text stringlengths 571 40.6k | label int64 0 1 |
|---|---|
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 note is devoted to an expository argument concerning a physical implication of the concept of moduli. The basic idea is the quantum adiabatic theorem. We give a sketchy argument how the adiabatic connection plays a role in discussing the mathematical phenomena that gives a response for the change of a point of moduli space. Two examples are considered; the connection over the Siegel space (moduli space of a family of abelian varieties) that is given in terms of the theta functions, and the other is concerning the connection of a space of the period vector defined over the moduli space of Riemann surfaces. quantum adiabatic theorem; Siegel space Applications of deformations of analytic structures to the sciences, Quantum field theory; related classical field theories, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Quantum connection over moduli 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 We consider the canonical map from the Calogero-Moser space to symmetric powers of the affine line, sending conjugacy classes of pairs of \(n\times n\)-matrices to their eigenvalues. We show that the character of a natural \(\mathbb{C}^*\)-action on the scheme-theoretic zero fiber of this map is given by Kostka polynomials. A similar result is proved for a cyclic version of the Calogero-Moser space. Calogero-Moser space; symmetric powers; affine lines; pairs of matrices; scheme-theoretic zero fiber; Kostka polynomials DOI: 10.1006/aima.2002.2083 Group actions on varieties or schemes (quotients), Combinatorial aspects of representation theory, Relationships between algebraic curves and integrable systems, Grassmannians, Schubert varieties, flag manifolds Calogero-Moser space and Kostka 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 \(C^0\subset \mathbb{C}^2\) be a reduced pure dimension one curve with one singular point \(p\). A conjecture of Oblomkov and Shende relates the HOMFLY polynomial of the link of the singularity at \(p\) with topological invariants of the punctual Hilbert scheme parametrizing zero-dimensional subschemes of \(C^0\) with topological support at \(p\). This conjecture has a natural physical interpretation in terms of the so-called large \(N\) duality for conifold transitions. An interesting question concerns the construction of counting invariants which would correspond to general knots via large \(N\) duality. The paper under review proposes such a construction.
To be more precise, let \(X\) be a projective Calabi--Yau threefold with a single conifold singularity \(q\), whose formal neighbourhood is isomorphic to the formal neighbourhood of the origin in the singular hypersurface \(xz-yw=0\) in \(\mathbb{C}^4\), assume that there exists a Weil divisor \(\Delta\cong \mathbb{P}^2\) containing \(q\) which is locally determined by the equation \(z=0\), and let \(\Gamma \subset X_0\) be a reduced irreducible plane curve contained in \(\Delta\) and passing through \(q\). Suppose furthermore that \(\Gamma\) is singular in \(q\) and smooth otherwise. Blowing up \(X_0\) along \(\Delta\) gives a crepant resolution \(X\to X_0\). Denote by \(C\) the strict transform of \(\Gamma\) and by \(C_0\) the exceptional \((-1,-1)\)-curve. The authors consider so-called \(C\)-framed stable pairs, which by definition are pairs \((F,s)\) consisting of a sheaf \(F\) on \(X\) topologically supported on \(C\cup C_0\) and a section \(s: \mathcal{O}_X \to F\) with zero-dimensional kernel subject to a further technical condition. There is a scheme \(P(X,C,r,n)\) parametrizing \(C\)-framed stable pairs such that \(\chi(F)=n\) and \(\text{ch}_2(F)=[C]+r[C_0]\) (\(r\geq 0\)). The main result of the paper expresses, roughly speaking, the generating function of the topological Euler numbers of the spaces \(P(X,C,r,n)\) as a product of the generating function of the similarly defined spaces \(P(X,C_0,r,n)\) and some further functions. The idea is to use the wall-crossing formalism developed by Joyce--Song and Kontsevich--Soibelman for a certain one-parameter family, depending on \(b\in \mathbb{R}\), of weak stability conditions on the heart \(\mathcal{A}\) of a certain perverse t-structure on \(D^b(X)\), the bounded derived category of coherent sheaves on \(X\). More precisely, the authors show that the results obtained for usual stable pairs by \textit{S. Toda} in [Saito, Masa-Hiko (ed.) et al., New developments in algebraic geometry, integrable systems and mirror symmetry. Papers based on the conference ``New developments in algebraic geometry, integrable systems and mirror symmetry'', Kyoto, Japan, January 7--11, 2008, and the workshop ``Quantum cohomology and mirror symmetry'', Kobe, Japan, January 4--5, 2008. Tokyo: Mathematical Society of Japan. Advanced Studies in Pure Mathematics 59, 389--434 (2010; Zbl 1216.14009)] and [Duke Math.\ J.\ 149, No.\ 1, 157--208 (2009; Zbl 1172.14007)] carry over to their framed setting.
Section 2 contains some facts concerning slope limit stability and the \(C\)-framed subcategory \(\mathcal{A}^C\) which is the analogue of the above \(\mathcal{A}\) in this setting. In Section 3 the main result is proved. The first step is a wall-crossing formula relating invariants for \(b\ll 0\) to small \(b>0\) invariants. This is in fact established in Appendix A.1. The second step is to find a connection between moduli spaces of stable \(C\)-framed objects for small \(b>0\) with the Hilbert scheme invariants. This is roughly done as follows. The authors construct a certain moduli stack \(Q(X,C,r,n)\) of decorated sheaves on \(X\) which turns out to be a \(\mathbb{C}^*\)-gerbe over a relative Quot-scheme where the latter is geometrically bijective with a certain nested Hilbert scheme. On the other hand, \(Q(X,C,r,n)\) is equipped with a geometric bijection to the moduli space of \(C\)-framed pairs. A stratification computation then concludes the proof.
Section 4 is concerned with motivic invariants. More precisely, the authors compare a certain motivic Hilbert scheme series with the motivic invariants one gets when a (partly) conjectural construction employing motivic vanishing cycles for formal functions due to Kontsevich and the third author is applied. plane curve singularities; stable pairs; framed sheaves; conifold transitions; Donaldson-Thomas invariants; wall-crossing Jiang, Y.: The moduli space of stable coherent sheaves via non-archimedean geometry. arXiv:1703.00497 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Calabi-Yau manifolds (algebro-geometric aspects), Representations of quivers and partially ordered sets, Singularities of curves, local rings HOMFLY polynomials, stable pairs and motivic Donaldson-Thomas invariants | 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 class of nonlinear transformations acting on many variables. These transformations have their origin in the theory of quantum integrability: they appear in the description of the symmetries of the Yang-Baxter equations and their higher dimensional generalizations. They are generated by involutions and act as birational mappings on various projective spaces. We present numerous figures, showing successive iterations of these mappings. The existence of algebraic invariants explains the aspect of these figures. We also study deformations of our transformations. iterated mappings; dynamical systems; Coxeter groups; birational transformations; Cremona transformations; inversion relations; Yang- Baxter equations; automorphisms of algebraic varieties; elliptic curves; resonant tori; Plücker variables; quantum integrability; involutions; birational mappings; deformations M.P. Bellon, J.-M. Maillard, C.-M. Viallet, Dynamical systems from quantum integrability, in: J.-M. Maillard (Ed.), Proceedings of the Conference Yang--Baxter Equations in Paris, World Scientific, Singapore, 1993, pp. 95--124, Int. J. Mod. Phys. Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems, Rational and birational maps, Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics, Automorphisms of surfaces and higher-dimensional varieties, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act Dynamical systems from quantum integrability | 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 homogenized quantum \(\mathfrak{sl}(2,\mathbb C)\) is the \(\mathbb C\)-algebra with generators \(E,F,K,K'\) and relations
\[
\begin{aligned} KE = q EK, & KF = q^{-1} FK, KK' = K'K,\\ K'E = q^{-1}EK', & K'F = q F K', [E,F] = \frac{K^2-K^{\prime 2}}{q-q^{-1}},\end{aligned}
\]
for some \(q \in \mathbb C \setminus \{0,\pm 1, \pm i\}\). The authors show that these algebras are Artin-Schelter regular degenerations of 4-dimensional Sklyanin algebras. As such, these algebras are examples of homogeneous coordinate rings of noncommutative \(\mathbb P^3\)'s.
The authors study the linear modules and the fat points of these algebras using the fact that \(K\) and \(K'\) are normal degree one elements. They connect these modules either to quotients which are 3-dimensional Artin-Schelter regular or to simple representations and Verma modules of quantized \(\mathfrak{sl}(2,\mathbb C)\).
This paper is an excellent example for people working in noncommutative geometry, as it demonstrates the connection between ordinary representation theory of affine algebras and the study of graded modules of their homogenized versions. noncommutative algebraic geometry; quantum groups; quantum 2 Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, Graded rings and modules (associative rings and algebras), Quantum groups (quantized enveloping algebras) and related deformations Noncommutative geometry of homogenized quantum \(\mathfrak{sl}(2, \mathbb{C})\) | 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 smooth projective variety over \(\mathbb{C}\). The quantum cohomology of \(X\) satisfies the axioms of Froebenius manifolds formulated by Dubrovin. In particular the third derivative of the Frobenius potential which is the generating function of the genus zero Gromov-Witten invariants of \(X\) give the structure constants of the product in the quantum cohomology. Dubrovin formulated a conjecture relating the monodromy data of the Frobenius manifold associated to the quantum cohomology of \(X\) and the structure of the bounded derived category of coherent sheaves on \(X\). In particular, the conjecture asserts that the Frobenius manifold is semisimple if and only if the derived category admits a full exceptional collection of \(\sum h^{p,p}(X)\) elements, and moreover, for any semisimple point of the Frobenius manifold, the Stokes matrix of the first structure connection is identified with the matrix of the Euler pairing of an exceptional collection. The conjecture is proven in some examples.
The paper under review generalizes Dubrovin's conjecture for a class of orbifolds given by the weighted projective line \(\mathbb{P}^1(a_1,a_2,a_3)\) such that \(\sum 1/a_i>1\). The existence of an exceptional collection for the derived category and the semisimplicity of the Frobenius manifold is already known for this class of orbifolds. The second part of the conjecture regarding the equivalence of the Stokes and Euler matrices is proven in this paper. The proof is based on the homological mirror symmetry. Stokes Matrices; quantum cohomology; Frobenius manifolds; mirror symmetry Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Stokes matrices for the quantum cohomologies of a class of orbifold projective 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 Let \(E\) be the bundle defined by applying a polynomial functor to the tautological bundle on the Hilbert scheme of \(n\) points in the complex plane. By a result of \textit{M. Haiman} [Invent. Math. 149, No. 2, 371--407 (2002; Zbl 1053.14005)], the Čech cohomology groups \(H^i(E)\) vanish for all \(i > 0\). It follows that the equivariant Euler characteristic with respect to the standard two-dimensional torus action has nonnegative integer coefficients in the torus variables \(z_1\), \(z_2\), because they count the dimensions of the weight spaces of \(H^0(E)\). We derive a formula for this Euler characteristic using residue formulas for the Euler characteristic coming from the description of the Hilbert scheme as a quiver variety [\textit{A. Neguţ}, ``Moduli of flags of sheaves and their K-theory'', Preprint, \url{arXiv:1209.4242}; \textit{N. A. Nekrasov}, Adv. Theor. Math. Phys. 7, No. 5, 831--864 (2003; Zbl 1056.81068)]. We evaluate this expression using \textit{N. Jing}'s Hall-Littlewood vertex operator with parameter \(z_1\) [Adv. Math. 87, No. 2, 226--248 (1991; Zbl 0742.16014)], and a new vertex operator formula given in Proposition 1 below. We conjecture that the summand in this formula is a polynomial in \(z_1\) with nonnegative integer coefficients, a special case of which was known to \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. A 286, 323--324 (1978; Zbl 0374.20010)]. representation theory; combinatorics; symmetric functions; Hilbert scheme; Hall-Littlewood polynomials Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations Hall-Littlewood polynomials and vector bundles on the Hilbert 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 \(K\) is the field of fractions of a complete discrete valuation ring \(R\) and \(A_K\) an abelian variety over \(K\). Write \(A'_K\) for its dual, \(A\) and \(A'\) for the Néron models of \(A_K\) and \(A'_K\) and \(\phi _A\), \(\phi _{A'}\) for the corresponding component groups. Grothendieck has constructed a pairing \(\phi _A\times \phi _{A'}\rightarrow {\mathbb Q}/{\mathbb Z}\) and conjectured that this pairing is perfect. This conjecture is already known in some cases, e.g., semi-stable reduction. The paper proves the conjecture in the case that the residue field of \(R\) is perfect and \(A\) has potentially multiplicative reduction. There is a hint for a possible proof in a more general situation. The highly technical proof uses arithmetic theory of abelian varieties, sheaves for the étale or smooth topology and a rigid analytic description of the groups \(\phi _A\), \(\phi _{A'}\) obtained from rigid uniformization of the abelian variety \(A_K\). abelian varieties; Néron model; component group; Grothendieck's pairing; complete discrete valuation ring; rigid uniformization; dual variety Bosch, S, Component groups of abelian varieties and grothendieck's duality conjecture, Ann. Inst. Fourier, 47, 1257-1287, (1997) Arithmetic ground fields for abelian varieties, Local ground fields in algebraic geometry, Valuation rings Component groups of abelian varieties and Grothendieck's duality 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 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 The author draws on, and builds, connections between concepts in singularity theory, geometry, and polyhedral combinatorics, to show that the spectrum at infinity of a tame Laurent polynomial can be used to count weighted lattice points in polytopes. Using this, the author derives an algorithm to compute the Ehrhart polynomial of a simplex containing the origin in its interior.
More precisely, the \textit{Newton polytope} of a Laurent polynomial on \((\mathbb C^*)^n\) is the convex hull of the points of \(\mathbb Z^n\) which occur as degree vectors of monomials of the Laurent polynomial. The author shows that the spectrum at infinity of a Laurent polynomial that is convenient and non-degenerate with respect to its Newton polytope (defined as in [\textit{A. G. Kushnirenko}, Invent. Math. 32, 1--31 (1976; Zbl 0328.32007)], can be expressed using the Newton filtration of the Newton polytope \(P\) of the polynomial. This is referred to as the \textit{Newton spectrum} of \(P\). The author then extends the above definition of Newton spectrum from Newton polytopes to any full-dimensional polytope containing the origin in its interior.
The main result -- essentially available [\textit{A. Stapledon}, Adv. Math. 219, No. 1, 63--88 (2008; Zbl 1174.52007)], though not in the framework of this paper -- is that the Newton spectrum of a full-dimensional polytope \(P\) with vertices in \(\mathbb Z^n\) and having the origin in its interior, is equal to the \textit{weighted} \(\delta\)-vector of the polytope. Here the notion of \(\delta\)-vector comes from Ehrhart theory. As a corollary, the author shows that the \(k\)-th coefficient of the \(\delta\)-vector of \(P\) is equal to the number of entries in its Newton spectrum having values in the interval \(]k-1,k]\). The last corollary above straightforwardly yields a general idea to compute the \(\delta\)-vector and Ehrhart polynomial of a full-dimensional lattice polytope containing the origin in its interior -- viz. to compute the Newton spectrum of the polytope and use the corollary.
The author has also proved closed form expressions for the Newton spectrum of \textit{reduced} simplices, i.e. \(n+1\)-vertex polytopes in \(\mathbb R^n\) with no common divisor between the volumes of the polytopes obtainable by replacing the \(i\)-th vertex vector, by its corresponding unit vector, \(1\). The closed form expression allows to implement the general idea described previously, into an effective algorithm for computing the Ehrhart polynomial and \(\delta\)-vector for a reduced simplex.
Finally, some additional results are proved for the Newton spectrum of reflexive polytopes and some remarks stated about the distribution of their \(\delta\)-vectors. toric varieties; polytopes; Ehrhart theory; spectrum of polytopes; spectrum of regular functions; Laurent polynomial; Newton polytope; Newton spectrum Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Exact enumeration problems, generating functions, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Mirror symmetry (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Ehrhart polynomials of polytopes and spectrum at infinity of Laurent 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 No review copy delivered. Hansen, Johan P., Quantum codes from toric surfaces, IEEE Trans. Inform. Theory, 0018-9448, 59, 2, 1188-1192, (2013) Quantum coding (general), Geometric methods (including applications of algebraic geometry) applied to coding theory, Toric varieties, Newton polyhedra, Okounkov bodies Quantum codes from toric 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 We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig's canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties. categorification; quantum groups; canonical basis H. Zheng, ``Categorification of integrable representations of quantum groups'', Acta Mech. Sinica (English Ed.)30 (2014) no. 6, p. 899-932 ##img## Creative Commons License BY-ND ISSN : 2429-7100 - e-ISSN : 2270-518X Quantum groups (quantized enveloping algebras) and related deformations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Double categories, \(2\)-categories, bicategories, hypercategories Categorification of integrable representations of quantum 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 This paper presents key ideas which unify the algebra-geometric methods in the theory of soliton equations and recent development in the theory of \(N=2\) supersymmetric gauge models. The author shows that the Seiberg-Witten theory of \(N=2\) supersymmetric gauge theories can be considered on one hand as a part of the Whitham theory and at the same time leads to a new general approach to the Hamiltonian theory of soliton equations. Baker-Akhiezer functions; integrable systems; gauge models Krichever, Igor: Baker--akhiezer functions and integrable systems, , 1-22 (2000) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between algebraic curves and integrable systems, Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems, KdV equations (Korteweg-de Vries equations), Supersymmetric field theories in quantum mechanics, Yang-Mills and other gauge theories in quantum field theory Baker-Akhiezer functions and integrable 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 \(\mathrm{Td}\) denote the Todd power series \(\prod_{j\geq 1}x_j/(1- e^{-x_j})\) viewed as a formal power series in the variables \(c_i\) (= the \(i\)th elementary symmetric functions of \(x_j\)), and let \(\mathrm{ch}\) denote the Chern power series \(r+ \sum_{j\geq 1} (e^{x_j'}- 1)\) viewed as a formal power series in the variables \(c_i'\) (= the \(i\)th elementary symmetric functions of \(x_i\)). Let \(T_m= \prod_p p^{[m/(p- 1)]}\), where the bracket denotes the integral part and the product is over prime numbers. Note that the integer \(T_m\) is the denominator of the degree \(m\) part of Td. Let \({\mathcal C}{\mathcal T}_m= T_m(\text{ch}\cdot\text{Td})_m\), where \((\bullet)_m\) denotes the degree \(m\) part of \((\bullet)\), so that \({\mathcal C}{\mathcal T}_m\in\mathbb{Z}[c_1,\dots, c_m, r, c_1',\dots, c_m']\). For a variety \(Y\) over a field \(k\) of characteristic zero, and a locally free coherent \({\mathcal O}_Y\)-sheaf \({\mathcal F}\) on \(Y\), let \({\mathcal C}{\mathcal T}_m({\mathcal F}, Y)={\mathcal C}{\mathcal T}_m(c_1(T_Y),\dots, c_m(T_Y), \text{rank}({\mathcal F}), c_1'({\mathcal F}),\dots, c_m'({\mathcal F}))\) in \(\text{CH}^m(Y)\).
