and algebras close to them. In particular, we present a way of introducing a \(q\)-differential calculus via Koszul type complexes. The elements of the \(q\)-calculus are applied to defining \(q\)-analogs of some relativistic wave operators. D. Gurevich and P. Saponov, Braided affine geometry and q-analogs of wave operators, J. Phys. A: Math., Theor., 42 (2009), 313001, 51pp.Zbl 1202.81108 MR 2521298 Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Research exposition (monographs, survey articles) pertaining to quantum theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Geometry and quantization, symplectic methods Braided affine geometry and \(q\)-analogs of wave operators | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In his fundamental paper [``Techniques de construction et théorèmes d'existence en géométrie algébrique. IV: Les schemes de Hilbert'', Sémin. Bourbaki, exp. 221 (1961; Zbl 0236.14003)], \textit{A. Grothendieck} introduced the so called Hilbert scheme, which is an object which parametrizes all projective subschemes of the projective space with fixed Hilbert polynomial.
In the paper under review, the authors present a more general object, called the multigraded Hilbert scheme, parametrizing all homogeneous ideals with fixed Hilbert function in a graded polynomial ring \(S\).
As in the case of Hilbert schemes, the multigraded Hilbert scheme is a projective scheme (quasi-projective if the grading of \(S\) is not positive), and, when the ground ring is a field, its tangent space at a point corresponding to an ideal \(I\) has a simple description: it is canonically isomorphic to the degree \(0\) piece of \(\Hom(I,S/I)\).
The construction of the multigraded Hilbert scheme is obtained in a great generality, and it enables the authors to prove a conjecture from \textit{D. Bayer}'s thesis [The division algorithm and the Hilbert scheme, Ph.D. thesis, Harvard University (1982)] on equations defining the Hilbert scheme, and to construct a natural morphism from the toric Hilbert scheme to the toric Chow variety, resolving Problem 6.4 appearing in the paper of \textit{B. Sturmfels} [The geometry of A-graded algebras, preprint, \texttt{http://arXiv.org/abs/math.AG/9410032}]. graded polynomial ring; Hilbert function; Chow morphism M. Haiman - B. Sturmfels, Multigraded Hilbert schemes. J. Algebraic Geom., 13 (4) (2004, pp. 725-769. Zbl1072.14007 MR2073194 Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Grassmannians, Schubert varieties, flag manifolds Multigraded Hilbert schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors introduce the notion of Zwiebach invariants that generalize Gromov-Witten invariants and homotopical algebra structures. One of the main results in the paper is a new interpretation of the Barannikov-Kontsevich construction for the WDVV equations. The authors gives a graph theoretical description (via trivalent trees) of the solution of Barannikov and Kontsevich. Among the other applications, constructions in this work give a solution to Getzler's elliptic equation. To show this, a new axiom, 1/12 axiom, is introduced. Zwiebach invariants; Getzler relation; 1/12 axiom Losev A., Shadrin S.: From Zwiebach invariants to Getzler relation. Commun. Math. Phys. 271(3), 649--679 (2007) Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) From Zwiebach invariants to Getzler relation | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Hilbert scheme and the Quot scheme, both constructed by Grothendieck, are fundamental objects in algebraic geometry: the first one parametrizes the subschemes of a projective space with fixed Hilbert polynomial, the second one the quotients with fixed Hilbert polynomial of a fixed coherent sheaf on a projective space.
Here the author considers an affine variety \(X\), with an action of a reductive group \(G\), and a coherent sheaf \(\mathcal M\) on \(X\), that is supposed to be \(G\)-linearized. First of all he proves the existence of a quasi-projective scheme, the invariant Quot scheme, parametrizing the quotients \(\mathcal L\) of \(\mathcal M\), such that the space of global sections \(H^0(X,\mathcal L)\) is a direct sum of simple \(G\)-modules with fixed finite multiplicities: the datum of these multiplicities is here the analogous of the Hilbert polynomial. The invariant Quot scheme is a natural generalization of the invariant Hilbert scheme recently introduced by \textit{V. Alexeev} and \textit{M. Brion} [J. Algebr. Geom. 14, 83--117 (2005; Zbl 1081.14005)]. The construction relies on the multigraded Quot scheme of Haiman and Sturmfels, corresponding to the case in which \(G\) is a torus [cf. \textit{M. Haiman} and \textit{B. Sturmfels}, J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)].
In the second part of the article, the author focuses on a special example, the cone \(X\) of primitive vectors of a simple \(G\)-module, with a free sheaf \(\mathcal M\) generated by another simple \(G\)-module. He proves that, in this case, the invariant Quot scheme has only one point, and that it is reduced unless \(X\) is the cone of the primitive vectors of a quadratic vector space \(V\) of odd dimension \(2n+1\) and \(G= \text{Spin}(2n+1)\times H\), for a connected reductive group \(H\). The Quot scheme of this example is isomorphic to \(\text{Spec}({\mathbb C}[t]/\langle t^2\rangle)\). Hilbert scheme; Quot scheme; reductive group; primitive vector Jansou, S., Le schéma quot invariant, J. Algebra, 306, 2, 461-493, (2006) Parametrization (Chow and Hilbert schemes), Geometric invariant theory, Group actions on varieties or schemes (quotients) The invariant Quot scheme | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials By \textit{P. Deligne} [Publ. Math., Inst. Hautes Étud. Sci. 44(1974), 5-77 (1975; Zbl 0237.14003)] and \textit{U. Jannsen} [Beilinson's conjectures on special values of \(L\)-functions, Meet. Oberwolfach 1986, Perspect. Math. 4, 305-372 (1988; Zbl 0701.14019)] it is known that for each complex algebraic variety \(Y\), the cohomology (also with compact support) \(H^k(Y, \mathbb{Q})\) carries a natural mixed Hodge structure, which coincides with the classical pure Hodge structure in the case of \(Y\) is smooth projective. Then one can define the virtual Hodge polynomial of \(Y\), \(e(Y) =e(Y : x,y)\), to be the polynomial \(\Sigma_{pq} e^{p,q} (Y)x^py^q\), where \(e^{p,q} (Y)= \Sigma_k (-1)^k h^{p,q} (H^k (Y,\mathbb{Q}))\) is the \((p,q)\)-virtual Hodge number of \(Y\). Notice that the virtual Hodge polynomial gives the Euler characteristic \(\chi (Y)\) of \(Y\) since \(\chi (Y)= e(Y : 1,1)\) and, when \(Y\) is smooth projective, \(e(Y)\) gives the Poincaré polynomial \(P(Y:z)\) of \(Y\) since \(P(Y:z)= e(Y:-z,-z)\). -- On the other hand, for any smooth complex variety \(X\), \textit{W. Fulton} and \textit{R. MacPherson} [Ann. Math., II. Ser. 139, No. 1, 183-225 (1994; Zbl 0820.14037)] constructed a smooth compactification \(X[n]\) of the configuration space of \(n\) distinct labeled points of \(X\), which is projective if \(X\) is.
