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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study quantum and Floer type cohomology on cosymplectic manifolds and show that the quantum type cohomology and the Floer type cohomology are naturally isomorphic. Gromov-Witten type invariant; quantum type cohomology; Maslov index; symplectic type action functional Cho, Y.S.: Quantum type cohomologies on cosymplectic manifolds, (Preprint) Symplectic aspects of Floer homology and cohomology, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum and Floer type cohomologies on cosymplectic manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a new proof that three families of polynomials coincide: the double Schubert polynomials of \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [Lett. Math. Phys. 10, 111--124 (1985; Zbl 0586.20007)] defined by divided difference operators, the pipe dream polynomials of \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, No. 4, 257--269 (1993; Zbl 0803.05054)], and the equivariant cohomology classes of matrix Schubert varieties. All three families are shown to satisfy a ``co transition formula'' which we explain to be some extent projectively dual to Lascoux' transition formula. We comment on the \(K\)-theoretic extensions. Classical problems, Schubert calculus, Representations of finite symmetric groups, Combinatorial aspects of algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry Schubert polynomials, pipe dreams, equivariant classes, and a co-transition formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathbb P ^{n}\) be the \(n\)-dimensional projective space over a field \(k\). A homogeneous polynomial \(F\) is called homaloidal if the polar map \(\partial F:\mathbb P ^{n}\longrightarrow \mathbb P ^{n}\) defined by partial derivatives of \(F\) is a birational transformation of \(\mathbb P ^{n}\). The author considers special polynomials \(F= \prod_{i=0}^r L_{i}^{m_i}\), where \(L_0,\dots , L_{r}\) are linear form, \(m_0,\dots , m_{r}\) are positive integers, and obtains a necessary and sufficient condition such polynomials to be homaloidal. Additional assumptions for this result are either \(\text{char} k=0\) or resolution of singularities holds in \(\text{char} k=p\) and dimension \(n\). projective spaces; birational transformations Bruno, A.: On homaloidal polynomials. Mich. Math. J. 55(2), 347--354 (2007) Birational automorphisms, Cremona group and generalizations, Rational and birational maps On homaloidal polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study algebraic properties of the small quantum homology algebra of symplectic toric Fano manifolds. A finite-dimensional commutative algebra over a field is called semisimple if it decomposes into a direct sum of fields. The quantum homology algebra of a symplectic manifold of real dimension \(2d\) is called semisimple if its \(2d\)-graded part is a semisimple algebra. Many examples of symplectic manifolds with a semisimple quantum homology are known.
Semisimplicity also holds in the case of a smooth \(2d\)-dimensional toric Fano variety with respect to a symplectic form satisfying some genericity assumption. Entov and Polterovich posed the question whether the quantum homology is semisimple for any symplectic toric manifold. The authors answer this question negatively by constructing an example of a toric symplectic manifold whose quantum homology is not semisimple. For a toric Fano manifold, there exists a distinguished symplectic form \(\omega_0\), the unique symplectic form for which the associated moment polytope is reflexive. The authors show that this distinguished symplectic form is the worst: If for a toric Fano manifold the quantum homology with respect to \(\omega_0\) is semisimple, then it is semisimple for any choice \(\omega\) of a toric symplectic form. A weaker condition is to require that the \(2d\)-graded part contains a field as a direct summand. The analogue of the above also holds for this case: If for a \(2d\)-dimensional toric Fano manifold the \(2d\)-graded part of quantum homology with respect to \(\omega_0\) contains a field as a direct summand, then so does the \(2d\)-graded part of quantum homology with respect to any toric symplectic form \(\omega\).
There is no example of a symplectic toric Fano manifold of dimension less than or equal to \(8\) whose \(2d\)-graded part of quantum homology does not contain a field as a direct summand. The property of having a field as a direct summand is equivalent to the existence of a non-degenerate critical point of the Landau-Ginzburg superpotential associated to the symplectic manifold. The authors use these results to construct new examples of symplectic manifolds admitting Calabi quasi-morphisms and symplectic quasi states. With these examples they can answer another question of Entov and Polterovich negatively, namely the question of uniqueness of Calabi quasi-morphisms. quantum homology; semisimplicity; toric Fano manifolds; Landau-Ginzburg superpotential; Calabi quasi-morphisms Ostrover, Y., Tyomkin, I.: On the quantum homology algebra of toric Fano manifolds. Selecta Math. (N.S.) \textbf{15}(1), 121-149 (2009) Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Toric varieties, Newton polyhedra, Okounkov bodies On the quantum homology algebra of toric Fano manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that the generating series of the Poincaré polynomials of quasihomogeneous Hilbert schemes of points in the plane has a beautiful decomposition into an infinite product. We also compute the generating series of the numbers of quasihomogeneous components in a moduli space of sheaves on the projective plane. The answer is given in terms of characters of the affine Lie algebra \(\widehat{sl}_m\). Parametrization (Chow and Hilbert schemes), Combinatorial aspects of representation theory, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Algebraic moduli problems, moduli of vector bundles Generating series of the Poincaré polynomials of quasihomogeneous Hilbert schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum plane is the non-commutative polynomial algebra in variables \(x\) and \(y\) with \(xy=qyx\). In this paper, we study the module variety of \(n\)-dimensional modules over the quantum plane, and provide an explicit description of its irreducible components and their dimensions. We also describe the irreducible components and their dimensions of the GIT quotient of the module variety with respect to the conjugation action of \(\mathrm{GL}_n\). varieties of modules; irreducible components; quantum plane Representations of quivers and partially ordered sets, Special varieties Varieties of modules over the quantum 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 In this paper, we prove a conjecture of Alexandrov that the generalized Brézin-Gross-Witten tau-functions are hypergeometric tau functions of BKP hierarchy after re-scaling. In particular, this shows that the original BGW tau-function, which has enumerative geometric interpretations, can be represented as a linear combination of Schur Q-polynomials with simple coefficients. Brézin-Gross-Witten \(\tau\)-function; Schur Q-polynomial; BKP hierarchy; Virasoro constraints Connections of hypergeometric functions with groups and algebras, and related topics, Enumerative problems (combinatorial problems) in algebraic geometry, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Q-polynomial expansion for Brézin-Gross-Witten tau-function | 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 Virasoro conjecture states that the generating function of GW invariants is annihilated by infinitely many differential operators which form a part of the Virasoro algebra.
In this expository work, the author explains how to apply ideas from his previous work [Adv. Math. 169, No. 2, 313--375 (2002; Zbl 1105.14075)] to study the Virasoro conjecture in all genera. The author also explains how to use a quantum product on the big phase space to interpret topological recursion relations and the Virasoro conjecture. quantum product; Virasoro conjecture; recursion relations Xiaobo Liu, Quantum product, topological recursion relations, and the Virasoro conjecture, Surveys on geometry and integrable systems, Adv. Stud. Pure Math., vol. 51, Math. Soc. Japan, Tokyo, 2008, pp. 235 -- 257. Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum product, topological recursion relations, and the Virasoro 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 The Grothendieck ring $K_0(\mathrm{Var}/{\mathbb C})$ is defined as the quotient of the group of formal linear combinations with integer coefficients modulo the relations $[Z]-[U]-[Z\backslash U] $ for all open subvarieties $U\subseteq Z.$ This is a ring with the product induced by the Cartesian product. The main result of the paper asserts that the class of the affine line is a zero divisor in the Grothendieck ring of varieties over ${\mathbb C}.$ The question is important since it has significant consequences. One of them is that a rational smooth cubic fourfold in ${\mathbb P}^5$ must have its Fano variety of lines birational to a symmetric square of a $K3$ surface (cf. [\textit{S. Galkin} and \textit{E. Shinder}, ``The Fano variety of lines and rationality problem for a cubic hypersurface'', Preprint, \url{arXiv:1405.5154}]). The other consequence is the fact that cut-and-paste conjecture of Larsen and Lunts fails (cf. Zbl [\textit{M. Larsen} and \textit{V. A. Lunts}, Mosc. Math. J. 3, No. 1, 85--95 (2003; Zbl 1056.14015)]). The proof of the main theorem is based on the Pfaffian-Grassmannian double mirror correspondence. Grothendieck ring; affine line; Pfaffian-Grassmanian double mirror correspondence Applications of methods of algebraic \(K\)-theory in algebraic geometry The class of the affine line is a zero divisor in the Grothendieck ring | 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 algorithm for computing a Gröbner basis of an ideal of polynomials whose coefficients are taken from a ring with zero divisors, is presented; such rings include \(\mathbb {Z}_n\) and \(\mathbb {Z}_n[i]\), where \(n\) is not a prime number. The algorithm is patterned after (1) Buchberger's algorithm for computing a Gröbner basis of a polynomial ideal whose coefficients are from a field and (2) its extension developed by Kandri-Rody and Kapur when the coefficients appearing in the polynomials are from a Euclidean domain. The algorithm works as Buchberger's algorithm when a polynomial ideal is over a field and as Kandri-Rody-Kapur's algorithm when a polynomial ideal is over a Euclidean domain. The proposed algorithm and the related technical development are quite different from a general framework of reduction rings proposed by Buchberger in 1984 and generalized later by Stifter to handle reduction rings with zero divisors. These different approaches are contrasted along with the obvious approach where for instance, in the case of \(\mathbb {Z}_n\), the algorithm for polynomial ideals over \(\mathbb {Z}\) could be used by augmenting the original ideal presented by polynomials over \(\mathbb {Z}_n\) with \(n\) (similarly, in the case of \(\mathbb {Z}_n[i]\), the original ideal is augmented with \(n\) and \(i^{2} + 1\)). Gröbner basis; ring with zero divisors; polynomial ideal; ideal membership; canonical form; algebraic geometry D. Kapur, Y. Cai, An algorithm for computing a Gröbner basis of a polynomial ideal over a ring with zero divisors. Math. Comput. Sci. 2, 601--634 (2009) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Effectivity, complexity and computational aspects of algebraic geometry An algorithm for computing a Gröbner basis of a polynomial ideal over a ring with zero divisors | 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 Grassmannian is known to be a graded quantum algebra with a straightening law when the poset of generating quantum minors is endowed with the standard partial ordering. In this paper it is shown that this result remains true when the ordering is subjected to cyclic shifts. The method involves proving that noncommutative dehomogenization is possible at any consecutive quantum minor. quantum groups; quantum matrices; quantum Grassmannians; quantum minors; quantum determinants; straightening laws; noncommutative dehomogenisation Lenagan, Cyclic orders on the quantum Grassmannian, Arab. J. Sci. Eng. Sect. C Theme Issues 33 pp 337-- (2008) Ring-theoretic aspects of quantum groups, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Rings arising from noncommutative algebraic geometry, Noetherian rings and modules (associative rings and algebras) Cyclic orders on the quantum Grassmannian. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors construct \(q\)-deformations of the affine Grassmann and flag varieties over the quantum general and special linear groups. Here \(q\)- deformations mean deformations of the coordinate rings of the appropriate varieties. This is done by deforming the relations given by the second author [in J. Algebra 47, 80-104 (1977; Zbl 0358.15033) and ibid. 61, 414-462 (1979; Zbl 0437.14030)]. Grassmann variety; \(q\)-deformations; flag varieties Taft, E.; Towber, J., Quantum deformation of flag schemes and Grassmann schemes, I: a \textit{q}-deformation of the shape-algebra for \(\operatorname{GL}(n)\), J. Algebra, 142, 1-36, (1991) Quantum groups (quantized enveloping algebras) and related deformations, Homogeneous spaces and generalizations Quantum deformation of flag schemes and Grassmann schemes. I: A \(q\)- deformation of the shape-algebra for \(GL(n)\) | 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 In the paper under review, it is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights, their Schubert varieties, big cells, and determinantal varieties are AS-Cohen-Macaulay. The notion of a quantum graded algebra with a straightening law, introduced by [\textit{T. H. Lenagan} and \textit{L. Rigal}, J. Algebra 301, No. 2, 670--702 (2006; Zbl 1108.16026)] is effectively used as a main tool. Using Stanley's Theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are AS-Gorenstein. quantum flag manifolds; straightening laws; Cohen-Macaulay; Gorenstein S. Kolb, The AS-Cohen-Macaulay property for quantum flag manifolds of minuscule weight. J. Algebra 319 (2008), 3518-3534. Quantum groups (quantized enveloping algebras) and related deformations, Rings with straightening laws, Hodge algebras, Grassmannians, Schubert varieties, flag manifolds, Ring-theoretic aspects of quantum groups The AS-Cohen-Macaulay property for quantum flag manifolds of minuscule weight | 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 the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety \(V\) or a Calabi-Yau hypersurface \(M\subset|V\). In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth \(V\), our results reproduce and clarify an algebraic solution of the \(V\) model due to Batyrev. In addition, we find an algebraic relation determining the solution for \(M\) in terms of that for \(V\). Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the \(M\) model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured ``monomial-divisor mirror map'' of \textit{P. S. Aspinwall}, \textit{B. R. Greene} and \textit{D. R. Morrison} [Int.Math. Res. Not. 1993, No. 12, 319-337 (1993; Zbl 0798.14030)]. moduli space; mirror symmetry; toric varieties; Calabi-Yau hypersurface D.R. Morrison and M.R. Plesser, \textit{Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties}, \textit{Nucl. Phys.}\textbf{B 440} (1995) 279 [hep-th/9412236] [INSPIRE]. Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems Summing the instantons: Quantum cohomology and mirror symmetry in 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 paper is concerned with the computation of the number \(N_{d,\alpha}\) of rational curves on \({\mathbb P}^2\) of degree \(d\) that pass through \(r\) given points \(x_i\) with multiplicities \(\alpha_i\) and \(3d-\sum_i\alpha_i-1\) other general points. These numbers appear as coefficients in the quantum cohomology potential function for the blow-up of \({\mathbb P}^2\) in \(r\) points. In the first part of the paper the associativity relations for the quantum product are used to derive recursion formulas for \(N_{d,\alpha}\). This formal computation is analogous to the computations of \textit{M. Kontsevich} and \textit{Yu. Manin} [Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020)] for the plane itself.
The major part of the paper is devoted to the question when this virtual number \(N_{d,\alpha}\) correctly counts rational curves with the required properties. This is shown to be the case under the following hypotheses: \(d>0\), \(\alpha_i\geq 0\) for all \(i\) and either \(n_{d,\alpha}>0\) or \(n_{d,\alpha}\geq 0\) and \(\alpha_i\in\{1,2\}\) for some \(i\). In this case each solution is an immersion of \({\mathbb P}^1\) into the blow-up. Finally, using Cremona transformations the authors slightly extend the range of parameters \((d,\alpha)\) where \(N_{d,\alpha}\) has a correct enumerative interpretation beyond the limits of the main theorem.
The paper concludes with tables for \(N_{d,\alpha}\) in the range \(d\leq 7\). These computations correct a number of \textit{P. Di Francesco} and \textit{C. Itzykson} [in: The moduli space of curves, Proc. Conf., Texel Island 1994, Prog. Math. 129, 81-148 (1995; Zbl 0868.14029)] for \(d=6\). number of rational curves; enumerative geometry; blow-up of projective plane; quantum cohomology potential function L. Göttsche, R. Pandharipande, The quantum cohomology of blow-ups of \(\mathbb{P}\)2 and enumerative geometry. \textit{J. Differential Geom}. \textbf{48} (1998), 61-90. MR1622601 Zbl 0946.14033 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, String and superstring theories; other extended objects (e.g., branes) in quantum field theory The quantum cohomology of blow-ups of \(\mathbb{P}^2\) and enumerative geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper continues the classification of quadratic Artin-Schelter regular algebras of global dimension 4 which map onto a twisted homogeneous coordinate ring of a quadric hypersurface in \(\mathbb{P}^3\). It is a continuation of papers by \textit{M. Vancliff} [J. Algebra 165, No. 1, 63-90 (1994; Zbl 0837.16023)], \textit{K. Van Rompay} [ibid. 180, No. 2, 483-512 (1996; Zbl 0853.17012)], \textit{M. Vancliff}, \textit{K. Van Rompay} [ibid. 195, No. 1, 93-129 (1997; Zbl 0910.16013)]. In the present paper the cases when the quadric has rank 3 are considered. quadratic Artin-Schelter regular algebras; global dimensions; twisted homogeneous coordinate rings; quadric hypersurfaces Shelton, B.; Vancliff, M., Embedding a quantum rank three quadric in a quantum \(\mathbb{P}^3\), Comm. Algebra, 27, 6, 2877-2904, (1999) Quadratic and Koszul algebras, Graded rings and modules (associative rings and algebras), Noncommutative algebraic geometry, Twisted and skew group rings, crossed products, Quantum groups (quantized enveloping algebras) and related deformations Embedding a quantum rank three quadric in a quantum \(\mathbb{P}^3\) | 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 Let \(k\) be a base field and \(G\) be an algebraic group over \(k\). J.-P. Serre defined \(G\) to be special if every \(G\)-torsor \(T \to X\) is locally trivial in the Zariski topology for every reduced algebraic variety \(X\) defined over \(k\). In recent papers an a priori weaker condition is used: \(G\) is called special if every \(G\)-torsor \(T \to \operatorname{Spec}(K)\) is split for every field~\(K\) containing \(k\). We show that these two definitions are equivalent. We also generalize this fact and propose a strengthened version of the Grothendieck-Serre conjecture based on the notion of essential dimension. algebraic group; torsor; special group; local ring; Grothendieck-Serre conjecture; essential dimension Representation theory for linear algebraic groups, Formal groups, \(p\)-divisible groups, Cohomology theory for linear algebraic groups, Linear algebraic groups over arbitrary fields, Linear algebraic groups over adèles and other rings and schemes Special groups, versality and the Grothendieck-Serre 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 The author reviews some aspects of the theory of noncommutative two-tori with real multiplication, focusing on the role played by Heisenberg groups in the definition of algebraic structures associated to these noncommutative spaces. Noncommutative geometry (à la Connes), Noncommutative algebraic geometry, Noncommutative differential geometry, Rings arising from noncommutative algebraic geometry, Noncommutative geometry in quantum theory Heisenberg modules over real multiplication noncommutative tori and related algebraic structures | 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 As a general principle, the rigidity of quantum algebras, especially those with generic parameters, means that these algebras have much less symmetry than their classical counterparts. In particular, their automorphism groups are usually far smaller, and classical group actions usually do not extend to actions on quantum algebras. Some extensions have been constructed, at the expense of additional structure, and the present paper achieves this for dihedral actions on Grassmannians.
The natural action of the symmetric group \(S_n\) on the Grassmannian \(G(m,n)\), by permutation of columns, extends to the coordinate ring \(\mathcal O(G(m,n))\) but not to the quantized coordinate ring \(\mathcal O_q(G(m,n))\). Even the cycle \(c=(1\,2\,\cdots\,n)\) does not induce an automorphism of \(\mathcal O_q(G(m,n))\). \textit{S. Launois} and \textit{T. H. Lenagan} showed that this lack can be repaired as follows: There is a cocycle twist \(T(\mathcal O_q(G(m,n)))\) which supports an isomorphism \(\theta\colon T(\mathcal O_q(G(m,n)))\to\mathcal O_q(G(m,n))\) that sends any \(m\times m\) quantum minor in \(T(\mathcal O_q(G(m,n)))\) with column index set \(I\) to the one in \(\mathcal O_q(G(m,n))\) with column index set \(c(I)\), up to a power of \(q\) [Proc. Am. Math. Soc. 139, No. 1, 99-110 (2011; Zbl 1219.16036)].