The main result of this paper is stated as follows. Let \(X\), \(S\) be smooth quasi-projective varieties over \(k\) and let \(f: X\to S\) be a projective morphism over \(k\). Set \(d= \dim X- \dim S\) and suppose that \({\mathcal F}\) is a coherent \({\mathcal O}_X\)-sheaf on \(X\). Then the identity \((T_{d+ n}/T_n){\mathcal C}{\mathcal T}_n(f_*[{\mathcal F}], S)= f_*({\mathcal C}{\mathcal T}_{d+n}({\mathcal F}, X))\) holds if \(d\geq 0\), and the identity \({\mathcal C}{\mathcal T}_n(f_*[{\mathcal F}], S)= (T_n/T_{n+d}) f_*({\mathcal C}{\mathcal T}_{n+ d}({\mathcal F},X))\) holds if \(d< 0\). The crucial ingredients for his proof are Hironaka's resolution of singularities and the weak factorization theorem for birational maps due to \textit{D. Abramovich}, \textit{K. Karu}, \textit{K. Matsuki} and \textit{J. Włodarczyk} [J. Am. Math. Soc. 15, No. 3, 531--572 (2002; Zbl 1032.14003)]; the use of these restricts the result to characteristic zero cases. Riemann-Roch theorems; birational map; characteristic classes Pappas, G.: Integral Grothendieck-Riemann-Roch theorem, Invent. math. 170, No. 3, 455-481 (2007) Riemann-Roch theorems, Rational and birational maps, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Integral Grothendieck-Riemann-Roch theorem | 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 the weighted projective variety defined by a weighted homogeneous ideal \(J\) and \(C\) a maximal cone in the Gröbner fan of \(J\) with \(m\) rays. We construct a flat family over \(\mathbb{A}^m\) that assembles the Gröbner degenerations of \(V\) associated with all faces of \(C\). This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base \(X_C\) (the toric variety associated to \(C)\) along the universal torsor \(\mathbb{A}^m \to X_C\). We apply this construction to the Grassmannians \(\mathrm{Gr}(2,\mathbb{C}^n)\) with their Plücker embeddings and the Grassmannian \(\mathrm{Gr}\big( 3, \mathbb{C}^6 \big)\) with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for \(\mathrm{Gr}(2, \mathbb{C}^n)\) we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation. cluster algebras; Gröbner basis; Gröbner fan; Grassmannians; flat degenerations; Newton-Okounkov bodies Cluster algebras, Fibrations, degenerations in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Families of Gröbner degenerations, Grassmannians and universal cluster 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 studies representations of positive polynomials on semialgebraic sets. If the set \(S\subseteq\mathbb{R}^n\) is basic semialgebraic, i.e., \(S=\{x\in\mathbb{R}^n\mid f_1(x)\geq 0, \dots, f_r(x)\geq 0\}\), \(f_1, \dots, f_r\in \mathbb{R}[X_1, \dots, X_n]\), then all polynomials in the cone
\[
P(f_1,\dots, f_r)=\biggl\{f \in \mathbb{R}[ X_1,\dots,X_r] \,\Bigl|\, \exists a_\varepsilon\in \sum\mathbb{R}[X_1, \dots,X_r]^2: f=\sum_{\varepsilon\leq \{0,1\}^r}a_\varepsilon \cdot f_1^{\varepsilon_1} \cdot \dots\cdot f_r^{\varepsilon_r}\biggr\}
\]
are trivially nonnegative on \(S\). (The set \(\sum A^2\) is the sums of squares in the ring \(A.)\) The question is whether, or when, every polynomial that is nonnegative on \(S\) belongs to the cone. Using methods of functional analysis, \textit{K. Schmüdgen} proved [Math. Ann. 289, 203--206 (1991; Zbl 0744.44008)] that if \(S\) is compact and if \(f|_S>0\), then \(f \in P(f_1,\dots,f_r)\). The present author gives the first purely algebraic proof of this remarkable result. His method is to relate compactness of the set \(S\) to the condition that the cone \(P(f_1,\dots,f_r)\) is Archimedean, and then to use the representation theorem of Kadison-Dubois. The author studies these questions in greater generality -- many of his results are true for algebras over subfields of \(\mathbb{R}\) and for preorderings of higher level. Major developments have taken place since the author wrote this thesis. An account of the current state-of-the art was written by \textit{C. Scheiderer} [Positivity and sums of squares: A guide to some recent results, \texttt{http://www.math.univ-rennes1.fr/geomreel/raag01/Surveys.html\#Scheiderer}]. semialgebraic sets; nonnegative polynomials; Archimedean cone T. Wörmann, Strikt positive Polynome in der semialgebraischen Geometrie, Dissertation, Universität Dortmund (1998). Semialgebraic sets and related spaces, Real algebra Strictly positive polynomials in semialgebraic 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 \(C\) be a reduced, irreducible, non-degenerate curve of degree \(d\) and arithmetic genus \(g\) in projective space \(\mathbb P^n\), \(n \geq 2\). A very old question is to determine what combinations \((d,g)\) can occur in \(\mathbb P^n\). An important step was obtained by Castelnuovo, who gave an explicit upper bound \(\pi_0(d,n)\) on \(g\). Furthermore, curves that attain this bound (\textit{Castelnuovo curves}) are well understood, and if \(d \geq 2n+1\) they must lie on a surface of minimal degree \(n-1\). Eisenbud and Harris [cf. \textit{J. Harris}, Curves in Projective Space. Universite de Montreal. Montreal, Quebec, Canada: Les Presses de l'Universite de Montreal. (1982; Zbl 0511.14014)] began a systematic study to extend this result. They defined explicit numbers \(\pi_\alpha(d,n)\) and conjectured that if \(C\) is a reduced, irreducible, non-degenerate curve of genus \(g\) and degree \(d \geq 2n+2\alpha-1\) in \(\mathbb P^n\), and if \(g > \pi_\alpha(d,n)\), then \(C\) lies on a surface of degree at most \(n+\alpha-2\). The case \(\alpha = 0\) is a reformulation of Castelnuovo's result. Eisenbud and Harris proved the case \(\alpha = 1\). They also proved that it holds for \(d\) sufficiently large, and they gave some observations for low values of \(d\). They also proved that if the conjecture is true, the lower bound on \(d\) is sharp.
The present paper proves the case \(\alpha = 2\), and proves the following partial results for \(\alpha = 3\) and \(\alpha = 4\): If \(g > \pi_3(d,n)\) (resp.\ \(g > \pi_4(d,n)\)) and \(2n+5 \leq d \leq 4n+6\) (resp.\ \(2n+7 \leq d \leq 3n+3\)) then \(C\) lies on a surface of degree \(\leq n+1\) (resp.\ \(\leq n+2\)).
As usual, the proof proceeds via a study of the Hilbert function of a general hyperplane section of \(C\). He defines a condition stronger than uniform position, namely \textit{symmetric position}, notes that a general hyperplane section of \(C\) has this property, and makes a study of the possible Hilbert functions that arise for such a set of points, drawing geometric consequences for extremal values. He then draws his conclusions for the main results. His main tool is a study of a Gröbner basis for the homogeneous ideal of the hyperplane section. Castelnuovo theory; arithmetic genus; projective curve; uniform position; symmetric position; Gröbner basis I. Petrakiev, Castelnuovo theory via Gro\"bner bases, J. Reine Angew. Math., 619:49- 73, 2008. Curves in algebraic geometry, Projective techniques in algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Castelnuovo theory via Gröbner bases | 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 considers a linear differential operator \(P\) of order \(n\) defined over \(\mathbb{Q}\), called a \(CY(n)\)-operator. Such an operator has the following properties:
(1) \(P\) has maximal unipotent monodromy at \(0\) (MUM)
(2) \(P\) is self-dual, and
(3) \(P\) has a convergent power series solution \(f_0(x)\in\mathbb{Z}[[x]]\) with \(f_0(0)=1\).
The question addressed in this paper is:
Given a \(CY(4)\)-operator \(P\) of a family of Calabi-Yau threefolds \(f : X\to \mathbb{P}^1\) defined over \(\mathbb{Z}\), is there a way to calculate the Frobenius polynomials \(P_s(T)\)?
The paper describes a method to solve this problem. \(CY(4)\)-operators arise from families of Calabi-Yau threefolds with \(h^{1,2}=1\) (so that \(B_3=4\)). In the paper, it is assumed that the \(CY(4)\)-operator is the Picard-Fuchs operator on a rank \(4\) submodule in \(H^3_{dR}\) of some family of smooth Calabi--Yau threefolds. Here the Frobenius polynomial is referred to the characteristic polynomial of the Frobenius morphism, which is of the form
\[
P_s(T)=1+aT+bpT^2+ap^3T^t+p^6T^4
\]
It has four different roots \(r_1, pr_2, p^2/r_2, p^3/r_1\) where \(r_1\) and \(r_2\) are \(p\)-adic units. Therefore, giving a formula for \(P_s(T)\) is equivalent to determining \(r_1\) and \(r_2\). The paper under review gives \(p\)-adic analytic formulas for the unit roots \(r_1\) and \(r_2\) assuming some conjectures on Dwork's congruences. Many examples are produced computing the Frobenius polynomials. At singular points, the Frobenius polynomial splits into a product of two linear factors and a quadratic part of the form \(1-a_pT+p^3T^2\). In the examples, the coefficients \(a_p\) are identified with the Fourier coefficients of modular forms of weight \(4\). This suggests that there is a rigid Calabi-Yau motive corresponding to the quadratic factor. Calabi-Yau differential equation; unit root; maximal unipotent monodromy; Frobenius polynomial; modular form Samol, K., van Straten, D.: Frobenius polynomials for Calabi-Yau equations. Commun. Number Theory Phys. \textbf{2}(3), 537-561 (2008) Calabi-Yau manifolds (algebro-geometric aspects) Frobenius polynomial for Calabi-Yau equations | 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 Using the WDVV equation, the author first proves that all Gromov-Witten invariants of blowups of surfaces can be computed from the Gromov-Witten invariants of the surface by some recursive relations. Furthermore, one may determine the quantum product on blowups. We also prove that there is some degree of functoriality of the big quantum cohomology for a blowup. DOI: 10.1016/S0252-9602(06)60100-8 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Quantum cohomology of blowups of surfaces and its functoriality property | 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 introducing the noncommutative differential calculus on the function space of the infinite/finite set and construct a homotopy operator, we prove the analogue of the Poincaré lemma for the difference complex. As an application of the differential calculus, a two dimensional integral model can be derived from the noncommutative differential calculus. noncommutative geometry; noncommutative differential calculus; Poincaré lemma; Toda lattice equation Noncommutative algebraic geometry, Classical or axiomatic geometry and physics, Applications of local differential geometry to the sciences Noncommutative differential calculus and its application on the lattice | 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 \(\mathrm{GL}(n)\) of Faddeev, Reshetikhin, and Takhtajan, and that of Dipper and Donkin are realized geometrically by using double partial flag varieties. As a consequence, the difference of these two Hopf algebras is caused by a twist of a cocycle in the multiplication.
It is natural to ask if the quantum \(\mathrm{GL}(n)\) itself admits a geometric realization. Since quantum \(\mathrm{GL}(n)\) is, in principle, dual to quantum \(\mathfrak{gl}(n)\), one may expect to get an answer from the dual construction of \textit{A. A. Beilinson} et al. [Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012)]. However, the answer to such a question is subtle, because the group \(\mathrm{GL}(n)\) admits several quantizations: one by \textit{L. D. Faddeev, N. Yu. Reshetikhin} and \textit{L. A. Takhtadzhan} [Algebraic analysis, Pap. Dedicated to Prof. Mikio Sato on the Occas. of his Sixtieth Birthday, Vol. 1, 129--139 (1989; Zbl 0677.17010)], one by \textit{R. Dipper} and \textit{S. Donkin} [Proc. Lond. Math. Soc., III. Ser. 63, No. 1, 165--211 (1991; Zbl 0734.20018)], one by \textit{M. Takeuchi} [Proc. Japan Acad., Ser. A 66, No. 5, 112--114 (1990; Zbl 0723.17012)], and one by \textit{M. Artin, W. Schelter}, and \textit{J. Tate} [Commun. Pure Appl. Math. 44, No. 8-9, 879--895 (1991; Zbl 0753.17015)].
In this paper, we show that the dual construction of \textit{A. A. Beilinson} et al. [loc. cit.] together with the coproduct defined by \textit{I. Grojnowski} [The coproduct for quantum \(\mathrm{GL}_n\). Preprint] and \textit{G. Lusztig} [Asian J. Math. 3, No. 1, 147--177 (1999; Zbl 0983.17011)] is isomorphic to the quantum \(\mathrm{GL}(n)\) of Dipper and Donkin. The quantum \(\mathrm{GL}(n)\) of Faddeev, Reshetikhin, and Takhtajan is also obtained from this setting by twisting a cocycle on the multiplication. In the geometric realization of both quantizations, the comultiplication is the same. This shows that the two quantizations are isomorphic as coalgebras, which was proved by \textit{J. Du} et al. [J. Lond. Math. Soc., II. Ser. 44, No. 3, 420--436 (1991; Zbl 0694.22014)] by a different method twenty years ago. A closer look at the geometric construction yields that the basis \(E^M\) in the quantum \(\mathrm{GL}(n)\) is the same as the basis consisting of all characteristic functions of certain orbits up to a twist. One interesting fact in the geometric realization is that the quantum determinants in both quantizations get identified with a certain Young symmetrizer. This symmetrizer gives rise to the determinant representation of \(\mathrm{GL}(n)\), and the parameter in the quantization process does not appear. quantum linear groups; partial flag varieties; Hopf algebras; geometric realizations; quantizations; coalgebras Quantum groups (quantized function algebras) and their representations, Quantum groups (quantized enveloping algebras) and related deformations, Ring-theoretic aspects of quantum groups, Schur and \(q\)-Schur algebras, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On quantum \(\mathrm{GL}(n)\). | 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 relationship between global and local is an eternal theme of mathematics. The classical Künneth formula expresses the cohomology of a product space as a tensor product of the cohomologies of the direct factors. Furthermore, if \(\pi: E\rightarrow B\) is a fiber bundle with fiber \(F\), the celebrated Leray-Hirsch theorem states that the cohomology of fiber bundle \(E\) is equal to the tensor product of the cohomologies of the base and fiber, i.e. \(H^* (E)\cong H^*(B)\otimes H^*(F)\). With the development of Gromov-Witten theory, a natural question is to ask : What is quantum geometry picture for the Leray-Hirsch theorem?
The paper under review tries to answer this question and studies the quantum version of Leray-Hirsch theorem in the context of orbifold Gromov-Witten theory of a gerbe \(\mathcal{Y}\rightarrow \mathcal{B}\) banded by a finite group \(G\), where \(\mathcal{B}\) is a smooth proper (i.e. compact) Deligne-Mumford stack and a \(G\)-gerbe \(\mathcal{Y}\rightarrow \mathcal{B}\) can be regarded as a fiber bundle over complex orbifold \(\mathcal{B}\) with fiber \(BG\). The main result (i.e. quantum Leray-Hirsch theorem) states that the Gromov-Witten theory of \(\mathcal{Y}\) can be expressed in terms of the Gromov-Witten theory of base \(\mathcal{B}\) and the information of fiber \(BG\). The key point of the proof is to analyze the properties and degree for the pushforward map \(\pi: \overline{\mathcal{M}}_{g, n}(\mathcal{Y}, \beta)\rightarrow \overline{\mathcal{M}}_{g, n}(\mathcal{B}, \beta)\).
In conclusion, the paper under review presents an important structural decomposition theorem for a banded \(G\)-gerbe over complex orbifold \(\mathcal{B}\) which can be regarded as a quantum Leray-Hirsch style theorem. The combinatorial degree formula and virtual pushforward formula are impressive in the proof. Gromov-Witten invariants; Leray-Hirsch theorem; Gerbe Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Differential geometric aspects of gerbes and differential characters, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds A quantum Leray-Hirsch theorem for banded gerbes | 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 0755.00012.]
The full dual Hopf algebra of \(U_ q(\text{sl}(n))\) is studied. \(U_ q(\text{sl}(n))\) and \(k[\text{SL}_ q(n)]\) are naturally paired. If \(q\) is not a root of 1 and \(\text{char}(k)\neq 2\) then all finite-dimensional representations of \(U_ q(\text{sl}(n))\) are completely reducible, \(U_ q(\text{sl}(n))^ 0\) is cosemisimple and decomposes into a semidirect product \(U_ q(\text{sl} (n))^ 0\cong k[\text{SL}_ q(n)] \rtimes \langle \gamma_ 1,\dots, \gamma_{n-1}\rangle\) where the \(\gamma_ i\) are Hopf algebra automorphisms of order 2 of \(k[\text{SL}_ q(n)]\) commuting with each other. The case of \(\text{char} (k)=2\) resp. \(q\) a root of 1 is studied, too, using the quantum hyperalgebra \(\widetilde{U}_ q (\text{sl} (n))\). In the case of \(n=2\) the structure of the Hopf algebras involved, of the integral and of the irreducible representations is given explicitly. An interesting and illuminating survey of results mainly from the literature showing which of the various notions fit together best. quantum group; deformation; dual Hopf algebra; quantum hyperalgebra Mitsuhiro Takeuchi, Hopf algebra techniques applied to the quantum group \?_{\?}(\?\?(2)), Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990) Contemp. Math., vol. 134, Amer. Math. Soc., Providence, RI, 1992, pp. 309 -- 323. Quantum groups (quantized enveloping algebras) and related deformations, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Affine algebraic groups, hyperalgebra constructions, Twisted and skew group rings, crossed products Hopf algebra techniques applied to the quantum group \(U_ q(sl(2))\) | 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 Hodge constructed explicit bases for the homogeneous co-ordinate rings of Grassmannians and their Schubert varieties, in terms of ``standard monomials''. This result of Hodge was then generalized [by the reviewer, \textit{C. Musili} and \textit{C. S. Seshadri}] to semisimple algebraic groups in the series ``Geometry of \(G/P\)'' [see, e.g., Bull. Am. Math. Soc., New Ser. 1, 432-435 (1979; Zbl 0466.14020)] wherein bases are constructed in terms of ``standard monomials''. Sturmfels and White exhibited a correspondence between Gröbner bases and the standard monomial bases in the case of the Grassmannians. In this paper, the authors extend this correspondence to the case of minuscule \(G/P\). This paper makes a valuable contribution to invariant theory and representation theory. straightening law; Gröbner basis; standard monomial basis; Grassmannians A. M.~Cohen and R. H.~Cushman, Gröbner Bases and Standard Monomial Theory Computational Algebraic Geometry, Progress in Mathematics~109 (Birkhäuser, Boston, Boston, MA, 1993)~pp. 41--60. Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Rings with straightening laws, Hodge algebras, Grassmannians, Schubert varieties, flag manifolds Gröbner bases 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 Suppose that \(S=\bigoplus_{d\geqslant 0}S_d\) is a graded associative algebra generated by \(S_1\) with an automorphism \(\sigma\) of \(S_1\) as a linear space which can be extended to an automorphism \(\sigma\) of \(S\). Then there is a twisted algebra \(S^\sigma\) with multiplication defined as \(a*b=a\cdot\sigma(b)\) for homogeneous elements \(a,b\in S\).