In the present paper the author obtains a generating function which expresses the virtual Hodge polynomials of the spaces \(X[n]\) in terms of the virtual Hodge polynomial of \(X\). This enables the author to read off the Euler characteristic of \(X[n]\) in terms of that of \(X\) and, in the projective case, the Hodge polynomial and the Poincaré polynomial of \(X[n]\) in terms of those of \(X\). virtual Hodge polynomial; Euler characteristic; Poincaré polynomial; configuration space Cheah J.: The Hodge polynomial of the Fulton--MacPherson compactification of configuration spaces. Amer. J. Math. 118(5), 963--977 (1996) Transcendental methods, Hodge theory (algebro-geometric aspects), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Topological properties in algebraic geometry, Embeddings in algebraic geometry, Compactification of analytic spaces The Hodge polynomial of the Fulton-MacPherson compactification of configuration spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Grassmannian \(\mathrm{Gr}(d,n)\) parametrizing the \(d\)-planes inside \(\Bbbk^n\) can be seen as a subvariety of some projective space via the Plücker embedding. Its intersection \(\mathrm{Gr}_0(d,n)\) with the main strata of the projective space consists of \(d\)-planes with non-zero Plücker coordinates. The tropicalization \(T\mathrm{Gr}_0(d,n)\) of \(\mathrm{Gr}_0(d,n)\) is a fan inside \(\Lambda^d\mathbb{R}^n\) that can be seen from several points of view:
\begin{itemize}
\item First, it is the image by the valuation of the \(\mathbb{K}\)-points of \(G_0(d,n)\) for \(\mathbb{K}\) some algebraically closed valued field with surjective valuation,
\item Then, it is the moduli space of linear tropical space that are realizable over valued extensions of \(\Bbbk\),
\item Finally, it consists of the vectors \(w\in\Lambda^d\mathbb{R}^n\) such that the initial degeneration \(\mathrm{in}_w \mathrm{Gr}_0(d,n)\) is non-empty. The goal of the paper is to study the properties of some of the initial degenerations of the Grassmannian.
\end{itemize}
The cones of \(T \mathrm{Gr}(d,n)\) are in bijection with some subdivisions \(\Delta_w\) of the hypersimplex \(\Delta(d,n)\) into matroid polytopes. Here, \(w\) is some point that belongs to the relative interior of the considered cone. As the ideals giving the initial degenerations \(\mathrm{in}\mathrm{Gr}_0(d,n)\) can be quite difficult to handle, even with a computer, the paper studies them using a closed embedding inside some varieties constructed from \(\Delta_w\) which are easier to manipulate. The description of this variety is as follows.
To each \(d\)-plane \(L\) inside \(\Bbbk^n\) is associated a matroid obtained via the hyperplane arrangement that the coordinate hyperplanes of \(\Bbbk^n\) realize inside the subspace \(L\). Such a matroid is called realizable. Conversely, to each realizable matroid \(M\), we can consider the \textit{thin Schubert cell} \(\mathrm{Gr}_M\subset\mathrm{Gr}(d,n)\) consisting of subspaces inducing the matroid \(M\).
Consider the subdivision \(\Delta_w\) of \(\Delta(d,n)\) into matroid polytopes. Let \(Q\) be one of the matroid polytopes and let \(M_Q\) be the corresponding matroid. A face \(Q'\) of \(Q\) corresponds to a matroid \(M_{Q'}\) which is the direct sum of a quotient and a contraction of \(M_Q\). Using this description, the paper gives a way to lift the inclusion \(Q'\subset Q\) to an application on the level of thin Schubert cells corresponding to \(M_Q\) and \(M_{Q'}\): \(\mathrm{Gr}_{M_Q}\rightarrow \mathrm{Gr}_{M_{Q'}}\). Therefore, it is possible to consider the inverse limit \[\varprojlim_{Q\in\Delta_w}\mathrm{Gr}_{M_Q},\] associated to the subdivision obtained from a cone of \(T\mathrm{Gr}_0(d,n)\).
For an element \(w\) in the fan \(T \mathrm{Gr}_0(d,n)\), the paper then constructs a closed immersion \[\mathrm{in}_w\mathrm{Gr}_0(d,n)\rightarrow\varprojlim_{Q\in\Delta_w}\mathrm{Gr}_{M_Q}.\] It then uses it to deduce results such as the smoothness and irreducibleness of the initial degenerations of \(\mathrm{Gr}_0(3,7)\), the fact that \(\mathrm{Gr}_0(3,9)\) has a non-connected initial degeneration.
They also use the setting to prove the \(n=7\) case of the following conjecture of Keel and Tevelev [26]. Let \(X(3,n)\) be the normalization of the quotient of \(\mathrm{Gr}(3,n)\) by a maximal torus \(H\subset \mathrm{PGL}(d)\), and \(X_0(3,n)\) the same quotient with \(\mathrm{Gr}_0(3,n)\) instead. As it is already known that \(X(3,n)\) is not log-canonical for \(n\geqslant 9\), the conjecture states that \(X(3,n)\) is a schön and log canonical compactification for \(X_0(3,n)\) when \(n=6,7\) and \(8\). The conjecture was already proven for \(n=6\) by
\textit{M. Luxton} [The log canonical compactification of the moduli space of six lines in \({{\mathbb{P}}}^2\). Texas: University of Texas at Austin (PhD Thesis) (2008)]. Grassmannian; initial degeneration; tropical; matroid Combinatorial aspects of tropical varieties, Grassmannians, Schubert varieties, flag manifolds, Embeddings in algebraic geometry Initial degenerations of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the global approach to Conformal Field Theory on higher genus Riemann surfaces initiated by Krichever and Novikov one replaces the Virasoro algebra and the related algebras by the corresponding algebras which are given by meromorphic objects with poles only at two points (see the review Zbl 0820.17036 for more background information). Starting from any simple Lie algebra \({\mathfrak g}\) (with associated group \(G)\) the generalized affine Lie algebra \(\widehat {\mathfrak g}\) was introduced by the author [Funct. Anal. Appl. 27, 266-272 (1993; Zbl 0820.17036)].
In the article under review the author defines and studies highest weight representations of these affine type Lie algebras. To study them he introduces the notion of Bloch weights which are analogs of the Bloch spectrum in the theory of completely integrable systems. An analog of the Weyl-Kac character formula is proven. Kac-Moody algebras; Krichever-Novikov algebras; affine algebras; almost graded algebras; central extensions; highest weight representations; Bloch weights; Weyl-Kac character formula O.K. Sheinman, ''Highest-Weight Modules for Affine Lie Algebras on Riemann Surfaces,'' Funct. Anal. Appl. 29(1), 44--55 (1995). Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Loop groups and related constructions, group-theoretic treatment, Differentials on Riemann surfaces, Riemann surfaces; Weierstrass points; gap sequences Modules with highest weight for affine Lie algebras on Riemann surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In [Contemp. Math. 186, 111-171 (1995; Zbl 0957.11047)] the first author introduced modular towers. This paper concentrates on exactly one application of these to the inverse Galois problem.
Let \(G\) be a finite (possibly simple) group, and let \(p\) be a prime dividing the order of \(G\). The characteristic finite quotients \({}_p^k \widetilde{G}\) of the universal \(p\)-Frattini cover of \(G\) are strikingly similar groups. It takes an effort to distinguish them for finding \(\mathbb{Q}\) regular realizations for the inverse Galois problem. This paper starts a program to show one can't realize all these groups as Galois groups of extensions \(L/\mathbb{Q}(x)\) with at most \(r\) (fixed) branch points. Let \({\mathcal C}\) be an \(r\)-tuple of \(p\)-regular conjugacy classes of \(G\). To compare realizations of these groups we use a sequence of varieties -- a modular tower -- attached to \((G,p,{\mathcal C})\). The notation for this sequence is \({\mathcal H}({}_p^k\widetilde{G}, {\mathcal C})\), \(k=0,1,\dots:{\mathcal H}({}_p^k\widetilde{G}, {\mathcal C})\) is the \(k\) level of the modular tower. Crucial properties of level \(k\) translate the properties of the characteristic modular representation of \({}_p^k \widetilde{G}\). Properties of these representations support the following statement.
Conjecture. For each \(r\) there exists \(k_r\) so that for \(k> k_r\), \(\mathbb{Q}\) regular realization of \({}_p^k\widetilde{G}\) requires more than \(r\) branch points.
For \(r=4\) this reduces to showing two pieces of geometric information.
(a) There is a uniform (with \(k\)) bound on the number of absolutely irreducible components at the \(k\)th level.
(b) For \(k\) large \({\mathcal H}({}_p^k\widetilde{G},{\mathcal C})\) has no obstructed components.