The present authors construct a quantum analog of the action on \(G(m,n)\) by the dihedral subgroup of \(S_n\) generated by the cycle \(c\) and the longest element \(w_0\). This analog takes the form of a groupoid based on a family of cocycle twists of \(\mathcal O_q(G(m,n))\). As a consequence, an action of the mentioned dihedral group on the poset of torus-invariant prime ideals of \(\mathcal O_q(G(m,n))\) is obtained. Moreover, there are actions of this dihedral group on the totally nonnegative and totally positive Grassmannians \(G^{\text{tnn}}(m,n)\) and \(G^{\text{tp}}(m,n)\). quantum algebras; coordinate rings; quantum Grassmannians; twistings; dihedral groups; torus-invariant prime ideals; totally nonnegative Grassmannians; totally positive Grassmannians Ring-theoretic aspects of quantum groups, Rings arising from noncommutative algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations, Ideals in associative algebras, Group actions on varieties or schemes (quotients) A quantum analogue of the dihedral action on Grassmannians. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An algebraic approach is developed to define and study infinite-dimensional Grassmannians. Using this approach, a quantum deformation (i.e. a deformation of the coordinate ring) is obtained for both the ind-variety union of all finite-dimensional Grassmannians \(G_{\infty}\), and the Sato Grassmannian \(\widetilde{UGM}\) introduced by Sato. They are both quantized as homogeneous spaces, that is together with a coaction of a quantum infinite dimensional group. At the end, an infinite-dimensional version of the first theorem of invariant theory is discussed for both the infinite-dimensional special linear group and its quantization. R. Fioresi and C. Hacon, \textit{On infinite-dimensional Grassmannians and their quantum deformations}, Rend. Sem. Mat. Univ. Padova, 111 (2004), pp. 1--24. Grassmannians, Schubert varieties, flag manifolds, Group structures and generalizations on infinite-dimensional manifolds, Geometry of quantum groups On infinite-dimensional Grassmannians and their quantum deformations | 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 announce a construction of quantum deformations of the Schubert varieties for semisimple algebraic groups (which are either classical or of types \(E_ 6\) or \(G_ 2\)). More precisely, the authors consider the coordinate ring \({\mathbf C}[X]\) of a Schubert variety \(X\) and then give a construction of a quantum analogue \({\mathbf C}_ q[X]\). This is done by means of the socalled admissible quadruples and standard monomials [cf. \textit{V. Lakshmibai} and \textit{K. N. Rajeswari}, Contemp. Math. 88, 449-578 (1989; Zbl 0682.14035)]. quantum deformations; Schubert varieties for semisimple algebraic groups V. Lakshmibai and N. Reshetikhin, ''Quantum deformations of flag and Schubert schemes,''C. R. Acad. Sci. Paris Ser. I Math.,313, No. 3, 121--126 (1991). Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Quantum deformations of flag and Schubert schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite, and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned. mutually unbiased bases; Galois fields and rings; maximally entangled states Planat, M., Rosu, H.C., Perrine, S.: A survey of finite algebraic geometrical structures underlying mutually unbiased measurements. Found. Phys. 36, 1662--1680 (2006) Quantum computation, Quantum measurement theory, state operations, state preparations, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Galois theory A survey of finite algebraic geometrical structures underlying mutually unbiased quantum measurements | 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 scheme over \({\mathbb Z}\) whose generic fibre \(X_{\mathbb Q}\) is smooth. Using exterior powers of hermitian bundles on \(X\) on the one hand and the natural grading of the real vector space \(\widetilde{A}(X):= \bigoplus_{p\geq 0} A^{p,p}(X({\mathbb C}))/(\text{im}(\partial) +\text{im}(\bar{\partial}))\) on the other hand, \textit{H. Gillet} and \textit{Ch. Soulé} have defined a natural pre-\(\lambda\)-ring structure on the arithmetic Grothendieck group \(\widehat{K}_0(X)\) [Ann. Math., II. Ser. 131, No. 2, 205-238 (1990; Zbl 0715.14006). The goal of the very well written paper under review is to show that this pre-\(\lambda\)-ring structure is in fact a (special) \(\lambda\)-ring structure which means to verify certain axioms for the composition of any \(\lambda\)-operation on \(\widehat{K}_0(X)\) with multiplication and with any other \(\lambda\)-operation.
The strategy applied is to ``compactify'' the universal ring for natural operations on classical Grothendieck groups in the spirit of Arakelov geometry, more precisely: He defines a ``compactification'' \(\widehat{R}\) of the representation ring of the group scheme \(\text{GL}_n \times \text{GL}_m\) over Spec(\({\mathbb Z}\)) and then shows that it is a (special) \(\lambda\)-ring and that the sub-pre-\(\lambda\)-ring of \(\widehat{K}_0(X)\) generated by any two hermitian bundles on \(X\) is the image of a pre-\(\lambda\)-ring homomorphism from \(\widehat{R}\) to \(\widehat{K}_0(X)\).
The main theorem of this paper plays a crucial role in the proof of the arithmetic Adams-Riemann-Roch theorem [\textit{D. Roessler}, Duke Math. J. 96, No. 1, 61-126 (1999; Zbl 0961.14006)] and in the proof of the invertibility of the Todd element in a Lefschetz fixed point formula [see \textit{K. Köhler} and \textit{D. Roessler}, Invent. Math. 145, 333-396 (2001; Zbl 0999.14002)]. Another approach to it based on the classical splitting principle has been carried out by \textit{A. Meissner} is his thesis [``Arithmetische \(K\)-Theorie, Univ. Regensburg (1993; Zbl 0833.19001)]. arithmetic Grothendieck group; lambda-ring; exterior powers; universal representation ring; hermitian metric; Adams operation; Arakelov geometry; arithmetic Adams-Riemann-Roch theorem D. Rössler, Lambda structure on arithmetic Grothendieck groups, Israel J. Math. 122 (2001), 279-304. Grothendieck groups and \(K_0\), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Value distribution theory in higher dimensions, Riemann-Roch theorems, Arithmetic varieties and schemes; Arakelov theory; heights, \(K\)-theory of schemes Lambda-structure on Grothendieck groups of Hermitian vector bundles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(w \colon \mathbb{C}^n \to \mathbb{C}\) be a quasi-homogeneous polynomial, which defines a smooth hypersurface \(Q_w\) in an appropriate weighted projective space, and \(G\) be a finite group, that acts on \(\mathbb{C}^n\) fixing \(w\). Assuming that the pair \((w, G)\) satisfies certain natural conditions, the associated Fan-Jarvis-Ruan-Witten (FJRW) invariants are cohomological invariants, constructed by studying the moduli stack \(X\) of (rigidified) \(G\)-spin curves, which satisfy the axioms of a cohomological field theory. Besides their physical meaning, they constitute a fundamental curve counting theory, as they encode the Gromov-Witten invariants of \(Q_w\), while being typically easier to compute. In this paper, the authors give an algebraic geometric interpretation of the FJRW cohomological field theories by showing that they can be expressed as integral, Fourier-Mukai type transforms whose kernels are algebraic virtual fundamental cycles. Their approach complements earlier constructions in the literature which were of analytic or more algebraic nature or worked in more restrictive cases. To produce the algebraic virtual cycle kernels, the authors apply the cosection localization principle to the stack \(X\). In the broad sector case, it is necessary to perform a suitable blowup as a first step before applying cosection localization. Fan-Jarvis-Ruan-Witten theory; virtual cycle; cosection localization Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Algebraic virtual cycles for quantum singularity theories | 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 survey paper is devoted to the explicit computation of Thom polynomials of singularities [\textit{R.~Thom}, Ann. Inst. Fourier 6, 43--87 (1956; Zbl 0075.32104)] in the form of linear combinations of Schur functions. In particular, the authors recall the notion of Schur functions and Thom polynomials, sketch the proof of positivity of coefficients in the Schur function expansion of a Thom polynomial [\textit{P.~Pragacz} and \textit{A.~Weber}, Fundam. Math. 195, No. 1, 85--95 (2007; Zbl 1146.05049)], the Rimanyi method of computation of Thom polynomials [\textit{R. Rimanyi}, Invent. Math. 143, No. 3, 499--521 (2001; Zbl 0985.32012)], overview Thom polynomials of the singularities \(A_i\), \((xy, x^2, y^3)\) and \((xy, x^2+y^2)\), and give some new computational results for the next simplest \((2,0)\)-Thom-Boardman singularities \((xy, x^2+y^3)\) and \((xy, x^3, y^3)\). The survey is well written and can serve as a quick introduction to the subject and an overview of preceding works by the authors. Thom polynomial; Schur function; singularity; Grassmanian; cotangent map; degeneracy locus; resultant Global theory of complex singularities; cohomological properties, Singularities of differentiable mappings in differential topology, Global theory of singularities, Symmetric functions and generalizations, Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus On Schur function expansions of Thom polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a recent conjecture by \textit{ Gyenge} et al. [Int. Math. Res. Not. 2017, No. 13, 4152--4159 (2017; Zbl 1405.14010); Eur. J. Math. 4, No. 2, 439--524 (2018; Zbl 1445.14007)] giving a formula of the generating function of Euler numbers of Hilbert schemes of points \(\operatorname{Hilb}^N(\mathbb{C}^2/ \Gamma )\) on a simple singularity \({\mathbb{C}^2}/ \Gamma \), where \(\Gamma\) is a finite subgroup of \(\operatorname{SL}(2)\). We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with \(\Gamma\) at \(\zeta =\exp (\frac{2\pi \sqrt{-1}}{2({h^{\vee }}+1)})\) are always 1, which is a special case of an earlier conjecture by Kuniba. Here \({h^{\vee }}\) is the dual Coxeter number. We also prove the claim, which was not known for \({E_7}\), \({E_8}\) before. Hilbert schemes of points; quantum affine algebras; quantum dimensions; simple surface singularities Parametrization (Chow and Hilbert schemes), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Quantum groups (quantized enveloping algebras) and related deformations Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give an account of the theory of dessins d'enfants which is both elementary and self-contained. We describe the equivalence of many categories (graphs embedded nicely on surfaces, finite sets with certain permutations, certain field extensions, and some classes of algebraic curves), some of which are naturally endowed with an action of the absolute Galois group of the rational field. We prove that the action is faithful. Eventually we prove that Gal\((\overline {\mathbb Q}/\mathbb Q)\) embeds into the Grothendieck-Teichmüller group \(\widehat {\mathcal {GT}_0}\) introduced by Drinfeld. There are explicit approximations of \(\widehat {\mathcal {GT}_0}\) by finite groups, and we hope to encourage computations in this area. Our treatment includes a result which has not appeared in the literature yet: the action of Gal\((\overline {\mathbb Q}/\mathbb Q)\) on the subset of \textit{regular} dessins -- that is, those exhibiting maximal symmetry -- is also faithful. dessins d'enfants; Grothendieck-Teichmüller; regular maps; absolute Galois group Guillot, Pierre, An elementary approach to dessins d'enfants and the Grothendieck-Teichmüller group, Enseign. Math., 60, 3-4, 293-375, (2014) Arithmetic aspects of dessins d'enfants, Belyĭ theory, Dessins d'enfants theory, Galois theory An elementary approach to dessins d'enfants 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 mostly expository article explores recent developments in the relations between the three objects in the title from an algebro-combinatorial perspective.
We prove a formula for Whittaker functions of a real semisimple group as an integral over a geometric crystal in the sense of Berenstein-Kazhdan. We explain the connections of this formula to the program of mirror symmetry of flag varieties developed by Givental and Rietsch; in particular, the integral formula proves the equivariant version of Rietsch's mirror symmetry conjecture.
We also explain the idea that Whittaker functions should be thought of as geometric analogues of irreducible characters of finite-dimensional representations. Lam, Th.: Whittaker functions, geometric crystals, and quantum Schubert calculus (2013). arXiv:1308.5451 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Whittaker functions, geometric crystals, and quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Iwahori-Hecke algebra \(H_n\) associated to the symmetric group \(S_n\) has a faithful representation as an algebra of operators of the polynomial algebra in \(n\) variables such that the symmetric polynomials are treated as scalars under the action of \(H_n\). These symmetrizing operators are the Newton divided differences and their deformations. The simple operators satisfy the Yang-Baxter relations. The authors give several expressions of the operators corresponding to a maximal permutation and recover the generalized Euler-Poincaré characteristic defined by Hirzebruch in the geometry of flat manifolds. Restricting the action of the Hecke algebra to weight spaces, the authors also recover one of the usual descriptions of its representations. They also obtain \(q\)-idempotents and give a \(q\)-analogue of the Specht representations as orbits of products of \(q\)-Vandermonde functions. The authors study different constructions of irreducible representations corresponding to hook partitions and describe them in terms of Kazhdan-Lustig graphs. This interpretation is applied to the diagonalization of the Hamiltonian of a quantum spin chain with the quantum superalgebra \(U_q ({\mathfrak su} (1/1))\) as a symmetry algebra. Iwahori-Hecke algebra; symmetric group; Newton divided differences; Yang- Baxter relations; Euler-Poincaré characteristic; flat manifolds; irreducible representations; hook partitions; Kazhdan-Lustig graphs; quantum spin chain; quantum superalgebra; symmetry algebra Duchamp, G., Krob, D., Lascoux, A., Leclerc, B., Scharf, T., Thibon, J.Y.: Euler-Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras. Publ. RIMS \textbf{31}, 179-201 (1995) Combinatorial aspects of representation theory, Representations of finite symmetric groups, Grassmannians, Schubert varieties, flag manifolds, Applications of linear algebraic groups to the sciences Euler-Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be an algebraically closed field of characteristic zero. The Grothendieck ring \(K_0({\mathcal{V}}/k)\) of algebraic varieties over \(k\) is generated (as an abelian group) by the isomorphism classes of schemes of finite type over \(k\) subject to the relations \([X]=[X\backslash Z] + [Z],\) where \(Z\subset X\) is a closed subscheme with the reduced structure. The product is defined as \([X]\cdot [Y]=[X\times Y].\) The main result of the paper asserts that for a pair of closed subschemes cut out (in certain way depending on a non-zero global section \(s\) of the appropriate homogenous variety) from the pair of Grassmanians of type \(G_2\) one has \(([X]-[Y])\cdot {\mathbb L} =0.\) Moreover, for the general choice of \(s\) one has \([X]\neq [Y] \) and both \(X\) and \(Y\) are smooth Calabi-Yau \(3\)-folds. Grothendieck ring; Grassmanian; Dynkin diagram; global section Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grothendieck groups and \(K_0\), Varieties and morphisms, Grassmannians, Schubert varieties, flag manifolds, Calabi-Yau manifolds (algebro-geometric aspects) The class of the affine line is a zero divisor in the Grothendieck ring: via \(G_2\)-Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a second in a series of three papers where the authors develop structural results and computational techniques for the higher genus Gromov-Witten potentials of quintic Calabi-Yau \(3\)-folds. The first paper introduced so called NMSP (N-Mixed-Spin-P) fields, and proved that their invariants can be separated into contributions from Gromov-Witten invariants of the quintic Calabi-Yau \(3\)-folds and Fan-Jarvis-Ruan-Witten invariants of the Fermat quintic. In this paper they use NMSP fields to prove the polynomial structure conjecture of Yamaguchi-Yau for the quintic Calabi-Yau \(3\)-folds, and develop an algorithm for calculating their Gromov-Witten potentials \(F_g\). One corollary is that the said potentials are analytic functions of the Novikov variable near \(0\). In the third paper of the series the authors prove the ``Feynman rule'' of Bershadsky-Cecotti-Ooguri-Vafa for computing the invariants, originally derived conjecturally from mirror symmetry.
The algorithm reduces the calculation of \(F_g\) to that of lower genus potentials \(F_h\) with \(h<g\), computable twisted point potentials, and a degree \(g-1\) polynomial determined by previously introduced NMSP-\([0,1]\) invariants representing the ambiguity. The latter correspond to NMSP virtual localization graphs all of whose vertices are labeled by \(0\) or \(1\) (not \(\infty\)). The stabilization of the virtual localization formula is rephrased in terms of the \(R\)-matrix action on the Cohomological Field Theory introduced by Givental.
The authors identify a ring of five generators that the normalized Gromov-Witten potentials \(P_{g,n}\) belong to when \(2g-2+n>0\), and give their canonical presentation in terms of the generators, thus proving the polynomial structure conjecture. The proof is based on two structure theorems. The first one gives explicit relations between global and local generating functions via the \(R\)-matrix, and the second one establishes polynomiality of the NMSP-\([0,1]\) correlators by building on the results of the first paper. quintic Calabi-Yau 3-folds; polynomial structure conjecture; Gromov-Witten invariants; mirror symmetry; BCOV theory; mixed spin moduli; cohomological field theory; R-matrix Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Polynomial structure of Gromov-Witten potential of quintic \(3\)-folds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a scheme \(X\), let \(D(qc/X)\) (resp. \(D^+(qc/X))\) denote the derived category of complexes of coherent sheaves (resp. bounded below complexes of coherent sheaves) on \(X\). The traditional Grothendieck duality theorem states that in many cases (say when the schemes involved are noetherian), for a proper map \(f:X\to Y\), the functor \(\mathbb{R} f_*:D^+ (qc/X) \to D^+(qc/Y)\) has a right adjoint \(f^!\). There is a short elegant proof of this due to Deligne [see the appendix to \textit{R. Hartshorne}'s book: ``Residues and duality'' (1966; Zbl 0212.26101)]. If \(Y\) is the spectrum of a field \(k\), we only have to produce \(f^!k\), and Deligne's proof is particularly transparent in this case as the reader can verify from what follows. For an open affine set \(U=\text{Spec} A\) of \(X\), consider the quasi-coherent sheaf \({\mathcal I}_U\) on \(X\) obtained by sheafifying the \(A\)-module \(\Hom_k(A,k)\) on \(U\), and then pushing it forward to all of \(X\). One checks that \({\mathcal I}_U\) is an injective \({\mathcal O}_X\)-module, and that for any quasi-coherent \({\mathcal O}_X\)-module \({\mathcal F}\), \(\Hom_{{\mathcal O}_X} ({\mathcal F}, {\mathcal I}_U)\) is dual, as a \(k\)-vector space, to \(\Gamma (U,{\mathcal F})\). Now let \({\mathcal U}= \{{\mathcal U}_i\}\) be an affine open cover of \(X\) and set \({\mathcal I}^{-p}= \bigoplus_{i_0< \cdots<i_p} {\mathcal I}_{U_{i_0} \cap \cdots \cap U_{i_p}}\) for \(p\geq 0\). The \({\mathcal I}^q\) string themselves together into a complex \({\mathcal I}^\bullet\) (the coboundary maps being ``dual'' to that obtained from the Čech construction) and one has \(\Hom^\bullet_{{\mathcal O}_X} ({\mathcal F}^\bullet, {\mathcal I}^\bullet) = \Hom_k^\bullet ({\mathcal C}^\bullet ({\mathcal U}, {\mathcal F}^\bullet),k)\), for all bounded below complexes \({\mathcal F}^\bullet\) of quasi-coherent sheaves, giving Grothendieck duality with \(f^!k={\mathcal I}^\bullet\). Digging deeper one finds that what is involved in the general case is a version of the Special Adjoint Functor theorem. The procedure gives right adjoints at the sheaf level which is transferred to the homotopy category via injective resolutions, and finally to the derived category. The procedure has been modified to work for maps of arbitrary quasi-compact quasi-separated schemes, for unbounded complexes (using Spaltenstein's results), and recently for maps of noetherian formal schemes. The first two results are unpublished works of Lipman, and the last is a recent joint work of \textit{Alonso Tarrío}, \textit{Jeremías Lopez} and \textit{Lipman}.
The approach of the paper under review is radically different. Ideas from homotopy theory are adapted (via work of Thomason on perfect complexes) to the problem on hand. Consequently, the adjoint to \(\mathbb{R} f_*\) is obtained directly at the derived category level, and the generalizations (with the exception of the result of formal schemes) mentioned above are obtained. The author gives a necessary and sufficient condition for the natural map \(\mathbb{L} f^*x \otimes^\mathbb{L} f^! {\mathcal O}_Y\to f'x\) to be an isomorphism, viz., that \(f^!\) commutes with co-products. The traditional sufficient condition is that \(f\) should be of ``finite Tor dimension''. Further, in the bounded below situation, the author shows that \(f^!\) behaves well with base changes by open immersions, giving the sheafified version of Grothendieck duality -- even in the non-noetherian situation. A counter-example for this is provided when one works with unbounded complexes.