The author considers the case when \(S\) is the complex polynomial algebra on \(Y_1,\dots,Y_n\) and \(\sigma\) is defined as \(\sigma(Y_1)=Y_1\) and \(\sigma(Y_{i+1})=Y_{i+1}+Y_i\) for \(i>0\). There is given a classification of the primitive ideals in \(S^\sigma\) which do not contain \(Y_1\). In particular, every primitive ideal in \(S^\sigma\) is generated by a regular sequence of homogeneous elements \(g_1,\dots,g_t\) where each \(g_i\) is irreducible and \(\sigma\)-invariant modulo the ideal generated by the preceding elements. So there exists a one to one correspondence between primitive ideals of \(S^\sigma\) and symplectic leaves of the Poisson structure induced by \(\sigma\). This result can be generalized to the case when an automorphism of \(S\) has a single eigenvalue. In this case the leaves are algebraic and realizable by orbits of an algebraic group. deformations of algebras; primitive ideals; twisted homogeneous coordinate rings; symplectic leaves; Poisson manifolds; complex polynomial rings; Poisson structures; primitive spectra Deformations of associative rings, Ordinary and skew polynomial rings and semigroup rings, Simple and semisimple modules, primitive rings and ideals in associative algebras, Graded rings and modules (associative rings and algebras), Ideals in associative algebras, Derivations, actions of Lie algebras, Noncommutative algebraic geometry Primitive and Poisson spectra of single-eigenvalue twists of polynomial 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 purpose of this paper is to give a decorated version of the Eynard-Orantin topological recursion using a 2D Topological Quantum Field Theory. We define a kernel for a 2D TQFT and use an algebraic reformulation of a topological recursion to define how to decorate a standard topological recursion by a 2D TQFT. The A-model side enumerative problem consists of counting cell graphs where in addition vertices are decorated by elements in a Frobenius algebra, and which are a decorated version of the generalized Catalan numbers. We show that the function that counts these decorated graphs, which is a decoration of the counting function of the generalized Catalan numbers by a Frobenius algebra, satisfies a topological recursion with respect to the edge-contraction axioms. The path we follow to pass from the A-model side to the remodeled B-model side is to use a discrete Laplace transform as a mirror symmetry map. We show that a decorated version by a 2D TQFT of the Eynard-Orantin differentials satisfies a decorated version of the Eynard-Orantin recursion formula. We illustrate these results using a toy model for the theory arising from the orbifold cohomology of the classifying space of a finite group. In this example, the graphs are orbifold cell graphs (graphs drawn on an orbifold punctured Riemann surface) defined out of the moduli space \(\overline {\mathcal M}_{g,n}(BG)\) of stable morphisms from twisted curves to the classifying space of a finite group \(G\). In particular we show that the cotangent class intersection numbers on the moduli space \(\overline {\mathcal{M}}_{g,n}(BG)\) satisfy a decorated Eynard-Orantin topological recursion and we derive an orbifold DVV equation as a consequence of it. This proves from a different perspective the known result which states that the \(\psi\)-class intersection numbers on \(\overline{\mathcal{M}}_{g,n}(BG)\) satisfy the Virasoro constraint condition. topological quantum field theory; topological recursion; Frobenius algebras; ribbon graphs; orbifold cohomology; Gromov-Witten invariants Relationships between algebraic curves and physics, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Applications of graph theory, Orbifold cohomology, Topological quantum field theories (aspects of differential topology), Topological field theories in quantum mechanics Topological recursion, topological quantum field theory and Gromov-Witten invariants of BG | 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 classical differential geometry there is the well-known Frobenius theorem stating that a smooth constant dimensional distribution on a manifold is integrable if and only if it is involutive. In this work, some variant of this theorem concerning the distributions over the algebra of truncated polynomials \({\mathcal O}_ n\) is obtained.
From the author's summary: The author describes the equivalence classes of TI-distributions, i.e., of those distributions \({\mathcal L}\) with respect to which the algebra \({\mathcal O}_ n\) has no nontrivial \({\mathcal L}\)- invariant ideals; he shows that over a perfect field any TI-distribution is equivalent to a general Lie algebra of Cartan type \(W_ s({\mathcal F})\), and he finds all the forms of the Zassenhaus algebra, in the process making essential use of the theory of representations of the chromatic quiver of Kronecker. prime characteristic; K-scheme; involutive distribution; integrable distributions; Zassenhaus algebra; Kronecker quiver; Frobenius theorem; algebra of truncated polynomials; TI-distributions; Lie algebra of Cartan type M. I. Kuznetsov, ''Distributions over a truncated polynomial algebra,'' Mat. Sb. 136(2), 187--205 (1988). [Sb. Math. 64 (1), 187--205 (1989). Modular Lie (super)algebras, Algebraic theory of abelian varieties, Representation theory of associative rings and algebras, Automorphisms, derivations, other operators for Lie algebras and super algebras Distributions over an algebra of truncated 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 \((R_0,{\mathfrak m}_0,k_0)\) be a local ring and \(R=R_0[R_1]\) a commutative Noetherian graded ring. Let \(M\) and \(N\) be two \(\mathbb{Z}\)-graded finitely generated \(R\)-modules and \(R_+\) denote the irrelevant ideal of \(R\). The paper under review studies the asymptotic behaviour of the graded pieces of the graded generalized local cohomology modules \[\text{H}_{R_+}^i(M,N):=\underset{n} {\varinjlim}\ \text{Ext}^i_R \left(\frac{M}{(R_+)^n M},N \right); \ i\in \mathbb{N}_0.\]
It is known that \(\text{H}_{R_+}^i(M,N)\) has a \(\mathbb{Z}\)-grading in a natural way and that the \(R_0\)-modules \(\text{H}_{R_+}^i(M,N)_n\) is finitely generated for all \(n\in \mathbb{Z}\) and \(\text{H}_{R_+}^i(M,N)_n=0\) for all \(n\gg 0\). So, the study of the asymptotic behaviour of \(\text{H}_{R_+}^i(M,N)_n\) for \(n\rightarrow -\infty\) is of great interest. Especially, in the case \(\dim R_0=0\), by [17, Theorem 2.2], the \(R\)-module \(\text{H}_{R_+}^i(M,N)\) is Artinian, and so there exists a polynomial \(P(X)\in \mathbb{Q}[X]\) such that \(P(n)=\ell_{R_0}(\text{H}_ {R_+}^i(M,N)_n)\) for all \(n\ll 0\).
In this paper, the authors establish some partial results in the case \(\dim R_0>0\). In particular, they show that \(\Gamma_{{\mathfrak m}_0R}(\text{H}_{R_+}^i(M,N))\) is Artinian for all \(i\leq g(M,N)\), where \(g(M,N):=\inf\{i\in \mathbb{N}_0\mid \sharp \{n\in \mathbb{Z} \mid \ell_{R_0}(\text{H}_{R_+}^i(M,N)_n)=\infty \}\) \\
\(=\infty\}.\) generalized local cohomology; Hilbert-Kirby polynomial; irrelevant ideal Local cohomology and commutative rings, Local cohomology and algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Commutative Artinian rings and modules, finite-dimensional algebras Hilbert-Kirby polynomials in generalized local cohomology modules | 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 \(k\) be a perfect field of characteristic different from two. We show that the filtration on the Grothendieck-Witt group \(GW(k)\) induced by the slice filtration for the sphere spectrum in the motivic stable homotopy category is the \(I\)-adic filtration, where \(I\) is the augmentation ideal in \(GW(k)\). Levine, M.: The slice filtration and Grothendieck-Witt groups. Pure Appl. Math. Q. \textbf{7}(4) , 1543-1584 (2011). Special Issue, In memory of Eckart Viehweg Motivic cohomology; motivic homotopy theory, Algebraic cycles, \(K\)-theory of schemes, Stable homotopy theory, spectra The slice filtration and Grothendieck-Witt 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 \(K_0\) groups of simply connected semisimple algebraic groups are calculated. The triviality of the Chow groups \(CH^1\) and \(CH^2\) of such groups is obtained as a consequence. simply connected semisimple algebraic groups; Chow groups \(K\)-theory of schemes, \(K\)-theory and homology; cyclic homology and cohomology, (Equivariant) Chow groups and rings; motives On Grothendieck group of simply connected semisimple algebraic 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 An invertible polynomial is a weighted homogeneous polynomial with the number of monomials coinciding with the number of variables and such that the weights of the variables and the quasi-degree are well defined. In the framework of the search for mirror symmetric orbifold Landau-Ginzburg models, P. Berglund and M. Henningson considered a pair \((f, G)\) consisting of an invertible polynomial \(f\) and an abelian group \(G\) of its symmetries together with a dual pair \((\widetilde{f}, \widetilde{G})\). We consider the so-called orbifold E-function of such a pair \((f, G)\) which is a generating function for the exponents of the monodromy action on an orbifold version of the mixed Hodge structure on the Milnor fibre of \(f\). We prove that the orbifold E-functions of Berglund-Henningson dual pairs coincide up to a sign depending on the number of variables and a simple change of variables. The proof is based on a relation between monomials (say, elements of a monomial basis of the Milnor algebra of an invertible polynomial) and elements of the whole symmetry group of the dual polynomial. mirror symmetry; singularity; mixed Hodge structure; monodromy; orbifold Ebeling, W; Gusein-Zade, SM; Takahashi, A, Orbifold E-functions of dual invertible polynomials, J. Geom. Phys., 106, 184-191, (2016) Complex surface and hypersurface singularities, Mixed Hodge theory of singular varieties (complex-analytic aspects), Mirror symmetry (algebro-geometric aspects), Group actions on varieties or schemes (quotients) Orbifold E-functions of 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 The main object of this paper is semisimplicity of quantum cohomology of varieties. The variety is said to have semisimple quantum cohomology if its quantum cohomology is a semisimple algebra. The main result of this paper (Theorem 3.1) states that if quantum cohomology of the variety \(X\) is semisimple, then the same is true for the blow up of \(X\) at the smooth points. For the proof, \textit{A. Gathmann}'s
[J. Algebr. Geom. 10, No.3, 399--432 (2001; Zbl 1080.14064)] description of Gromov-Witten invariants of such blow ups is used (Theorem 3.2). The main result is motivated by the conjecture on the equivalence between the semisimplicity of quantum cohomology of variety and existence of full exceptional collection on it. The version of this conjecture that agrees with the main theorem (Conjecture 4.3, proposed in \textit{A. Bayer} and \textit{Yu. I. Manin} [in: The Fano conference, Torino, 2002 (Torino: Università di Torino, Dipartimento di Matematica) 143--173 (2004; Zbl 1077.14082)]) is wrong (one can see it on the example of five-dimensional cubic, which is semisimple by \textit{G. Tian} and \textit{G.Xu} [Math. Res. Lett. 4, No.4, 481--488 (1997; Zbl 0919.14024)] but have not exceptional collection of maximal length). But nevertheless, the main result of this paper is interesting. Using this, one can produce a lot of examples of semisimple varieties (in particular, with arbitrary Picard rank). Moreover, this means that one can consider the relations between semisimplicity and exceptional systems not only for Fano varieties but also for the other types of them. semisimplicity; exceptional collections, blow up A. Bayer, ''Semisimple Quantum Cohomology and Blowups,'' Int. Math. Res. Not., No. 40, 2069--2083 (2004); arXiv:math/0403260. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories Semisimple quantum cohomology and blowups | 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 numerical polynomial is a polynomial with coefficients in \(\mathbb Q\) which takes integer values at the integers. Also, instead of integers one can consider any subring of \(\mathbb Q\) with unit (like, say \(\mathbb Z[1/2]\)). Furthermore, define a Laurent polynomial \(f(x)\in\mathbb Q\) to be stable numerical if \(x^kf(x)\) is numerical for some \(k\).
It turns out to be that there are many cohomology theories \(h\) whose Hopf algebra \(A_h\) of degree 0 stable cooperations are stable numerical. A well-known example is the complex \(K\)-theory. The author shows many examples of elliptic cohomology \(E\) whose algebra \(E_h\) is stable numerical. Elliptic cohomology; formal groups; Bernoulli numbers K. Johnson,, Numerical polynomials and endomorphisms of formal group laws. London Math. Soc. Lect. Notes 342, 204-213 (2007) Formal groups, \(p\)-divisible groups, Bernoulli and Euler numbers and polynomials, Polynomials over commutative rings, Polynomial rings and ideals; rings of integer-valued polynomials, Generalized (extraordinary) homology and cohomology theories in algebraic topology Numerical polynomials and endomorphisms of formal group laws | 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 See the joint review of the entire collection in Zbl 0588.14028. Illusie, L.; Szpiro, L. (ed.), Déformations de groupes de Barsotti-Tate, (1985), London Formal groups, \(p\)-divisible groups, Elliptic curves over local fields, Abelian varieties of dimension \(> 1\), Class field theory; \(p\)-adic formal groups Deformations of Barsotti-Tate groups (after A. Grothendieck) | 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 is an outstanding work which suggest a series of real enumerative invariants of new type and provides a new, deep and unexpected explanation of the geometry behind the refined tropical enumerative invariants. The enumerative problem stated and solved in the paper is as follows: count real rational curves in the plane (or another real toric surface) having a given degree and passing through a fixed configuration of points on the coordinate axes such that the fixed points have either real or purely imaginary coordinates. The key observation is that, in such a case, the signed area of the complex amoeba (the image under the map \((z,w)\mapsto(\log|z|,\log|w|)\)) of a chosen component of the set of non-real points of the curve equals \(\frac{\pi^2}{2}\) times an integer, called the quantum index of the curve with a chosen component of the set of non-real points (or, equivalently, with a given orientation of the real point set). The first main result of the paper is that the number of the real plane curves of a given degree and of a given quantum index, and obeying point constraints as above, does not depend on the choice of the constraint when counting curves with the Welschinger sign (see [\textit{J.-Y. Welschinger}, Invent. Math. 162, No. 1, 195--234 (2005; Zbl 1082.14052)]). The second main theorem states that the generating Laurent polynomial for the above invariants equals the refined Block-Göttsche invariant (multiplied with a standard polynomial) for the corresponding tropical enumerative problem. Namely, the tropical enumerative problem reads as the count of plane rational tropical curves of a given (tropical) degree that pass through a fixed configuration of points on the boundary of the tropical toric surface and are counted with the Block-Göttsche weights. The tropical curves in count appear to be trivalent trees, and the Block-Göttsche weight then equals the product of the expressions \(\frac{y^{\mu/2}-y^{-\mu/2}}{y^{1/2}-y^{-1/2}}\) over all the trivalent vertices, where \(\mu\) is Mikhalkin's weight of the vertex (see [\textit{F. Block} and \textit{L. Göttsche}, Compos. Math. 152, No. 1, 115--151 (2016; Zbl 1348.14125)]). These two results uncover a nice geometry hidden in the formally defined Block-Göttsche invariants. One of the exciting consequences is that the count of real curves in the considered problem determines the count of all complex curves in the complexified problem. This, in particular, explains the phenomenon of the logarithmic asymptotic equivalence of the real and complex count. real plane rational curves; complex amoebas; tropical curves; quantum index; Block-Göttsche invariants Enumerative problems (combinatorial problems) in algebraic geometry, Plane and space curves, Real algebraic sets, Combinatorial aspects of tropical varieties Quantum indices and refined enumeration of real plane 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 Let \(\underline a=(0<a_1<a_2< \cdots <a_k<n)\) be a sequence of integers and consider the general flag variety \(\mathbb{F}(\underline a, \mathbb{C}^n)\). Its homology is generated by the Schubert varieties. The Pieri formula gives a description of the cup product when one of the factors is a Chern class of a tautological bundle.
The (small) quantum cohomology of \(\mathbb{F}(\underline a, \mathbb{C}^n)\) is studied. The main approach to compute the quantum products is a quantum Pieri formula from \textit{I. Ciocan-Fontanine} [Duke Math. J. 98, No.3, 485--524 (1999; Zbl 0969.14039)]. This formula is reproven here by a geometrical argument which reduces it to the usual Pieri formula. The algorithm for computing a general quantum product is thus to use a quantum Giambelli formula (which expresses a quantum cohomology class as a polynomial in the Chern classes of tautological bundles) and then applying the quantum Pieri formula. quantum cohomology; flag manifold; quantum Pieri formula; quantum Giambelli formula Buch A.S.: Quantum cohomology of partial flag manifolds. Trans. Am. Math. Soc. 357, 443--458 (2005) (electronic) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum cohomology of partial flag 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 The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by \textit{A. Knutson} and \textit{E. Miller} [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, \textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] obtained alternative combinatorics for the class of ``vexillary'' matrix Schubert varieties. We initiate a study of general diagonal degenerations, relating them to a neglected formula of \textit{A. Lascoux} [``Chern and Yang through ice'', Preprint] in terms of the 6-vertex ice model (recently rediscovered by \textit{T. Lam} et al. [Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)] in the guise of ``bumpless pipe dreams''). Schubert polynomial; bumpless pipe dream; matrix Schubert variety Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Gröbner geometry of Schubert polynomials through ice | 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 so-called Schubert polynomials are significant in combinatorics and in geometry.
There are two combinatorial ways of writing Schubert polynomials as sums of products of linear factors. One is parameterized by ``pipe dreams'' [\textit{A. Knutson} and \textit{E. Miller}, Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)], the other is parameterized by ``bumpless pipe dreams'' [\textit{A. Lascoux}, ``Chern and Yang through ice'', Preprint; \textit{T. Lam} et al., Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)].
In geometry Schubert polynomials are fundamental classes of matrix Schubert varieties [\textit{L. M. Fehér} and \textit{R. Rimányi}, Cent. Eur. J. Math. 1, No. 4, 418--434 (2003; Zbl 1038.57008); Knutson-Miller, loc. cit.]. Under favorable (e.g. Gröbner) degenerations the fundamental class of a variety does not change. Hence if a matrix Schubert variety is degenerated to a union of linear spaces, then its fundamental class is written as a sum of product of linear factors. Such geometric interpretation for the ``pipe dream'' formula is known, via the anti-diagonal Gröbner degeneration.