The main example of this paper and \textit{M. Fried} and \textit{Y. Kopeliovich} [\(A_5\) modular towers, 30 page preprint (1996)] is \(G= A_5\), \(p=2\) and \({\mathcal C}= {\mathcal C}_{3^r}\) with \(r=4\) repetitions of the conjugacy class of 3-cycles. It allows full explanation and illustration of the significance of obstructed components. modular towers; inverse Galois problem Fried, M.D., Kopeliovich, Y.: Applying modular towers to the inverse Galois problem. In: Geometric Galois Actions, 2. London Math. Soc. Lecture Note Ser., vol. 243, pp. 151--175. Cambridge University Press, Cambridge (1997) Inverse Galois theory, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Arithmetic theory of algebraic function fields Applying modular towers to the inverse Galois problem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(H=K[X_1,\dots,X_n]/(X_1^{p^s_1},\dots,X_n^{p^s_n})\) (\(n\geq 1\), \(s_1\geq\cdots\geq s_n\geq 1\)) be the truncated polynomial algebra in \(n\) variables over a field \(K\) of characteristic \(p>0\). In the paper under review the authors study the coactions of the Hopf algebra \(H_m\) of the multiplicative group with underlying algebra \(H\) on a commutative \(K\)-algebra \(A\). Without further assumptions the multiplication rules of the additive endomorphisms of the coaction are determined. For the rest of the paper the authors consider only \(s_1=\cdots=s_n\geq 1\). In this case an explicit expression of any coaction on \(A\) is given in terms of \(n\) derivations of \(A\).
The paper concludes with some applications of this description and results of the second author and \textit{H.-J. Schneider} [J.\ Algebra 261, No.\ 2, 229-244 (2003; Zbl 1019.16026)] to the local case. The authors give necessary and sufficient conditions for a local algebra being a faithfully flat \(H_m\)-Galois extension over the subalgebra of its \(H_m\)-coinvariants. Finally, in the case that \(A\) is local of dimension \(d>0\) satisfying certain technical conditions, it is shown that the existence of a finite \(p\)-basis of \(A\) implies the same for the subalgebra of \(H_m\)-coinvariants in \(A\). truncated polynomial algebras; commutative Hopf algebras; coactions; right comodule algebras; derivations; algebras of coinvariants; local algebras; faithfully flat Hopf-Galois extensions; \(p\)-bases Hopf algebras and their applications, Actions of groups and semigroups; invariant theory (associative rings and algebras), Derivations and commutative rings, Group schemes Coactions of Hopf algebras on algebras in positive characteristic. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Double Hurwitz numbers enumerating weighted \(n\)-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of \(S_n\) generated by transpositions are determined by an associated weight generating function. A uniquely determined 1-parameter family of 2D Toda \(\tau\)-functions of hypergeometric type is shown to consist of generating functions for such weighted Hurwitz numbers. Four classical cases are detailed, in which the weighting is uniform: Okounkov's double Hurwitz numbers for which the ramification is simple at all but two specified branch points; the case of Belyi curves, with three branch points, two with specified profiles; the general case, with a specified number of branch points, two with fixed profiles, the rest constrained only by the genus; and the signed enumeration case, with sign determined by the parity of the number of branch points. Using the exponentiated quantum dilogarithm function as a weight generator, three new types of weighted enumerations are introduced. These determine quantum Hurwitz numbers depending on a deformation parameter \(q\). By suitable interpretation of \(q\), the statistical mechanics of quantum weighted branched covers may be related to that of Bosonic gases. The standard double Hurwitz numbers are recovered in the classical limit. double Hurwitz numbers Guay-Paquet, M.; Harnad, J., Generating functions for weighted Hurwitz numbers, J. Math. Phys., 58, 083503, (2017) Exact enumeration problems, generating functions, Relationships between algebraic curves and physics, Coverings of curves, fundamental group, Time-dependent Schrödinger equations and Dirac equations, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics Generating functions for weighted Hurwitz numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We compute the Poincaré-Hodge polynomial of a symmetric product of a compact Kähler manifold, following the method used by \textit{I. G. Macdonald} [Proc. Camb. Philos. Soc. 58, 563--568 (1962; Zbl 0121.39601)] in the topological case, to compute the Poincaré polynomial of a compact polyhedron, and we give some applications, in particular to the case of curves. product of curves; Poincaré-Hodge polynomial; symmetric product of a compact Kähler manifold J. Burillo, ''The Poincaré-Hodge Polynomial of a Symmetric Product of Compact Kähler Manifolds,'' Collect. Math. 41, 59--69 (1990). Transcendental methods, Hodge theory (algebro-geometric aspects), Curves in algebraic geometry, Global differential geometry of Hermitian and Kählerian manifolds, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The Poincaré-Hodge polynomial of a symmetric product of compact Kähler manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe a direct connection between the representation theory of the general linear group and classical Schubert calculus on the Grassmannian, which goes via the Chern-Weil theory of characteristic classes. We also explain why the analogous constructions do not give the same result for other Lie groups. representations of general linear groups; Schubert calculus on the Grassmannian; polynomial representation ring; cohomology ring Tamvakis, H, The connection between representation theory and Schubert calculus, Enseign. Math., 50, 267-286, (2004) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Semisimple Lie groups and their representations, Connections (general theory) The connection between representation theory and Schubert calculcus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights \(\lambda\) and \(\mu\) for a Lie algebra \(\mathfrak g\), which we will assume is simply laced. In this paper, we relate the category \(\mathcal O\) over truncated shifted Yangians to categorified tensor products: For a generic integral choice of parameters, category \(\mathcal O\) is equivalent to a weight space in the categorification of a tensor product of fundamental representations defined by the third author using KLRW algebras. We also give a precise description of category \(\mathcal O\) for arbitrary parameters using a new algebra which we call the parity KLRW algebra. In particular, we confirm the conjecture of the authors that the highest weights of category \(\mathcal O\) are in canonical bijection with a product monomial crystal depending on the choice of parameters.
This work also has interesting applications to classical representation theory. In particular, it allows us to give a classification of simple Gelfand-Tsetlin modules of \(U(\mathfrak{gl}_n )\) and its associated W-algebras. Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized function algebras) and their representations, Quantum groups (quantized enveloping algebras) and related deformations On category \(\mathcal O\) for affine Grassmannian slices and categorified tensor products | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Each cohomology ring of a Grassmannian or flag variety has a basis of Schubert classes indexed by the elements of the corresponding Weyl group. Classical Schubert calculus computes the cohomology rings of Grassmannians and flag varieties in terms of the Schubert classes. This paper is ``doing Schubert calculus'' in the equivariant cohomology rings of Peterson varieties.
The Peterson variety is a subvariety of the flag variety \(G/B\) parameterized by a linear subspace \(H_{\mathrm{Pet}} \subseteq \mathfrak g\) and a regular nilpotent operator \(N_0 \in \mathfrak g\). We can define the Peterson variety as
\[
\mathrm{Pet}=\{gB\in G\backslash B:\mathrm{Ad}(g^{-1})N_0 \in H_{\mathrm{Pet}}\}.