The author's techniques are general enough to give adjoints for the functor \(f_+\) between derived categories of complexes of \(D\)-modules. How does all this compare with Deligne's approach (modified and extended by Lipman et al.)? Both approaches need a nice set of generators for the relevent category. Since Deligne first establishes adjointness at the sheaf level, coherent sheaves form generators. Neeman's generators are perfect complexes, since he works entirely in the derived category. For this one needs the results of Thomason [see \textit{R. W. Thomason} and \textit{T. Trobaugh} in: The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 247-435 (1990; Zbl 0731.14001)], a work that can overwhelm by its length. It should however be mentioned that the author has simplifications in a previous paper [see Ann. Sci. Écol. Norm. Supér., IV. Sér. 25, No. 5, 547-566 (1992)]. Deligne's approach generalizes to give duality for maps of noetherian formal schemes. Since it is not clear that perfect complexes in this situation extend from open subsets, it is not clear that the author's approach works here. On the other hand, the author has proved that the functor \(f_+\) mentioned above has a right adjoint.
The above comparison is not meant to diminish the very real achievements of the paper. The game is not for the shortest possible proof, but for conceptual clarification (or else, one would hold, with the Luddites, that Grothendieck's proof of Stein factorization and Zariski's main theorem are much too long, involving so much of homological algebra -- a point of view which is surely untenable). This paper has brought fresh ideas into the game, and the reviewer has certainly learnt a lot. derived categories of complexes of \(D\)-modules; Grothendieck duality theorem; adjoints A. Neeman, ``The Grothendieck duality theorem via Bousfield's techniques and Brown representability'', \textit{J. Amer. Math. Soc. }9(1) (1996), 205--236. Étale and other Grothendieck topologies and (co)homologies, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Stable homotopy theory, spectra The Grothendieck duality theorem via Bousfield's techniques and Brown representability | 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 \(\mathcal F\) be the unique functor, which associates to compact complex algebraic varieties the group of constructible functions, with \(f_ *(1_ W(p)) = \chi(F^{-1}(p) \cap W)\). MacPherson showed that there exists a natural transformation from \(\mathcal F\) to homology, which on nonsingular varieties assigns to the constant function 1 the Poincaré dual of the total Chern class. The author generalizes the theory to a natural transformation \(C_{t_ *}: {\mathcal F}^ t \to H_ *(-;\mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{Z}[t]\). The construction is based upon a twisted push-forward for the local Euler obstruction: if \(f_ *Eu_ W = \sum_ Sn_ SEu_ S\), then \(f^ t_ *Eu_ W = \sum_ Sn_ St^{\dim W- \dim S}Eu_ S\). The author goes on to prove some results related to the twisted theory. The last paragraph of the Introduction reads: `At the moment we do not have reasonable applications of our twisted DGM-theory \(C_{t^*}\), but we just remark that Professor M. Kashiwara pointed out that the idea of the twisted DGM-theory might be applicable to the \(q\)- analogue of the universal enveloping algebra, which remains to be seen'. compact complex algebraic varieties; group of constructible functions; Poincaré dual of the total Chern class; twisted push-forward for the local Euler obstruction Yokura, S.: An extension of delign--Grothendieck--macpherson's theory C\(\ast \)of Chern classes for singular varieties. Publ. R.I.M.S. Kyoto univ. 27, 745-762 (1991) Characteristic classes and numbers in differential topology, (Co)homology theory in algebraic geometry, Categories in geometry and topology, Singularities in algebraic geometry An extension of Deligne-Grothendieck-MacPherson's theory \(C_ *\) of Chern classes for singular 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 ``Introduction to polynomial formal groups and Hopf algebras''; ``Dimension one polynomial formal groups'', by L. Childs, D. Moss, J. Sauerberg and K. Zimmermann; ``Dimension two polynomial formal groups and Hopf algebras'', by. L. Childs, D. Moss, J. Sauerberg and K. Zimmermann; ``Degree two formal groups and Hopf algebras'', by L. Childs and J. Sauerberg; ``\(p\)-elementary group schemes -- constructions and Raynaud's theory'', by C. Greither and L. Childs.
Let \(p\) be a prime number and let \(R\) be the valuation ring of a local field \(K\) containing \(\mathbb Q_p\). The main theme of this monograph is to construct and classify finite, \(p\)-elementary \(R\)-group schemes, or equivalently, commutative, co-commutative, \(p\)-power rank \(R\)-Hopf algebras of exponent \(p\).
In 1970, \textit{J. Tate} and \textit{F. Oort} [Ann. Sci. Éc. Norm. Supér. (4) 3, 1--21 (1970; Zbl 0195.50801)] classified \(R\)-Hopf algebras of rank \(p\): they are in bijective correspondence with \(\{b\in R: b\) divides \(p\}/R^{*\,p-1}\) with \(b\in R\) corresponding to a Hopf algebra order \(H_b=R[x]/(x^p-bx)\) in the dual \((KG)^*\) (\(G=C_p\) the cyclic group of order \(p\) if and only if \(b\in K^{*\,p-1}\)). (In what follows, \(H_b\) is referred to as the Tate-Oort Hopf algebra.)
In 1974, \textit{M. Raynaud} [Bull. Soc. Math. Fr. 102, 241--280 (1974; Zbl 0325.14020)] constructed a class of group schemes of exponent \(p\) and order \(p^n\), \(n\geq 1\) over \(R\), generalizing the classification result of Tate and Oort (which corresponds to the \(n=1\) case). Another generalization was given by \textit{R. G. Larson} [J. Algebra 38, 414--452 (1976; Zbl 0407.20007)] using group valuations, and this method was further systematically exploited by \textit{C. Greither} [Math. Z. 210, 37--68 (1992; Zbl 0737.11038)] and by \textit{N. P. Byott} [Proc. Lond. Math. Soc. (3) 67, 277--304 (1993; Zbl 0795.16026)].
In this monograph, formal groups are utilized to seek insight into the structure and classification of local \(p\)-power commutative, co-commutative \(R\)-Hopf algebras, by studying isogenies of \(p\)-divisible formal groups. The formal groups considered here are polynomial formal groups of degree two and dimension \(n\) linearly isomorphic to the additive group G\(_a^n\) or to the multiplicative group G\(_m^n\). (Polynomial formal groups \({\mathcal F}\) and \({\mathcal G}\) of degree two and dimension \(n\) are linearly isomorphic if there is an invertible \(n\times n\) matrix \(\Theta\) such that \({\mathcal G}=\Theta^{-1}{\mathcal F}\Theta\).)
In paper I, a classification of polynomial formal groups of dimension one, up to linear isomorphism, is given: any polynomial formal group law of dimension one must be of degree two, and over \(K\) must be linearly isomorphic to G\(_m\) or G\(_a\). Further, using isogenies of these formal groups, certain \(p\)-power rank \(R\)-Hopf algebras are constructed. Also, the Kummer theory of formal groups is applied to classify the principal homogeneous spaces over these Hopf algebras.
In paper II, degree two dimension two polynomial formal groups are studied. In particular, when a formal group \({\mathcal F}\) is linearly isomorphic to G\(_m^2\) over \(K\), the endomorphism \([p]_{\mathcal F}\), under a certain assumption, can be made into an isogeny whose kernel is represented by a Hopf order over \(R\) in \(KC_p^2\). Such orders are extensions of one Tate-Oort algebra by another, and they form a special proper subset of the set of all extensions of the two Tate-Oort algebras, which have already been classified by Greither and by Byott.
In paper III, degree two polynomial formal groups and Hopf algebras are studied, e.g., formal groups which are linearly isomorphic to G\(_m^n\) over \(K\): any such formal group is of the form \({\mathcal F}=\Theta^{-1}\)G\(_m^n\Theta\), where \(\Theta\) is a lower triangular matrix in \(M_n(K)\). If a linear isomorphism \(f\) between \({\mathcal F}\) and G\(_m^2\) is defined over a finite Galois extension \(L\) of \(K\), then the endomorphism \([p]_{\mathcal F}\), under a certain hypothesis, may be made into an isogeny \(f\) on \({\mathcal F}\) so that \(H_f=R[\overline x]/(f)\) is a form over \(R\) of an order in \(LC_p^n\). Then the Hopf algebras are explicitly described as nontrivial iterated extensions of Tate-Oort algebras. The constructions yield Raynaud orders, i.e., Hopf orders in \(KC_p^n\) which admit an action by \(F_q\subset R/{\mathfrak m}\), the residue field of \(R\), when the matrices are made up of symmetric functions of roots of the minimal polynomial of a field generator of \(F_q/ F_p\).
The last paper, paper IV, describes how Raynaud orders relate to the orders defined by dimension \(n\), degree two formal groups. The paper presents a new ``exponential'' construction of Hopf orders over \(R\) in \(KC_p^n\) as iterated extensions of Tate-Oort Hopf algebras, complementing the formal groups approach. (In fact, the formal groups approach may be viewed as a linearization of the exponential approach.) The constructions yield all Raynaud orders: they are associated to lower triangular matrices of symmetric functions of roots of a generator of \(F_q\) over \(F_p\), and to ``admissible'' \(n\)-tuples: \( i=(i_0,\cdots, i_{n-1})\) with \(0\leq i_0\leq i_1\leq\cdots \leq i_{n-1}\leq e/(p-1),\,i_{n-1}\leq pi_0\) and \(e/(p-1)-i_0\leq p(e/(p-1)-i_{n-1})\). Here \(e=\text{ord }(p)\) is the absolute ramification index of \(K\).
The articles of this volume will be reviewed individually. Hopf algebras; Polynomial formal groups; Raynaud orders Childs, L.N.; Greither, C.; Moss, D.J.; Sauerberg, J.; Zimmermann, K.: Hopf algebras, polynomial formal groups, and raynaud orders. Mem. amer. Math. soc. 136 (1998) Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Formal groups, \(p\)-divisible groups, Group schemes, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Collections of articles of miscellaneous specific interest Hopf algebras, polynomial formal groups, and Raynaud orders | 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 real semialgebraic geometry it is common to represent a polynomial \(q\) which is positive on a region \(R\) as a weighted sum of squares. Serious obstructions arise when \(q\) is not strictly positive on the region \(R\). Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails.
Specifically, we treat a ``symmetric'' polynomial \(q(x, h)\) in noncommuting variables, \(\left\{ x_1, \dots, x_{g_x} \right\}\) and \(\left\{ h_1,\dots,h_{g_h}\right\}\) for which \(q(X,H)\) is positive semidefinite whenever
\[
X= (X_1), \dots, X_{g_x}\quad \text{and}\quad H= (H_1), \dots, H_{g_h}
\]
are tuples of selfadjoint matrices with \(\|X_{j}\| \leq 1\) but \(H_{j}\) unconstrained. The representation we obtain is a Gram representation in the variables \(h\)
\[
q(x,h) = V(x)[h]^T P_q(x)[h],
\]
where \(P_{q}\) is a symmetric matrix whose entries are noncommutative polynomials only in \(x\) and \(V\) is a ``vector'' whose entries are polynomials in both \(x\) and \(h\). We show that one can choose \(P_{q}\) such that the matrix \(P_{q}(X)\) is positive semidefinite for all \(||X_{j}|| \leq 1\). The representation covers sum of square results [\textit{J.~W.\ Helton}, Ann.\ Math.\ (2) 156, No.~2, 675--694 (2002; Zbl 1033.12001); \textit{S.~McCullough}, Linear Algebra Appl.\ 326, No.\ 1--3, 193--203 (2001; Zbl 0980.47024); \textit{S.~McCullough} and \textit{M.~Putinar}, Pac.\ J.\ Math.\ 218, No.~1, 167--171 (2005; Zbl 1177.47020)] when \(g_{x} = 0\). Also it allows for arbitrary degree in \(h\), rather than degree two, in the main result of [\textit{J.~F.\ Camino, J.~W.\ Helton, R.~E.\ Skelton} and \textit{J.--P.\ Ye}, Integral Equations Oper.\ Theory 46, No.~4, 399--454 (2003; Zbl 1046.68139)] when restricted to \(x\)-domains of the type \(\|X_{j}\| \leq 1\). J. William Helton, Scott McCullough, and Mihai Putinar, Matrix representations for positive noncommutative polynomials, Positivity 10 (2006), no. 1, 145-163. Several-variable operator theory (spectral, Fredholm, etc.), Semialgebraic sets and related spaces, Positive matrices and their generalizations; cones of matrices, Noncommutative function spaces Matrix representations for positive noncommutative 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 multi-dimensional \(q\)-deformed bosonic Newton oscillator algebra with \textit{SU(d)}-symmetry is considered. In this framework, we first introduce some new properties concerning the \(q\)-deformed calculus related to the algebra, and we then discuss possible consequences of applying these deformed oscillators in some quantum optical issues such as in the construction of coherent states and their effects on the photon statistics. Second, we investigate the role of \(q\)-deformation on both the energy levels and the wave functions of the bosonic Newton oscillators by constructing the \(q\)-deformed Hermite polynomials. The results obtained in this work might have some implications for studies on quantum information based technologies such as in photonic quantum computing.
{\copyright 2021 American Institute of Physics} Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, Bosonic systems in quantum theory, Finite-dimensional groups and algebras motivated by physics and their representations, Quantum optics, Coherent states, Formal methods and deformations in algebraic geometry, Quantum computation Multi-dimensional \(q\)-deformed bosonic Newton oscillators and the related \(q\)-calculus, \(q\)-coherent states, and Hermite \(q\)-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 Givental's work on equivariant Gromov-Witten invariants has established the relationship between quantum cohomology and hypergeometric series [see \textit{A. B. Givental'}, Int. Math. Res. Not. 1996, No. 13, 613-663 (1996; Zbl 0881.55006)]. Consider a toral action on a compact Kähler manifold \(X\) with finitely many fixed points \(x_w\). The solutions of the differential equations arising from quantum cohomology are related to equivariant correlators \(Z_w\) associated with the \(x_w\). The correlators \(Z_w\) are the hypergeometric series associated with equivariant quantum cohomology, and can be uniquely determined by linear recursion relations. The main result in the paper under review is the explicit determination of the recursion relations for the \(Z_w\) on flag spaces \(X=G/B\). Here \(G\) is the simply connected algebraic group associated with a finite root system \(R\). Hence \(X\) is a homogeneous space for the action of the maximal torus of \(G\). The set of fixed points is in this case finite and parametrized by the Weyl group of \(R\). A simple explicit formula for the \(Z_w\) is presented in the case \(G=SL(3)\). Givental' recursion relations; hypergeometric functions; quantum cohomology; equivariant Gromov-Witten invariants; flag spaces Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Other hypergeometric functions and integrals in several variables, Quantization in field theory; cohomological methods, Grassmannians, Schubert varieties, flag manifolds On hypergeometric functions connected with quantum cohomology of flag 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 author gives a simplified exposition focusing on the variety of complete flags. She explains how the quantum cohomology ring for the full flag variety can be used to study totally positive uni-triangular Toeplitz matrices. total positive uni-triangular Toeplitz matrices Rietsch, K.C.: Total positivity and real flag varieties. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge (1998) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Grassmannians, Schubert varieties, flag manifolds Total positivity, flag varieties 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 Alexandre Grothendieck (1928--2014) was a towering figure in contemporary mathematics whose revolutionary contributions to the development of functional analysis, algebraic geometry, homological algebra, and arithmetic geometry profoundly changed the face of pure mathematics during the nearly twenty years of his active research work, that is, in the 1950s and 1960s. In 1970, at the age of 42, he suddenly quit his prestigeous position at the IHES, ruptured with his mathematical colleagues and friends, withdrew from mathematical research and started devoting himself exclusively to his own individual spirituality and eccentricity. Finally, in the early 1990s, the great mathematician Alexandre Grothendieck decided to live as a hermit from this time on, hiding himself somewhere in the French Pyrenees and wishing no contact with the outside world.
These aspects of A. Grothendieck's remarkable life have been vividly depicted in the books by \textit{W. Scharlau} [Wer ist Alexander Grothendieck? Anarchie, Mathematik, Spiritualität. Eine Biographie. Teil 1: Anarchie. Havixbeck: Scharlau (2007; Zbl 1129.01018); Teil 3: Spiritualität. Norderstedt: Books on Demand (2010; Zbl 1234.01007)]. Also, the article [Am. Math. Mon. 113, No. 9, 831--846 (2006; Zbl 1215.01014)] by \textit{P. Pragacz} gives an enlightening impression of both A. Grothendieck's personality and his mathematical achievements.
Anyway, Alexandre Grothendieck has left behind his legendary fame in mathematics, on the one hand, and a wealth of pioneering ideas, concepts, methods, techniques and approaches, on the other hand, that will still foster the development of mathematics for many decades.
The book under review is a collection of thirteen articles written by authors who knew Alexandre Grothendieck personally, including former friends, colleagues, students, and collaborators. Their aim is to provide some explanation of what made Grothendieck such a unique mathematician, what the nature of his mathematical thought was, and how his overall mathematical impact shaped their own research work. Accordingly, these articles do not concentrate on specific mathematical contents of Grothendieck's gigantic work, but instead on the characteristic traits of his tackling of problems: maximum generality, structuralism, fundamental simplicity of new approaches, ingenious generalizations of classical concepts, technical prowess, etc. Each of the articles is devoted to one or another aspect of Grothendieck's contributions, accompanied with personal impressions, memories, anecdotes, and biographical insertions.
More precisely, the book comprises the following articles:
1. \textit{Joe Diestel}, ``Grothendieck and Banach space theory'' (pp. 1--12): This note describes Grothendieck's pioneering work in and around Banach space theory between 1953 and 1955, especially concerning topological tensor products and nuclear spaces. The focus is on explaining how his approaches profoundly changed the landscape of the subject.
2. \textit{Max Karoubi}, ``L'influence d'Alexandre Grothendieck en \(K\)-théorie'' (pp. 13--23): The author discusses Grothendieck's introduction of the \(K\)-group of algebraic vector bundles on a smooth, quasi-projective variety in the context of his generalization of the Riemann-Roch theorem, on the one hand, and some further developments in (higher) algebraic \(K\)-theory based on Grothendieck's ideas, on the other hand.
3. \textit{Michel Raynaud}, ``Grothendieck et la théorie des schémas'' (pp. 25--34): Here, one of Grothendieck's most fundamental contributions is discussed, namely the new foundation of algebraic geometry via the theory of algebraic schemes, with particular reference to the new conceptual framework as developed in his EGA and SGA volumes.
4. \textit{Steven L. Kleiman}, The Picard scheme (pp. 35--74): As the author's abstract aptly indicates, this article introduces, informally, the substance and the spirit of Grothendieck's theory of the Picard scheme, highlighting its elegant simplicity, natural generality, and ingenious originality against the larger historical record.
5. \textit{David Mumford}, ``My introduction to schemes and functors'' (pp. 75--81): In this essay, Fields medalist David Mumford talks about how Grothendieck's re-foundation of algebraic geometry profoundly affected his own understanding of the subject at Harvard University in 1958.
6. \textit{Carlos T. Simpson}, ``Descent'' (pp. 83--141): Grothendieck's so-called ``theory of descent'', i.e., piecing together a global picture out of local pieces and gluing data, permeates his work in a ubiquitous way. This is explained in the present article, with particular emphasis on those directions in which the theory of decent is or will be developing, in relation with higher categories, stacks, derived stacks, and higher non-abelian cohomology.
7. \textit{Jacob P. Murre}, ``On Grothendieck's work on the fundamental group'' (pp. 143--167): In this lecture, the author outlines the key ideas of Grothendieck's work on the algebraic fundamental group as developed in [\textit{A. Grothendieck}, Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1). Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0234.14002)].
8. \textit{Robin Hartshorne}, ``An apprenticeship'' (pp. 169--174): The author narrates how he got acquainted with A. Grothendieck, and how his own lecture notes [Residues and duality. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0212.26101)] were influenced by Grothendieck's ideas on the subject.