The paper under review searches for such a geometric interpretation of the ``bumpless pipe dream'' formula for Schubert polynomials. Namely, the authors conjecture that the ``bumpless pipe dream'' formula corresponds naturally to any diagonal Gröbner degeneration. They prove this conjecture for a large class of permutations they call banner permutations. Basic steps of their recursive arguments rely on Lascoux-Schützenberger transitions. Schubert polynomial; bumpless pipe dream; matrix Schubert variety; Gröbner bases Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Gröbner geometry of Schubert polynomials through ice | 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 \(\mathbb C_q[ \text{Mat}_{k\times m}]\) be the \(q\)-deformation of the coordinate ring of the space of \(k\times m\) complex matrices where \(k\leq m\). This is the \(\mathbb C(q)\)-algebra with unity generated by indeterminates \(x_{i,j}\) for \(i\in [1\dots k]\) and \(j\in [1\dots m]\) subject to the Faddeev-Reshetikhin-Takhtadzhyan relations. The article is concerned with a special family of elements \(\Delta_{I,J}\in \mathbb C_q[\text{Mat}_{k\times m}]\) indexed by pairs of non-empty subsets \(I\) and \(J\) of \([1\dots k]\) and \([1\dots m]\) respectively with \(| I| =| J| =l\), defined by
\[
\Delta_{IJ}:= \sum_{\sigma\in S_l} (-1)^{-l(\sigma)} x_{i_1,j_{\sigma(1)}} \dots x_{i_l, j_{\sigma(l)}},
\]
where \(I=\{i_1<\cdots<i_l\}, J=\{j_1<\cdots<j_l\}\), and \(l(\sigma)\) is the length of the \(l\)-permutation \(\sigma\). The elements \(\Delta_{I,J}\) are called quantum minors.
Two quantum minors \(\Delta_{A,B}\) and \(\Delta_{C,D}\) are said to quasi-commute if \(\Delta_{C,D}\Delta_{A,B} =q^c\Delta_{A,B}\Delta_{C,D}\) for some integer \(c\). This integer is uniquely determined by \(\Delta_{A,B}\) and \(\Delta_{C,D}\) and is denoted by \(c(\Delta_{A,B}| \Delta_{C,D})\).
The main result of the article it to find necessary and sufficient conditions for two quantum minors to quasi-commute and to explicitely compute \(c(\Delta_{A,B}| \Delta_{C,D})\) in terms of \(A,B,C,D\).
The techniques used are developed in [\textit{B. Leclerc} and \textit{A. Zelevinsky}, in: Kirillov's seminar on representation theory, Transl., Ser. 2, Am. Math. Soc. 181(35), 85--108 (1998; Zbl 0894.14021)]. In [loc. cit.] only the quantum flag variety is considered, so the problem in the general case is much more difficult, however the authors demonstrate that the problem can de reduced to a special case of the problem treated in [loc. cit.]. The criterion for quasi-commutativity is described in terms of the notion of ``weak separability'' of [loc. cit.]. quantum groups; quantum Plücker coordinates; quantum Grassmannians; quantum flag manifolds J. Scott, \textit{Quasi-commuting families of quantum minors}, \textit{J. Algebra}\textbf{290} (2005) 204 [math/0008100]. Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized function algebras) and their representations Quasi-commuting families of quantum minors | 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 first three sections of the paper under review have a survey's character. The first one is devoted to recall some sufficient conditions for a polynomial vector field \(f:\mathbb{R}^n\to\mathbb{R}^n\) to be proper. The interested reader is also refered to the work by \textit{C. Ueno} and the reviewer [in: Real algebraic geometry, Lect. Notes Math. 1524, 240--256 (1992; Zbl 0793.14038)]. The second one collects today's classical results by \textit{E. Becker} and \textit{T. Wörmann} [in: Recent advances in real algebraic geometry and quadratic forms. Contemp. Math. 155, 271--291 (1994; Zbl 0835.11016)] and by \textit{P. Pedersen}, \textit{M.-F. Roy} and \textit{A. Szpirglas} [in: Computational algebraic geometry. Prog. Math. 109, 203--224 (1993; Zbl 0806.14042)] which lead to effective formulae to compute the cardinal of the fibers and the topological degree of some proper polynomial maps. The particular case of nondegenerated homogeneous polynomial vector fields with fixed multi-degree is explained in Section 3, by following, essentially, two papers of \textit{G. N. Khimshiashvili} [Soobshch. Akad. Nauk Gruz. SSR 85, 309--312 (1977; Zbl 0346.55008); Georgian Math. J. 8, No. 1, 97--109 (2001; Zbl 0987.32013)]. A very nice and clear article on this topic is due to \textit{A. Lecki} and \textit{Z. Szafraniec} [in: Topology in nonlinear analysis. Banach Cent. Publ. 35, 73--83 (1996; Zbl 0872.55004)]. The most original part of the paper is Section 4, which concerns the bidimensional case, with emphasis on the effectiveness of the proposed methods. Sometimes along the paper the conditions imposed to the vector field \(f\) are unclear for the reviewer. First, notice that in the real case properness does not imply finiteness of the fibers; for example, many fibers of the proper polynomial map
\[
f: \mathbb{R}^2\to\mathbb{R}^2: (x,y)\to (x^2+y^2,x^2+y^2)
\]
are circumferences. So, it has no meaning to compute ``the cardinal of the fibers'' in such a case. On the other hand, the author seems to correct this in the last line of page 5329 by imposing \(f\) to be smooth. The reviewer has not found the meaning of smooth in this context along the paper; perhaps it could mean that the Jacobian determinant of \(f\) does not vanish at any point of \(\mathbb{R}^n\). But this, together with the properness of \(f\), implies that \(f\) is a diffeomorphism, by Hadamard theorem, and so the fibers are singletons. proper endomorphism; topological degree; signature formula T. Aliashvili, ''Counting real roots of polynomial endomorphisms,'' J. Math. Sci., 118, 5325--5346 (2003). Effectivity, complexity and computational aspects of algebraic geometry, Real algebraic and real-analytic geometry, Polynomials and rational functions of one complex variable, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) Counting real roots of polynomial endomorphisms. | 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 is an introduction to a new development in classical enumerative geometry based on the theory of quantum cohomology. The authors discuss the construction of the moduli space \(M_{g,n}(X,\beta)\) of isomorphism classes of pointed maps \((C,p_1,\ldots,p_n,\mu)\), where \(C\) is a smooth projective curve of genus \(g\), with fixed distinct \(n\) points on it, and \(\mu\) is a map \(C\to X\) to a fixed smooth algebraic variety such that \(\mu_*([C])\) is equal to the fixed algebraic 1-cycle \(\beta\) on \(X\). This moduli space admits a natural compactification \(\overline M_{g,n}(X,\beta)\) which corresponds to adding stable degenerate curves and stable maps. Based on these spaces, one can define the Gromov-Witten invariants \(I_\beta(\gamma_1,\ldots,\gamma_n)\). They assign to arbitrary algebraic cycles \(\gamma_1,\ldots,\gamma_n\) the top dimensional part of the cohomology class \(\rho_1^*(\gamma_1)\cup\ldots\cup\rho_n^*(\gamma_n)\) on \(\overline M_{0,n}(X,\beta)\), where \(\rho_i:\overline M_{0,n}(X,\beta)\to X\) is defined by sending \((C,p_1,\ldots,p_n,\mu)\) to \(\mu(p_i)\). When \(g=0\) and \(n =3\) the Gromov-Witten invariants lead to the definition of quantum cohomology of \(X\). When the target space is a homogeneous space with automorphism group \(G\) (e.g. projective space) and \(\gamma_i = [\Gamma_i]\) for some pure-dimensional subvarieties of \(X\) with \(\sum_{i=1}^n \text{ codim}(\Gamma_i) = \dim(X)+\int_\beta c_1(T_X)+n-3\), one can interpret \(I_\beta(\gamma_1,\ldots,\gamma_n)\) as the number of rational curves \(R\) on \(X\) which intersect general translates of the \(\Gamma_i\) at a fixed point \(q_i\in R\). The property of the associativity of the quantum cohomology gives some non-trivial relations for enumerative problems on \(X\). For example, the authors discuss the solutions of the following classical problems: Find the degree of the locus of plane rational curves of given degree; find the number of degree \(d\) rational curves in \(\mathbb{P}^3\) meeting \(a\) general lines and \(b\) general points, where \(a=2b=3d\). enumerative geometry; quantum cohomology; moduli space; Gromov-Witten invariants Fulton, W., and Pandharipande, R.: Notes on stable maps and quantum cohomology. \textit{Algebraic geometry}, Santa Cruz 1995, Proc. Sympos. Pure Math, vol. 62, Amer. Math. Soc., Providence, RI (1997), 45--96. Enumerative problems (combinatorial problems) in algebraic geometry, Families, moduli of curves (algebraic), Birational geometry, Quantum field theory on curved space or space-time backgrounds, Quantization in field theory; cohomological methods Notes on stable maps and 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 \(c\) be an element of the Weyl algebra \(\mathcal W(d)\) which is given by a strictly positive operator in the Schrödinger representation. It is shown that, under some conditions, there exist certain elements \(b_1,\dots,b_d\) from \(\mathcal W(d)\) such that \(\sum^d_{j=1} b_j cb^*_j\) is a finite sum of squares. Weyl algebra; Hermitean element; Stone-von Neumann theorem; \(m\)-admissible wedge; Fock-Bergman realization; \(^*\)-representation; algebraically bounded part of a \(^*\)-algebra Konrad Schmüdgen, A strict Positivstellensatz for the Weyl algebra, Math. Ann. 331 (2005), no. 4, 779 -- 794. Linear operators in \({}^*\)-algebras, Rings of differential operators (associative algebraic aspects), Sums of squares and representations by other particular quadratic forms, Semialgebraic sets and related spaces, Rings with involution; Lie, Jordan and other nonassociative structures, Representations of (nonselfadjoint) operator algebras, Algebras of unbounded operators; partial algebras of operators A strict Positivstellensatz for the Weyl algebra | 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 \(w \in S_n\) be a permutation, \(X_w\) the corresponding Schubert variety, and \(P_{id,w}(q)\) the corresponding Kazhdan-Lusztig polynomial. It is known that \(P_{id,w}(1)\) equals \(1\) if and only if \(X_w\) is smooth. (Since Kazhdan-Lusztig polynomials have non--negative integer coefficients and constant term \(1\), \(P_{id,w}(1)=1\) is equivalent to \(P_{id,w}(q)=1\)). In this paper, the author gives necessary and sufficient conditions for \(P_{id,w}(1)=2\): he shows that this is equivalent to a geometric condition, namely that the singular locus of \(X_w\) has exactly one irreducible component, plus a combinatorial one, namely that the permutation \(w\) avoids a list of six patterns. He further shows that when \(P_{id,w}(1)=2\), \(P_{id,w}(q)=1+q^h\) where \(h\) is computed combinatorially from \(w\).
An appendix by Sara Billey and Jonathan Weed gives a characterization of \(P_{id,w}(1)=2\) purely in terms of pattern avoidance, the number of patterns used being \(66\). Schubert variety; Kazhdan-Lusztig Woo, A, Permutations with Kazhdan-Lusztig polynomial \(P_{Id, w}(q)=1+q^h\), with an appendix by sara billey and jonathan weed, Electron. J. Combin., 16, r10, (2009) Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects) Permutations with Kazhdan-Lusztig polynomial \(P_{id,w}(q)=1+q^{h}\). With an appendix by Sara Billey and Jonathan Weed | 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 short text was originally written as a contribution to the ``Grothendieck day'' which took place in Utrecht on April 12, 1996. It is brief and informal, and was intended to give the audience some partial idea of what is contained in \textit{A. Grothendieck}'s long manuscript ``La longue marche à travers la théorie de Galois'' (in preparation by J. Malgoire), see also \textit{A. Grothendieck} in: Geometric Galois actions. 1, Lond. Math. Soc. Lect. Note Ser. 242, 5-48 and 49-58 (1997; Zbl 0901.14001 and Zbl 0901.14002). anabelian geometry; arithmetic fundamental group; Galois theory Coverings of curves, fundamental group, Families, moduli of curves (algebraic), Homotopy theory and fundamental groups in algebraic geometry Grothendieck's ``Long March through Galois 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 This Ph.D. thesis presents a mathematically rigorous, purely algebro-geometric contribution to a very recent development in both mathematics (complex geometry) and physics (topological quantum field theory). The key words here are ``Witten theory'' (in greater generality) and ``quantum cohomology theory'' (in a more specific sense). -- However, the subjects studied in this doctoral dissertation, and the methods developed for investigating them, are of purely algebro-geometric nature. The background of the research conducted in this thesis is provided by the fact that, during the past ten years, there has been an enormous progress in the enumerative geometry of moduli spaces of algebraic curves, which was essentially promoted by physical reasoning in topological quantum field theory. Especially the investigation of Witten's supersymmetric sigma model has produced deep algebro-geometric conjectures and predictions on the cohomology rings and Chow rings of various moduli spaces for algebraic curves, based on physical intuition and axiomatic assumptions which have been lacking mathematical affirmation and rigor for quite a time. The concept of quantum cohomology, independently proposed by E. Witten and C. Vafa in 1991, and the postulation of the existence of the so-called Gromov-Witten invariants, has been a great source of challenge and inspiration for algebraic geometers ever since. In the meantime, there are mathematically rigorous approaches to establishing quantum cohomology rings and Gromov-Witten invariants for large classes of complex phase spaces (manifolds), and in special cases some of the (low-degree) Gromov-Witten invariants have been efficiently calculated and interpreted. The essential break-through in modern enumerative algebraic geometry has begun, about four years ago, with the pioneering work of \textit{M. Kontsevich} and \textit{Yu. Manin} [cf. Commun. Math. Phys. 164, 525-562 (1994; Zbl 0853.14020)]. Their basic idea for defining the right objects (quantum cohomology rings, Gromov-Witten classes) was to utilize the concept of stable maps from \(n\)-pointed stable algebraic curves to the target variety, a notion that was ingeniously introduced by M. Kontsevich shortly before. In his thesis under review, the author takes up the construction of the Gromov-Witten classes by Kontsevich and Manin, as well as another recent construction due to \textit{R. Vakil} [cf. ``The enumerative geometry of rational and elliptic curves in projective space'' (Preprint alg-geom/9709007)], which leads to the so-called degeneration invariants of the cohomology of mouli spaces for stable curves, and he gives explicit computations and enumerative interpretations of them for special cases of the ambient manifold (phase space).
The work is subdivided into three chapters. After a thorough introductory preface, chapter I gives a brief but very clear overview of the basic concepts and facts used in the sequel: Gromov-Witten invariants after Kontsevich-Manin, gravitational descendants as extensions of the Gromov-Witten invariants and, as an application of the latter ones, computations of virtual numbers of certain rational curves (i.e., higher-order-contact curves) in projective space. -- Chapter II provides calculations of Gromov-Witten invariants of blowing-up's of points in the so-called convex varieties, together with a remarkable vanishing theorem for Gromov-Witten invariants under special conditions. The enumerative significance of the computed Gromov-Witten invariants is then discussed in general, as well as in the case of blowing-up's of certain projective spaces. This chapter concludes with a method of counting curves in projective space of given homology class and prescribed tangency conditions with respect to a given point, followed on by a few explicit numerical examples. -- Chapter III deals with the degeneration invariants initiated by \textit{L. Caporaso} and \textit{J. Harris} [cf. Invent. Math. 131, No. 2, 345-392 (1998)], and having been extensively studied in the very recent paper by \textit{R. Vakil} (cited above). The underlying degeneration methods, originally developed with respect to hyperplanes in projective space, are here extended towards arbitrary hypersurfaces in \(\mathbb{P}^n\). This leads to an extension of the results of R. Vakil. However, since hypersurfaces in \(\mathbb{P}^n\) are in general not convex, this requires the very subtle use of certain virtual fundamental classes on the moduli spaces of curves under consideration, and that makes the author's result particularly significant. Subsequently, an explicit formula for some degeneration invariants of degree one is derived separately, and this is then applied to establishing explicit relations between Gromov-Witten invariants of projective spaces and their hypersurfaces, at least up to degree two and not involving any unknown correction terms.
In the concluding section of this thesis, the latter results are applied to the case of the quintic threefold in \(\mathbb{P}^4\), which provides a highly nontrivial illustrating example for the techniques developed in this chapter. As the author points out, similar degeneration methods are supposed to work for arbitrary degree (of the degeneration invariants), and perhaps also for higher-genus curves, which indicates that the degeneration invariants (after R. Vakil) are possibly more suitable for computing numbers of higher-genus curves in varieties, quite so in contrast to the applicability of the Gromov-Witten theory in this context.
Altogether, this is an excellently written Ph.D. thesis which contains a lot of original ideas, extended methods and techniques, and deep-going results, and that with regard to a very recent and central topic in contemporary algebraic geometry. Without any doubt, this work is a great contribution to the development of the subject, with a promising prospective as for further elaboration. Witten theory; quantum cohomology theory; Gromov-Witten invariants; higher-order-contact curves; quintic threefold A Gathmann, Gromov-Witten and degeneration invariants: computation and enumerative significance, PhD thesis, University of Hannover (1998) Families, moduli of curves (algebraic), Topological properties in algebraic geometry, Special varieties, Special algebraic curves and curves of low genus, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Quantization in field theory; cohomological methods Gromov-Witten- and degeneration invariants: Computation and enumerative significance | 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 make a correction to Remark 4.3 and the proof of Theorem 4.2 (Peterson's Theorem) in the paper cited in the heading [ibid. 16, No. 2, 363--392 (2003; Zbl 1057.14065)] which identifies \(qH^*(\text{SL}_n/P)\) with the coordinate ring \(\mathcal O(\mathcal Y_P)\) of a certain affine stratum of the Peterson variety \(\mathcal Y\). Explicitly, we introduce additional coordinates to obtain a complete coordinate system on \(B^+w_P B^-/B^-\) and then show that they lie in the defining ideal of the Peterson variety \(\mathcal Y_P\), hence play no role in the presentation of \(\mathcal O(\mathcal Y_P)\). flag varieties; quantum cohomology; total positivity; cell decomposition Rietsch, K.: Errata to: ''totally positive Toeplitz matrices and quantum cohomology of partial flag varieties''. J. amer. Math. soc. 21, 611-614 (2008) Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over the reals, the complexes, the quaternions, Positive matrices and their generalizations; cones of matrices, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Errata to ``Totally positive Toeplitz matrices and quantum cohomology of partial flag 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 Quantum entanglement was first recognized as a feature of quantum mechanics in the famous paper of \textit{A. Einstein, B. Podolsky} and \textit{N. Rosen} [Phys. Rev., II. Ser. 47, 777--780 (1935; Zbl 0012.04201)]. Recently it has been realized that quantum entanglement is a key ingredient in quantum computation, quantum communication, and quantum cryptography. In this paper, we introduce algebraic sets, which are determinantal varieties in the complex projective spaces or the products of complex projective spaces, for the mixed states on bipartite or multipartite quantum systems as their invariants under local unitary transformations. These invariants are naturally arised from the physical consideration of measuring mixed states by separable pure states. Our construction has applications in the following important topics in quantum information theory: (1) separability criterion, it is proved that the algebraic sets must be a union of the linear subspaces if the mixed states are separable; (2) simulation of Hamiltonians, it is proved that the simulation of semipositive Hamiltonians of the same rank implies the projective isomorphisms of the corresponding algebraic sets; (3) construction of bound entangled mixed states, examples of the entangled mixed states which are invariant under partial transpositions (thus PPT bound entanglement) are constructed systematically from our new separability criterion. DOI: 10.1063/1.2194629 Quantum measurement theory, state operations, state preparations, Special varieties, Quantum computation Quantum entanglement and geometry of determinantal 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 Motivated by a result of \textit{P. Fiebig} [Adv. Math. 217, No. 2, 683--712 (2008; Zbl 1140.14044)], we categorify some properties of Kazhdan-Lusztig polynomials via sheaves on Bruhat moment graphs. In order to do this, we develop new techniques and apply them to the combinatorial data encoded in these moment graphs. Bruhat graphs; moment graphs; Kazhdan-Lusztig polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Graph theory, Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Moment graphs and Kazhdan-Lusztig 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 paper gives some results on the generalized Hamming weights of duals of BCH codes. Let \(w(A)= |\{i:a_i \neq 0\) for some \(a= (a_1, \dots, a_n) \in A\}\) denote the support of the subcode \(A\). The main result is the following:
Consider the primitive \(t\)-error-correcting BCH code \(\text{BCH} (t)\) of length \(n= p^m-1\), over \(F_p\) with designed distance \(d= 2t+1\). Let
\[
s= \begin{cases} 2t-1, \quad & \text{if } p |2t, \\ 2t, \quad & \text{if } p \nmid 2t. \end{cases}
\]
Assume that \(2t\leq p^{m \over 2}\) and \(m\equiv 0 \pmod 2\). Then the following holds:
(a) If \(p^{m \over 2} \equiv -1 \pmod s\), then there exists, for \(1\leq r\leq {m \over 2}\), an \(r\)-dimensional subcode \(A\subseteq\text{BCH}(t)^\perp\) of weight
\[
w(A)=(1-p^{-r}) \bigl(p^m- (s-1) p^{m\over 2} \bigr).