\]
Peterson varieties were introduced by Peterson in the 1990s. Peterson constructed the small quantum cohomology of partial flag varieties from what are now Peterson varieties. \textit{B. Kostant} [Sel. Math., New Ser. 2, No. 1, 43--91 (1996; Zbl 0868.14024)] used Peterson varieties to describe the quantum cohomology of the flag manifold and \textit{K. Rietsch} [Nagoya Math. J. 183, 105--142 (2006; Zbl 1111.14048)] gave the totally non-negative part of type A Peterson varieties. \textit{E. Insko} and \textit{A. Yong} [Transform. Groups 17, No. 4, 1011--1036 (2012; Zbl 1267.14066)] explicitly identified the singular locus of type A Peterson varieties and intersected them with Schubert varieties. \textit{M. Harada} and \textit{J. Tymoczko} [Proc. Lond. Math. Soc. (3) 103, No. 1, 40--72 (2011; Zbl 1219.14065)] proved that there is a circle action \(\mathbb S^1\) which preserves Peterson varieties. In this paper the authors study the equivariant cohomology of the Peterson variety with respect to this action and also they use GKM theory as a model for studying equivariant cohomology, but Peterson varieties are not GKM spaces under the action of \(\mathbb S^1\). Using work by Harada and Tymoczko [Zbl 1219.14065] and \textit{M. Precup} [Sel. Math., New Ser. 19, No. 4, 903--922 (2013; Zbl 1292.14032)], they construct a basis for the \(\mathbb S^1\)-equivariant cohomology of Peterson varieties in all Lie types. This construction gives a set of classes which we call Peterson Schubert classes. The name indicates that the classes are projections of Schubert classes, they do not satisfy all the classical properties of Schubert classes.
Classical Schubert calculus asks how to multiply Schubert classes; here, the authors asks how to multiply in the basis of Peterson Schubert classes. She gives a Monk's formula for multiplying a ring generator and a module generator, and a Giambelli's formula for expressing any Peterson Schubert class in the basis in terms of the ring generators. Peterson variety; equivariant cohomology; Monk's rule; Giambelli's formula; Schubert calculus Drellich, Elizabeth, Monk's rule and Giambelli's formula for Peterson varieties of all Lie types, J. Algebraic Combin., 41, 2, 539-575, (2015) Classical problems, Schubert calculus, Equivariant homology and cohomology in algebraic topology, Grassmannians, Schubert varieties, flag manifolds Monk's rule and Giambelli's formula for Peterson varieties of all Lie types | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors' goal is to compute the Chern-Schwartz-MacPherson and Segre-Schwartz-MacPherson classes of the orbits of a certain representation called the matrix Schubert cells. More precisely, let \(k\le n\) be nonnegative integers. Let us consider the group \(GL_k(\mathbb C)\times B_n^-\) acting on \(\text{Hom}(\mathbb C^k; \mathbb C^n)\) by \((A,B)\cdot M = BMA^{-1}\), where \(B_n^-\) denotes the Borel subgroup of \(n\times n\) lower triangular matrices. The finitely many orbits of this action are parametrized by \(d\)-element subsets \(J = \{j_1 < \dots < j_d\} \subset \{1, \dots, n\}\) where \(d \le k\). The corresponding orbits, denoted by \(\Omega_J\), are called matrix Schubert cells and their closures are usually called matrix Schubert varieties (see [\textit{L. M. Fehér} and \textit{R. Rimányi}, Cent. Eur. J. Math. 1, No. 4, 418--434 (2003; Zbl 1038.57008)]). Let us fix \(I\subset \{1,\dots,k\}\). The authors prove that the equivariant Chern-Schwartz-MacPherson class of \(\Omega_I\) is equal to the value of the weight function \(W_I(\alpha, \beta)\), considered in [\textit{V. Tarasov} and \textit{A. Varchenko}, Invent. Math. 128, No. 3, 501--588 (1997; Zbl 0877.33013)] or [\textit{R. Rimányi} and \textit{A. Varchenko}, Impanga 15. EMS Series of Congress Reports 225--235 (2018; Zbl 1391.14108)], where \(\alpha = (\alpha_1, \dots, \alpha_k)\) and \(\beta = (\beta_1, \dots, \beta_n)\) are suitable partitions. In turn, the weight function \(W_I(\alpha, \beta)\) can be expressed in terms of appropriated symmetric functions, which can be computed as values of the ``iterated residues'' of some generating functions parameterized by partitions, and so on. In a similar same way it is possible to compute the Segre-Schwartz-MacPherson classes. As an application, the authors perform the corresponding calculations and write out the exact formulas for these classes in the case of \(A_2\) quiver representation. characteristic classes; equivariant cohomology; Borel subgroup; symmetric functions; fundamental class; degeneracy loci; weight functions; Schubert cells; Schur expansion; iterated residues Grassmannians, Schubert varieties, flag manifolds, Global theory of complex singularities; cohomological properties, Global theory and resolution of singularities (algebro-geometric aspects), Classical problems, Schubert calculus, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Characteristic classes and numbers in differential topology Chern-Schwartz-MacPherson classes of degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(A_n\) be the \(n\)-th Weyl algebra, i.e., the associative algebra generated over \(\mathbb C\) by the commuting variables \(x_1,\dots,x_n\) and the \(n\) partial derivatives \(\partial_i=\partial/\partial x_i\), with the natural commutator relations between them. The classical Dixmier conjecture \(\text{D}_n\) (open for all \(n>0\)) states that \(\Aut(A_n)=\text{End}(A_n)\), i.e., every algebra endomorphism of \(A_n\) is an automorphism. It is known that \(\text{D}_n\) implies the \(n\)-th Jacobian conjecture \(\text{J}_n\) (for the endomorphisms of \(\mathbb C[x_1,\dots,x_n]\) with invertible Jacobian matrix and open for all \(n>1\)). It is also known that \(\text{J}_{2n}\) implies \(\text{D}_n\). Let \(P_n=\mathbb C[r_1,\dots,r_n,s_1,\dots,s_n]\) be the polynomial algebra in \(2n\) variables with its canonical Poisson bracket \(\{\,,\,\}\), where \(\{r_i,s_j\}=\delta_{ij}\), \(\{r_i,r_j\}=\{s_i,s_j\}=0\). Conjecturally \(\Aut(P_n)=\text{End}(P_n)\) (where the endomorphisms preserve both the multiplication and the Poisson bracket in \(P_n\)) and this is equivalent to \(\text{J}_n\). \textit{A. Belov-Kanel} and \textit{M. Kontsevich} [Lett. Math. Phys. 74, No. 2, 181-199 (2005; Zbl 1081.16031)] stated the \(\text{BK}_n\)-conjecture that the groups \(\Aut(A_n)\) and \(\Aut(P_n)\) are canonically isomorphic. This is known for \(n=1\) only.
In the proof of the equivalence between the Dixmier and Jacobian conjectures and the approach of Belov-Kanel and Kontsevich to their own conjecture an essential moment is that the Weyl algebra in positive characteristic is Azumaya and the conditions which guarantee that an endomorphism of an Azumaya algebra is an automorphism. In the paper under review the author proposes that instead of using Weyl algebras in positive characteristic to use quantized Weyl algebras defined over \(\mathbb C\) at roots of unity. The author discusses the disadvantages and the advantages of the root of unity method to the \(\text{BK}_n\)-conjecture versus the reduction modulo prime method. The main advantage is that one never leaves the base field \(\mathbb C\). As an illustration the case \(n=1\) is considered in detail. quantized Weyl algebras; Azumaya algebras; Dixmier conjecture; Jacobian conjecture; Belov-Kanel-Kontsevich conjecture; algebra endomorphisms; algebra automorphisms Backelin, E., Endomorphisms of quantized Weyl algebras, Lett. Math. Phys., 97, 3, 317-338, (2011) Automorphisms and endomorphisms, Rings of differential operators (associative algebraic aspects), Jacobian problem, Derivations, actions of Lie algebras, Ring-theoretic aspects of quantum groups, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations Endomorphisms of quantized Weyl algebras. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a semisimple complex algebraic group with a Borel subgroup \(B\) and the associated Weyl group \(W\). Then the cohomology with integral coefficients \(H^*(G/B)\) of the flag variety \(G/B\) has the Schubert basis \(\{\varepsilon^w\}_{w\in W}\). Now, as it is well known, the cup product in this basis has nonnegative coefficients. The aim of this note is to extend this `nonnegativity' result to the flag variety of an arbitrary (not necessarily even symmetrizable) Kac-Moody group \(G\). To our knowledge, this nonnegativity result was not known for any Kac-Moody group beyond the (finite-dimensional) semisimple group. The main difficulty in extending the proof from the finite-dimensional case to an arbitrary Kac-Moody case lies in the fact that (unlike the finite-dimensional case) there is no algebraic group which acts transitively on the flag variety of an infinite-dimensional Kac-Moody group \(G\). cohomology with integral coefficients; flag variety; Schubert basis; Kac-Moody group Kumar, S.; Nori, M. V.: Positivity of the cup product in cohomology of flag varieties associated to Kac-Moody groups. Int. math. Res. not. IMRN, No. 14, 757-763 (1998) Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Infinite-dimensional Lie groups and their Lie algebras: general properties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Homogeneous spaces and generalizations Positivity of the cup product in cohomology of flag varieties associated to Kac-Moody groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors define Schubert cells and Schubert varieties for the classical ind-groups \(GL(\infty)\), \(O(\infty)\) and \(Sp(\infty)\).