9. \textit{Luc Illusie}, ``Grothendieck et la cohomologie étale'' (pp. 175--192): In this note, Grothendieck's concept of étale cohomology and its influence on the development of arithmetic geometry are thoroughly explained, thereby reviewing the related aspects elaborated in Grothendieck's various SGA volumes and thereafter.
10. \textit{Leila Schneps}, ``The Grothendieck-Serre correspondence'' (pp. 193--230): The correspondence between Alexandre Grothendieck and Jean-Pierre Serre was published in [\textit{P. Colmez} (ed.) and \textit{J.-P. Serre} (ed.), Correspondance Grothendieck-Serre. Paris: Société Mathématique de France (2001; Zbl 0986.01019)]. The aim of the present article is to explain the contents and history of the many mathematical events discussed in these letters over many years, thereby rendering the flavor of the most important results and notions through short and informal explanations. Also, the author has tried to place the letters in the context of the personalities and the lives of the two great mathematicians.
11. \textit{Frans Oort}, ``Did earlier thoughts inspire Grothendieck?'' (pp. 231--268): As the author points out in the introduction to this paper, the general question of where inspiration does come from is particularly fascinating in connection with the mathematical work of Alexandre Grothendieck. This question of whether Grothendieck's brilliant ideas had simply occurred to him or whether they had some connection to earlier thought is analyzed in the present note, thereby discussing his very characteristic way of both approaching and doing mathematics as well as possible ``roots'' of his mathematical thinking.
12. \textit{Pierre Cartier}, ``A country of which nothing is known but the name: Grothendieck and `motives''' (pp. 269--298): In this note, the author discusses the interactions between Alexandre Grothendieck's scientific work and his extraordinary personality, including many biographical elements and numerous personal impressions, analyses, and conclusions. Particular attention is paid to Grothendieck's spirituality and obsession in his later years.
13. \textit{Yuri I. Manin}, ``Forgotten motives: the varieties of scientific experience'' (pp. 299--307): Here, the author describes his personal contact with A. Grothendieck in 1967, when the grandmaster himself gave him some private tutoring on his then brand-new project of motives and the so-called ``standard conjectures'' on those. This gives an enlightening picture of the early history of motives as well as an outlook on their recent applications in mathematical physics.
At the end of the book, there is a collection of photographs of the contributors to this volume, taken at a time when A. Grothendieck had not completely disappeared yet.
Altogether, this book makes the reader see Alexandre Grothendieck through the eyes of people who knew him and his mathematics very well. The mathematical portrait painted here provides a better understanding of what his unique contribution to mathematics really was, in particular with a view toward further generations of mathematicians. biography; Alexandre Grothendieck; history of mathematics; history of algebraic geometry; history of functional analysis; schools of mathematics Schneps, L. (ed.): Alexandre Grothendieck: A Mathematical Portrait. International Press, Somerville (2014) Proceedings, conferences, collections, etc. pertaining to history and biography, Biographies, obituaries, personalia, bibliographies, Schools of mathematics, History of mathematics in the 20th century, History of algebraic geometry, History of functional analysis, Collections of articles of miscellaneous specific interest Alexandre Grothendieck: a mathematical portrait | 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 devoted to the study of the (small) quantum cohomology ring of the \(d\)-th symmetric power of a smooth projective curve \(C\). Considering such a variety in the context of quantum cohomology is particularly interesting in view of the isomorphism between the Seiberg-Witten-Floer cohomology of the real \(3\)-manifold \(C\times S^1\) and the (ordinary) cohomology of a suitable \(d\)-th symmetric power of \(C\), where the index \(d\) depends on the spin-\(c\) structure on \(C\times S^1\) chosen in Seiberg-Witten theory [\textit{S.~K.~Donaldson}, Bull. Am. Math. Soc., New Ser. 33, No. 1, 45--70 (1996; Zbl 0872.57023)]. It is indeed expected that the quantum multiplication on the symmetric power \(C_d\) corresponds to a natural product in the Seiberg-Witten-Floer cohomology of \(C\times S^1\) [\textit{S.~Piunikhin}, \textit{D.~Salamon, D.} and \textit{M. Schwarz}, in: Contact and symplectic geometry, Publ. Newton Inst. 8, 171--200 (1996; Zbl 0874.53031)]. In Donaldson's language, this correspondence can be formulated by saying that \(QH^*(C_d)\) is the base ring of the quantum category in Seiberg-Witten theory.
Being concerned with small quantum cohomology, the central object of the paper are the genus zero \(3\)-point Gromov-Witten invariants of \(C_d\). These are computed by algebraic geometrical methods, namely using ideas from Brill-Noether theory. More precisely, Bertram and Thaddeus consider the Abel-Jacobi map \(C_d\to \text{Jac}_d(C)\) and look at the strata where the dimension of the fiber is constant. The enumerative geometry of these strata is then studied by means of the Harris-Tu formula for the Chern numbers of determinantal varieties [\textit{J.~Harris} and \textit{L.~Tu}, Invent. Math. 75, 467--475 (1984; Zbl 0542.14015)], applied to suitable locally free sheaves on \(\text{Jac}_d(C)\).
The results obtained by the authors are several and remarkable; we will only list the most relevant here:
-- By the general theory of quantum cohomology, the number of possible deformation parameters in the quantum cohomology ring of \(C_d\) is the second Betti number of \(C_d\), but it is shown that the quantum product in \(QH^*(C_d)\) actually depends nontrivially on a single parameter \(q\).
- Explicit formulas for the coefficients of \(q\) and \(q^2\) in the quantum product are determined.
-- Let \(g\) be the genus of \(C\). The coefficients of \(q^e\) in the quantum product are shown to vanish if \(d<g-1\) and \(e>(d-3)/(g-1-d)\) or if \(d>g-1\) and \(e>1\). The ``discriminating value'' \(d=g-1\) is particularly interesting: it is a relative version of the Aspinwall-Morrison computation of Gromov-Witten invariants counting rational curves with normal bundle \({\mathcal O}(-1)\oplus{\mathcal O}(-1)\) in a Calabi-Yau threefold [\textit{P.~S.~Aspinwall} and \textit{D.~R.~Morrison}, Commun. Math. Phys. 151, No. 2, 245--262 (1993; Zbl 0776.53043)].
These facts together completely determine the quantum product on \(C_d\) in all cases exept for \(d\) in the interval \([(3/4)g,g-1)\) and a presentation of \(QH^*(C_d)\) by means of generators and relations is obtained under the weaker hypotesis \(d\) not in \([(4/5)g-3/5,g-1)\). This presentation is a deformation of the classical Macdonald presentation of \(H^*(C_d)\) [\textit{I.~G.~Macdonald}, Topology 1, 319--343 (1962; Zbl 0121.38003)]. Moreover, in the particular case \(C={\mathbb P}^1\), the Bertram-Thaddeus presentation of \(QH^*(({\mathbb P}^1)_d)\) coincides with the well known presentation of \(QH^*({\mathbb P}^d)\).
The final part of the paper describes an analogy with the Givental's work on the Gromov-Witten invariants of a quintic threefold [\textit{A.~B.~Givental}, Int. Math. Res. Not. 1996, No. 13, 613--663 (1996; Zbl 0881.55006)]. The point is that Givental's formulas for equivariant Gromov-Witten invariants can be seen as universal formulas for Gromov-Witten invariants of complete intersections in projective bundles and, if \(d\leq 2g-2\), the \(d\)-th symmetric power \(C_d\) can be naturally realized as a complete intersection in a \({\mathbb P}^{g-1}\)-bundle over \(\text{Jac}_{2g-1}(C)\).
The paper also contains a very neat introduction to basic concepts of quantum cohomology and to the classical algebraic geometry of symmetric powers of curves, enriched with several punctual references. quantum cohomology; Gromov-Witten invariants; symmetric products of algebraic curves; Brill-Noether theory Bertram, A.; Thaddeus, M., On the quantum cohomology of a symmetric product of an algebraic curve, Duke Math. J., 108, 329-362, (2001) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Special divisors on curves (gonality, Brill-Noether theory) On the quantum cohomology of a symmetric product of an algebraic 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 The author proves that a degeneration of a basic finite dimensional algebra over a field preserves the rank of the Grothendieck group if and only if it is idempotent rigid. algebra; locally bounded category; algebraic variety; degeneration of algebras; degeneration order DOI: 10.1007/s10468-012-9381-z Varieties and morphisms Degenerations of algebras and ranks of Grothendieck groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author uses Gröbner basis algorithm to obtain certain algebraic relations between a finite family of polynomials. More precisely, let \({\mathcal F}=\{f_ 1(X),...,f_ m(X)\}\subset K[X]\), where \(X=(X_ 1,...,X_ n)\) is a vector of variables. Then the author deals with two situations:
(a) A final polynomial for \({\mathcal F}\) is a polynomial P(X,Y)\(\in K[X,Y]\), \(Y=(Y_ 1,...,Y_ m)\), such that \(P(X,f_ 1(X),...,f_ m(X))=0\) and \(P(X,0,...,0)=1\). Notice that if \(K={\mathbb{C}}\), then a final polynomial for \({\mathcal F}\) exists if and only if the zero set of \({\mathcal F}\) is empty, or equivalently 0 is not in the image of the map \({\mathcal F}:{\mathbb{C}}^ n\to {\mathbb{C}}^ n:\;x\mapsto (f_ 1(x),...,f_ m(x)).\) In this case a final polynomial gives an explicit representation of 1 as an algebraic combination of \(f_ 1(X),...,f_ m(X).\)
(b) A syzygy for \({\mathcal F}\) is a polynomial Q(Y)\(\in K[Y]\) such that \(Q(f_ 1(X),...,f_ m(X))=0\). A syzygy is called final if it is also a final polynomial. Then, for \(K={\mathbb{C}}\) the author remarks that a final syzygy for \({\mathcal F}\) exists if and only if 0 is not in the Zariski closure of the image of the map \({\mathcal F}.\)
The method proposed to produce final polynomials and syzygies for \({\mathcal F}\) is the following: consider \(\hat {\mathcal F}=\{\hat f_ 1(X,Y),...,\hat f_ m(X,Y)\}\), where \(\hat f_ i(X,Y)=f_ i(X)-Y_ i\). Let \(\hat {\mathcal G}\) be a Gröbner basis for \(\hat {\mathcal F}\) with respect to the purely lexicographic ordering induced by \(Y_ 1<...<Y_ m<X_ 1<...<X_ n\). Then a final polynomial for \({\mathcal F}\) (resp. syzygy) exists if and only if \(\hat {\mathcal G}\) contains a final polynomial (resp. syzygy).
The author works out some examples and application of this method to different situations. In particular he shows that the Sylvester-Gallai- theorem (among any n points in the real euclidean plane, not all in a line, there exist two points such that no other point is collinear with them) fails in the complex plane. Gröbner basis algorithm; final polynomial; syzygy; Sylvester-Gallai- theorem Sturmfels, B.: Computing final polynomials and final syzygies using buchberger's Gröbner bases method. Results math 15, 551-560 (1989) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials, Relevant commutative algebra, Polynomials over commutative rings, Syzygies, resolutions, complexes and commutative rings, Computational aspects in algebraic geometry Computing final polynomials and final syzygies using Buchberger's Gröbner bases method | 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 study set-theoretical complete intersection varieties, defined over fields of arbitrary characteristic. For a projective variety \(Y\), the authors define the invariant \(\mathrm{ara}(Y)\) (the arithmetic rank of \(Y\)) as the minimal cardinality of a set of polynomials \(S\) whose zero-locus coincides with \(Y\). Thus, \(Y\) is set-theoretical complete intersection exactly when \(\mathrm{ara}(Y)\) equals the codimension of \(Y\). Using an algebraic approach to the Grothendieck-Lefschetz theory, the authors determine several necessary conditions on the Picard group of a variety, to be set theoretical-complete intersection. Using these characterizations, the authors are able to produce many examples of surfaces, which are not set-theoretical complete intersection. The authors point then their attention to rational normal scrolls. Rational normal scrolls of dimension \(2\) are known to be set-theoretical complete intersection. For any dimension, the authors show that a rational normal scroll embedded in \(\mathbb P^N\) has \(\mathrm{ara}(Y)=N-2\). It follows that rational normal scrolls of dimension greater than two are not set-theoretical complete intersection. set-theoretic complete intersection Lucian Bădescu and Giuseppe Valla, Grothendieck-Lefschetz theory, set-theoretic complete intersections and rational normal scrolls, J. Algebra 324 (2010), no. 7, 1636 -- 1655. Complete intersections, Rational and unirational varieties Grothendieck-Lefschetz theory, set-theoretic complete intersections and rational normal scrolls | 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 investigate the theta vector and quantum theta function over noncommutative \({\mathbb T}^4 \) from the embedding of \({\mathbb R} \times {\mathbb Z}^2 \). Manin has constructed the quantum theta functions from the lattice embedding into vector space (\(\times\) finite group). We extend Manin's construction of the quantum theta function to the embedding of vector space \(\times\) lattice case. We find that the holomorphic theta vector exists only over the vector space part of the embedding, and over the lattice part we can only impose the condition for the Schwartz function. The quantum theta function built on this partial theta vector satisfies the requirement of the quantum theta function. However, two subsequent quantum translations from the embedding into the lattice part are nonadditive, contrary to the additivity of those from the vector space part. Noncommutative geometry in quantum theory, Theta functions and curves; Schottky problem, Theta functions and abelian varieties Quantum thetas on noncommutative \({\mathbb T}^4 \) from embeddings into 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 author discusses and establishes fundamental relationships among two kinds of Grothendieck-Teichmüller groups, multiple zeta values and certain Galois images. These materials were introduced and studied intensively by \textit{V. G. Drinfel'd} [Leningr. Math. J. 2, No. 4, 829--860 (1991); translation from Algebra Anal. 2, No. 4, 149--181 (1990; Zbl 0728.16021)], \textit{Y. Ihara} [in: Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 99--120 (1991; Zbl 0757.20007)] and other authors.
One of the main results of this paper states that the spectrum of the multiple zeta values modulo \(\pi^2\) -- \(\text{Spec} \mathbb{Z}./(\pi^2)\) -- is canonically embedded into the (unipotent) graded Grothendieck-Teichmüller group \(\underline{GRT}_1\). On the other hand, a notion of the rational \(l\)-adic Galois image \(\underline{Gal}^{(l)}_{\mathbb Q}\) is introduced as a subgroup of the (unipotent) Grothendieck-Teichmüller group \(\underline{GT}_1\). Both embeddings \(\Phi_{DR}: \text{Spec} \mathbb{Z}./(\pi^2)\hookrightarrow \underline{GRT}_1\) and \(\Phi^{(l)}_{\mathbb Q}: \underline{Gal}^{(l)}_{\mathbb Q} \hookrightarrow \underline{GT}_1\) (conjectured to be isomorphisms) are related to each other in twofold ways through the middle Grothendieck-Teichmüller torsor \(\underline{M}\) (of the group-like universal associators) which is a left \(\underline{GRT}\)- and right \(\underline{GT}\)-torsor. First, using the Drinfeld associator \(\varphi_{KZ}\in M({\mathbb C})\) as the quotient of two basic solutions to the Knizhnik-Zamolodchikov equation, one obtains a standard isomorphism \(\underline{GT}_1\times_{\mathbb Q} {\mathbb C}\cong \underline{GRT}_1\times_{\mathbb Q} {\mathbb C}\), under which the \({\mathbb Q}\)-structures of \(\underline{GT}\) and of \(\underline{GRT}\) are conjugate but not equal.
Second, taking graded quotients by the weight filtration in \(\underline{GT}_1\), one obtains a canonical \({\mathbb Q}\)-isomorphism \(Gr\underline{GT}_1 \cong \underline{GRT}_1\) through which the graded \(l\)-adic Galois image coincides over \({\mathbb Q}_l\) with the spectrum of multiple zeta values modulo \(\pi^2\).
Various related topics and important conjectures are also discussed in the above frameworks. Especially, one can construct a ``universal assoicator'' as a non-commutative two variable power series in the coefficients in the ring of functions on \(\underline{GRT}_1\) that dominates both the \(l\)-adic Ihara associator in Galois side and the Drinfeld associator in Hodge side. Tsunogai's question (raised in Problem 2.1.16 of this paper) on implication of 5-cyclic relation to 3-cyclic relation has been answered affirmatively in a recent brilliant work of the author [``Pentagon and hexagon equations'', to appear in Ann. Math. (2)]. Drinfeld associator; negatively weighted pro-algebraic group H. Furusho, ''Multiple zeta values and Grothendieck-Teichmüller groups,'' Amer. Math. Sci. Contemp. Math., vol. 416, pp. 49-82, 2006. Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Other Dirichlet series and zeta functions, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act Multiple zeta values and Grothendieck-Teichmüller 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 Let \(G\) be a complex semisimple simply-connected algebraic group and let \(P\) be a parabolic subgroup. Consider the flag variety \(X_:=G/P\) and the Schubert subvarieties \(X_w:= \overline{BwP/P}\subset G/P\) for any \(w\in W/W_P\), where \(W\) is the Weyl group of \(G\), \(W_P\) is the Weyl group of \(P\) and \(B\) is a Borel subgroup of \(G\) contained in \(P\). Then, the classes \([{\mathcal O}_{X_w}]\) of structure sheaves of \(X_w\) form a \(\mathbb{Z}\)-basis of the \(K\)-group \(K(X)\) of \(X\). Write, under the product in \(K(X)\), for any \(u,\,v\in W/W_P\):
\[
[{\mathcal O}_{X_u}]= \sum_{w\in W/W_P} c^w_{u,v}[{\mathcal O}_{X_w}],
\]
for some \(c^w_{u,v}\in \mathbb{Z}\).
Then, the main result of the paper under review asserts that
\[
c^w_{u,v}(-1)^{\text{codim\,}X_u+ \text{codim\,}X_v+ \text{codim\,}X_w}\geq 0.
\]
This was conjectured by \textit{A. Buch} [Acta Math. 189, 37--78 (2002; Zbl 1090.14015)] and proved by him for the Grassmannians.
The author, in fact, proves the following more general result asked by \textit{W. Graham} [Duke Math. J. 102, 599--614 (2001; Zbl 1069.14055)]: Let \(Y\subset X\) be a closed subvariety with rational singularities. Express
\[
[{\mathcal O}_Y]= \sum_{w\in W/W_P} c^w_Y[{\mathcal O}_{X_w}],\text{ for }c^w_Y\in \mathbb{Z}.