\]
(b) If \(m\equiv 0 \pmod 4\) and \(p^{m \over 4} \equiv -1 \pmod s\), then there exists, for \(1\leq r\leq {m\over 4}\), an \(r\)-dimensional subcode \(A' \subseteq \text{BCH} (t)^\perp\) with
\[
w(A') =(1-p^{-r}) \bigl(p^m+ (s-1) p^{m \over 2} \bigr).
\]
algebraic function fields; generalized Hamming weights 5.I. Duursma, H. Stichtenoth, and C. Voß, ''Generalized Hamming weights for duals of BCH codes and maximal algebraic function fields'', preprint 1993. Geometric methods (including applications of algebraic geometry) applied to coding theory, Special algebraic curves and curves of low genus Generalized Hamming weights for duals of BCH codes, and maximal algebraic function fields | 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 associate a polynomial to any diagram of unit cells in the first quadrant of the plane using Kohnert's algorithm for moving cells down. In this way, for every weak composition one can choose a cell diagram with corresponding row-counts, with each choice giving rise to a combinatorially-defined basis of polynomials. These \textit{Kohnert bases} provide a simultaneous generalization of Schubert polynomials and Demazure characters for the general linear group. Using the monomial and fundamental slide bases defined earlier by the authors, we show that Kohnert polynomials stabilize to quasisymmetric functions that are nonnegative on the fundamental basis for quasisymmetric functions. For initial applications, we define and study two new Kohnert bases. The elements of one basis are conjecturally Schubert-positive and stabilize to the skew-Schur functions; the elements of the other basis stabilize to a new basis of quasisymmetric functions that contains the Schur functions. Schubert polynomials; Demazure characters; key polynomials; fundamental slide polynomials Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Symmetric functions and generalizations Kohnert 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 earlier work of the first and the third name authors [J. Am. Math. Soc. 31, No. 3, 661--697 (2018; Zbl 1387.05265)] introduced the algebra \(\mathbb{A}_{q,t}\) and its polynomial representation. In this paper we construct an action of this algebra on the equivariant \(K\)-theory of certain smooth strata in the flag Hilbert scheme of points on the plane. In this presentation, the fixed points of the torus action correspond to generalized Macdonald polynomials, and the matrix elements of the operators have an explicit presentation. Graded rings and modules (associative rings and algebras), Combinatorial aspects of groups and algebras, Parametrization (Chow and Hilbert schemes), Derived categories and associative algebras, Hermitian \(K\)-theory, relations with \(K\)-theory of rings, Orthogonal polynomials and functions associated with root systems, Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) The \(\mathbb{A}_{q,t}\) algebra and parabolic flag 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 Let \(V\) be a symplectic vector space and \(LG\) the Lagrangian Grassmannian which parameterizes the maximal isotropic subspaces in \(V\). In the paper under review the author studies the quantum cohomology ring \(QH^*(LG)\) and proves that its multiplicative structure is determined by the ring of \(Q\)-polynomials. quantum cohomology; Lagrangian Grassmannian; \(Q\)-polynomials. Kresch A., Tamvakis H.: Quantum cohomology of the Lagrangian Grassmannian. J. Algebraic Geom. 12, 777--810 (2003) Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology of the Lagrangian 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 This paper is the third episode of the trilogy ``Quantum cohomology of minuscule homogeneous spaces''. A (co)minuscule homogeneous space is a variety of the form \(X:=G/P_\omega\) where \(G\) is a simple complex algebraic group and \(P_\omega\) is the parabolic subgroup associated to a (co)minuscule fundamental weight associated to the pair \((G,T)\), where \(T\) is a maximal torus of \(G\). The first episode [Transform. Groups 13, No. 1, 47--89 (2008; Zbl 1147.14023)] was about a very uniform presentation of the quantum cohomology ring of (co)minuscule homogeneous spaces, ranging from the usual and Lagrangian Grassmannians, quadrics, spinor varieties up to exceptional hermitian spaces like the Freudenthal variety or the Cayley plane. The second episode [Int. Math. Res. Not. 2007, No. 22, Article ID rnm107, 29 p. (2007; Zbl 1142.14033)], instead, was on hidden symmetries of Grassmannians. There, the main result was about a \textsl{strange duality} statement in the localization \(QA^*(X)_{loc}:=QA^*(X)\otimes_{{\mathbb Z}[q]}{\mathbb Z}[q,q^{-1}]\) of the quantum cohomology \(QA^*(X)\) of a minuscule homogeneous space \(X\). This new third episode aims to give an explanation of the aforementioned \textsl{strange duality} by showing its relationship with the semi-simplicity of the quantum cohomology specialized at \(q=1\).
Such a semi-simplicity was essentially known for classical groups, as a byproduct of the explicit presentation found in the first episode of the trilogy. The paper under review completes the picture for the exceptional cases. If \({\mathfrak g}\) is the Lie algebra of \(G\), denote by \(Z({\mathfrak g})\) the zero scheme in a Cartan sub-algebra \({\mathfrak t}\) of \({\mathfrak g}\) defined by the ideals of relations of the presentation for the quantum cohomology ring of \(G/P\), determined in the first episode of the trilogy. A first result of the article, Proposition~1.1., says that for any simple Lie algebra \({\mathfrak g}\), with the possible exception of two exceptional cases, the scheme \(Z({\mathfrak g})\) is reduced and is a free \(W\)-orbit, where \(W\) is the Weil group of \(G\).
The main result is, however, the beautiful Theorem 1.3. To fully appreciate its flavour, recall that any commutative, semisimple, finite-dimensional algebra \(H\) has a real vector space decomposition of the form \({\mathbb R}^n\oplus {\mathbb C}^p\), from which a natural automorphism of \(H\) is sending the complex part onto its conjugate. This is oviously an involution and Theorem 1.3. proves that for any minuscule or cominuscule homogeneous space \(X=G/P\) (including the exceptional cases, see below), complex conjugation and strange duality define the same involution. The result is proven by showing that strange duality and complex conjugation coincide on a set of generators of the quantum cohomology ring of \(X\). Compare the strength of such a result with the fact that only painful computations were able to show, in the second episode, that strange duality was an automorphism: its identification with complex conjugations, in the sense described above, makes this fact evident.
The authors also get some Vafa-Intrilligator type formulas [\textit{B. Siebert} and \textit{G. Tian}, Asian J. Math. 1, No. 4, 679--695 (1997; Zbl 0974.14040)] for the Gromov-Witten invariant of \(X=G/P\). The main ingredient is the \textsl{quantum Euler class} of \(X\) introduced in an intriguing paper by \textit{L. Abrams} [Isr. J. Math. 117, 335--352 (2000; Zbl 0954.53048)].
The paper is very well organized, respecting the tradition of the first two episodes. It is not an easy paper, but it challenges the reader to get rid of the non trivial technicalities solely appealing on the beauty of the subject. The first two sections are very preliminary: the former about complex conjugation and the latter about Grassmannians, recalling its small quantum cohomology. More special varieties like orthogonal Grassmannians and quadrics are in Section 4 and Lagrangian Grassmannians in Section 5. The main dish comes with the discussions of the exceptional cases that the interested readers have already encountered in the previous episodes: the Cayley plane and the Freudenthal variety. Here, Theorem 1.3 is proven for these cases as well. The Vafa-Intriligator formula, as well as the computation of Gromov-Witten invariants, is in Section 8 and a precise useful list of references concludes the paper.
The review is not finished yet, as the reviewer would like to spend few more words on the introduction. There the authors explain that the hypothesis of \(X=G/P\) being (co)minuscule is essential for the validity of Theorem 1.3, and, in fact, they quote the example of the Grassmannian of isotropic planes on a six-dimensional complex symplectic space (whose quantum cohomology is not semi-simple).
In spite of the indications coming from computations, the authors admit that is not at all clear, a priori, that the complex conjugate of a Schubert class is again a (multiple of) a Schubert class, though in the second episode this was granted by the very definition of strange duality. It would be then truly interesting, quoting the authors word by word, ``to have a conceptual explanation of that phenomenon (which no longer holds true for non-minuscule or cominuscule spaces)''.
It is hard to make predictions on the way any research develops. It appears rather clear, however, that there is still room for embedding the present trilogy, ``Quantum cohomology of minuscule homogeneous spaces'', into a tetralogy, by writing a new exciting episode able to supply that explanation necessary to make the mystery less mysterious. Coming next? quantum cohomology; minuscule homogeneous spaces; Schubert calculus; quantum Euler class Chaput, P. E.; Manivel, L.; Perrin, N., \textit{quantum cohomology of minuscule homogeneous spaces III. semi-simplicity and consequences}, Canad. J. Math., 62, 1246-1263, (2010) Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology of minuscule homogeneous spaces. III. Semi-simplicity and consequences | 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 bivariate polynomials of degree \(n\leq 5\) we give fast numerical constructions of determinantal representations with \(n\times n\) matrices. Unlike some other existing constructions, our approach returns matrices of the smallest possible size \(n\times n\) for all (not just generic) polynomials of degree \(n\) and does not require any symbolic computation. We can apply these linearizations to numerically compute the roots of a system of two bivariate polynomials by using numerical methods for the two-parameter eigenvalue problems. bivariate polynomial; determinantal representation Numerical computation of eigenvalues and eigenvectors of matrices, Numerical computation of roots of polynomial equations, Computational methods for sparse matrices, Solving polynomial systems; resultants, Determinantal varieties, Computational aspects of algebraic curves Explicit determinantal representations of up to quintic bivariate 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 \(R\) be a UFD containing \(\mathbb{Q}\) and \(B=R[Y_1,\dots,Y_m]\) be a polynomial ring over \(R\). A derivation \(d:B\to B\) is an \textit{\(R\)-derivation} if \(d(r)=0\) for any \(r\in R\). An \(R\)-derivation is called elementary one may choose generators \(Y_i\) is such a way that \(d(Y_i)\in R\) for any \(i=1,\dots,m\). In this case we have \(d=\sum_{i=1}^m a_i\frac{\partial}{\partial Y_i}\) for some \(a_i\in R\). Clearly, the elements \(L_{ij}=\frac{a_i}{g_{ij}}Y_j-\frac{a_j}{g_{ij}}Y_i\), where \(g_{ij}=\text{GCD}(a_i,a_j)\), are contained in the kernel of the derivation \(d\). Let us say that \(d\) is \textit{standard} if the elements \(L_{ij}\) generate \(\text{Ker}(d)\) as a subalgebra.
Let \(K\) be a field of characteristic zero, and \(R=K[X_1,\dots,X_n]\) be a polynomial ring. In [\textit{A.~Nowicki}, Polynomial derivations and their rings of constants. Wydawnictwo Uniwersytetu Mikolaja Kopernika, Torun (1994; Zbl 1236.13023)], it was conjectured that the derivation \(\sum_{i=1}^n X_i\frac{\partial}{\partial Y_i}\) of the ring \(R[Y_1,\dots,Y_n]\) is standard. This conjecture is proved in the present article even for more general class of derivations \(\sum_{i=1}^n X_i^{t_i}\frac{\partial}{\partial Y_i}\), \(t_i\in\mathbb{Z}_{\geq 0}\). The proof is based on the explicit calculation of a Groebner basis in an ideal associated with the generators \(L_{ij}\).
In [\textit{H.~Kojima} and \textit{M.~Miyanishi}, J. Pure Appl. Algebra 122, No. 3, 277--292 (1997; Zbl 0887.13009)], a particular case of the above result was obtained. In the last section the author gives some comments and corrections to the proof of Kojima and Miyanishi. locally nilpotent derivations; elimination theory Brouwer, A.: http://www.win.tue.nl/~aeb/math/invar.html Derivations and commutative rings, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Group actions on affine varieties A Gröbner basis approach to solve a conjecture of Nowicki | 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 part of Grothendieck's program for studying the Galois group \(G_Q\) of the field of all algebraic numbers \(\overline{Q}\) emerged from his insight that one should lift its action upon \(\overline{Q}\) to the action of \(G_Q\) upon the (appropriately defined) profinite completion of \(\pi_1(\mathrm{P}^1\{0,1,\infty\})\). The latter admits a good combinatorial encoding via finite graphs, ``dessins d'enfant''.
This part was actively developing during the last decades, starting with foundational works of A. Belyi, V. Drinfeld and Y. Ihara.
This chapter concerns another part of Grothendieck's program, in which its geometric environment is extended to moduli spaces of algebraic curves, more specifically, stable curves of genus zero with marked/labeled points. Our main goal is to show that dual graphs of such curves may play the role of ``modular dessins'' in an appropriate operadic context. Polycategories/dioperads, properads, PROPs, cyclic operads, modular operads, Operads (general), Quantum equilibrium statistical mechanics (general), Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Dessins d'enfants theory Dessins for modular operads and the Grothendieck-Teichmüller group | 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 a review of studies about the encoding of properties of an algebraic variety \(X\) in its algebraic fundamental group \(\pi _1(X\otimes _k \overline k)\), the profinite group which classifies finite étale coverings, defined by A. Grothendieck. The interesting case is when \(k\) is a proper subfield of \(\overline k \), since we can view the algebraic fundamental group as an extension
\[
0 \rightarrow \pi _1(X\otimes _k \overline k) \rightarrow \pi _1(X) \rightarrow \text{ Gal}(\overline k /k) \rightarrow 0.
\]
A result of \textit{S. Mochizuchi} [see Invent. Math. 138, No. 2, 319-423 (1999)] shows that for hyperbolic curves \(X_1\), \(X_2\) over \(k\), where \(k\) is a finitely generated extension of \({\mathbb Q}_p\), any open map \( \pi _1(X_1) \rightarrow \pi _1(X_2)\), as \(\pi _1(X_2\otimes _k \overline k)\)-conjugacy class of map of extensions, is induced by a unique and dominant map \(X_1 \rightarrow X_2\). The main tool used here is \(p\)-adic Hodge theory. hyperbolic curve; algebraic fundamental group; étale coverings; \(p\)-adic Hodge theory Faltings, G., Curves and their fundamental groups (following Grothendieck, Tamagawa and mochizuki), Astérisque, 252, pp., (1998) Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry Curves and their fundamental groups [following Grothendieck, Tamagawa and Mochizuki] | 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 paper under review, the author studies particular loci of the Hilbert scheme \(\mathcal{H}\mathrm{ilb}^{r}_{n}\) of \(r\) points in the affine space \(\mathbb{A}^n\). In a previous paper [J. Commut. Algebra 3, No. 3, 349--404 (2011; Zbl 1237.14012)], the author introduced the functor \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta}_{n,k}: (k\mathrm{-Alg}) \rightarrow (\mathrm{Sets})\) that associates to any algebra \(B\) over a ring \(k\) the set of reduced Gröbner bases in the ring \(B[x_1,\ldots,x_n]\) with respect to the lexicographic order with a given standard set \(\Delta\) of \(r\) monomials. He proved that this functor is representable and represented by a locally closed subscheme of \(\mathcal{H}\mathrm{ilb}^{r}_{n}\) called Gröbner stratum. In this paper, the author studies the subfunctor \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta,\text{ét}}_{n,k}\) of \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta}_{n,k}\) that considers only the reduced Gröbner bases of ideals defining reduced points. The main result of the paper is that \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta,\text{ét}}_{n,k}\) is representable and, in the case of a ring \(k\) such that \(\mathrm{Spec}\, k\) is irreducible, all the connected components of the representing scheme have the same dimension. Moreover, the number of connected components and their dimension are nicely described in terms of combinatorial properties of the standard set \(\Delta\). Hilbert scheme of points; Gröbner stratum; lexicographic order; reduced points Mathias Lederer (2014). Components of Gröbner strata in the Hilbert scheme of points. \textit{Proc. Lond. Math. Soc}. (3) \textbf{108}(1), 187-224. ISSN 0024-6115. URL http://dx.doi.org/10.1112/plms/pdt018. Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomials over commutative rings, Enumerative problems (combinatorial problems) in algebraic geometry Components of Gröbner strata in the Hilbert scheme of points | 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 determines an explicit presentation of the the small quantum cohomology of the Hilbert scheme of \(2\) points on the surface \(\mathbb P^1\times \mathbb P^1\). We remind the reader that the small quantum cohomology is obtained as a deformation of the usual cohomology ring by including corrections coming from the genus \(0\) three point Gromov-Witten invariants of the target. Furthermore, the author connects certain genus \(0\) invariants of the Hilbert scheme with a count of hyperelliptic curves on \(\mathbb P^1\times \mathbb P^1\) of fixed bidegree, and with appropriate incidence conditions (Theorem \(4.12\)). Note that the results of this paper are reminiscent of previous computations due to \textit{T. Graber}, who considered the quantum cohomology of the Hilbert scheme of two points on \(\mathbb{P}^2\), as well as its connection with the enumerative geometry of plane hyperelliptic curves [J. Algebr. Geom. 10, No. 4, 725--755 (2001; Zbl 1079.14536)]. quantum cohomology; Gromov-Witten invariants; Hilbert scheme; hyperelliptic curves Pontoni, D.: Quantum Cohomology of \({\mathrm Hilb}^2({\mathbb{P}}^1 \times {\mathbb{P}}^1)\) and enumerative applications. PhD Thesis, Padova (2003) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry Quantum cohomology of Hilb\(^2(\mathbb{P}^1 \times\mathbb{P}^1)\) and enumerative \newline applications | 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 we construct Laurent polynomial Landau-Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in [Rie08]. The Laurent polynomial takes a similar shape to the one given in [Giv96] for projective complete intersections, i.e., it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in [CMP08], associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in [PRW16] and [PR13] for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau-Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety. Laurent polynomial Landau-Ginzburg models for cominuscule homogeneous 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 Let \(G\) be a complex simply connected semisimple Lie group, let \(B\) be a Borel subgroup and let \(X=G/B\) be the associated flag manifold. Inside \(X\) is a certain affine subvariety \(Q\), determined by the choice of a principal nilpotent element \(f\) of \(\text{Lie}(G)\), whose closure is denoted \(P_f\) and whose affine coordinate ring, denoted \(A(Q)\), is a polynomial ring. The intersection of \(P_f\) and the Schubert cell defined by \(B\) is denoted \(R\). Furthermore, there is an isomorphism of affine varieties which identifies \(R\) and a certain subset \(Y_0\) of \(\text{Lie}(G)\); under this isomorphism \(Q\cap R\) gets identified with the Toda leaf \(Y^*_0\). The author describes the affine algebras \(A(Q)\), \(A(Y_0)\), \(A(R)\), and \(A(Q\cap R)\) in terms of polynomial generators and relations.