Flag varieties and Schubert varieties are very important geometric object attached to a given reductive algebraic group. For the ind-groups \(G=GL(\infty)\), \(O(\infty)\) and \(Sp(\infty)\) there are infinitely many conjugacy classes of splitting Borel subgroups \(B\), and hence there are infinitely many flag ind-varieties \(G/B\).
In [\textit{I. Dimitrov} and \textit{I. Penkov}, Int. Math. Res. Not. 2004, No. 55, 2935--2953 (2004; Zbl 1069.22009)], these smooth ind-varieties has been described explicitly as the ind-variety of certain generalized flags in the natural representation \(V\) of \(G\). These varieties has been studied also in [\textit{I. Dimitrov} and \textit{I. Penkov}, Int. Math. Res. Not. 1999, No. 5, 223--249 (1999; Zbl 0917.17002)] and [\textit{I. Dimitrov} et al., Am. J. Math. 124, No. 5, 955--998 (2002; Zbl 1016.22012)].
The authors define Schubert decomposition of a more general ind-variety \(G/P\), where \(P\) is a splitting parabolic subgroup of \(G\). They define the Schubert cells as the \(B\)-orbits for any Borel ind-subgroup \(B\) which contains a common splitting maximal ind-torus with \(P\). The essential difference with the finite-dimensional case is that \(B\) is not necessarily conjugate to a Borel subgroup of \(P\). This leads to the existence of many non-conjugate Schubert decompositions of a given \(G/P\). The authors compute the dimensions of the cells of all Schubert decompositions of \(G/P\) for any splitting Borel subgroup \(B\) of \(G\). Moreover, a Bruhat decomposition of \(G/P\) induces a Bruhat decomposition of the ind-group \(G\) into double cosets.
Finally the authors studies the smoothness of Schubert varieties, namely the closures of Schubert cells. In particular, they establish a criterion for smoothness which allows to conclude that certain known criteria for smoothness of finite-dimensional Schubert varieties pass to the limit at infinity.
Schubert subvarieties of \(G/B\) are defined also in [\textit{H. Salmasian}, J. Algebra 320, No. 8, 3187--3198 (2008; Zbl 1172.14034)]. In such paper they are defined as arbitrary direct limits of Schubert varieties on finite-dimensional flag subvarieties of \(G/B\). With such definition Schubert varieties may be singular at all of its points. Instead with the definition of the present paper, which takes into account the natural action of \(G\) on \(G/B\), all Schubert varieties have a smooth big cell. classical ind-group; Bruhat decomposition; Schubert decomposition; generalized flag; homogeneous ind-variety Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Linear algebraic groups and related topics Schubert decompositions for ind-varieties of generalized flags | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors recall in Theorems 2.1 and 2.2 their results with \textit{S. Gitler} in [Bol. Soc. Mat. Mex. II Ser. 30, No. 1, 1--11 (1985; Zbl 0639.57013)] considering generalized higher dimensional versions of the Riemann-Hurwitz formula. In the first part of the paper they apply those results to rational maps of projective varieties. Then, in the second part, they apply them to the study of determinantal varieties realized as the degenerate loci of morphisms of complex vector bundles over a complex projective variety \(X\). They highlight two examples, the general symmetric bundle maps and the flagged bundles. Riemann-Hurwitz formula; rational maps; iterated maps; degeneracy locus; determinantal variety Determinantal varieties, Low-dimensional topology of special (e.g., branched) coverings, Algebraic topology on manifolds and differential topology, Rational and birational maps, Meromorphic mappings in several complex variables, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Rational and iterated maps, degeneracy loci, and the generalized Riemann-Hurwitz formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, the authors study the relationship between the quantum cohomology of a monotone closed symplectic manifold \(M\) and the symplectic cohomology of the complement of a simple normal crossings (or SC) divisor \(D\), mainly under the assumption that the pair \((M,D)\) is logarithmic effective. They prove that the former is a deformation of the latter in a suitable sense. They also prove some rigidity results about the isotropic skeleton of \(X=M-D\).
A closed symplectic manifold \((M,\omega)\) is called monotone if the cohomology class \([\omega]\) is a multiple of \(c_1(TM)\). In particular, \([\omega]\) is a rational cohomology class, and Donaldson's construction of a (smooth) symplectic divisor Poincare dual to a large multiple of \([\omega]\) applies. In this paper, the authors consider an SC divisor \(D=\bigcup_{i=1}^N D_i\) such that \(c_1(TM)\) is a weighted sum
\[
c_1(TM)=\textbf{D}:=\sum_{i=1}^N a_i D_i
\]
with positive rational weights (Note that \(\lambda_i=2a_i\) in the paper's notation). The last identity and the monotonicity imply that \(X\) is a convex exact symplectic manifold with a vanishing first Chern class. Furthermore, if \([\omega]\) is a positive multiple of \(c_1(TM)\), the positivity of \(D\) implies that \(X\) is a Weinstein domain and retracts to an isotropic skeleton. In Hypothesis A, the authors assume that \(a_i \leq 1\), which means
\[
K_M+D=D-\textbf{D}=\sum_{i=1}^n (1-a_i)D_i
\]
is effective (in the algebraic geometric language). Here, \(K_M\) is the canonical line bundle of \((M,\omega)\) and \(K_M+D\) is the logarithmic canonical line bundle of the pair \((M,D)\). The borderline case is \(a_i\equiv 1\), where \((X,D)\) is a symplectic log Calabi-Yau pair.
The vague conjecture in the literature has been that under certain assumptions on \((M,D)\), the Novikov field-valued quantum cohomology \(QH^*(M,\Lambda)\) of \(M\) coincides the cohomology of a natural deformation \(\big(SC^*(X;\Lambda),\partial\big)\) of the symplectic cochain complex \(\big(SC^*(X;\Lambda),d\big)\). The former is defined using closed pseudoholomorphic curves, and the latter involves periodic orbits of a Hamiltonian function on \(X\) and ceratin pseudoholomorphic-type curves connecting them. In Theorem C, the authors prove this conjecture when \(M\) is monotone and \((M,D)\) is log-effective. Furthermore, they prove that the spectral sequence of the deformed filtered complex converges on the first page to the quantum cohomology. As a corollary (Corollary 1.7), they conclude that \(X\) has non-vanishing symplectic cohomology.