\]
Then, \((-1)^{\text{codim\,}X_w+ \text{codim\,}Y} c^w_Y\geq 0\). Schubert calculus \beginbarticle \bauthor\binitsM. \bsnmBrion, \batitlePositivity in the Grothendieck group of complex flag varieties, \bjtitleJ. Algebra \bvolume258 (\byear2002), no. \bissue1, page 137-\blpage159. \endbarticle \OrigBibText Michel Brion, Positivity in the Grothendieck group of complex flag varieties , J. Algebra 258 (2002), no. 1, 137-159. \endOrigBibText \bptokstructpyb \endbibitem Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Grothendieck groups, \(K\)-theory and commutative rings, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Positivity in the Grothendieck group of complex 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 We introduce and study a deformation of commutative polynomial algebras in even numbers of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [\textit{W. Zhao}, J. Algebra 324, No.~2, 231--247 (2010)] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture [\textit{O. H. Keller}, Monatsh. Math. Phys. 47, 299--306 (1939; Zbl 0021.15303; JFM 65.0713.02)] is reduced to an open problem on this deformation of polynomial algebras. the generalized Laguerre polynomials; total symbols of differential operators; the image conjecture; the Jacobian conjecture Wenhua Zhao, A deformation of commutative polynomial algebras in even numbers of variables, Cent. Eur. J. Math. 8 (2010), no. 1, 73 -- 97. Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Differential operators in several variables, Jacobian problem A deformation of commutative polynomial algebras in even numbers of variables | 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 construct deformations of the small quantum cohomology rings of homogeneous spaces \(G / P\), and obtain an irredundant set of inequalities determining the multiplicative eigen polytope for the compact form \(K\) of \(G\). quantum cohomology; multiplicative eigenvalue problem; deformation theory P. Belkale, S. Kumar, \textit{The multiplicative eigenvalue problem and deformed quantum cohomology}, Adv. Math. (2015), DOI: 10.1016/j.aim.2015.09.034, arXiv:1310.3191 (2013). Selfadjoint operator theory in quantum theory, including spectral analysis, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Homogeneous spaces and generalizations The multiplicative eigenvalue problem and deformed 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 We study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of \(2\times n\) quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thought of as giving an enumeration of the points that are invariant under the induced action of the torus in the ``variety of \(2\times n\) quantum matrices''. primitive ideals; quantum matrices; quantised enveloping algebras; Cauchon diagrams; perfect matchings; Pfaffians; rational torus actions; Gelfand-Kirillov dimension; prime ideals Bell, J., Launois, S., Nguyen, N.: Dimension and enumeration of primitive ideals in quantum algebras. J. Algebr. Comb. 29(3), 269--294 (2009) Ring-theoretic aspects of quantum groups, Ideals in associative algebras, Growth rate, Gelfand-Kirillov dimension, Rings arising from noncommutative algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations, Group actions on varieties or schemes (quotients) Dimension and enumeration of primitive ideals in quantum 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 We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in \(\text{GL}_n\) forms a real semi-algebraic cell of dimension \(n-1\). Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of \(\text{GL}_n(\mathbb{C})\) relying in particular on the positivity of the structure constants, which are enumerative Gromov-Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterson's which we explain with proofs in the type \(A\) case. flag varieties; quantum cohomology; total positivity; cell decomposition K. Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties, J. Amer. Math. Soc. 16 (2003), no. 2, 363--392 (electronic). 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) 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 The classification of noncommutative quadric hypersurfaces in quantum \(\mathbb{P}^{n-1}\)'s is a big project in noncommutative algebraic geometry and it is far away from complete. In this note, we mainly give some result about noncommutative conics in quantum \(\mathbb{P}^2\)'s. noncommutative quadric hypersurfaces Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry, Derived categories and associative algebras, Abelian categories, Grothendieck categories Algebras associated to noncommutative conics in quantum projective planes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is now a well-known fact that a polynomial automorphism \(F = (F_ 1, \dots, F_ n) : k^ n \to k^ n\), where \(k\) is a field, \(n \geq 2\) is completely determined by its face polynomials \(F_ i |_{X_ j = 0}\), \(i,j=1,\dots,n\). Several proofs were given, by McKay and Wang and Li. Explicit formulae to reconstruct the automorphism for the case \(n=2\) were given by McKay and Wang and Adjamagbo and van den Essen. In J. Pure Appl. Algebra 80, No. 3, 327-336 (1992; Zbl 0763.14004), \textit{A. van den Essen} and the author show that these formulae do not always work in higher dimensions. Instead, an algorithm for reconstructing the inverse, based on a theorem from \textit{A. van den Essen} [Commun. Algebra 18, No. 10, 3183-3186 (1990; Zbl 0718.13008)], is given there. That algorithm involves \(n+1\) Gröbner basis computations. The main theorem presented here expresses the inverse of \(F\) in terms of the reduced Gröbner basis of an ideal constructed from face polynomials. Thus, it leads to an algorithm for the computation of \(F\) which involves only two Gröbner basis computations. inverse of polynomial automorphism; Gröbner basis; face polynomials M. Kwieciński, Automorphisms from face polynomials via two Gröbner bases, J. Pure Appl. Algebra, to appear. Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials, Automorphisms of curves Automorphisms from face polynomials via two 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 is motivated by the problem of constructing the Grothendieck ring of equivalence classes of triangulated categories. However, difficulties such as the fact that the cone of a morphism is not a functorial operation, could not be overcome, and another context had to be chosen. An important part of the paper deals with the presentation of the concept of pretriangulated category. These are DG categories with special properties which imply that the cone of a morphism is a functor. Denote by \({\mathcal P}{\mathcal I}\) the abelian group generated by quasi-equivalence classes of pretriangulated categories with relations coming from semiorthogonal decompositions of corresponding triangulated categories.
The authors introduce a \(\bullet\)-product which makes \({\mathcal P}{\mathcal I}\) into a commutative ring. Two applications are given. The first is concerned with the representability of standard functors between derived categories of coherent sheaves on a smooth projective variety. The second is the construction of a motivic measure, i.e., a homomorphism from the Grothendieck ring of varieties to the Grothendieck ring of pretriangulated categories. triangulated categories; DG categories; enhanced triangulated categories; projective varieties; motivic measure A. Bondal, M. Larsen and V. Lunts, \textit{Grothendieck ring of pretriangulated categories}, International Mathematics Research Notices \textbf{2004}, 1461-1495. Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grothendieck groups (category-theoretic aspects) Grothendieck ring of pretriangulated categories | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a Macdonald polynomial analogue of the celebrated Nekrasov-Okounkov hook-length formula from the theory of random partitions. As an application we obtain a proof of one of the main conjectures of \textit{T. Hausel} and \textit{F. Rodriguez-Villegas} [Invent. Math. 174, No. 3, 555--624 (2008; Zbl 1213.14020)] from their work on mixed Hodge polynomials of the moduli space of stable Higgs bundles on Riemann surfaces. hook-length formulas; Macdonald polynomials Combinatorial identities, bijective combinatorics, Symmetric functions and generalizations, Combinatorial aspects of representation theory, Families, moduli, classification: algebraic theory A Nekrasov-Okounkov formula for Macdonald 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 is a set of lecture notes for the first author's lectures on the difference equations in 2019 at the Institute of Advanced Study for Mathematics at Zhejiang University. We focus on explicit computations and examples. The convergence of local solutions is discussed. quantum \(K\)-theory; \(q\)-difference equation Difference equations, scaling (\(q\)-differences), Other functions coming from differential, difference and integral equations, \(q\)-gamma functions, \(q\)-beta functions and integrals, \(q\)-calculus and related topics, Binomial coefficients; factorials; \(q\)-identities, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum \(K\)-theory and \(q\)-difference 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 The aim of this work is to give a more detailed exposition of the author's results (see the preceding review) in the simplest nontrivial case. This case amounts to the study of the category of perverse sheaves on a complex plane which are smooth along the stratification induced by the three lines intersecting at one point.
In Section 1 we give (using the Beilinson glueing theorem) the description of this category in terms of linear algebra data. In Section 2 we derive some consequences. Namely, we show how to interpret, in terms of quantum groups, the linear algebra data corresponding to certain standard sheaves.
One sould remark that the results in Section 1 are not new. The main result, Lemma 1.4.6, seems to be a particular case of \textit{R. MacPherson} and \textit{K. Vilonen} [Comment. Math. Helv. 63, 89-103 (1988; Zbl 0638.32014)]. However, our approach is different. vanishing cycles representation; intersection cohomology; Kac-Moody-type quantum groups; perverse sheaves; Beilinson glueing theorem; linear algebra data; quantum groups V. Schechtman, Quantum groups and perverse sheaves: an example , The Gelfand Mathematical Seminars, 1990-1992, Birkhäuser, Boston, 1993, pp. 203-216. Quantum groups (quantized enveloping algebras) and related deformations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Representation theory of associative rings and algebras Quantum groups and perverse sheaves: An example | 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 der Arbeit wird die Vermutung von Tate über die Gleichheit der Dimensionen des Raumes der Galois-Invarianten in der zweiten \(\ell\)- adischen Kohomologiegruppe und des von den Kohomologieklassen von Divisoren aufgespannten Raumes, und der Polordnung in \(s=2\) der entsprechenden L-Funktion für die nichtsingulären Kompaktifizierungen einer Hilbert-Blumenthal-Fläche betrachtet. Insbesondere wird die Vermutung über abelschen Erweiterungen von \({\mathbb{Q}}\) bewiesen. (Die Aussage wird auf eine entsprechende Aussage über Hilbert-Blumenthal- Flächen zurückgeführt.)
Dazu wird die mit der zweiten Schnittkohomologiegruppe gebildete L- Funktion mit einem Produkt von automorphen L-Funktionen identifiziert. Es wird gezeigt, daß eine unendlich-dimensionale automorphe Darstellung \(\pi\) bei der Zerlegung der zweiten (Schnitt-)Kohomologiegruppe unter der Aktion der Heckealgebra einen höchstens eindimensionalen Beitrag zu den Invarianten über einer zyklotomischen Erweiterung liefert, und daß dieser Beitrag genau dann nichttrivial ist, wenn \(\pi\) ''ausgezeichnet'' (im Sinne der Definition 2.7. der Arbeit) ist. Analoges gilt für die Polordnung der entsprechenden L-Funktion von \(\pi\). Schließlich wird der Raum der ''Hirzebruch-Zagier-Zyklen'' konstruiert, der von gewissen algebraischen Zyklen über zyklotomischen Körpern aufgespannt wird (i.w. Transformierte der Diagonalen unter den Heckeoperatoren), und es wird gezeigt, daß der Beitrag von \(\pi\) genau dann einen Hirzebruch- Zagier-Zykel enthält, wenn \(\pi\) ausgezeichnet ist. Es wird auch gezeigt, daß, falls der Beitrag von \(\pi\) Invarianten erst über einer nichtabelschen Erweiterung enthält, so ist \(\pi\) vom CM-Typ. Für den entsprechenden Teil der Kohomologie bleibt die Frage nach der Gültigkeit der Tate'schen Vermutungen ungeklärt [Vgl. aber: \textit{C. Klingenberg}, Dissertation (Bonn)]. Tate conjecture; compactification of Hilbert-Blumenthal surface; L- function; algebraic cycles; Hirzebruch-Zagier cycles Harder, G.; Langlands, R. P.; Rapoport, M., Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. Reine Angew. Math., 0075-4102, 366, 53-120, (1986) Cycles and subschemes, Special surfaces, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Algebraische Zyklen auf Hilbert-Blumenthal-Flächen | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article gives a self-contained treatment of the theory of Kazhdan-Lusztig polynomials with special emphasis on affine reflection groups. There are only a few new results but several new proofs. We close with a conjectural character formula for tilting modules, which formed the starting point of these investigations. Kazhdan-Lusztig polynomials; affine reflection groups; character formula; tilting modules Soergel, Wolfgang, Kazhdan-Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln, Represent. Theory, 1088-4165, 1, 37-68 (electronic), (1997) Algebraic combinatorics, Quantum groups (quantized enveloping algebras) and related deformations Kazhdan-Lusztig polynomials and a combinatorics for tilting modulus | 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 strings are not discussed in the paper; they are subject of physical papers given in references. They, however, season with the flavour of being administered though the mathematics involved is of interest per se and this is one of the intended messages.
On \({\mathbb{C}}P^{1,N}\), the projective superspace with \(N=1\) or 2, realized as a quotient of the orthosymplectic supergroups, or, rather, their conformal versions, there are considered the cross-ratio of 4 points and some additional, purely ``super'', similar invariants. The correct analogue of the ``imaginary part'' is derived for the upper half- space \(H^{1,N}\), \(N=1\). The classical Schottky uniformization procedure is generalized to \(N=1\), and, to an extent, \(N=2.\)
A relation of the structures described above with the contact structure on \({\mathbb{C}}P^{1,N}\), i.e. preserving the form dz-\(\sum_{1\leq i\leq N}\theta_ id\theta_ i \), is established (for \(N=1)\). In particular, the Schwarz derivative is derived as a byproduct of interpretation of the supervariety of contact structures on \(P^{1,1}\) as the subvariety in the superspace of differential operators of certain type. superalgebraic curves; superstrings; superspace; cross-ratio of 4 points; Schottky uniformization; Schwarz derivative; supervariety of contact structures Yu. I. Manin, ''Superalgebraic curves and quantum strings,'' Trudy Mat. Inst. Steklov,183, 126--138 (1990). Theta functions and curves; Schottky problem, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Supervarieties, Complex supergeometry Superalgebraic curves and quantum strings | 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 review in Zbl 0729.14023. superalgebraic curves; superstrings; superspace; cross-ratio of 4 points; Schottky uniformization; Schwarz derivative; supervariety of contact structures Theta functions and curves; Schottky problem, Supervarieties, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Complex supergeometry Superalgebraic curves and quantum strings | 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 theta functions were introduced by \textit{Yuri I. Manin} [Prog. Theor. Phys., Suppl. 102, 219--228 (1990; Zbl 0796.14005)]. They are certain elements in the function rings of quantum tori. By definition, they satisfy a version of the classical functional equations involving shifts by multiplicative periods. This contribution addresses two interrelated questions: (a) What is the connection between quantum theta functions and theta vectors? (b) Does there exists a quantum analog of the classical functional equation for thetas (related to the action of the metaplectic group)?
Briefly, (partial) answers the author gives here look as follows.
(i) Schwarz's theta vectors are certain elements of projective modules over \(C^{\infty}\)- or \(C^{*}\)-rings of unitary quantum tori. When such a module is induced from the basic Heisenberg representation by a lattice embedding into a vector Heisenberg group, the respective theta vectors \(f_{T}\) are parametrized by the points \(T\) of the Siegel upper half space, and in different models of the basic representation take the form of a ``quadratic exponent'' \(e^{\pi i x ^{t}T x},\) a classical theta, or Fock vacuum state;
(ii) The classical functional equation relates two thetas considered as sections of line bundles over two isomorphic complex tori (Fourier series). Bundles and sections are lifted to the universal covers which are then identified compatibly with period lattices. Consider a classical theta function of \(z \in {\mathbb C}^{N}\)
\[
\theta(z, \Omega) = \sum_{n \in {\mathbb Z}^{N}} e^{\pi i n ^{t}\Omega n + 2\pi i n^{t}z}, \tag{0.1}
\]
where \(\Omega\) is a symmetric complex matrix with positive defined imaginary part. This function satisfies two sets of functional equations. Firstly, for all \(m \in {\mathbb Z}^{N}\)
\[
\theta(z + m, \Omega) = \theta(z, \Omega),
\]
\[
\theta(z + \Omega m, \Omega) = e^{-\pi i m ^{t}\Omega m - 2\pi i m ^{t}z} \theta(z, \Omega).
\]
Secondly,
\[
\theta(\Omega^{-1} z, -\Omega^{-1}) = (det(\Omega/i))^{1/2} e^{\pi i z ^{t}\Omega^{-1} z } \theta(z, \Omega).\tag{0.4}
\]
In fact, (0.4) is the most important special case of a more general modular functional equation related to the action of \(Sp(2N, {\mathbb Z}).\) In this paper, the author proposes an analog of the functional equation (0.4) corresponding this time to the change \(A \rightarrow A^{-1}\) of the quantization matrix. The noncommutative geometric context replacing the classical isomorphism of triples \(({\mathcal T}, {L}, \theta),\) consisting of a complex torus, a line bundle, and its section, involves now the (strong) Morita equivalence of the relevant quantum tori, compatible complex structures on these quantum tori, and theta vectors in the respective projective bimodule. Let \({K}\) (resp. \({Z}\)) be an Abelian group written additively (resp. multiplicatively). Morever, describe \({G}\) as a central extension of \({K}\) by \({Z}\):
\[
1 \rightarrow {Z} \rightarrow {G} \rightarrow {K} \rightarrow 1.
\]
Theorem 2.2: The completion of \({H}_{\infty}\) with respect to the norm
\[
\| \Phi\| ^{2}: = \| _{D}\langle \Phi, \Phi \rangle\| = \| \sum_{h \in D} \langle \Phi, e_{D,\alpha}(h)\Psi \rangle e_{D,\alpha}(h)\| \;\text{(the norm in } C^{\infty}(D, \alpha))
\]
is a finitely generated projective left \(C^{*}(D, \alpha)\)-module \(P.\) The scalar product has a natural extension to \(_{D}\langle , \rangle : P \times P \rightarrow C^{*}(D, \alpha).\) The properties : (i) Symmetry: \(_{D}\langle \Phi, \Psi \rangle^{*} = _{D}\langle \Psi, \Phi \rangle;\) (ii) Bilinearity: \( _{D}\langle a \Psi, \Phi \rangle = a _{D} \langle \Psi, \Phi \rangle\) for any \(a \in C^{\infty}(D, \alpha);\) (iii) Positivity: \(_{D}\langle \Phi, \Phi \rangle\) belongs to the cone of positive elements of \(C^{\infty}(D, \alpha).\) Moreover, if \(_{D}\langle \Phi, \Phi \rangle = 0\) then \(\Phi = 0;\) (iv) Density: The image of \(\langle , \rangle\) is dense in \(C^{\infty}(D, \alpha),\) hold for this extension as well.
Theorem 3.2.1: (i) We have
\[
_{D}\langle f_{T}, f_{T} \rangle = \frac{1}{\sqrt{2^{N}\det \operatorname{Im} T}} \sum_{h \in D} e^{-\frac{\pi}{2} H(\underline{h}, \underline{h})} e_{D,\alpha}(h).
\]
Moreover,
\[
\Theta_{D}: = \sum_{h \in D} e^{-\frac{\pi}{2} H(\underline{h}, \underline{h})} e_{D,\alpha}(h)
\]
is a quantum theta function in the ring \(C^{\infty}(D, \alpha)\) satisfying the following functional equations:
\[
\forall g \in D: C_{g} e_{D,\alpha}(g) x^{*}_{g}(\Theta_{D}) = \Theta_{D},
\]
where
\[
C_{g} = e^{-\frac{\pi}{2} H(\underline{g}, \underline{g})}, x^{*}_{g}(e_{D,\alpha}(h)) = e ^{-\pi H(\underline{g}, \underline{h})} e_{D,\alpha}(h). \tag{3.10}
\]
(ii) We have
\[
\Theta_{D} {\mathbf 1} = \sum_{h \in D} e^{ -\pi H(\underline{h}, \underline{h}) - \pi H(\underline{x}, \underline{h})},
\]
where \textbf{1} is the vacuum vector in the model \(\text{II}_{T}\) represented by the function identically equal to 1.
Theorem 3.5.1: Let \(T(D,\alpha)\) be a quantum torus over \({\mathbb C},\) with unitary quantization form \(\alpha.\) Let \({L} : B \rightarrow {G}(D,\alpha)\) be an ample multiplier such that the left representative projection \(B \rightarrow D\) is an isomorphism which we will use to identify \(B\) with \(D.\) Assume moreover that one can define a Kähler structure on \({\mathbb R} \times D\) such that
\[
\langle g, h \rangle = e^{-\pi H(\underline{g}, \underline{h})}, \alpha(g, h) = e^{\pi i \Im H(\underline{g}, \underline{h})}
\]
for all \(g, h \in D\) holds (\(\langle \,,\, \rangle\) being the structure form of \({L}).\) Then there exists a real space \({K}\) endowed with a bicharacter \(\psi,\) an compatible Kähler structure, and a lattice embedding \(D \subset {K}\) such that \(\psi\) induces \(\alpha\) on \(D,\) and an appropriate generator of \(\Gamma({L})\) is of the form \(\Theta_{D}\) as above.