As the author explains in the first sections of the paper, this work was inspired by some conjectures in the case \(G=SL_n(\mathbb{C})\) where the quantum cohomology algebra \(CH(X,\mathbb{C})\) was conjectured and then proven to have a certain description in terms of generators and relations. The author shows this algebra is isomorphic to \(A(Y_0)\) in the \(SL_n\) case.
There is much more in the paper than this brief review can summarize. Although it relies extensively on previous work of the author and others, the paper features examples and exposition making it largely self-contained. flag manifold; affine algebras Kostant, Bertram, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight \textit{\({\rho}\)}, Selecta Math. (N.S.), 2, 1, 43-91, (1996) Grassmannians, Schubert varieties, flag manifolds, Applications of linear algebraic groups to the sciences Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight \(\rho\) | 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 the quantum sheaf cohomology of flag manifolds with deformations of the tangent bundle and use the ring structure to derive how the deformation transforms under the biholomorphic duality of flag manifolds. Realized as the OPE ring of A/2-twisted two-dimensional theories with (0,2) supersymmetry, quantum sheaf cohomology generalizes the notion of quantum cohomology. Complete descriptions of quantum sheaf cohomology have been obtained for abelian gauged linear sigma models (GLSMs) and for nonabelian GLSMs describing Grassmannians. In this paper we continue to explore the quantum sheaf cohomology of nonabelian theories. We first propose a method to compute the generating relations for (0,2) GLSMs with (2,2) locus. We apply this method to derive the quantum sheaf cohomology of products of Grassmannians and flag manifolds. The dual deformation associated with the biholomorphic duality gives rise to an explicit IR duality of two A/2-twisted (0,2) gauge theories. Yang-Mills and other gauge theories in quantum field theory, Model quantum field theories, Supersymmetric field theories in quantum mechanics, Correspondence, duality, holography (AdS/CFT, gauge/gravity, etc.), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Formal methods and deformations in algebraic geometry, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) Quantum sheaf cohomology and duality of flag 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 The author provides an intellectually pleasing treatment of that part of Hodge theory centered around the Hodge conjecture. In a rather comprehensive, albeit user-friendly style, the reader is introduced in the first section to the Betti and de Rham cycle class maps, and the classical Hodge conjecture. The next section gives a nice treatment of Hodge loci, the deep work of Cattani, Deligne and Kaplan in this direction, absolute Hodge classes, as well as the author's approach towards reducing the classical Hodge conjecture to smooth projective varieties defined over number fields. Section three gives a thorough discussion of the (Grothendieck amended) general Hodge conjecture and example situations. Section four deals with the Bloch conjecture and some generalizations, as well as the earlier works partially leading to this conjecture due to Mumford, as well as a different approach due to Bloch (which is actually based on an earlier idea of J.-L. Colliot-Thelene). Finally, section five deals with the so-called deep nilpotent conjecture and Kimura's theorem, and how this relates to the Bloch conjecture, as well as further topics centred around the general Hodge conjecture. Hodge conjecture; Grothendieck-Hodge conjectures C. Voisin, ''Lectures on the Hodge and Grothendieck-Hodge conjectures,'' Rend. Semin. Mat. Univ. Politec. Torino, vol. 69, iss. 2, pp. 149-198, 2011. Algebraic cycles Lectures on the Hodge and Grothendieck-Hodge conjectures | 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 \(S\) be a fixed Noetherian scheme and \(\mathcal S\) the category of separated, essentially finite-type, finite tor-dimension schemes \(x : X \rightarrow S\) over \(S\). For any such scheme \(X\), let \(\text{D}_{\text{qc}}(X)\) denote the derived category of the category of complexes of \({\mathcal O}_X\)-modules with quasi-coherent cohomology sheaves. For \(A\), \(B\) in \(\text{D}_{\text{qc}}(X)\), consider the graded abelian group:
\[
\text{E}_X(A,B) := { \bigoplus_{i \in {\mathbb Z}}} \text{Hom}_{\text{D}(X)}(A,B[i])\, .
\]
For \(A\), \(B\), \(C\) in \(\text{D}_{\text{qc}}(X)\) there exists an obvious graded bilinear composition map:
\[
\text{E}_X(B,C)\times \text{E}_X(A,B) \longrightarrow \text{E}_X(A,C)\, .
\]
Via this composition map, \(H_X := \text{E}_X({\mathcal O}_X,{\mathcal O}_X) \simeq \bigoplus_{i \geq 0}\text{H}^i(X,{\mathcal O}_X)\) becomes a commutative-graded ring and \(\text{E}_X(A,B)\) a symmetric (left and right) graded \(H_X\)-module. Putting \(H := H_S\), the category \(\text{D}_X\) whose objects are the objects of \(\text{D}_{\text{qc}}(X)\) and whose Hom-groups are \(\text{E}_X(A,B)\) is a \(H\)-\textit{graded category}. If \(f : X \rightarrow Y\) is a morphism in \(\mathcal S\) then the pseudofunctors \(\text{R}f_\ast : \text{D}_{\text{qc}}(X) \rightarrow \text{D}_{\text{qc}}(Y)\) and \(\text{L}f^\ast : \text{D}_{\text{qc}}(Y) \rightarrow \text{D}_{\text{qc}}(X)\) induce pseudofunctors of \(H\)-graded categories \(\text{R}f_\ast : \text{D}_X \rightarrow \text{D}_Y\) and \(\text{L}f^\ast : \text{D}_Y \rightarrow \text{D}_X\).
\textit{S. Nayak} [Adv. Math. 222, No. 2, 527--546 (2009; Zbl 1175.14003)], unifying the local and global Grothendieck duality theories, constructed, under the above hypotheses, a \textit{twisted inverse image} pseudofunctor \(f_+^! : \text{D}^+_{\text{qc}}(Y) \rightarrow \text{D}^+_{\text{qc}}(X)\) which is pseudofunctorially right-adjoint to \(\text{R}f_\ast : \text{D}^+_{\text{qc}}(X) \rightarrow \text{D}^+_{\text{qc}}(Y)\) if \(f\) is proper, and such that \(f_+^! = f^\ast\) if \(f\) is essentially étale (which means that \({\mathcal O}_{X,x}\) is formally étale over \({\mathcal O}_{Y,f(x)}\), \(\forall \, x \in X\)). One can show that \(f_+^!\) can be extended to a pseudofunctor of \(H\)-graded categories \(f^! : \text{D}_Y \rightarrow \text{D}_X\) such that \(f^!C = f_+^!{\mathcal O}_Y\otimes_{{\mathcal O}_X}^{\text{L}}\text{L}f^\ast C\) for \(C \in \text{D}_{\text{qc}}(Y)\).
Finally, for any object \(X \rightarrow S\) of \(\mathcal S\), consider the \textit{pre-Hochschild complex} \({\mathcal H}_X := \text{L}\delta_X^\ast \text{R}\delta_{X\ast}{\mathcal O}_X\), where \(\delta_X : X \rightarrow X\times_SX\) is the diagonal morphism. Using the properties of these complexes (which can be found in the paper of \textit{R.-O. Buchweitz} and \textit{H. Flenner} [Adv. Math. 217, No. 1, 205--242 (2008; Zbl 1140.14015)]) and of the twisted inverse image pseudofunctor \(f^!\), the authors of the paper under review develop a \textit{bivariant theory} (in the sense of \textit{W. Fulton} and \textit{R. MacPherson} [``Categorical framework for the study of singular spaces'', Mem. Am. Math. Soc. 243, 165 p. (1981; Zbl 0467.55005)]) on the category \(\mathcal S\), with values in the \(H\)-graded categories \(\text{D}_X\), \(X \in {\mathcal S}\), for the pseudofunctors \((-)^\ast\), \((-)_\ast\) and \((-)^!\), with proper morphisms as \textit{confined maps}, and with cartesian squares with flat bottom as \textit{independent squares}. This theory associates to a morphism \(f : (X \overset{x}\rightarrow S) \rightarrow (Y \overset{y}\rightarrow S)\) the graded \(H\)-module:
\[
\text{HH}^\ast(f) := \text{E}_X({\mathcal H}_X,f^!{\mathcal H}_Y) = { \bigoplus_{i\in {\mathbb Z}}}\text{Hom}_{\text{D}(X)}({\mathcal H}_X, f^!{\mathcal H}_Y[i])
\]
so that the associated cohomology groups are:
\[
\text{HH}^i(X/S) := \text{HH}^i(\text{id}_X) = \text{Ext}^i_{{\mathcal O}_X}({\mathcal H}_X,{\mathcal H}_X)
\]
and the associated homology groups are:
\[
\text{HH}_i(X/S) := \text{HH}^{-i}(x) = \text{Ext}^{-i}_{{\mathcal O}_X}({\mathcal H}_X,x^!{\mathcal O}_S)\, .
\]
Most of the proofs consist of the (non-trivial) verification of the commutativity of certain diagrams.
These results lay the foundation for the construction, in a sequel of this paper, of the \textit{fundamental class} of a \textit{flat} \(f\) as above, which is a natural functorial map:
\[
\text{c}_f : \text{L}\delta^\ast_X\text{R}\delta_{X\ast}\text{L}f^\ast \longrightarrow f^!\text{L}\delta^\ast_Y\text{R}\delta_{Y\ast}
\]
satisfying a transitivity relation with respect to the composition of morphisms in \(\mathcal S\). Hochschild homology; bivariant theory; Grothendieck duality; fundamental class Tarrío, L. Alonso; Lopez, A. Jeremías; Lipman, J.; Bivariance: Grothendieck duality and Hochschild homology I: Construction of a bivariant theory, Asian J. Math. 15, 451-498 (2011) (Co)homology theory in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Bivariance, Grothendieck duality and Hochschild homology. I: Construction of a bivariant 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 We introduce categories of homogeneous strict polynomial functors, \( \mathsf {Pol}^{\mathrm {I}}_{d,\Bbbk }\) and \( \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk }\), defined on vector superspaces over a field \( \Bbbk \) of characteristic not equal 2. These categories are related to polynomial representations of the supergroups \( GL(m| n)\) and \( Q(n)\). In particular, we prove an equivalence between \( \mathsf {Pol}^{\mathrm {I}}_{d,\Bbbk }\), \( \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk }\) and the category of finite dimensional supermodules over the Schur superalgebra \( \mathcal {S}(m| n,d)\), \( \mathcal {Q}(n,d)\) respectively provided \( m,n \geq d\). We also discuss some aspects of Sergeev duality from the viewpoint of the category \( \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk }\). Axtell, J, \textit{spin polynomial functors and representations of Schur superalgebras}, Represent. Theory, 17, 584-609, (2013) Enriched categories (over closed or monoidal categories), Group schemes, Module categories in associative algebras, Superalgebras, Representation theory for linear algebraic groups, Schur and \(q\)-Schur algebras Spin polynomial functors and representations of Schur superalgebras | 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 show that all toric noncommutative crepant resolutions (NCCRs) of affine GIT quotients of ``weakly symmetric'' unimodular torus representations are derived equivalent. This yields evidence for a non-commutative extension of a well known conjecture by Bondal and Orlov stating that all crepant resolutions of a Gorenstein singularity are derived equivalent. We prove our result by showing that \textit{all} toric NCCRs of the affine GIT quotient are derived equivalent to a fixed Deligne-Mumford GIT quotient stack associated to a generic character of the torus. This extends a result by Halpern-Leistner and Sam which showed that such GIT quotient stacks are a geometric incarnation of a family of \textit{specific} toric NCCRs constructed earlier by the authors. Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Noncommutative algebraic geometry, Derived categories, triangulated categories On the noncommutative Bondal-Orlov conjecture for some toric 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 The Abhyankar-Sathaye problem asks whether any biregular embedding \(\varphi:\mathbb{C}^k\hookrightarrow\mathbb{C}^n\) can be rectified, that is, whether there exists an automorphism \(\alpha\in{\operatorname{Aut}}\,\mathbb{C}^n\) such that \(\alpha\circ\varphi\) is a linear embedding. Here we study this problem for the embeddings \(\varphi:\mathbb{C}^3\hookrightarrow \mathbb{C}^4\) whose image \(X=\varphi(\mathbb{C}^3)\) is given in \(\mathbb{C}^4\) by an equation \(p=f(x,y)u+g(x,y,z)=0\), where \(f\in\mathbb{C}[x,y]\backslash\{0\}\) and \(g\in\mathbb{C}[x,y,z]\). Under certain additional assumptions we show that, indeed, the polynomial \(p\) is a variable of the polynomial ring \(\mathbb{C}^{[4]}=\mathbb{C}[x,y,z,u]\) (i.e., a coordinate of a polynomial automorphism of \(\mathbb{C}^4\)). This is an analog of a theorem due to \textit{A. Sathaye} [Proc. Am. Math. Soc. 56, 1--7 (1976; Zbl 0345.14013)] which concerns the case of embeddings \(\mathbb{C}^2\hookrightarrow\mathbb{C}^3\). Besides, we generalize a theorem of \textit{M. Miyanishi} [Am. J. Math. 106, 1469--1485 (1984; Zbl 0595.14025)] giving, for a polynomial \(p\) as above, a criterion for when \(X=p^{-1}(0)\simeq\mathbb{C}^3\). affine space; affine modification; birational extension; linearization problem; Abhyankar-Sathaye problem; polynomial automorphism S. Kaliman, S. Vénéreau and M. Zaidenberg, Simple birational extensions of the polynomial algebra \(\mathbb{ C}^{[3]}\), Trans. Amer. Math. Soc. 356 (2004), 509--555 (electronic). Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Birational automorphisms, Cremona group and generalizations, Affine fibrations Simple birational extensions of the polynomial algebra \(\mathbb{C}^{[3]}\) | 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 work we study the quantisation of the Seiberg-Witten curve for the E-string theory compactified on a two-torus. We find that the resulting operator expression belongs to the class of elliptic quantum curves. It can be rephrased as an eigenvalue equation with eigenvectors corresponding to co-dimension 2 defect operators and eigenvalues to co-dimension 4 Wilson surfaces wrapping the elliptic curve, respectively. Moreover, the operator we find is a generalised version of the van Diejen operator arising in the study of elliptic integrable systems. Although the microscopic representation of the co-dimension 4 defect only furnishes an SO(16) flavour symmetry in the UV, we find an enhancement in the IR to representations in terms of affine \(E_8\) characters. Finally, using the Nekrasov-Shatashvili limit of the E-string BPS partition function, we give a path integral derivation of the quantum curve. Quantization in field theory; cohomological methods E-string quantum curve | 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 \(p\neq 2\) be a prime number, \({\mathbb F}_p\) the finite field with \(p\) elements, \({\mathbb P}^1\) the projective line over \({\mathbb F}_p\), \({\mathbb G}_m={\mathbb P}^1\setminus \{0,\infty\}\), \(\text{Kl}_2\) the Kloosterman sheaf of rank \(2\) over \({\mathbb G}_m\). The authors study the \(L\)-function
\[
M_k(p,T):=L({\mathbb P}^1,j_*(\text{Sym}(\text{Kl}_2)),T),
\]
\(k>0\) an integer, \(j:{\mathbb G}_m\to {\mathbb P}^1\) the inclusion. Let \(c\) be the constant in its functional equation. If \(k\) is even, then
\[
c=p^{(k+1)(\left[\frac{k-2}{4}\right]-\left[\frac{k}{2p}\right])}.
\]
If \(k\) is odd, then
\[
c=(-1)^{\frac{k-1}{2}+\left[\frac{k}{2p}+\frac12\right]}p^{\frac{k+1}{2}(\frac{k-1}{2}-\left[\frac{k}{2p}+\frac12\right])}\left(\frac{-2}{p}\right)^{\left[\frac{k}{2p}+\frac12\right]}\prod_{{j\in\left\{0,1,\ldots,\left[\tfrac{k}{2}\right]\right\}}\atop {p\nmid 2j+1}}\left(\frac{(-1)^j(2j+1)}{p}\right).