Over the course of the proof, they study compact subsets of \(X\) and their symplectic homology relative to \(M\). For instance, if \((M,D)\) is log-effective, they prove that the isotropic skeleton \(L\) of \(X\) is SH-full, meaning that the symplectic homology of any compact set \(K\) in \(X-L\) is zero. They conclude that \(L\) can not be displaced from any ``Floer-theoretically essential monotone'' Lagrangian in \(X\). symplectic cohomology; quantum cohomology; divisor complement; spectral sequences Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Symplectic aspects of Floer homology and cohomology Quantum cohomology as a deformation of symplectic cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the series `Geometry of \(G/P\)', \textit{V. Lakshmibai}, \textit{C. Musili} and \textit{C. S. Seshadri} [see Bull. Am. Math. Soc., New. Ser. 1, 432-435 (1979; Zbl 0466.14020)] developed a standard monomial theory for semisimple algebraic groups as a generalization of the Hodge-Young standard monomial theory for \(GL(n)\). Standard monomial theory consists in constructing explicit bases for spaces of sections of effective line bundles on the generalized flag variety. Standard monomial theory has led to very many important geometric and representation-theoretic consequences.
In this article, the author gives a different approach to standard monomial theory (which avoids the case by case consideration) through path models of representations and their associated bases. The bases are constructed using the theory of quantum groups at a root of unity. Using these bases, geometric and representation-theoretic consequences -- vanishing theorems for higher cohomology for effetive line bundles on Schubert varieties, a proof of the Demazure character formula, projective normality of Schubert varieties, good filtration property -- are deduced.
It should be added that the path model theory has also led to a Littlewood-Richardson rule for symmetrizable Kac-Moody algebras proved by the author [cf. \textit{P. Littelmann}, Invent. Math. 116, No. 1-3, 329-346 (1994; Zbl 0805.17019)]. flag variety; standard monomial theory; path models of representations; quantum groups; vanishing theorems; Schubert varieties; Demazure character formula; projective normality; good filtration property Peter Littelmann, The path model, the quantum Frobenius map and standard monomial theory, Algebraic groups and their representations (Cambridge, 1997) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 517, Kluwer Acad. Publ., Dordrecht, 1998, pp. 175 -- 212. Homogeneous spaces and generalizations, Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds The path model, the quantum Frobenius map and standard monomial theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Mirror symmetry has made a lot of surprising predictions in algebraic geometry. There are still many wonderful problems to work on. From a mathematical point of view, mirror symmetry is formulated as the correspondence of the A-model and B-model geometry.
This paper first studies the mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces \(\mathbb{P}(w):=\mathbb{P}(w_0,w_1,\dots,w_n)\) (A-model) in the framework of differential equations. Following [\textit{H. Iritani}, Adv. Math. 222, 1016--1079 (2009; Zbl 1190.14054)], one can attach a quantum differential system to any proper smooth Deligne-Mumford stack using the quantum orbifold cohomology. By the result \textit{T. Coates} et al., [Acta. Math. 202(2), 139--193 (2009; Zbl 1213.53106)], this construction can be done explicitly in the case of weighted projective spaces and yields a quantum differential system
\[
\mathcal{Q}^A=(\mathcal{M}_A,\tilde{H}^{A,\text{sm}},\tilde{\nabla}^{A,\text{sm}},\tilde{S}^{A,\text{sm}},n)
\]
where \(\mathcal{M}_A\simeq \mathbb{C}^*\), the metric \(\tilde{S}^{A,\text{sm}}\) being constructed with the help of the orbifold Poincaré duality. This quantum differential system is called the ``small A-model quantum differential system'' by the authors. On the other hand, using the method developed in [\textit{E. Mann}, J. Alg. Geom. 17, 137--166 (2008; Zbl 1146.14029)], from the Gauss-Manin system of the function \(F: U\times \mathcal{M}_B\rightarrow\mathbb{C}\) defined by
\[
F(u_1,\dots,u_n,x)=u_1+\cdots+u_n+\frac{x}{u_1^{w_1}\cdots u_n^{w_n}}
\]
where \(U=(\mathbb{C}^*)^n\) and \(\mathcal{M}_B=\mathbb{C}^*\), the authors obtain a quantum differential system
\[
\mathcal{Q}^B=(\mathcal{M}_B,H^{B},\nabla^{B},S^{B},n)
\]
which is called the ``B-model quantum differential system''. Then the authors prove that the quantum differential systems \(\mathcal{Q}^A\) and \(\mathcal{Q}^B\) are isomorphic. Identifying these two models under this isomorphism, the authors finally obtain a quantum differential system
\[
\mathcal{S}_w=(\mathcal{M},H,\nabla,S,n)
\]
where \(\mathcal{M}=\mathbb{C}^*\)(the index \(w\) recalls the weights \(w_0,\dots,w_n\)). Then the associated Frobenius type structure \(F_w\) on \(\mathcal{M}\) is a tuple
\[
\mathbb{F}_w=(\mathcal{M},E,R_0,R_\infty,\Phi,\nabla,g).
\]
In the second part of this paper, the authors study the behavior of these structures at the origin. They construct a limit quantum differential system
\[
\overline{\mathcal{S}}_w=(\overline{H},\overline{\nabla},\overline{S},n)
\]
on \(\mathbb{P}^1\), and also a limit Frobenius type structure \(\overline{\mathbb{F}}_w\).
The last part of this paper is devoted to the construction of classical, limit and logarithmic Frobenius manifolds. quantum cohomology; Frobenius manifolds; weighted projective spaces; Brieskorn lattices; mirror symmetry Douai, A; Mann, E, The small quantum cohomology of a weighted projective space, a mirror \(D\)-module and their classical limits, Geom. Dedicata, 164, 187-226, (2013) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms The small quantum cohomology of a weighted projective space, a mirror \(D\)-module and their classical limits | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Algebras and their representations in geometric representation theory can often be constructed geometrically in terms of convolution of cycles using Borel-Moore homology and constructible sheaves, e.g., the Springer correspondence describes how irreducible representations of a Weyl group can be realized in terms of a convolution action on the free vector spaces spanned by irreducible components of Springer fibers.
The authors establish the foundations of a motivic Springer theory using Chow groups and motivic sheaves (Theorem 4.9, page 209). Motivic sheaves are a relative version of triangulated category of mixed motives and their Hom-spaces are governed by Chow groups. As established in the setting of flag varieties, motivic sheaves are a graded version of constructible sheaves that are mathematically advantageous over the mixed \(\ell\)-adic sheaves or mixed Hodge modules.
They show that representations of convolution algebras, such as Lusztig's graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in type \(A\) and \(\widetilde{A}\), can be realized in terms of Springer motives (Theorem 5.2, page 211 and Theorem 6.5, page 215). The authors lay foundations to a motivic Springer theory and prove formality results using weight structures. They also express Koszul and Ringel duality in terms of a weight complex functor and show that partial quiver flag varieties in type \(\widetilde{A}\) with cyclic orientation admit an affine paving (Theorem 6.3, page 213). representation theory; convolution algebras; motives; weight structures; Springer resolution (Equivariant) Chow groups and rings; motives, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Representations of quivers and partially ordered sets, Hecke algebras and their representations Motivic Springer theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we give rigorous justification of the ideas put forward in {\S}20, Chapter 4 of Schubert's book; a section that deals with the enumeration of conics in space. In that section, Schubert introduced two degenerate conditions about conics, i.e., the double line and the two intersection lines. Using these two degenerate conditions, he obtained all relations regarding the following three conditions: conics whose planes pass through a given point, conics intersecting with a given line, and conics which are tangent to a given plane. We use the language of blow-ups to rigorously treat the two degenerate conditions and prove all formulas about degenerate conditions stemming from Schubert's idea.
For Part I, see [the author, ibid. 41, No. 1, 97--113 (2021; Zbl 07556683)]. Hilbert problem 15; enumeration geometry; blow-up Plane and space curves, Classical problems, Schubert calculus Understanding Schubert's book. II. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be an algebraic group over the complex and let \(P\) be a parabolic subgroup. A Schubert class \(\sigma\) in cohomology of \(X=G/P\) is called rigid if the only projective subvarieties of \(X\) representing \(\sigma\) are Schubert varieties. A Schubert class \(\sigma\) is called multi rigid if the only projective subvarieties of \(X\) representing a positive integral multiple \(k\sigma\) are unions of \(k\) Schubert varieties.