Theorem 3.6.1: (i) The scalar products \(_{D}\langle f_{T,a}, f_{T,b} \rangle\) are quantum theta functions belonging to the space \(\Gamma({L})\) where \({L}\) is the multiplier
\[
D_{0} \rightarrow {G}(D,\alpha),\;g \mapsto [C_{g}; x_{g}, g],
\]
\(C_{g}, x_{g}\) being defined by (3.10), with \(H\) is lifted to \({\mathbb R}^{N} \times F \times \widehat{F}\) via projection. (ii) The scalar products \(_{D}\langle f_{T,a}, f_{T,b} \rangle\) form a basis of \(\Gamma({L}).\) quantum theta functions; the function rings of quantum tori; metaplectic group; projective modules over \(C^{\infty}\)- or \(C^{*}\)-rings of unitary quantum tori; Heisenberg representation vector; Heisenberg group; Siegel upper half space. Y.I. Manin, Functional equations for quantum theta functions, Kyoto University. Research Institute for Mathematical Sciences. Publications 40 (2004), 605--624. Deformation quantization, star products, Theta functions and abelian varieties Functional equations for quantum theta functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using the torus action method, we construct a one-variable polynomial representation of quantum cohomology rings for degree \(k\) hypersurfaces in \(\mathbb{C} P^{N-1}\). The results interpolate the well-known result of \(\mathbb{C} P^{N-2}\) models and the one of Calabi-Yau hypersurfaces in \(\mathbb{C} P^{N-1}\). In the case \(k\leq N-2\) we find that the principal relation of this ring has a very simple form compatible with the toric compactification of the moduli space of holomorphic maps from \(\mathbb{C} P^1\) to \(\mathbb{C} P^{N-1}\). torus action; quantum cohomology rings; toric compactification; moduli space of holomorphic maps Jinzenji, M., On quantum cohomology rings for hypersurfaces in \(\mathbf {CP}^{N-1}\), J. Math. Phys., 38, 6613-6638, (1997) Applications of deformations of analytic structures to the sciences, Hypersurfaces and algebraic geometry, Quantization in field theory; cohomological methods On quantum cohomology rings for hypersurfaces in \(\mathbb{C} P^{N-1}\) | 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 objective of this paper is to clarify the relationships between the quantum \(D\)-module and equivariant Floer theory. Equivariant Floer theory was introduced by \textit{A. Givental} in his paper ``Homological Geometry'' [Sel. Math., New Ser. 1, No. 2, 325--345 (1995; Zbl 0920.14028)]. He conjectured that the quantum \(D\)-module of a symplectic manifold is isomorphic to the equivariant Floer cohomology for the universal cover of the free loop space. First, motivated by the work of Guest, we formulate the notion of abstract quantum \(D\)-module which generalizes the \(D\)-module defined by the small quantum cohomology algebra. Second, we define the equivariant Floer cohomology of toric complete intersections rigorously as a \(D\)-module, using Givental's model. This is shown to satisfy the axioms of abstract quantum \(D\)-module. By Givental's mirror theorem \textit{A. Givental} [A mirror theorem for toric complete intersections. Prog. Math. 160, 141--175 (1998; Zbl 0936.14031)], it follows that the equivariant Floer cohomology defined here is isomorphic to the quantum cohomology \(D\)-module. symplectic manifold H. Iritani, Quantum \(D\)-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (2006), 577--622. Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symplectic aspects of Floer homology and cohomology Quantum \(D\)-modules and equivariant Floer theory for free loop 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 This is the publication of a greater part of the correspondence by letters between Jean-Pierre Serre (born in 1926) and Alexander Grothendieck (born in 1928), when they lived far apart (in Princeton, Harvard, etc.). The period covered is 1955 (with Grothendieck's letter from Lawrence, Kansas) to 1969, and a few letters from 1984 to 1987. There are 47 letters by Grothendieck (one of them to Cartan), and 39 by Serre. The letters are slightly linguistically edited, and some personal comments are omitted. One can enjoy watching the intellectual exchanges between the two men, related, for instance to the field of algebraic geometry as well as the corresponding cast of characters; this can not only help a historian of mathematics but also a researcher, for these juicy letters are quite inspiring in their own right by offering diverse thought meanderings absent from finished products, such as published papers. Of course we can read here about the happenings related to Bourbaki activities as well.
The letters also offer a glimpse into the two men's personal (academic) lives; for instance, in one of the letters of 1955, Grothendieck is inquiring with Serre about a possibility of a position for himself in relation to a hundred of new positions to be opened in France (as he had heard). He says that he is enormously interested in this, for he would rather stay in France (or perhaps in Germany or South America), rather than in the USA. Serre replies that they have no details about these positions and how many will be available for mathematicians. ``In Paris, Cartan has a candidate that everybody wants: Chevalley (confidential!).'' In case that he is interested in a position in a province, Grothendieck need not worry for there is a good possibility for a position in Strasbourg. Moreover, Serre understands that Spencer is recommending him for a good position at Stanford, or a place like that. Serre tells Grothendieck that Grothendieck's paper on algebraic homology has converted the whole world to his point of view (including Dieudonné who seems to be completely functorial!).
There are humorous points, such as when Grothendieck says that he is pleased that ``your formula with Ext pleases you'' with Serre's comment that ``your'' is an obvious error that should be replaced by ``my''. An interesting point is Grothendieck's letter from the same year (before the reviewer's time) where he introduces the Mittag-Leffler condition, that is to be found again in his \textit{Éléments de Géometrie Algébrique}, Publ. Math., Inst. Hautes Étud. Sci. 11, 349-511 (1962; Zbl 0118.36206). One finds many topics discussed, such as sheaf theory, the zeta function, Rieman-Roch, etc. The letters are teaming with crucial mathematical personalities and their results of the period (including the Russians; or a Serre's ``Izvestiya'' paper) and one can follow the major players feeling as though one is participating on an equal footing.
Not knowing either Grothendieck or Serre personally, the reviewer had got an impression of Serre as an aloof royal, contrasted (or complemented) by an emotional and colorful Grothendieck. For instance in one of his letters, Grothendieck speaks of the mathematical atmosphere at Harvard as rich and full of fresh air in comparison to Paris that gets more stale every year; or his passionate letter to Cartan against military service (for mathematicians), etc. Grothendieck has retreated from the public mathematical view in 1971 and the first letter to Serre after this year is in 1984; there Grothendieck announces his ``reflexion-retrospective'' work ``Récoltes et Semailles,'' (``Harvests and Crops'').
Comments at the end of the book provide further clarifications of the letters and the references are useful. Translating the book into English (or other languages) would be a good idea and adding a (much more detailed) subject and name indexes would increase the book's value. algebraic geometry; French mathematics; Serre; Grothendieck; correspondence; Récoltes et Semailles; US mathematics Colmez, P., Serre, J.-P. (eds.): Correspondance Grothendieck-Serre. Documents Mathématiques (Paris), vol. 2. Société Mathématique de France, Paris (2001) Biographies, obituaries, personalia, bibliographies, History of algebraic geometry, History of mathematics in the 20th century, Research exposition (monographs, survey articles) pertaining to history and biography Correspondence Grothendieck-Serre | 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 note studies the quantum cohomology algebra of the Grassmannian of \(l\)-dimensional planes in \(N\)-dimensional space. Take the coefficients to be \(q\)-polynomials over a cyclotomic field, the extension \(\mathbb{Q}(\zeta)\) by a primitive \(N\)-th root of \((-1)^{l+1}\). The main result is that when \(q\) is specialized to \(1\) the quantum cohomology algebra is a direct sum of its minimal prime ideals each isomorphic to \(\mathbb{Q}(\zeta)\). It follows that the rational quantum cohomology algebra with \(q=1\) is also semisimple and the eigenvalues of multiplications by Schubert classes lie in \(\mathbb{Q}(\zeta)\). Galkin, S; Golyshev, V, Quantum cohomology of Grassmannians and cyclotomic fields, Russ. Math. Surv., 61, 171, (2006) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Exterior algebra, Grassmann algebras Quantum cohomology of Grassmannians and cyclotomic 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 Let \(G\) be a connected semi-simple complex Lie group, \(B\) its Borel subgroup, \(T\) a maximal complex torus contained in \(B\), and Lie \((T)\) its Lie algebra. This setup gives rise to two constructions; the generalized nonperiodic Toda lattice and the flag manifold \(G/B\). The Toda lattice for \((G,B,T)\) is the dynamical system on the cotangent bundle \(T^* \text{Lie}(T)\) endowed with the canonical holomorphic symplectic form and the holomorphic Hamiltonian function
\[
H(p,q)=(p,p)-\sum_{\text{simple roots } \alpha_i}(\alpha_i,\alpha_i) \exp\bigl(\alpha_i (q)\bigr),
\]
where \((,)\) is any fixed nonzero multiplication of the Killing form on each simple component of Lie \((G)\) and the simple roots are given by the roots of \(B\) with respect to \(T\). This system is known to he completely integrable [\textit{B. Kostant}, Sel. Math., New Ser. 2, 43--91 (1996; Zbl 0868.14024) and Adv. Math. 34, 195--338 (1979; Zbl 0433.22008)]. Therefore the variety defined by the ideal generated by the integrals of motions is the Lagrangian analytic submanifold of \(T^*\text{Lie} (T)\). On the other hand, for the flag manifold \(G/B\) we have the small quantum cohomology ring \(QH^*(G/B,\mathbb{C})\) which is generated by second cohomology classes and parameters. Denoting as \(q_i\) the coordinates of the parameter space \(H^2(G/B,\mathbb{C})\) defined by \(q_i(\sum a_jp_j)= \exp (a_i)\) (here \(p_j\) is the cohomology class corresponding to the fundamental weights), the author describes the ring structure of \(QH^* (G,B)\):
Theorem 1. The small quantum cohomology ring \(QH^*(G/B, \mathbb{C})\) is canonical isomorphic to \(\mathbb{C}[p_1,\dots, p_l,q_1, \dots, q_l]/I\), where \(I\) is the ideal generalized by the nonconstant complete integrals of motions of the Toda lattice for the Langlands-dual Lie group \((G^v,B^v,T^v)\) o \((G,B,T)\).
In fact the author proves more in this paper. Using the quantum hyperplane section principle it is possible to compute the virtual numbers of rational curves in Calabi-Yau 3-fold complete intersections in homogeneous spaces with the knowledge of the quantum \({\mathcal D}\)-module structure of the ambient spaces. The author shows that the \({\mathcal D}\)-module structure for \(G/B\) is governed by the conservation laws of quantum Toda lattices which are the quantizations of the Toda lattices and still integrable [\textit{B. Kostant}, Invent. Math. 48, 101--184 (1978; Zbl 0405.12013) and London Math. Soc. Lect. Note Ser. 34, 287--316 (1979; Zbl 0474.58010)], \textit{A. Reyman} and \textit{M. Semenov-Tiam-Shansky}, Invent. Math. 54, 81--100 (1979; Zbl 0403.58004)]. The Hamiltonian operator he considers is
\[
\widehat H=\Delta-\sum_{\text{simple roots }\alpha_i} (\alpha_i, \alpha_i) \exp \bigl(\alpha_i(q) \bigr),
\]
where \(\Delta\) is the Laplacian on Lie \(T\) associated with the invariant form \((,)\). Let \({\mathcal D}\) be the differential operator algebra over \(\mathbb{C}\), generated by \(\hbar\frac {\partial} {\partial t_i}\), multiplication by \(\hbar\) and \(\exp t_i\).
Theorem II. The quantum \({\mathcal D}\)-module of \(G/B\) is canonically isomorphic to \({\mathcal D}/{\mathcal I}\) where \({\mathcal I}\) is the left ideal generated by the nonconstant complete quantum integrals of motions of the quantum Toda lattice for the Langlands-dual Lie group \(G^\vee, B^\vee,T^\vee)\) of \((G,B,T)\). lagrangian submanifold; quantum \({\mathcal D}\) modules; mirror theorem; dynamical system; holomorphic hamiltonian function Kim, B., Quantum cohomology of flag manifolds \(G / B\) and quantum Toda lattices, Ann. of Math. (2), 149, 1, 129-148, (1999) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) Quantum cohomology of flag manifolds \(G/B\) and quantum Toda lattices. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(f\in \mathbb{C}[x_1,\ldots,x_n]\) be a weighted homogeneous polynomial of degree \(d\) with respect to weights \((w_1,\ldots,w_n).\) Denote by \(I_f:=(\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n})\) the Jacobian ideal of \(f\) and by \(M(f):=\frac{\mathbb{C}[x_1,\ldots,x_n]}{I_f}\) the Milnor algebra of \(f.\) We say that two weighted homogeneous polynomials \(f\) and \(g\) are right-equivalent if there exists a diffeomorphism \(\psi:\mathbb{C}^n\to \mathbb{C}^n\) such that \(f\psi=g.\)
\noindent The main result of this paper is that if \(f\) and \(g\) are weighted homogeneous polynomials of degree \(d\) with respect to weights \((w_1,\ldots,w_n)\) such that \(I_f=I_g,\) the they are right-equivalent. As a consequence it is shown that if \(f\) and \(g\) are as above and \(M(f)\) and \(M(g)\) are isomorphic graded \(\mathbb{C}\)-algebras, then \(f\) and \(g\) are right-equivalent. weighted homogeneous polynomial; Milnor algebra; right-equivalence Group actions on varieties or schemes (quotients), Singularities of surfaces or higher-dimensional varieties, Actions of groups and semigroups; invariant theory (associative rings and algebras) Weighted homogeneous polynomials with isomorphic Milnor 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 This well-written paper generalizes various results of \textit{E. Becker} and \textit{V. Powers} [J. Reine Angew. Math. 480, 71-109 (1996; Zbl 0922.12003)] on holomorphy rings, of Schmüdgen-Wörmann on the moment problem and the Positivstellensatz, and of Mahé on Pythagoras numbers of commutative rings. The author introduces the notions of ``totally archimedean ring'' and ``iterated holomorphy ring''. -- One of the main results reads:
Theorem 4.3: Let \(k\) be a totally archimedean field, let \(A\) be a totally archimedean commutative \(k\)-algebra of finite type, let \(f\in A\) be strictly positive on \(\text{Spec}_rA\) (the real spectrum of \(A)\). Then \(f\in\sum A^2\).
The paper also contains several illuminating examples and counter-examples, and some open problems. totally archimedean ring; holomorphy rings; Positivstellensatz Jean-Philippe Monnier, Anneaux d'holomorphie et Positivstellensatz archimédien, Manuscripta Math. 97 (1998), no. 3, 269 -- 302 (French, with English summary). Relevant commutative algebra, Real algebraic and real-analytic geometry, Algebraic theory of quadratic forms; Witt groups and rings, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Arithmetic rings and other special commutative rings Holomorphy rings and the archimedean Positivstellensatz | 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 an algebraically closed field, and let \(C\subset \mathbb P_k^n\) be a reduced closed subscheme with ideal sheaf \(\mathcal S\). Let \(\mathcal S^{\langle 2\rangle}\) be the second symbolic power of \(\mathcal S\). When \(C\) is an integral curve, we compute the Hilbert polynomial of \(\mathcal O_{\mathbb{P}^n}/\mathcal S^{\langle 2\rangle}\) in terms of invariants of \(C\). curve; Hilbert polynomial; hypersurface; Kähler differentials; singular locus; symbolic power Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Plane and space curves, Singularities of surfaces or higher-dimensional varieties The Hilbert polynomial of a symbolic square | 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 Schubert polynomials \({\mathfrak S}_ \sigma(x_ 1,x_ 2,\dots)\) indexed by permutations have been introduced and investigated by \textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand} [Russ. Math. Surveys 28, No. 3, 1-26 (1973; Zbl 0286.57025)], \textit{M. Demazure} [Ann. Sci. École Norm. Sup., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)], and by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)]; see also their paper [Symmetry and flag manifolds, Lect. Notes in Math. 996, 118-144 (1983; Zbl 0542.14031)].
In this paper the theory of Schubert polynomials is recovered using the nilCoxeter algebra \({\mathfrak C}_ n\) with the identity element \(e\), given by its generators and defining relations as the \(K\)-algebra
\[
\begin{multlined} {\mathfrak C}_ n=\Bigl\langle u_ 1,\dots, u_{n-1}\mid u^ 2_ i= 0\;(i\in I_{n-1}),\;u_ i u_ j= u_ j u_ i\\ (\text{for }| i- j|\geq 2),\text{ and } u_ i u_{i+1} u_ i= u_{i+1} u_ i u_{i+1} (\text{for } i\in I_{n-2})\Bigr\rangle\end{multlined}
\]
over any commutative ring \(K\); here \(I_ n= \{1,2,\dots, n\}\). This algebra can be faithfully represented by the algebra of operators generated by \(\Phi_ i\) \((i\in I_{n-1})\),
\[
\Phi_ i(\sigma)= \begin{cases} \sigma\tau_ i &\text{if } \ell(\sigma\tau_ i)= \ell(\sigma)+1;\\ 0 & \text{otherwise}.\end{cases}
\]
Here, \(\sigma\) is any permutation in the symmetric group \({\mathcal S}_ n\) defined on \(I_ n\), \(\tau_ i\) \((i\in I_{n-1})\) is the `adjacent' transposition \((i,i+1)\), and \(\ell(\sigma)\) is the length of \(\sigma\in {\mathcal S}_ n\) defined as the minimal \(p\) such that \(\sigma= \tau_{a_ 1}\cdot\tau_{a_ 2}\cdot\dots\cdot \tau_{a_ p}\) for some \(a_ j\in I_{n-1}\). A sequence \(a= (a_ 1,\dots, a_ p)\), \(a_ j\in I_{n-1}\) is called a reduced decomposition of \(\sigma\) if \(p= \ell(\sigma)\). \(R(\sigma)\) denote the set of all reduced decompositions for \(\sigma\). For any reduced decomposition \(a= (a_ 1,\dots, a_ p)\) let us identify the monomial \(u_{a_ 1} u_{a_ 2}\cdots u_{a_ k}\) in \({\mathfrak C}_ n\) with \(\tau_{a_ 1} \cdot \tau_{a_ 2}\cdot\dots\cdot \tau_{a_ k}\) in \({\mathcal S}_ n\); the defining relations for \({\mathfrak C}_ n\) guarantee the correctness of such notation, and we see that \({\mathcal S}_ n\) gives a \(K\)-basis for \({\mathfrak C}_ n\). As usual, denote by \(\langle f,\sigma\rangle\) the coefficient of \(\sigma\in {\mathcal S}_ n\) in the \(K\)- expression for \(f\in {\mathfrak C}_ n\). Further, denote
\[
A_ i(x)= (e+ xu_{n-1})\cdot (e+ xu_{n-2})\cdot\dots\cdot (e+ xu_ i)
\]
for any \(i\in I_{n-1}\), \(\bar x= (x_ 1,\dots, x_{n-1})\), \({\mathfrak S}(\bar x)= A_ 1(x_ 1)\cdot A_ 2(x_ 2)\cdot \dots\cdot A_{n-1}(x_{n- 1})\) and let \({\mathfrak S}_ \sigma(\bar x)= \langle{\mathfrak S}(\bar x),\sigma\rangle\). Among the results of this paper is Theorem 2.2 saying that \({\mathfrak S}_ \sigma(\bar x)\) is a Schubert polynomial. The authors prove also (Lemma 2.3) that in the case of \(\text{char } K= 0\),
\[
{\mathfrak S}_ \sigma(1,\dots, 1)= {1\over p!} \sum_{(a_ 1,\dots, a_ p)\in R(\sigma)} a_ 1\cdots a_ p.