\]
Kloosterman sheaf; \(L\)-function; functional equation Lei Fu and Daqing Wan, Functional equations of \?-functions for symmetric products of the Kloosterman sheaf, Trans. Amer. Math. Soc. 362 (2010), no. 11, 5947 -- 5965. Finite ground fields in algebraic geometry, Gauss and Kloosterman sums; generalizations Functional equations of \(L\)-functions for symmetric products of the Kloosterman sheaf | 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 define holomorphic structures on canonical line bundles of the quantum projective space \(\mathbb CP^\ell_q\) and identify their space of holomorphic sections. This determines the quantum homogeneous coordinate ring of the quantum projective space. We show that the fundamental class of \(\mathbb CP^\ell_q\) is naturally presented by a twisted positive Hochschild cocycle. Finally, we verify the main statements of the Riemann-Roch formula and the Serre duality for \(\mathbb CP^1_q\) and \(\mathbb CP^2_q\). noncommutative complex geometry; quantum projective space; quantum homogeneous coordinate ring; twisted cyclic cocycle; positive Hochschild cocycle; Serre duality Khalkhali M., Moatadelro A.: Noncommutative complex geometry of the quantum projective space. J. Geom. Phys. 61(12), 2436--2452 (2011) Noncommutative geometry in quantum theory, Noncommutative geometry methods in quantum field theory, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Projective analytic geometry Noncommutative complex geometry of the quantum projective space | 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 odd symplectic Grassmannian \(\text{IG}:=\text{IG}(k, 2n+1)\) parametrizes \(k\) dimensional subspaces of \({\mathbb {C}}^{2n+1}\) which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on \(\text{IG}\) with two orbits, and \(\text{IG}\) is itself a smooth Schubert variety in the submaximal isotropic Grassmannian \(\text{IG}(k, 2n+2)\). We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of \(\text{IG}\), i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case \(k=2\), and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Equivariant quantum cohomology of the odd symplectic 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 This paper is a review of the basic properties of effective actions of families of theories (i.e., the actions depending on additional non-perturbative moduli along with perturbative couplings), and their description in terms of operators (called check-operators), which act on the moduli space. This is a review of earlier results mostly described in [\textit{A. S. Alexandrov} et al., Fortschr. Phys. 53, No. 5--6, 512--521 (2005; Zbl 1065.81103); Int. J. Mod. Phys. A 21, No. 12, 2481--2517 (2006; Zbl 1098.81075); ibid. A 24, No. 27, 4939--4998 (2009; Zbl 1179.81127); ibid. A 19, No. 24, 4127--4163 (2004; Zbl 1087.81051)] and [\textit{D. Galakhov} et al.,``\(S\)-duality and modular transformation as a non-perturbative deformation of the ordinary \(pq\)-duality'', J. High Energy Phys. 2014, No. 6, Article ID 050, 24 p. (2014)]. In particular, the check-operators were introduced in [\textit{A. S. Alexandrov} et al., Fortschr. Phys. 53, No. 5--6, 512--521 (2005; Zbl 1065.81103); Int. J. Mod. Phys. A 21, No. 12, 2481--2517 (2006; Zbl 1098.81075)], where their properties were discussed. Among the check-operators, there is the main check-operator with a crucial property. In a different situation, this main check-operator was discussed in [Galakhov et al., loc. cit.]. The authors introduced and explained the very important notion of check-operator: the operator that acts on the moduli space of theories (or vacua/solutions). They constructed the operator manifestly in the simplest example of the Hermitian matrix model and in a more involved example of the two-dimensional conformal field theory, and demonstrated its use by deriving the corresponding Seiberg-Witten structures and the quantum spectral curves. They also illustrated the usefulness of the concept by a simple evaluation of the kernel of modular transformation of the conformal blocks done in terms of the check-operators. The calculation used a wonderful relation (a crucial property), which provides the impressive example of the properties and the relevance of check-operators for the quantization theory. The paper is divided into six sections entitled : Introduction, Multiple solutions to the Virasoro constraints, Check-operators, Seiberg-Witten (SW) like solutions and integrable properties, Quantum spectral curves, Quantum curves from degenerate conformal blocks, Modular kernels in conformal field theory and Conclusion. matrix models; check-operators; Seiberg-Witten theory; modular kernel in CFT A. Mironov and A. Morozov, \textit{Check-operators and Quantum Spectral Curves}, arXiv:1701.03057 [INSPIRE]. Relationships between algebraic curves and integrable systems, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Groups and algebras in quantum theory and relations with integrable systems, Yang-Mills and other gauge theories in quantum field theory Check-operators and quantum spectral 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 The paper under review establishes that the Poincaré polynomial, \(p_{{\mathbf w}}(t)= \sum_{{\mathbf v}\prec{\mathbf w}}t^{\ell({\mathbf v})}\), where \(\preceq\) is the Bruhat order, \({\mathbf v}\) and \({\mathbf w}\) are permutations of \(n\), and \(\ell({\mathbf v})\) is the rank or length of \({\mathbf v}\), can be factored into polynomials of the form \(1+t+t^2+\cdots+ t^r\) if and only if \({\mathbf w}\) is 4231- or 3412-avoiding. Poincaré polynomial; Bruhat order Gasharov, V, Factoring the Poincaré polynomials for the Bruhat order on \(S_n\), J. Combin. Theory Ser. A, 83, 159-164, (1998) Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Factoring the Poincaré polynomials for the Bruhat order on \(S_n\) | 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 present note is not intended in any way as an introduction to Grothendieck-Teichmüller
theory. It is essentially a concentrated list of questions in and around this theory, most of which
are open, although we have included some questions which are natural to ask but easy to answer,
and a few others which were open but are now settled. In order for the reader to appreciate the
relative depth, difficulty, and interest of these problems, and their position within the theory, some
previous knowledge is required. We do give some important facts and definitions, but they are
intended to remind the reader of relatively well-known elements of the theory, to give something of
the flavor of the objects concerned and to make statements unambiguous. They are not sufficient
to provide a deep understanding of the theory. Lochak, P., Schneps, L.: Open problems in Grothendieck-Teichmüller theory. In: Problems on mapping class groups and related topics. Proc. Sympos. Pure Math., vol. 74, pp. 165-186. Am. Math. Soc., Providence, RI (2006) Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Grothendieck groups (category-theoretic aspects), Fundamental groups and their automorphisms (group-theoretic aspects), Topological methods in group theory Open problems in Grothendieck-Teichmüller 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 The Bernstein-Sato polynomial (or global \(b\)-function) is an important invariant in singularity theory, which can be computed using symbolic methods in the theory of \(D\)-modules. After providing a survey of known algorithms for computing the global \(b\)-function, we develop a new method to compute the local \(b\)-function for a single polynomial. We then develop algorithms that compute generalized Bernstein-Sato polynomials of Budur-Mustaţă-Saito and Shibuta for an arbitrary polynomial ideal. These lead to computations of log canonical thresholds, jumping coefficients, and multiplier ideals. Our algorithm for multiplier ideals simplifies that of Shibuta and shares a common subroutine with our local \(b\)-function algorithm. The algorithms we present have been implemented in the \(D\)-modules package of the computer algebra system Macaulay2. \(D\)-modules; \(V\)-filtration; Bernstein-Sato polynomial; jumping coefficients; log-canonical threshold; multiplier ideals [BL10] C. Berkesch and A. Leykin, Algorithms for Bernstein--Sato polynomials and multiplier ideals, ISSAC 2010--Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, pp. 99--106, ACM, New York, 2010. Symbolic computation and algebraic computation, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Multiplier ideals, Computational aspects of higher-dimensional varieties, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) Algorithms for Bernstein-Sato polynomials and multiplier 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 For quasihomogeneous polynomials with isolated singularity, V. I. Arnold introduced the notion of inner modality and classified them with inner modality \(=0,1\). The next works were done for inner modality \(=2,3,4,5\) in [the author, Proc. Japan Acad., Ser. A 57, 160--163 (1981; Zbl 0495.14001); \textit{E. Yoshinaga} and the auhtor, Invent. Math. 55, 185--206 (1979; Zbl 0406.58008)] and further for inner modality \(=6\) in [\textit{S. J. Estrada} et al., An. Inst. Mat., Univ. Nac. Autón. Méx. 26, 21--39 (1986; Zbl 0642.58011)]. Recently the classification is developed for inner modality \(=7,8,9\) in [the author, Proc. Inst. Nat. Sci., Nihon Univ. 45, 333--340 (2010; Zbl 1220.58009)]. In this aricle we will classify quasihomogeneous polynomials of corank \(=3\) with inner modality \(\leq 14\). quasihomogeneous polynomials; quasihomogeneous singularities; inner modality Local complex singularities, Singularities in algebraic geometry Classification of quasihomogeneous polynomials of corank three with inner modality \(\leq 14\) | 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 Vénéreau polynomials are a sequence of polynomials
\[
b_{m}=y+x^{m}(xz+y(yu+z^{2}))\in\mathbb{C}[x][y,z,u],
\]
\(m\geq1\), which were proposed by S. Vénéreau as potential counterexamples to important conjectures in affine geometry: the Abhyankar-Sathaye embedding conjecture which asserts that every closed embedding of an affine space into another is equivalent to an embedding as a linear subspace, and the Dolgachev-Weisfeiler conjecture which asks whether every flat fibration from an affine space to another with all fibers also isomorphic to affine spaces is a trivial affine bundle.
It is known that the level hypersurfaces of Vénéreau polynomials are isomorphic to affine spaces of dimension \(3\) and that for every \(m\geq1\), the fibration \((b_{m},x):\mathbb{A}^{4}\rightarrow\mathbb{A}^{2}\) is flat with all fibers isomorphic to affine spaces \(\mathbb{A}^{2}\). So \(b_{m}\) provides a counterexample to the Abhyankar-Sathaye embedding conjecture unless it is a \textit{variable} of the polynomial ring \(\mathbb{C}[x,y,z,u]\), i. e., there exists polynomials \(f_{1},f_{2},f_{3}\) such that \(\mathbb{C}[x,y,z,u]=\mathbb{C}[b_{m},f_{1},f_{2},f_{3}]\), and a counterexample to the Dolgachev-Weisfeiler conjecture unless it has the stronger property to be a \(\mathbb{C}[x]\)-\textit{variable} of \(\mathbb{C}[x,y,z,u]\) in the sense that there exists polynomials \(g_{1},g_{2}\) such that \(\mathbb{C}[x,y,z,u]=\mathbb{C}[x,b_{m},g_{1},g_{2}]\). It was established by S. Vénéreau that \(b_{m}\) is indeed a \(\mathbb{C}[x]\)-coordinate for every \(m\geq3\) but the question for \(m=1,2\) remained open.
In the article under review, the author introduces a more general class of \textit{Vénéreau -type polynomials} of the form \(f_{Q}=y+xQ\) with \(Q\in\mathbb{C}[x][v,w]\) and he proves that \(b_{m}\) is a coordinate if and only if so is \(f_{Q}\) for \(Q=x^{2m-1}w\). This is applied to recover the fact that \(b_{m}\) is a \(\mathbb{C}[x]\)-coordinate for \(m\geq3\) and to prove the new result that \(b_{2}\) is a \(\mathbb{C}[x]\)-coordinate too. Other properties of polynomials \(f_{Q}\) in relation with the aforementioned conjectures are established. polynomial rings; Vénéreau polynomials; coordinates; Dolgachev-Weisfeiler conjecture Lewis, D., Vénéreau-type polynomials as potential counterexamples, J. Pure Appl. Algebra, 217, 5, 946-957, (2013) Polynomials over commutative rings, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Vénéreau-type polynomials as potential counterexamples | 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 any algebraic curve \(\mathcal{C}\) and \(n\geq 1\), Etingof introduced the global counterparts of rational Cherednik algebras, as natural deformations of the cross product \(\mathcal{D}(\mathcal{C}^n)\rtimes \mathbb{S}_n\) of the the algebra of differential operators on \(\mathcal{C}^n \) and the symmetric group. This paper studies ``a category of character \(\mathcal{D}\)-modules on a representation scheme associated with \(\mathcal{C}\)'' and provides a construction of an important spherical subalgebra of the global Cherednik algebra in terms of quantum Hamiltonian reduction of \(\mathcal{D}_{nk, \psi}(\text{rep}^n_\mathcal{C}\times\mathbb{P}^{n-1})\), a sheaf of twisted differential operators on \(\text{rep}^n_\mathcal{C}\times\mathbb{P}^{n-1}\). More precisely, it is constructed an exact functor from the category of modules over \(\mathcal{D}_{nk, \psi}(\text{rep}^n_\mathcal{C}\times\mathbb{P}^{n-1})\) to the category of modules over the global Cherednik algebra.
``In the special case of the curve \(\mathcal{C}=\mathbb{C}^\times\), the global Cherednik algebra reduces to the trigonometric Cherednik algebra of type \(\mathbb{A}_{n-1}\), and the character \(\mathcal{D}\)-modules become holonomic \(\mathcal{D}\)-modules on \(\text{GL}_n(\mathbb{C})\times\mathbb{C}^n\). The corresponding perverse sheaves are reminiscent of (and include as special cases) Lusztig's character sheaves.''
The subject studied and the results obtained in this work are extremely interesting but require a rather technical writing given the diversity of mathematical concepts involved, that the authors were careful to remember and explain, simplifying the reading. \(\mathcal D\)-modules; Character sheaves; Cherednik algebras M. Finkelberg, V. Ginzburg, \textit{Cherednik algebras for algebraic curves}, in: \textit{Representation Theory of Algebraic Groups and Quantum Groups}, Progr. Math., Vol. 284, Birkhäuser/Springer, New York, 2010, pp. 121-153. Noncommutative algebraic geometry, Hecke algebras and their representations, Linear algebraic groups and related topics Cherednik algebras for algebraic 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 The present paper provides new characterizations based on Gröbner bases for the Cohen-Macaulay property of a projective monomial curve. The projective monomial curve \(\overline{C}(\mathbf{a})\) is called arithmetically Cohen-Macaulay if its vanishing ideal \(I(\mathbf{a})^h\) is a Cohen-Macaulay ideal. Arithmetically Cohen-Macaulay curves are not rare among projective monomial curves [\textit{T. Vu}, J. Algebra 418, 66--90 (2014; Zbl 1317.13037)].
A main result of the present paper states that \(\overline{C}(\mathbf{a})\) is arithmetically Cohen-Macaulay if and only if the initial ideal \(\mathrm{in}(I(\mathbf{a})^h)\), respectively \(\mathrm{in}(I(\mathbf{a}))\) is a Cohen-Macaulay ideal, see Theorem 2,2, which gives other equivalent properties.
The monomial curves associated to a sequence \(\mathbf{a}\) and its dual \(\mathbf{a}'\) are isomorphic. Section 3 of the paper focuses on the ideals \(I(\mathbf{a})\) and \(I(\mathbf{a}')\) and their reduced Gröbner bases, giving criteria for the arithmetical Cohen-Macaulay property based on the relations between the reduced Gröbner bases of \(I(\mathbf{a})\) and \(I(\mathbf{a}')\) and their initial ideals, see Theorems 3.2 and 3.6.
The paper finishes by applying Theorem 2.2 to test the arithmetically Cohen-Macaulay property for some families of monomial curves in \(\mathbb{P}^3\) that have appeared in the literature. arithmetically Cohen-Macaulay; projective monomial curve; revlex; Gröbner basis; numerical semigroup; Apéry set Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Syzygies, resolutions, complexes and commutative rings, Commutative semigroups, Semigroup rings, multiplicative semigroups of rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Polynomial rings and ideals; rings of integer-valued polynomials, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes Cohen-Macaulay criteria for projective monomial curves via Gröbner bases | 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 surveys the different ideas concerning quantum cohomology of homogeneous Kähler manifolds. Lecture 1 treats `Moduli spaces of stable maps', Lecture `Gromov-Witten invariants', Lecture 3 deals with `\(QH^*(G/B)\) and quantum Toda lattices', where \(QH^*(x)\) denotes the quantum cohomology algebra. Lecture 4 surveys `Singularity theory' and finally Lecture 5 concerns (Toda lattices and the mirror conjecture). The survey is nice written, and contains many exercises and examples. Alexander Givental, A tutorial on quantum cohomology. In: \textit{Symplectic geometry} \textit{and topology (Park City, UT, 1997)}, IAS/Park City Math. Ser. 7 (1999) 231--264. Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) A tutorial on 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 On a category of representations of a simply laced affine Kac-Moody algebra \(\tilde{\mathfrak g}\) (category O, basically) a tensor structure is introduced and the obtained tensor category is equivalent to that of finite-dimensional representations of the quantum algebra corresponding to \({\mathfrak g}\). Via references relations with Gal(\({\bar {\mathbb{Q}}}/{\mathbb{Q}})\), intersection cohomology of infinite-dimensional Schubert varieties and physics, represented by Knizhnik-Zamolodchikov equations, can be detected. quantum groups; representations; simply laced affine Kac-Moody algebra; tensor category; intersection cohomology of infinite-dimensional Schubert varieties; Knizhnik-Zamolodchikov equations D.~Kazhdan and G.~Lusztig 1991 Affine Lie algebras and quantum groups \textit{Internat. Math. Res. Notices}1991 2 21--29 Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Affine Lie algebras and quantum 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 Eynard-Orantin invariants of a plane curve are multidifferentials on the curve. For a particular class of genus zero plane curves these invariants can be equivalently expressed in terms of simpler expressions given by polynomials obtained from an expansion of the Eynard-Orantin invariants around a point on the curve. This class of curves governs many interesting enumerative problems in geometry including counting lattice points in the moduli space of curves and the Gromov-Witten invariants of the 2-sphere. genus zero plane curves; Eynard-Orantin invariants Norbury, P., Scott, N.: Polynomials representing Eynard-Orantin invariants. Q. J. Math. (2012). [arXiv:1001.0449v1 [math.AG Compact Riemann surfaces and uniformization, Families, moduli of curves (analytic) Polynomials representing Eynard-Orantin invariants | 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 views Hopf algebras as the noncommutative counterpart to affine group varieties. Given an algebraic group \(B\). Then, in this noncommutative setup, comodule algebras are viewed as noncommutative \(B\)-varieties. A function invariant on orbits of an action of a group on a set can be viewed as a function on the set of orbits. Invariant functions are coinvariants in the algebra of all functions with respect to the coaction of a Hopf algebra of functions on the group. However, in most cases the space of orbits is only determined by coinvariants in a larger category. A generalization of the commutative geometric invariant theory starts with the idea of locally defined quotients.
This paper gives an overview of efforts to access and use the local information on noncommutative quotients, mainly in the case of noncommutative coset spaces. This includes: Noncommutative localizations used to replace Zariski open subsets, compatibility of noncommutative localization with the coaction on the quotient \(E/B\) which are analogues of \(B\)-equivariant (quasicoherent) sheaves on \(E\), localized coinvariants, corresponding to charts in a coset space, noncommutative Gauss decomposition for matrix bialgebras, and finally, calculations with quasideterminants.
The article includes detailed descriptions of Ore and Gabriel localizations, and the interplay between these in the covering by localizations. The (trivial) principal bundles are defined in the noncommutative setting, leading to commutative local triviality and torsors. The localized coinvariants and their compatibility are studied, so that quantum bundles can be defined. Matrices with noncommutative entries are studied, that is matrix bialgebras and Hopf algebras, quantum matrix groups and quantum determinants.
The article is compactly written, with a lot of words. However comparison with other approaches is indicated in the end of the paper, making the theory possible to understand for people working in these directions. Hopf algebra; noncommutative localization; quantum bundle; comodule Škoda, Z.: Localizations for construction of quantum coset spaces. Banach center publ. 61, 265-298 (2003) Noncommutative algebraic geometry, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Group actions on varieties or schemes (quotients), Geometry of quantum groups Localizations for construction of quantum coset 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 additive Grothendieck-Springer resolution (see diagram (1.0.2)) serves as a resolution of singularities in the study of du Val singularities for algebraic surfaces. Given the stack of principal \(G\)-bundles over an elliptic curve with a simply connected structure group \(G\), this article describes the singularities and log resolutions coming from the elliptic Grothendieck-Springer resolution (see diagram (1.0.3)), which is a stacky version of the additive one.