In recent years, there has been significant progress in determining multi rigid Schubert classes in compact complex Hermitian symmetric spaces by using differential geometric methods.
In [J. Differ. Geom. 87, No. 3, 493--514 (2011; Zbl 1232.14032)] the author characterized the rigid Schubert classes in Grassmannians, using algebro-geometric techniques. In this paper he extends such techniques to orthogonal Grassmannians. Remark that these varieties are not in general compact complex Hermitian symmetric spaces.
The algebro-geometric techniques have two main advantages over the differential geometric techniques. First, they can prove the rigidity of a class even when the class is not multi rigid. Second, they apply to homogeneous varieties other than compact complex Hermitian symmetric spaces. The main disadvantage is that studying multiples of Schubert classes gets successively more difficult as the multiple increases, making the technique less suitable for studying multi rigidity.
In a Grassmannian \(G(k,n)\) the Schubert classes are the classes of the Schubert varieties, which are the closure of the locus constructed by fixing the dimensions of the intersection of \(V\in G(k,n)\) with the subspaces of a fixed flag. To obtain the Schubert varieties of the odd orthogonal Grassmanian \(OG(k,2n+1)\) one has to start with an isotropic flag. The case of even orthogonal is slightly more technical; moreover the author allow \(OG(k,2k)\) do have 2 connected components. A Schubert class in \(OG(k,n)\) is of Grassmannian type (resp. of quadratic type) if every (resp. none) flag element in his definition is isotropic. Under the intersection pairing, the dual of a class of Grassmannian type is a class of quadratic type and vice versa.
The author characterizes rigid and multi rigid Schubert class of Grassmannian and quadratic type. He also characterize all the rigid classes in \(OG(2,n)\) if \(n>8\).
The study of rigidity is strongly motivated by the classical problem of determing whether a cohomology class can be represented by a smooth subvariety, respectively by a (positive) linear combination of classes of smooth subvarieties.
Rigid (respectively multi rigid) singular Schubert varieties cannot be represented by a smooth subvariety (respectively by a positive linear combination of classes of smooth subvarieties).
For a large set of non rigid Schubert classes, the author provides explicit smooth deformations of Schubert varieties using combinatorially defined varieties called restriction varieties. Schubert class; orthogonal Grassmannian Coskun, Izzet, Rigidity of Schubert classes in orthogonal Grassmannians, Israel J. Math., 200, 1, 85-126, (2014) Grassmannians, Schubert varieties, flag manifolds Rigidity of Schubert classes in orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a semisimple simply connected algebraic group over an algebraically closed field. Let \(T\) be a maximal torus and let \(B\) be a Borel subgroup of \(G\) containing \(T\). Let \(X\) be the wonderful compactification of the adjoint group \(G/Z(G)\) of De Concini and Procesi.
In a previous article [Transform. Groups 12, No. 2, 371--406 (2007; Zbl 1129.19003)] the author proved that the Grothendieck ring \(K(X)\) is a free \(K(G/B)\)-module and provided an explicit basis.
Here he constructs a different basis of \(K(X)\) with rational coefficients, as a free \(K(G/B)\)-module, lifting the Schubert classes in the \(T\)-equivariant Grothendieck ring \(K_T(G/B)\) to \(R(T)\otimes R(T)\), where \(R(T)\) is the representation ring of \(T\). The structure constants of this new basis are thus described in terms of the structure constants of the Schubert basis. equivariant K-theory; flag varieties; structure constants; wonderful compactification Equivariant \(K\)-theory, Compactifications; symmetric and spherical varieties, Grassmannians, Schubert varieties, flag manifolds Equivariant \(K\)-theory of flag varieties revisited and related results | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we construct a bar-cobar adjunction and a Koszul duality theory for protoperads, which are an operadic type notion encoding faithfully some categories of gebras with diagonal symmetries, like double Lie algebras \((\mathcal{DL}ie)\). We give a criterion to show that a binary quadratic protoperad is Koszul and we apply it successfully to the protoperad \(\mathcal{DL}ie\). As a corollary, we deduce that the properad \(\mathcal{DP}ois\) which encodes double Poisson algebras is Koszul. This allows us to describe the homotopy properties of double Poisson algebras which play a key role in noncommutative geometry.
For part I, see [the author, ``Protoperads I: combinatorics and definitions'', preprint, \url{arXiv:1901.05653}; High. Struct. 6, No. 1, 256--310 (2022; Zbl 07567921)]. properad; protoperad; Koszul duality; double Poisson Algebraic operads, cooperads, and Koszul duality, Poisson algebras, Noncommutative algebraic geometry Protoperads II: Koszul duality | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this paper is to compute a Gröbner basis for the ideal defining a union of schemes each given by northwest rank conditions with respect to an antidiagonal term order. Recall that a scheme defined by northwest rank conditions is any scheme whose defining equations are of the form ``all \(k\times k\) minors in the northwest \(i\times j\) submatrix of a matrix of variables'' where \(i,j,k \) are natural integers. These schemes appears naturally in many geometric situations. One geometrically important collection of such schemes are matrix Schubert varieties, closures of Schubert varieties lifted from the flag manifold to the space of matrices. The importance of Schubert varieties, and hence of matrix Schubert varieties, to other areas of geometry has increased in last times. Schubert varieties; Gröbner basis; determinantal ideals Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Generating the ideals defining unions of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the torus equivariant \(K\)-theory ring of a Grassmannian the classes of the structure sheaves of Schubert varieties form a natural, geometric basis. Understanding the structure constants with respect to this basis is equivalent to describing the ring structure. In an earlier work the authors presented the structure constants as the cardinalities of certain combinatorially defined tableaux. In the paper under review the authors show a bijection from those tableaux to certain puzzles (fillings of a triangle with certain permitted puzzle pieces). As a result they obtain a (mild modification of the) puzzle rule for the structure constants as the number of puzzles, originally conjectured by Knutson and Vakil.