\]
Also proved is the \(q\)-analogue of this last formula conjectured by \textit{I. Macdonald} [Notes on Schubert polynomials, LACIM, Université du Québec, Montréal (1991)]:
\[
{\mathfrak S}_ \sigma(1,q,\dots, q^{n-2})= {1\over [1]\cdot[2]\cdot\dots\cdot [p]}\sum_{(a_ 1,\dots, a_ p)\in R(\sigma)} [a_ 1]\cdot\dots\cdot [a_ p]q^{\sum_{\{i\mid a_ i\leq a_{i+1}\}}} i,
\]
where \([t]= 1+ q+\cdots+ q^{t-1}\). Schubert polynomials; nilCoxeter algebra; reduced decomposition S. Fomin and R. P. Stanley. ''Schubert polynomials and the NilCoxeter algebra''. Adv. Math. 103(1994), pp. 196--207.DOI. Symmetric functions and generalizations, Representations of finite symmetric groups, Grassmannians, Schubert varieties, flag manifolds, Combinatorial identities, bijective combinatorics Schubert polynomials and the nilCoxeter 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 We present the quantum plane from the viewpoint of ``noncommutative algebraic geometry'', and we show how, in this context, \(\text{GL}_ q (2)\) appears as its natural ``automorphism object-group''. Conversely, one can try to construct a ``quantum space'' from a linear representation of \(\text{GL}_ q\). This can be done by observing that the standard symmetry \(x\otimes y\to y\otimes x\) of the tensor product must be deformed for becoming equivariant. The corresponding deformation of the symmetric algebra gives the space we want. This text does not claim to be original and is inspired by \textit{A. Joyal} and \textit{R. Street} [An introduction to Tannaka duality and quantum groups, Lect. Notes Math. 1488, 413-492 (1991; Zbl 0745.57001)] and \textit{Yu. I. Manin} [Quantum groups (Montréal 1990; Zbl 0724.17006)]. quantum plane Quantum groups (quantized enveloping algebras) and related deformations, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Noncommutative algebraic geometry, Knots and links in the 3-sphere, Supervarieties Representations of \(GL_ q(2)\) and commutativity of the tensor product | 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 Grothendieck-Riemann-Roch theorem plays an essential role in the study of algebraic geometry. Roughly speaking, it describes how far the Chern character map between the algebraic \(K\)-theory and the algebraic Chow-theory to be commutative with the push-forward operations w.r.t. proper morphisms. This theorem vastly generalizes the classical Riemann-Roch theorem on compact Riemann surfaces, and hence it can be viewed as an Atiyah-Singer type index theorem.
In previous work, the Grothendieck-Riemann-Roch theorem was extended by several mathematicians to an arithmetic setting in the sense of Arakelov. In this situation, the generators of algebraic \(K\)-groups and algebraic Chow groups are equipped with additional Hermitian structures so that the fibers of an arithmetic variety over the infinite places can be treated as the same footing as the finite ones.
Historically, \textit{H. Gillet} and \textit{C. Soulé} [Invent. Math. 110, No. 3, 473--543 (1992; Zbl 0777.14008)] proved a degree one version of the arithmetic Grothendieck-Riemann-Roch theorem, they used the theory of Ray-Singer analytic torsion to define a reasonable push-forward on the level of arithmetic \(K\)-groups. Whence \textit{J.-M. Bismut} and \textit{K. Köhler} [J. Algebr. Geom. 1, No. 4, 647--684 (1992; Zbl 0784.32023)] developed the analytic torsion theory to higher degree case, a complete proof of the arithmetic Grothendieck-Riemann-Roch was given by Gillet, Rössler and Soulé [\textit{H. Gillet} et al., Ann. Inst. Fourier 58, No. 6, 2169--2189 (2008; Zbl 1152.14023)]. But, due to technical reason, these two works had to restrict to push-forward by morphisms which are smooth over generic fibers. The aim of the article under review is to break this limit, namely is to generalize the works of Gillet-Soulé and Gillet-Rössler-Soulé to arbitrary projective morphisms, not necessarily smooth over the generic fibers. This forces the authors to enlarge the arithmetic \(K\)-groups and the arithmetic Chow groups to afford corresponding push-forward functorialities. The strategy the authors followed was replacing smooth differential forms in the theory of Gillet-Soulé by currents with possibly non-empty wave front sets. As a consequence, by explaining the relationship between the topological correction term and the generalized analytic torsion classes, the authors obtained all the possible forms of the arithmetic Grothendieck-Riemann-Roch theorem. Arakelov theory; Grothendieck-Riemann-Roch theorem; projective morphism Burgos, J. I.; Freixas, G.; Litcanu, R., The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms, Ann. Fac. Sci. Toulouse Math. (6), 23, 3, 513-559, (2014) Arithmetic varieties and schemes; Arakelov theory; heights, Riemann-Roch theorems The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms | 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 the afternoon of Tuesday, January 30, 2007, Illusie met with University of Chicago mathematicians Alexander Beilinson, Spencer Bloch, and Vladimir Drinfeld, as well as a few other guests, at Beilinson's home in Chicago. Illusie chatted by the fireside, recalling memories of his days with Grothendieck. In this warm conversation one can learn a lot of things about Grothendieck, from mathematical career to the style of his life, as narrated by one of his closest friends and students, Illusie. Grothendieck had a very strong feeling for music. He liked Bach, and his most beloved pieces were the last quartets by Beethoven. His favorite tree was the olive tree. We learn that he had been a member of the Bourbaki group. Grothendieck was the father of K-theory and he was interested in loop spaces, iterated loop spaces; \(n\)-categories, Picard category, the cotangent complex, \(n\)-stacks and more \dots of course.
It was surprising that he was not allowed to speak on motives at the Bourbaki seminar. He asked for six or seven exposés, and the organizers considered it was too much. According to Illusie he was the first one to write a map vertically instead of from left to right.
That was one principle of Grothendieck: every assertion should be justified, either by a reference or by a proof. Even a ``trivial'' one. He hated such phrases as ``It's easy to see,'' ``It's easily checked.'' On the other hand he thought that some complements, even if they were not immediately useful, could prove important later and therefore should not be removed. He wanted to see all the facets of a theory. Grothendieck Illusie, Luc: Reminiscences of Grothendieck and his school, Notices amer. Math. soc. 57, No. 9 (October 2010) History of algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Cycles and subschemes Reminiscences of Grothendieck and his school | 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 \(q\)-Kostka polynomials \(K_{\lambda\mu}(q)\) expressing the Schur function \(S_{\lambda}(\mathbf{x})\) as a linear combination of Hall-Littlewood polynomials \(P_{\mu}(\mathbf{x},q)\) have been an object of intensive study in the last two decades at the crossroads of combinatorics, algebra and geometry. The topic of the paper under review comes from the work of Lascoux and Schützenberger presenting the variant polynomial \(\widetilde K_{\lambda\mu}(q)=q^{n(\mu)}K_{\lambda\mu}(1/q)\) as a sum of the atom polynomials \(R_{\lambda\nu}(q)\), \(\nu\geq\mu\), where \(R_{\nu\mu}(q)\) themselves have non-negative coefficients. The main purpose of the paper is to give a geometric interpretation of the atomic decomposition in the language of scheme-theoretic intersections of nilpotent orbit varieties introduced by Kraft and De Concini-Procesi in the early 80's. In particular, involving a recent result of Broer, and as a consequence of their approach, the authors obtain a new proof of the atomic decomposition of the \(q\)-Kostka polynomials. Kostka polynomials; atomic decomposition; nilpotent conjugacy classes; nilpotent orbit varieties William Brockman and Mark Haiman, Nilpotent orbit varieties and the atomic decomposition of the \?-Kostka polynomials, Canad. J. Math. 50 (1998), no. 3, 525 -- 537. Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Special varieties, Representation theory for linear algebraic groups Nilpotent orbit varieties and the atomic decomposition of the \(q\)-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 [For part I see \textit{D. Gaitsgory}, ``Operads, Grothendieck topologies and deformation theory'', preprint 1995 (alg-gem/9502010)]
Let \(\mathcal{A}\) be a quasi-coherent sheaf of algebras over a scheme \(X\). A site \(\mathcal{C}_{X}(\mathcal{A})\) is defined whose objects are of the form \((U, \mathcal{B}_{U},\phi)\) where \(U\) is Zariski open in \(X\), \(\mathcal{B}_{U}\) is a quasi-coherent sheaf of associative algebras on \(U\), and \(\phi : \mathcal{B}_{U}\to \mathcal{A}| _{U}\) is a map of sheaves of associative algebras. The morphisms (resp., topology) are (resp., is) defined with respect to the second (resp. first) coordinate of the object(s).
For \(X\) affine, the category \(\mathcal{S}\mathcal{H}^{qc}_{X}(\mathcal{A})\) of quasi-coherent sheaves on \(\mathcal{C}_{X}(\mathcal{A})\) is seen to be equivalent to the category of sheaves of abelian groups on an ``older category'' \(\mathcal{S}\mathcal{H}^{old}_{X}(\mathcal{A})\) defined in a similar way to \(\mathcal{S}\mathcal{H}^{qc}_{X}(\mathcal{A})\) but rather, for instance, considering algebraic maps to \(\mathcal{A}\), and cohomologies computed for \(\mathcal{S}{\mathcal H}^{qc}_{X}(\mathcal{A})\) are the same as those for \(\mathcal{S}{\mathcal H}_{X}^{old}(\mathcal{A})\).
For a general scheme \(X\), functors connecting the cohomology of \(\mathcal{S}{\mathcal H}^{qc}_{X}(\mathcal{A})\) to that of the category \({\mathcal A}_{qc}\)-mod of quasi-coherent sheaves of \(\mathcal{A}\)-bimodules on \(X\) are found. The category \(\mathcal{D}{\mathcal E}{\mathcal F}{\mathcal O}{\mathcal R}{\mathcal M}^{1}(\mathcal{A})\) of deformations of \(\mathcal{A} \) is seen to be equivalent to a category of torsors on \(\mathcal{C}_{X}(\mathcal{A})\). Such results are then interpreted in cohomological terms. Grothendieck topology; sheaves; deformations; derived category; gerbe; torsor Gaitsgory, D.: Grothendieck topologies and deformation theory. Compositio math. 106, 321-348 (1997) Grothendieck topologies and Grothendieck topoi, Étale and other Grothendieck topologies and (co)homologies, Formal methods and deformations in algebraic geometry Grothendieck topologies and deformation theory. II | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The coefficients of the Kazhdan-Lusztig polynomials \(P_{v,w}(q)\) are nonnegative integers that are upper semicontinuous relative to Bruhat order. Conjecturally, the same properties hold for \(h\)-polynomials \(H_{v,w}(q)\) of local rings of Schubert varieties. This suggests a parallel between the two families of polynomials. We prove our conjectures for Grassmannians, and more generally, covexillary Schubert varieties in complete flag varieties, by deriving a combinatorial formula for \(H_{v,w}(q)\). We introduce drift configurations to formulate a new and compatible combinatorial rule for \(P_{v,w}(q)\). From our rules we deduce, for these cases, the coefficient-wise inequality \(P_{v,w}(q) \preceq H_{v,w}(q)\). Kazhdan-Lusztig polynomials; Hilbert series; Schubert varieties Li L., Yong A.: Kazhdan-Lusztig polynomials and drift configurations. Algebra Number Theory 5(5), 595--626 (2011) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Hecke algebras and their representations Kazhdan-Lusztig polynomials and drift configurations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author shows that Schiffmann's formulas for counts of Higgs bundles over finite fields can be reduced to a simpler formula conjedtured by Mozgovoy.
Consider the Frobenius action on the first cohomology of \(C\) with eigenvalues \(\alpha_1, \dots, \alpha_{2g}\) with \(\alpha_{i+g} = q\alpha_i^{-1}\) for \(i = 1, \dots, g\), which gives
\[
\# C(\mathbb{F}_{q^k}) = 1 + q^k - \sum_{i=1}^{2g} \alpha_i^k.
\]
Schiffmann's result in [\textit{R. Fedorov} et al., Commun. Number Theory Phys. 12, No. 4, 687--766 (2019; Zbl 1409.14059)] says that the number of absolutely indecomposable bundles or rank \(r\) and degree \(d\) over a complete curve \(C\) of genus \(g\) over \(\mathbb{F}_q\) is given by a Laurent polynomial
\[
A_{g,r,d} (q, \alpha_1, \dots, \alpha_{g} ) \in \mathbb{Z} [q, \alpha_1^{\pm 1}, \dots, \alpha_{g}^{\pm 1} ]
\]
which is independent of \(C\), symmetric in \(\alpha_i\), and invariant under \(\alpha_o \to q \alpha_i^{-1}\).
The main result of this paper is:
Theorem 1.1. Let \(g \ge 1\). Let \(\Omega_g\) denote the series
\[
\Omega_g = \sum_{\mu \in \mathcal{P}} T^{|\mu|} \prod_{\square \in \mu} \frac{ \prod_{i=1}^g (z^{a(\square)+1} - \alpha_i q^{l (\square)} ) (z^{a(\square)} - \alpha_i^{-1} q^{l (\square)+1} ) }{ (z^{a(\square)+1} - q^{l (\square)} ) (z^{a(\square)} - q^{l (\square)+1} ) },
\]
where \(a(\square)\), \(l(\square)\), and \(\mathcal{P}\) are certain combinatorial notations from Young diagrams. Also let
\[
H_g = -(1-q) (1-z) \operatorname{Log} \Omega_g, H_g = \sum_{r=1}^{\infty} H_{g,r} T^r.
\]
Then for all \(r \ge 1\), \(H_{g,r}\) is a Laurent polynomial in \(q,z\) and \(\alpha, \dots, \alpha_g\), and for all \(d\), \(A_{g,r,d}\) is obtained by setting \(z=1\) in \(H_{g,r}\):
\[
A_{g,r,d} (q, \alpha_1, \dots, \alpha_g) = H_{g,r} (q, 1, \alpha_1, \dots, \alpha_g).
\]
As a corollary, the GL-version of the conjecture of \textit{T. Hausel} (Conjecture 3.2 in [Prog. Math. 235, 193--217 (2005; Zbl 1099.14026)]) is obtained:
Corollary 1.2. For \(r, d, d'\) satisfying \((r,d) = (r,d') =1\), the \(E\)-polynomials of \(\mathcal{M}(g,r,d)\) and \(\mathcal{M}(g,r,d')\) coincide.
In the next paper [Ann. Math. (2) 192, No. 1, 165--228 (2020; Zbl 1467.14086)], the author extends the methods of [\textit{O. Schiffmann}, Ann. Math. (2) 183, No. 1, 297--362 (2016; Zbl 1342.14076)] to the parabolic case. This gives a proof of the conjecture of \textit{T. Hausel} et al. [Duke Math. J. 160, No. 2, 323--400 (2011; Zbl 1246.14063)] on the Poincaré polynomials of character varieties with punctures. Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli, Character varieties Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures) | 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 some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations. Braid groups; Yang-Baxter groups; dynamical Yang-Baxter relations; classical Yang-Baxter relations; Kohno-Drinfeld algebras; 3-term relations algebras; Gaudin elements; Jucys-Murphy elements; small quantum cohomology; \(K\)-theory of flag varieties; Pieri rules; chromatic number; Tutte and Betti polynomials; reduced polynomials; Chan-Robbins-Yuen polytope; \(k\)-dissections of a convex \((n+k+1)\)-gon; Lagrange inversion formula; Richardson permutations; multiparameter deformations of Fuss-Catalan and Schröder polynomials; poly-Bernoulli numbers; Stirling numbers; Euler numbers; Brauer algebras; VSASM; CSTCPP; Birman-Ko-Lee monoid; Kronecker elliptic sigma functions Kirillov, Anatol N., On some quadratic algebras {I}~{\(\frac{1}{2}\)}: combinatorics of {D}unkl and {G}audin elements, {S}chubert, {G}rothendieck, {F}uss--{C}atalan, universal {T}utte and reduced polynomials, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 12, 002, 172~pages, (2016) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Yang-Baxter equations, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On some quadratic algebras. I \(\frac{1}{2}\): Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that categorified quantum \(\mathfrak{sl}_2\) is an inverse limit of flag 2-categories defined using cohomology rings of iterated flag varieties. This inverse limit is an instance of a 2-limit in a bicategory giving rise to a universal property that characterizes the categorification of quantum \(\mathfrak{sl}_2\) uniquely up to equivalence. As an application we characterize all bimodule homomorphisms in the flag 2-category and prove that on the homological level the categorified quantum Casimir of \(\mathfrak{sl}_2\) acts appropriately on these 2-representations. A. Beliakova, A. D. Lauda, \textit{Categorified quantum}[InlineMediaObject not available: see fulltext.]\textit{is an inverse limit of flag 2-categories}, Transform. Groups \textbf{19} (2014), no. 1, 1-26. Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Special properties of functors (faithful, full, etc.), Double categories, \(2\)-categories, bicategories, hypercategories Categorified quantum \(\mathfrak{sl}_2\) is an inverse limit of flag 2-categories | 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 preprojective cohomological Hall algebra (CoHA) introduced by the authors in [Proc. Lond. Math. Soc. (3) 116, No. 5, 1029--1074 (2018; Zbl 1431.17013)] for any quiver \(Q\) and any one-parameter formal group \({\mathbb {G}}\). In this paper, we construct a comultiplication on the CoHA, making it a bialgebra. We also construct the Drinfeld double of the CoHA. The Drinfeld double is a quantum affine algebra of the Lie algebra \(\mathfrak {g}_Q\) associated to \(Q\), whose quantization comes from the formal group \({\mathbb G}\). We prove, when the group \({\mathbb G}\) is the additive group, the Drinfeld double of the CoHA is isomorphic to the Yangian. quantum group; shuffle algebra; Hall algebra; Yangian; Drinfeld double Quantum groups (quantized enveloping algebras) and related deformations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Bordism and cobordism theories and formal group laws in algebraic topology Cohomological Hall algebras and affine 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 manifold \(\mathcal Fl_n\) of complete flags in the \(n\) dimensional vector space \(\mathbb C^n\) over the complex numbers is an object that, by its various definitions, is an object in the intersection of algebra and geometry. On the one hand it can be expressed as the quotient \(B\backslash \text{GL}_n\) of all invertible \(n\times n\)-matrices by its subgroup of lower triangular matrices, and on the other hand as fibers of certain bundles constructed universally from complex vector bundles. Combinatorics enter into the study via the cohomology ring \(H^\ast(\mathcal Fl_n) =H^\ast(\mathcal Fl_n;\mathbb Z)\) with integer coefficients, that can be described as the quotient of the polynomial ring \(\mathbb Z[x_1,\dots, x_n]\) modulo the ideal generated by all non-constant homogeneous functions invariant under permutations of \(x_1,\dots, x_n\). This ring is a free abelian group of rank \(n!\) with basis given by monomials dividing \(\prod_{i=1}^{n-1}x_i^{n-i}\). The ring also has a much more geometric basis given by the Schubert classes \([X_w]\) in the cohomology ring \(H^\ast(\mathcal Fl_n)\).
This article makes an important contribution to bridging the algebra and combinatorics of Schubert polynomials with the geometry of Schubert varieties. It brings new perspectives to problems in commutative algebra concerning ideals generated by minors of generic matrices, and provides a geometric context in which polynomial representatives for Schubert classes are uniquely singled out with no choices but a Borel subgroup of the general linear group \(\text{GL}_n\mathbb C\) in such a way that it is geometrically obvious that these representatives have nonnegative coefficients.
One of the main ideas in the article is to translate ordinary cohomological statements concerning Borel orbit closures on the flag manifold \(\mathcal Fl_n\) into equivariant-cohomological statements concerning double Borel orbit closures on the \(n\times n\) matrices \(M_n\). To be more precise, the preimage \(\widetilde X_w\subseteq \text{GL}_n\mathbb C\) of a Schubert variety \(X_w\in \mathcal Fl_n\) is an orbit closure for the action of the product \(B\times B^+\) of the lower and upper triangular subgroups of \(\text{GL}_n\mathbb C\) acting by multiplication on the left and right. When \(\overline X_w\subseteq M_n\) is the closure of \(\widetilde X_w\) and \(T\) is the torus in \(B\), the \(T\)-equivariant cohomology class \([\overline X_w]_T\in H_T^\ast(M_n)\) is the polynomial representative. It has positive coefficients because there is a \(T\)-equivariant flat (Gröbner) degeneration of \(\overline X_w\) to \(\mathcal L_w\) that is a union of coordinate subspaces \(L\subseteq M_n\). Each subspace \(L\subseteq \mathcal L_w\) has an equivariant cohomology class \([L]_T\in H_T^\ast(M_n)\) that is a monomial in \(x_1,\dots, x_n\), and the sum of these is \([\overline X_w]_T\). The formula is \([\overline X_w]_T = [\mathcal L_w]_T =\sum_{L\in \mathcal L_w}[L]_T\). More importantly, the authors identify a particularly natural degeneration of the matrix Schubert variety \(\overline X_w\) with a reduced and Cohen-Macaulay limit \(\mathcal L_w\) in which the subspaces have combinatorial interpretations and coincides with known combinatorial formulas for Schubert polynomials.