When \(G\neq SL_2\), Theorem 1.0.2 (proven in section 2) shows the existence of an equivariant slice through subregular unstable bundles with good properties. Theorems 1.0.3 (proven in section 3) and 1.0.6 (proven in section 4) gives explicit descriptions of the pullbacks to these slices, depending on the Dynkin type of the subregular unstable \(G\)-bundle, and the singular surfaces appearing on the fibrations.
This paper contains the results of chapters 5 and 6 of the author's PhD thesis. singularities; principal bundles; elliptic curves Vector bundles on curves and their moduli, Singularities of surfaces or higher-dimensional varieties, Classical groups (algebro-geometric aspects), Other algebraic groups (geometric aspects) On subregular slices of the elliptic Grothendieck-Springer resolution | 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 Young's raising operators to give short and uniform proofs of several well known results about Schur polynomials and symmetric functions, starting from the Jacobi-Trudi identity. Young's raising operators Symmetric functions and generalizations, Classical problems, Schubert calculus The theory of Schur polynomials revisited | 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 It is known that a linear Fuchsian \(2\times2\)-matrix ODE with three singular points \(0,1,\infty\) is solvable in terms of the Gauss hypergeometric function, and that the isomonodromy deformation of a linear Fuchsian \(2\times2\)-matrix ODE with four singular points \(0,1,t,\infty\) w.r.t.\ the complex parameter \(t\) is governed by the sixth Painlevé equation \(P_6\). It is also known that a rational change of the independent variable in the Fuchsian ODE for the Gauss hypergeometric function with some special values of its parameters supplemented by an appropriate gauge transformation yields a linear Fuchsian ODE with four singularities where the deformation parameter \(t\) takes a particular value. Thus the above transformation gives rise to a solution for \(P_6\) with quite special initial data. Miraculous, the particular initial data for one of such solutions derived in an earlier work by the author [see: the first author and \textit{A. V. Kitaev}, Commun. Math. Phys. 228, 151--176 (2002; Zbl 1019.34086)] coincide with those prescribed in quantum cohomology. Besides making this observation, the author computes the relevant monodromy matrices for this special solution and shows that all of them belong to SL\((2,{\mathbb Z})\). Painlevé equation; monodromy matrix; rational map; gauge transformation; hypergeometric function Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Rational and birational maps Direct computation of the monodromy data for \(P_6\) corresponding to the quantum cohomology of the projective plane | 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 Hamiltonian of the quantum Calogero-Sutherland model of \(N\) identical particles on the circle with \(1/r^{2}\) interactions has eigenfunctions consisting of Jack polynomials times the base state. By use of the generalized Jack polynomials taking values in modules of the symmetric group and the matrix solution of a system of linear differential equations one constructs novel eigenfunctions of the Hamiltonian. Like the usual wavefunctions each eigenfunction determines a symmetric probability density on the \(N\)-torus. The construction applies to any irreducible representation of the symmetric group. The methods depend on the theory of generalized Jack polynomials due to Griffeth, and the Yang-Baxter graph approach of Luque and the author. Calogero-Sutherland model on torus; generalized Jack polynomials; representations of the symmetric group Dunkl, Charles F., Vector-valued Jack polynomials and wavefunctions on the torus, Journal of Physics. A. Mathematical and Theoretical, 50, 24, 245201, 21~pages, (2017) Selfadjoint operator theory in quantum theory, including spectral analysis, Many-body theory; quantum Hall effect, Representations of finite symmetric groups, Toric varieties, Newton polyhedra, Okounkov bodies, Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable, Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices, Commutation relations and statistics as related to quantum mechanics (general) Vector-valued Jack polynomials and wavefunctions on the torus | 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 work gives the explicit Poisson structure of the manifold \(\Delta G\times K\), where \(G\) is a simple algebraic group over \(\mathbb{C}\), \(\Delta G\cong G\) is the diagonal, and \(K\) is a subgroup of \(B^+\times B^-\subset G\times G\). Under a certain identification, this Poisson structure exactly matches that of Semenov-Tyan-Shansky, a standard construction out of a Manin triple. Two recent works of the author [Adv. Math. 230, No. 4--6, 2235--2294 (2012; Zbl 1264.14067), Nagoya Math. J. 214, 1--52 (2014; Zbl 1311.14047)] are founded by the results of this paper. T. Tanisaki, Manin triples and differential operators on quantum groups , Tokyo J. Math., 36 (2013), 49-83. Quantum groups (quantized enveloping algebras) and related deformations, Poisson manifolds; Poisson groupoids and algebroids, Geometry of quantum groups, Grassmannians, Schubert varieties, flag manifolds Manin triples and differential operators on quantum 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 We show how the threshold level of affine fusion, the fusion of Wess-Zumino-Witten (WZW) conformal field theories, fits into the Schubert calculus introduced by Gepner. The Pieri rule can be modified in a simple way to include the threshold level, so that calculations may be done for all (non-negative integer) levels at once. With the usual Giambelli formula, the modified Pieri formula deforms the tensor product coefficients (and the fusion coefficients) into what we call threshold polynomials. We compare them with the \(q\)-deformed tensor product coefficients and fusion coefficients that are related to \(q\)-deformed weight multiplicities. We also discuss the meaning of the threshold level in the context of paths on graphs. Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Classical problems, Schubert calculus Schubert calculus and threshold polynomials of affine fusion | 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 construction of the moduli space of stable maps to the classifying stack \(B\mu_r\) of a cyclic group by a sequence of \(r\)-th root constructions on \({\overline{\mathcal{M}}}_{0, n}\) and prove a closed formula for the total Chern class of \(\mu_r\)-eigenspaces of the Hodge bundle. From this, linear recursions for all genus-zero Gromov-Witten invariants of the stacks \([{\mathbb{C}}^N/\mu_r]\) are deduced.
More precisely, let \({\overline{\mathcal{M}}}_{0, n}(e_1, \dots, e_n; B\mu_r)\) be the moduli space of twisted stable maps to \(B\mu_r \cong [0/\mu_r] \subset [{\mathbb{C}}^N/\mu_r]\), where the branching behavior at the \(i\)-th section is prescribed by \(e_i \in \mu_r\). For every proper subset \(T \subset \{1, \dots, n-1\}\) having at least 2 elements, let \(D^T\) be the boundary divisor of \({\overline{\mathcal{M}}}_{0,n}\) consisting of curves having a node which separates the marking labels \(1,\ldots, n\) into \(T\) and \(\{1,\dots,n\}\setminus T\), and let \(r_T=\prod_{i \in T} e_i\). The authors show that \({\overline{\mathcal{M}}}_{0, n}(e_1, \dots, e_n; B\mu_r)\) is a \(\mu_r\)-gerbe over the stack constructed from \({\overline{\mathcal{M}}}_{0, n}\) by successively doing the \(r_T\)-th root construction at the boundary divisor \(D^T\) for all proper subsets \(T \subset \{1,\dots,n-1\}\) having at least 2 elements.
To determine a formula for the Chern class of the obstruction bundle, the authors introduce an ad-hoc definition of a ``moduli space of weighted stable maps to \(B\mu_r\)'', inspired by by the notion of weighted stable curves [\textit{B.~Hassett}, Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)] and weighted stable maps [\textit{V.~Alexeev, G.~M.~Guy}, J. Inst. Math. Jussieu 7, No. 3, 425--456 (2008; Zbl 1166.14034); \textit{A.~M.~Mustaţă, A.~Mustaţă}, J. Reine Angew. Math. 615, 93--119 (2008; Zbl 1139.14043); \textit{A.~Bayer, Yu I.~Manin}, Mosc. Math. J. 9, No. 1, 3--32 (2009; Zbl 1216.14051)]. When the weights of the linear action of \(\mu_r\) on \({\mathbb{C}}^N\) are chosen such that all fibers of the universal curve are irreducible, the obstruction bundle can easily be computed from general facts about the \(r\)-th root construction [\textit{C.~Cadman}, Am. J. Math. 129, No. 2, 405--427 (2007; Zbl 1127.14002)]. In particular, this is done for the weight data that give a moduli space isomorphic to \({\mathbb{P}}^{n-3}\). Next, by a careful analysis of the wall--crossing for changing weights, the authors are able to lift this to a closed formula for the equivariant top Chern class of the obstruction bundle for the standard (non-weighted) stable maps.
Finally, by a generalized inclusion-exclusion principle, the Chern class formula leads to linear recursions for all Gromov-Witten invariants of \([{\mathbb{C}}^N/\mu_r]\) by a sum over partitions, where every partition corresponds to a moduli space of comb curves, and an explicit formula for the non-equivariant invariants of \([{\mathbb{C}}^3/\mu_3]\) is deduced. stable maps; root construction; Gromov-Witten invariants; stacks; quantum orbifold cohomology A. Bayer and C. Cadman, Quantum cohomology of \([\mathbb{C}^N/\mu_r]\), Compos. Math. 146 (2010), no. 5, 1291-1322. MR2684301 (2012d:14095) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Stacks and moduli problems, Families, moduli of curves (algebraic) Quantum cohomology of \([\mathbb C^N/ \mu _r]\) | 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 modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler's elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops which appear as coefficients of the generating function for the Faber polynomials.{
\copyright 2016 American Institute of Physics} Korteweg-de Vries hierarchy; isometry/isoenergy conditions; Euler's elastica Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), KdV equations (Korteweg-de Vries equations), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials), Special sequences and polynomials From Euler's elastica to the mKdV hierarchy, through the Faber 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 In order to gain better understanding of the multiplication in the integral cohomology of the complex Grassmann manifold \(G_{k, n}(\mathbb{C})\) (in the Borel picture) a minimal strong Gröbner basis for the ideal \(I_{k, n}\) determining this cohomology is obtained. These results are applied to obtain recurrence relations among Kostka numbers which completely determine these numbers. Corresponding results for real Grassmann manifolds are also presented. cohomology of Grassmannians; Gröbner bases; Kostka numbers; Chern classes; Stiefel-Whitney classes; Schubert classes; Pieri's formula; immersions Petrović, Z.Z.; Prvulović, B.I.; Radovanović, M., Multiplication in the cohomology of Grassmannians via Gröbner bases, J. algebra, 438, 60-84, (2015) Symmetric functions and generalizations, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Grassmannians, Schubert varieties, flag manifolds Multiplication in the cohomology of Grassmannians via Gröbner bases | 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 0707.00010.]
The authors give a view of the status of art of the theory of de Rham- Grothendieck coefficients and the methods of proofs. differential operator; de Rham-Grothendieck coefficients Mebkhout, Z.; Narvaez, L., Sur LES coefficients de de Rham-Grothendieck des variétés algébriques, (\textit{p}-adic Analysis, Lect. Notes Math., vol. 1454, (1990), Springer-Verlag Heidelberg), 267-309 de Rham cohomology and algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Sur les coefficients de De Rham-Grothendieck des variétés algébriques. (On the De Rham-Grothendieck coefficients of algebraic 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 The author introduces a new notion of quasi manifold structure on an algebraic stack, proves some important properties of this structure and constructs the fundamental class. These notions are introduced to prove that the moduli stack, \({\mathcal M}_{gn\beta}(V)\), classifying the stable maps from \(n\)-pointed prestable curves of genus \(g\) to an algebraic proper and smooth variety \(V\) (so that the image of the fundamental class of the curve is \(\beta\), a Néron-Severi class of dimension 1) has quasi manifold structure. This result and the properties of the quasi manifold structures allow the author to prove that the fundamental class of the moduli \({\mathcal M}_{gn\beta}(V)\) satisfy all the axioms of \textit{M. Kontsevich} and \textit{Yu. Manin} [Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020)] for the Gromov-Witten class. The author proves the deformation invariance and the Borel localization formula for the Gromov-Witten class. quantum cohomology; algebraic stack; moduli stack; Gromov-Witten class Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), General geometric structures on manifolds (almost complex, almost product structures, etc.), Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds The Gromov-Witten class and a perturbation theory in 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 The Drinfeld realization of quantum affine algebras has been tremendously useful since its discovery. Combining techniques of Beck and Nakajima with our previous approach, we give a complete and conceptual proof of the Drinfeld realization for the twisted quantum affine algebras using Lusztig's braid group action. DOI: 10.1155/2016/4843075 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Affine algebraic groups, hyperalgebra constructions, Braid groups; Artin groups Drinfeld realization of quantum twisted affine algebras via braid group | 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 computes the Poincaré polynomial of the pointed linear sigma quotient. This quotient is birational to the the moduli space of stable maps \(\overline{M}_{0,n}(\mathbb{P}^{r},d)\) which is used to calculate its Betti numbers.
Let \(f:\mathbb{P}^{1}\rightarrow \mathbb{P}^{r}\) be a degree \(d\) morphism given by \((r+1)\) homogeneous degree-\(d\) polynomials on two variables with no common zero, together with a choice of \(n\) distinct points on the source \(\mathbb{P}^{1}\). This object corresponds to the base point free locus
\[
((\mathbb{P}^{1})^{n}\,\backslash\, \Delta)\times (f_{0}:\ldots :f_{r})\subset (\mathbb{P}^{1})^{n}\times \mathbb{P}\big(\bigoplus_{0}^{r}H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(d))\big) :=(\mathbb{P}^{1})^{n}\times \mathbb{P}^{r}_{d}\; ,
\]
and the compactification on the right is called the \textit{pointed sigma model}. It comes with a natural \(G=\mathrm{SL}_{2}(\mathbb{C})\) action which can be linearized to an ample line bundle \(\mathcal{L}\) to form the GIT quotient \(((\mathbb{P}^{1})^{n}\times \mathbb{P}^{r}_{d})^{ss}(\mathcal{L})/\!\!/G\), called the \textit{pointed linear sigma quotient}. Based on results from a previous paper of the same author [Ill. J. Math. 51, No. 3, 1003--1025 (2007; Zbl 1166.14006)] about a GIT construction of the moduli space of stable maps, the GIT stable and the GIT semistable loci are computed.
The main result of the present article, Theorem 2.1, consists on exhibit an explicit formula for the Poincare polynomial of such GIT quotient (under some restrictions of \(n\), \(r\) and \(d\)) for which, both GIT stable and semistable loci, do coincide, in which case the quotient is a projective variety.
Using results from [loc. cit.], the pointed linear sigma quotient \(((\mathbb{P}^{1})^{n}\times \mathbb{P}^{r}_{d})^{ss}/\!\!/G\) is birational to the space of stable maps \(\overline{M}_{0,n}(\mathbb{P}^{r},d)\), via the Givental map. If both spaces have the same Picard number, it can be proved that the birational map is an isomorphism, and this leads to compute the Betti numbers of several moduli spaces of stable maps in section \(3\). moduli spaces; geometric invariant theory; Poincaré polynomial; Betti numbers; moduli of stable maps; pointed sigma model Algebraic moduli problems, moduli of vector bundles, Geometric invariant theory, Families, moduli of curves (algebraic), Rational and birational maps The Poincaré polynomial of the pointed linear sigma quotient | 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 Indian mathematicians \textit{H. Anand, V. C. Dumir} and \textit{H. Gupta} [Duke Math. J. 33, 757--769 (1966; Zbl 0144.00401)] investigated a combinatorial distribution problem and formulated some conjectures on the number of solutions. These conjectures were solved and extended by R. Stanley. His solution, based on methods of commutative algebra, was one of the starting points of combinatorial commutative algebra. In this article we describe the conjectures and their proofs and introduce the notions of commutative algebra on which the proofs are based. Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Toric varieties, Newton polyhedra, Okounkov bodies, Exact enumeration problems, generating functions Commutative algebra arising from the Anand-Dumir-Gupta conjectures | 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 adapt the Bender-Wu algorithm \textit{C. M. Bender} and \textit{T. T. Wu}, ``Anharmonic oscillator. 2: A study of perturbation theory in large order'' in [Phys. Rev. D 7, 1620--1636 (1973; \url{doi:1103/PhysRevD.7.1620})] to solve perturbatively but very efficiently the eigenvalue problem of ``relativistic'' quantum mechanical problems whose Hamiltonians are difference operators of the exponential-polynomial type. We implement the algorithm in the function BWDifference in the updated Mathematica package BenderWu. With the help of BWDifference, we survey quantum mirror curves of toric fano Calabi-Yau threefolds, and find strong evidence that not only are the perturbative eigenenergies of the associated 1d quantum mechanical problems Borel summable, but also that the Borel sums are exact. nonperturbative effects; resummation; Calabi-Yau threefold; Bender-Wu algorithm; solitons monopoles and instantons; topological strings Gu, J.; Sulejmanpasic, T., High order perturbation theory for difference equations and Borel summability of quantum mirror curves, JHEP, 12, 014, (2017) Topological field theories in quantum mechanics, String and superstring theories in gravitational theory, Geometry and quantization, symplectic methods, Soliton equations, Calabi-Yau manifolds (algebro-geometric aspects), Calabi-Yau theory (complex-analytic aspects) High order perturbation theory for difference equations and Borel summability of quantum mirror 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 Consider a chain of vector bundle maps over a base space. The points in the source, over which the ranks of the vector bundle maps and their compositions have certain values, are called quiver degeneracy loci. The authors study the classes these degeneracy loci represent in the cohomology and \(K\)-theory of the base space.
Namely, they calculate the so-called \(K\)-polynomials and multidegrees of the quiver loci corresponding to equioriented quivers of type \(A_n\). Four formulas are presented in the paper, all having their own advantages. The formulas are combinatorial, i.e. they express the result in terms of certain combinatorial objects (Zelevinsky permutations, lacing diagrams, Young-tableaux, pipe dreams). All the formulas are positive, which means that the result is presented as a sum with positive coefficients. Three of the formulas are geometric, that is, the terms correspond bijectively to torus-invariant schemes. The last formula is very similar in nature to the conjectured formula of Buch and Fulton for the same loci. multidegree; quiver polynomial; degeneraci loci; Schubert calculus Knutson, Allen; Miller, Ezra; Shimozono, Mark, Four positive formulae for type \(A\) quiver polynomials, Invent. Math., 166, 2, 229-325, (2006) Enumerative problems (combinatorial problems) in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Four positive formulae for type \(A\) quiver 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 This paper gives a complete and elementary proof of the Yomdin-Gromov algebraic lemma which states that some `differentiable size' of an semi-algebraic compact subset of a finite-dimensional Euclidian space may be bounded in terms of its dimension, degree and diameter. semi-algebraic sets; Yomdin-Gromov algebraic lemma D. Burguet, ''A proof of Yomdin-Gromov's algebraic lemma,'' Israel J. Math., 168, 291--316 (2008). Semialgebraic sets and related spaces A proof of Yomdin-Gromov's algebraic Lemma | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.