For part I, see [\textit{O. Pechenik} and \textit{A. Yong}, Forum Math. Pi 5, Article ID e3, 128 p. (2017; Zbl 1369.14060)]. Grassmannians; equivariant \(K\)-theory; puzzle; Schubert calculus; Littlewood-Richardson rule O. Pechenik and A. Yong. ''Equivariant K-theory of Grassmannians II: The Knutson-Vakil conjecture''. 2015. arXiv:1508.00446. Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Equivariant \(K\)-theory of Grassmannians. II: The Knutson-Vakil conjecture. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the present paper we prove that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line \(\mathbb{X}\) over finite fields. These polynomials are then used to define the generic Ringel-Hall algebra of \(\mathbb{X}\) as well as its Drinfeld double. Combining this construction with a result of Cramer, we show that Hall polynomials exist for tame quivers, which not only refines a result of Hubery, but also confirms a conjecture of Berenstein and Greenstein. weighted projective line; Hall polynomial; Ringel-Hall algebra; Green's formula Representations of quivers and partially ordered sets, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Quantum groups (quantized enveloping algebras) and related deformations Hall polynomials for tame type | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is based on the author's talk at the 2012 Workshop on Geometric Methods in Physics held in Białowieża, Poland. The aim of the talk is to introduce the audience to the Eynard-Orantin topological recursion. The formalism is originated in random matrix theory. It has been predicted, and in some cases it has been proven, that the theory provides an effective mechanism to calculate certain quantum invariants and a solution to enumerative geometry problems, such as open Gromov-Witten invariants of toric Calabi-Yau threefolds, single and double Hurwitz numbers, the number of lattice points on the moduli space of smooth algebraic curves, and quantum knot invariants. In this paper we use the Laplace transform of generalized Catalan numbers of an arbitrary genus as an example, and present the Eynard-Orantin recursion. We examine various aspects of the theory, such as its relations to mirror symmetry, Gromov-Witten invariants, integrable hierarchies such as the KP equations, and the Schrödinger equations. mirror symmetry; Laplace transform; higher-genus Catalan numbers; topological recursion 32. M. Mulase, The Laplace transform, mirror symmetry, and the topological recursion of Eynard-Orantin, Geometric Methods in Physics (Trends Math., Birkhäuser/Springer, Basel, 2013), pp. 127-142. genRefLink(16, 'S0129167X16500725BIB032', '10.1007%252F978-3-0348-0645-9_11'); Mirror symmetry (algebro-geometric aspects), Families, moduli of curves (analytic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Enumeration in graph theory, Lattice points in specified regions The Laplace transform, mirror symmetry, and the topological recursion of Eynard-Orantin | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give geometric descriptions of the category \(C_k(n,d)\) of rational polynomial representations of \(\mathrm{GL}_n\) over a field \(k\) of degree \(d\) for \(d\leq n\), the Schur functor and Schur-Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category \(C_k(n,d)\) and the Schur functor. modular Springer theory; Schur algebras; Schur functors; Schur-Weyl duality; perverse sheaves; nilpotent cones; affine Grassmannians; categories of polynomial representations; general linear groups; representations of symmetric groups Mautner, C., A geometric Schur functor, Selecta Math. (N.S.), 20, 4, 961-977, (2014) Schur and \(q\)-Schur algebras, Representation theory for linear algebraic groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Coadjoint orbits; nilpotent varieties, Representations of finite symmetric groups A geometric Schur functor. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(C\) be a smooth projective curve of genus \(g\geq 3\), and let \(J(C)\) be its Jacobian variety. Choosing a base point, one can embed \(C\) into \(J(C)\), and define \(W_k\) to be the natural image of the \(k\)-th symmetric product of \(C\). \textit{G. Ceresa} [Ann. Math. (2) 117, 285--291 (1983; Zbl 0538.14024)] showed that \(W_k-W_k^-\) is not algebraically trivial when \(C\) is generic and \(1\leq k\leq g-2\). In this article the author focuses on the Fermat curve of degree \(N\geq 4\) and gives a criterion for the non-vanishing of \(W_k-W_k^-\) modulo algebraic equivalence in terms of special values of generalized hypergeometric functions. algebraic cycle; iterated integral; hypergeometric function Algebraic cycles, Generalized hypergeometric series, \({}_pF_q\), Appell, Horn and Lauricella functions On the Abel-Jacobi maps of Fermat Jacobians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article is a continuation of the author's study of Schubert varieties related to a fusion product of \(\mathfrak{sl}(2,{\mathbb C})\)-modules \({\mathbb C}^{a_{i}}\) with \(A:=(1 < a_{1} \leq \cdots \leq a_{n}).\) Various constructions from the author's article [Schubert varieties and the fusion products, preprint, \texttt{math.QA/0305437}] are generalized in order to investigate the Schubert variety \(\text{sh}_{A}\) for a general sequence \(A.\) The author studies the Picard groups of these varieties, the surjections between two Schubert varieties of this type and proves some results on the singularities of these varieties. In particular, the only non-singular Schubert variety is \(\text{sh}_{A}\) where \(a_{1}, \dots, a_{n}\) are mutually distinct. Fusion products; Schubert varieties Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Lie algebras and Lie superalgebras Schubert varieties and the fusion products: the general case | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We discuss a class of vertex operator algebras \({\mathcal{W}}_{ m \vert n \times \infty}\) generated by a supermatrix of fields for each integral spin \(1, 2, 3, \dots\). The algebras admit a large family of truncations that are in correspondence with holomorphic functions on the Calabi-Yau singularity given by solutions to \( xy = z^m w^n\). We propose a free-field realization of such truncations generalizing the Miura transformation for \({\mathcal{W}}_N\) algebras. Relations in the ring of holomorphic functions lead to bosonization-like relations between different free-field realizations. The discussion provides a concrete example of a non-trivial interplay between vertex operator algebras, algebraic geometry and gauge theory. conformal and W symmetry; conformal field models in string theory; supersymmetric gauge theory; differential and algebraic geometry Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Calabi-Yau manifolds (algebro-geometric aspects), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, String and superstring theories in gravitational theory, Supersymmetric field theories in quantum mechanics, Applications of differential geometry to physics On extensions of \(\mathfrak{gl}\widehat{\left(m \vert n \right)}\) Kac-Moody algebras and Calabi-Yau singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The symmetric Grothendieck polynomials generalize Schur polynomials and are Schur-positive by degree. Combinatorially this is manifested as the generalization of semistandard Young tableaux by set-valued tableaux. We define a (weak) symmetric \(P\)-Grothendieck polynomial which generalizes \(P\)-Schur polynomials in the same way. Combinatorially this is manifested as the generalization of shifted semistandard Young tableaux by a new type of tableau which we call shifted multiset tableaux. symmetric Grothendieck polynomials; semistandard Young tableaux; set-valued tableaux Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds \(P\)-Schur positive \(P\)-Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We use classical Schubert calculus to give a direct geometric proof of the quantum version of Monk's formula [see \textit{S. Fomin, S. Gelfand} and \textit{A. Postnikov}, J. Am. Math. Soc. 10, No. 3, 565-596 (1997; Zbl 0912.14018)]. quantum Schubert polynomials Anders Skovsted Buch, A direct proof of the quantum version of Monk's formula, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2037 -- 2042. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds A direct proof of the quantum version of Monk's formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The goal of this book is to develop the connections between the representation theory of a complex semisimple Lie group \(G\) and the geometry of the flag variety. W. McGovern begins by introducing finite-dimensional representations, and then outlines the Beilinson-Bernstein construction of irreducible representations of \(G\) from closures of orbits of subgroups of \(G\) in \(X=G/B\). He also shows how to go the other way, from representations to orbit closures via the characteristic variety of a representation. This leads to studying which orbit closures lie in which others and the singularities of orbit closures at points of smaller orbits.
McGovern discusses Kazhdan-Lusztig and Lusztig-Vogan polynomials, connecting them to singularities and representations. He also provides combinatorial characterizations of nonsingular orbit closures. Furthermore, he investigates Springer fibers in \(X\) and the singularities of their components. This is a nicely written book, strongly recommended for anyone interested in the geometry of the flag variety. flag variety; nilpotent orbits; orbital varieties; Springer fibers; rationally smooth points; symmetric varieties; K-orbits; Hecke algebra; Kazhdan-Lusztig polynomials; Luszti-Vogan polynomials; Harish-Chandra modules; characteristic variety Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Singularities in algebraic geometry Representation theory and geometry of the flag variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider the infinite family of Feymman graphs known as the ``banana graphs'' and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern-Schwartz-MacPherson classes, using the classical Cremona transformation and the dual graph, and a blowup formula for characteristic classes. We outline the interesting similarities between these operations and we give formulae for cones obtained by simple operations on graphs. We formulate a positivity conjecture for characteristic classes of graph hypersurfaces and discuss briefly the effect of passing to noncommutative space-time. M. Marcolli, \textit{Feynman motives}, World Scientific, Singapore (2009). Feynman integrals and graphs; applications of algebraic topology and algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Birational automorphisms, Cremona group and generalizations, Feynman diagrams Feynman motives of banana graphs | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors give a combinatorial proof that the standard monomials (as developed in the series ``Geometry of \(G/P\). I-IX'' by \textit{V. Lakshmibai} et al.) give a basis for the multihomogeneous coordinate rings of Schubert varieties in \(GL_n/B\). standard monomials; Schubert varieties Reiner, V.; Shimozono, M., Straightening for standard monomials on Schubert varieties, J. Algebra, 195, 130-140, (1997) Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Straightening for standard monomials on Schubert varieties | 0 |
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