Instead of using equivariant classes associated to closed subvarieties of non-compact spaces the authors develop their theory in the context of multidegrees. The equivariant considerations for matrix Schubert varieties \([\overline X_w]\subseteq M_n\) are then done as multigraded commutative algebra for the Schubert determinantal ideals \(I_w\) cutting out the varieties \(\overline X_w\).
The Gröbner geometry of Schubert polynomials introduced provides a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. In fact the authors describe, for every matrix Schubert variety \(\overline X_w\);
(1) its multidegree and Hilbert series in terms of Schubert and Grothendieck polynomials
(2) a Gröbner basis consisting of minors in its defining ideal \(I_w\)
(3) the Stanley-Reisner complex \(\mathcal L_w\) of its initial ideal \(J_w\), which they prove is Cohen-Macaulay
(4) an inductive irredundant algorithm of weak Bruhat order for listing the facets of \(\mathcal L_w\).
The authors introduce a powerful inductive method that they call Bruhat induction, for working with determinantal ideals and their initial ideals. Bruhat induction as well as the derivation of the main theorems concerning Gröbner geometry rely on results concerning positivity of torus-equivariant cohomology classes represented by subschemes and shellability of certain simplicial complexes that reflect the nature of reduced subwords of words in Coxeter generators for Coxeter groups. The latter technique gives a new perspective, from simplicial topology, of the combinatorics of Schubert and Grothendieck polynomials.
Among the most important applications of the work is the geometrically positive formulae for Schubert polynomials, and connections with Fulton's theory of degeneracy loci, relations between multidegrees and \(K\)-polynomials on \(n\times n\) matrices with classical cohomological theories on the flag manifold, and comparisons with the commutative algebra of determinantal ideals. Stanley-Reisner complex; Coxeter group; Bruhat order; Cohen-Macaulay ideal; initial ideal; Bruhat group; equivariant cohomology; divided differences; Bruhat induction Knutson, [Knutson and Miller 05] A.; Miller, E., Gröbner Geometry of Schubert Polynomials., Ann. Math. (2), 161, 3, 1245-1318, (2005) Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Linkage, complete intersections and determinantal ideals, Classical problems, Schubert calculus, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Group actions on posets and homology groups of posets [See also 06A09] Gröbner geometry of Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct a ``spectral curve'' for the generalized Toda system, which allows efficiently finding its quantization. In turn, the quantization is realized using the technique of the quantum characteristic polynomial for the Gaudin system and an appropriate Alder-Kostant-Symes reduction. We also discuss some relations of this result to the recent consideration of the Drinfeld Zastava space, the monopole space, and corresponding symmetries of the Borel Yangian. quantization; integrable system; flag space Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Special quantum systems, such as solvable systems, Finite-dimensional groups and algebras motivated by physics and their representations, Groups and algebras in quantum theory and relations with integrable systems, Relationships between algebraic curves and integrable systems Quantum generalized Toda system | 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 objective of this paper is to establish a new connection between the Hermitian rank-1 projector solutions of the Euclidean \(\mathbb{C}P^{2S}\) sigma model in two dimensions and the particular hypergeometric orthogonal polynomials called Krawtchouk polynomials. We show that any Veronese subsequent analytical solutions of the \(\mathbb{C}P^{2S}\) model, defined on the Riemann sphere and having a finite action, can be explicitly parametrized in terms of these polynomials. We apply these results to the analysis of surfaces associated with \(\mathbb{C}P^{2S}\) models defined using the generalized Weierstrass formula for immersion. We show that these surfaces are homeomorphic to spheres in the \(\mathfrak{su}(2s+1)\) algebra and express several other geometrical characteristics in terms of the Krawtchouk polynomials. Finally, a connection between the \(\mathfrak{su}(2)\) spin-s representation and the \(\mathbb{C}P^{2S}\) model is explored in detail. Topological field theories in quantum mechanics, Differential geometric aspects of harmonic maps, Soliton equations, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Compact Riemann surfaces and uniformization, Riemann surfaces; Weierstrass points; gap sequences \(\mathbb{C}P^{2S}\) sigma models described through hypergeometric orthogonal 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 \(w : \mathbb C^N \to \mathbb C\) be a quasi-homogeneous polynomial whose total degree \(d\) is equal to the sum of the weights of each variable. Let \(G \le SL_N (\mathbb C)\) be a diagonal subgroup of automorphisms of \(w\). In the article [\textit{Y.-P. Lee} et al., Ann. Sci. Éc. Norm. Supér. (4) 49, No. 6, 1403--1443 (2016; Zbl 1360.14133)] the commutativity of a diagram called LG/CY square is proven. The top row vertices of this square are \(GWT_0([\mathbb C^N /G])\), the genus zero Gromov-Witten theory of \([\mathbb C^N /G]\), and \(GWT_0(\mathrm{tot}(\mathcal O_{\mathbb P(G)}(-d)))\), the genus zero Gromov-Witten theory of a partial crepant resolution of \([\mathbb C^N /G]\). The bottom row vertices are \(FJRW_0(w, G)\), the genus zero FJRW theory of the Landau-Ginzburg model given by the pair \((w, G)\), and \(GWT_0(\mathcal Z)\), the genus zero Gromov-Witten theory of a hypersurface \(\mathcal Z\) defined as the vanishing locus of \(w\) in an appropriate finite quotient of weighted projective space \(\mathbb P(G)\). The arrows in the square are the crepant transformation conjecture, quantum Serre duality, the LG/CY correspondence, and the local GW/FJRW correspondence.
The goal of the paper under review is to relate each of the above correspondences to an integral transform between appropriate derived categories, i.e. to lift the LG/CY square to the derived category to obtain a cube of relations. It is known that the crepant transformation conjecture (the top horizontal arrow of the LG/CY square) is compatible with a natural Fourier-Mukai transform. A similar result is known for the bottom horizontal arrow at least when \(G\) is cyclic.
The paper under review shows that there are natural derived functors corresponding to both of the vertical arrows of LG/CY square as well, after restricting to subcategories of \(D([\mathbb C^N /G])\) and \(D(\mathrm{tot}(\mathcal O(-d)))\) with proper support. This requires a reformulation of both the local GW/FJRW correspondence and quantum Serre duality in terms of narrow quantum \(D\)-modules, which turns out to be a more natural way of describing these correspondences.
Even though the corresponding square of derived functors does not commute in general the paper shows that the induced maps on \(K\)-theory commute. Gromov-Witten theory; FJRW theory; crepant resolution conjecture; LG/CY correspondence; wall crossing; Fourier-Mukai Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Integral transforms and quantum correspondences | 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 modules of the ring \(D_ 0\) of differential operators with power series coefficients. For \(D_ 0\)-modules, we introduce a new notion of \(F\)-Gröbner basis and present an algorithmic method to compute it. Our method is more algebraic than that of \textit{F. Castro} [in C. R. Acad. Sci., Paris, Sér. I 302, 487-490 (1986; Zbl 0606.32007) and in: Géométrie algébrique et applications, C. R. 2ième Conf. Int., La Rabida 1984, III: Géométrie Réelle. Systèmes différentielles et théorie de Hodge, Trav. Cours 24, 1-19 (1987; Zbl 0633.13009)] which is based on the Weierstrass-Hironaka division theorem. The essential point of our method consists in using a filtration of \(D_ 0\) introduced by \textit{M. Kashiwara} [``Systems of microdifferential equations'', Prog. Math. 34 (1983; Zbl 0521.58057)]. This enables us to extend some of the algorithmic methods for rings of power series to \(D_ 0\)-modules.
As applications, we can compute, in some cases, the characteristic variety, and the dimension of the space of solutions, of a system of linear differential equations via \(F\)-Gröbner bases. The relation to previously known methods is also stated. \(F\)-Gröbner basis; characteristic variety; linear differential equations Oaku, T; Shimoyama, T, A Gröbner basis method for modules over rings of differential operators, J. Symb. Comput., 18, 223-248, (1994) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Rings of differential operators (associative algebraic aspects), Abstract differential equations, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials A Gröbner basis method for modules over rings of differential operators | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give an explicit procedure which computes for degree \(d\leq 3\) the correlation functions of topological sigma model (A-model) on a projective Fano hypersurface \(X\) as homogeneous polynomials of degree \(d\) in the correlation functions of degree 1 (number of lines). We extend this formalism to the case of Calabi-Yau hypersurfaces and explain how the polynomial property is preserved. Our key tool is the construction of universal recursive formulas which express the structure constants of the quantum cohomology ring of \(X\) as weighted homogeneous polynomial functions of the constants of the Fano hypersurface with the same degree and dimension one more. We propose some conjectures about the existence and the form of the recursive laws for the structure constants of rational curves of arbitrary degree. Our recursive formulas should yield the coefficients of the hypergeometric series used in the mirror calculation. Assuming the validity of the conjectures we find the recursive laws for rational curves of degree four. small quantum cohomology rings; Fano hypersurface; Calabi-Yau hypersurfaces; recursive laws for rational curves Collino A., Jinzenji M.: On the structure of small quantum cohomology rings for projective hypersurfaces. Commun. Math. Phys. 206, 157--183 (1999) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Hypersurfaces and algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects), Quantization in field theory; cohomological methods On the structure of the small quantum cohomology rings of projective hypersurfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a scheme \(X\) let \(GW_{0}(X)\) denote the Grothendieck-Witt group of symmetric bilinear spaces over \(X\). This is the abelian group generated by isometry classes \([\mathcal V,\phi ]\) of vector bundles \(\mathcal V\) over \(X\) with a nonsingular symmetric bilinear form \(\phi: \mathcal V\otimes \mathcal V\to O_X\) subject to the relations \([(\mathcal V,\phi) \perp (\mathcal V ',\phi ' ]= [{\mathcal V},{\phi}]+[\mathcal V ',\phi ']\) and \([\mathcal M,\phi]=[\mathcal H(\mathcal N)]\) for every metabolic space \((\mathcal M, \phi)\) with Lagrangian subbundle \(\mathcal N=\mathcal N^{\perp} \subset \mathcal M\) and associated hyperbolic space \(\mathcal H(\mathcal N)\). The higher Grothendieck-Witt groups \(GW_{i}(X), \, i\in {\mathbb N}\) were defined by the author [``Higher Grothendieck-Witt groups of exact categories'', J. K-theory (to appear)].
In the paper the author proves the Mayer-Vietoris sequence for open covers (Theorem 1). This main theorem of the paper (in fact Theorem 16 of the paper is more general and includes versions for skew symmetric forms and coefficients in line bundles different than \(O_{X}\) ) is derived from the localization (Theorem 2) and Zariski excision (Theorem 3) theorems. The author proves also additivity, fibration and approximation theorems for the hermitian \(K\)-theory of exact categories with weak equivalences and duality. As the author noticed, \textit{P. Balmer} [K-Theory 23, No.~1, 15--30 (2001; Zbl 0987.19002)] and \textit{J. Hornbostel} [Topology 44, No.~3, 661--687 (2005; Zbl 1078.19004)] proved similar results to theorems 1--3. However, their assumptions are stricter than these of the author. Grothendieck-Witt groups; hermitian K-theory; Mayer-Vietoris sequence; localization; excision Berrick, J., Karoubi, M., Østvær Paul, A., Schlichting, M.: The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-Theory, arxiv:1011.4977v2 ( 2011) Hermitian \(K\)-theory, relations with \(K\)-theory of rings, \(K\)-theory of schemes, Witt groups of rings, Algebraic theory of quadratic forms; Witt groups and rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry The Mayer-Vietoris principle for Grothendieck-Witt groups of 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 \(k\) be an algebraically closed field of characteristic \(p>0\), \(W\) the ring of Witt vectors over \(k\) and \(R\) the integral closure of \(W\) in the algebraic closure \(\overline{K}\) of \(K:=\mathrm{Frac}(W)\); let moreover \(X\) be a smooth, connected and projective scheme over \(W\) and \(H\) a relatively very ample line bundle over \(X\). We prove that when \(\dim(X/W)\geq 2\) there exists an integer \(d_{0}\), depending only on \(X\), such that for any \(d\geq d_{0}\), any \(Y\in |H^{\otimes d}|\) connected and smooth over \(W\) and any \(y\in Y(W)\) the natural \(R\)-morphism of fundamental group schemes \(\pi _{1}(Y_{R},y_{R})\to \pi _{1}(X_{R},y_{R})\) is faithfully flat, \(X_{R}, Y_{R}, y_{R}\) being respectively the pull back of \(X\), \(Y\), \(y\) over \(\mathrm{Spec}(R)\). If moreover \(\dim(X/W)\geq 3\) then there exists an integer \(d_{1}\), depending only on \(X\), such that for any \(d\geq d_{1}\), any \(Y\in |H^{\otimes d}|\) connected and smooth over \(W\) and any section \(y\in Y(W)\) the morphism \(\pi _{1}(Y_{R},y_{R})\to \pi _{1}(X_{R},y_{R})\) is an isomorphism. fundamental group scheme; essentially finite vector bundles; Grothendieck-Lefschetz theorem Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Group schemes, Divisors, linear systems, invertible sheaves, Arithmetic varieties and schemes; Arakelov theory; heights On the Grothendieck-Lefschetz theorem for a family of 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 We construct an equivalence of graded Abelian categories from a category of representations of the quiver-Hecke algebra of type \(A_1^{(1)}\) to the category of equivariant perverse coherent sheaves on the nilpotent cone of type \(A\). We prove that this equivalence is weakly monoidal. This gives a representation-theoretic categorification of the preprojective K-theoretic Hall algebra considered by Schiffmann and Vasserot. Using this categorification, we compare the monoidal categorification of the quantum open unipotent cells of type \(A_1^{(1)}\) given by Kang, Kashiwara, Kim, Oh and Park in terms of quiver-Hecke algebras with the one given by Cautis and Williams in terms of equivariant perverse coherent sheaves on the affine Grassmannians. Coherent categorification of quantum loop algebras: the \(\operatorname{SL}(2)\) case | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(f\) be an invertible polynomial and \(G\) a group of diagonal symmetries of \(f\). This note shows that the orbifold Jacobian algebra Jac \((f,G)\) of \((f,G)\) defined by the authors et al. [``Orbifold Jacobian algebras for invertible polynomials'', Preprint, \url{arXiv:1608.08962}] is isomorphic as a \(\mathbb Z/2\mathbb Z\)-graded algebra to the Hochschild cohomology \(\mathsf{HH}^*(\mathrm{MF}_G(f))\) of the dg-category \(\mathrm{MF}_G(f)\) of \(G\)-equivariant matrix factorizations of \(f\) by calculating the product formula of \(\mathsf{HH}^*(\mathrm{MF}_G(f))\) given by \textit{D. Shklyarov} [``On Hochschild invariants of Landau-Ginzburg orbifolds'', Preprint, \url{arXiv:1708.06030}]. We also discuss the relation of our previous results to the categorical equivalence. Hochschild cohomology; \(G\)-equivariant matrix factorizations; \(G\)-equivariant Jacobian algebra Mirror symmetry (algebro-geometric aspects), Singularities in algebraic geometry, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) Hochschild cohomology and orbifold Jacobian algebras associated to 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 author identifies certain multiplicity spaces corresponding to the cohomology ring of a \(\text{GL}_N\) flag variety with multiplicity spaces of irreducible \(sl_n\)- modules in a fusion product. More specifically, if the \(m\)-tuple \(\mu=(\mu_1,\dots ,\mu _m)\) represents the highest weight of a symmetric power representation and \(\sum \mu _i =N\), then the cohomology ring \(R\) of the flag variety \(F_\mu\) of \(\text{GL}_N\) can be decomposed into irreducible \(S_N\) (symmetric group) modules as \(R=\bigoplus _{| \lambda| \leq n} W_\lambda\otimes M_{\lambda ,\mu }\), where \(W_\lambda\) is a Specht module and \(M_{\lambda ,\mu}\) is a graded multiplicity space. The author identifies the \(M_{\lambda ,\mu}\) with multiplicity spaces of irreducible \(sl _n\)-modules in a fusion produce under the special condition that the evaluation modules in the fusion product are symmetric power representations. fusion product; flag manifold; Kostka polynomial; cohomology ring Rinat Kedem, Fusion products of \?\?_{\?} symmetric power representations and Kostka polynomials, Quantum theory and symmetries, World Sci. Publ., Hackensack, NJ, 2004, pp. 88 -- 93. Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Fusion products, cohomology of \(\text{GL}_N\) flag manifolds, 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 \(G\) be a connected semisimple complex algebraic group, let \(V\) be a simple \(G\)-module, and let \({\mathbb P}(V)\) be the projective space of all hyperplanes in \(V\) endowed with the natural action of \(G\). The space \({\mathbb P}(V)\) contains the unique closed \(G\)-orbit \(X\). Using Weyl's dimension formula, the authors compute the Hilbert polynomial, the Hilbert series, the dimension, and the degree of \(X\), and consider several examples.
Reviewer's remark. These results are the special cases of the known results about Hilbert polynomials, degrees, and dimensions of arbitrary normal spherical varieties; see [\textit{M. Brion}, Duke Math. J. 58, No. 2, 397--424 (1989; Zbl 0701.14052); Lect. Notes Math. 1296, 177--192 (1987; Zbl 0667.58012); \textit{A. Yu. Okounkov}, Funct. Anal. Appl. 31, No. 2, 138--140 (1997); translation from Funkts. Anal. Prilozh. 31, No. 2, 82--85 (1997; Zbl 0928.14032); and also \textit{D. Panyushev}, Transform. Groups 2, No. 1, 91--115 (1997; Zbl 0891.22013)]. semisimple group; homogeneous projective variety; Hilbert polynomial Gross (B. H.); Wallach (N. R.).-- On the Hilbert polynomials and Hilbert series of homogeneous projective varieties, In: Arithmetic geometry and automorphic forms, Adv. Lect. Math. 19 p. 253-263 (2011). Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Semisimple Lie groups and their representations, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the Hilbert polynomials and Hilbert series of homogeneous projective varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A classical problem in invariant theory is to find a precise description of the subring of the coordinate ring for the \(n\times r\) matrices \(k[M_{nr}]\), \(r<n\), consisting of all the invariant elements with respect to the natural right action of the special linear group \(\text{SL}_r(k)\), where \(k\) is an algebraically closed field of characteristic \(0\). In this paper the authors present a solution for the analogous problem in the quantum case. Here \(k[M_{nr}]\) is replaced by the quantum matrix bialgebra and \(\text{SL}_r(k)\) by the quantum special linear group. This leads to the formulation of the first and second fundamental theorem for the quantum special linear group. Similarly to what happens for the commutative case, the first theorem of quantum coinvariant theory states that the ring of quantum coinvariants coincides with the ring generated by certain quantum minors in the quantum matrix bialgebra. This is precisely the ring of the so-called quantum Grassmannian. Using the results in the first author's earlier paper they are able to give a presentation of the ring of quantum coinvariants in terms of generators and relations. This is the content of the second fundamental theorem of quantum coinvariant theory. Both the first and the second theorem of quantum coinvariant theory reduce to the corresponding classical results when the indeterminate \(q\) is specialized to \(1\). At the end they use the given presentation of the quantum Grassmannian to define quantum Schubert varieties and to show that they are quantum homogeneous spaces; that is, they admit a coaction by a suitable quantum group. quantum special linear groups; quantum matrix bialgebras; rings of quantum coinvariants; quantum Grassmannians; presentations; quantum Schubert varieties; coactions R. Fioresi and C. Hacon, Quantum coinvariant theory for the quantum special linear group and quantum Schubert varieties. J. Algebra 242 (2001), 433-446. Quantum groups (quantized function algebras) and their representations, Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds, Actions of groups and semigroups; invariant theory (associative rings and algebras), Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Quantum groups (quantized enveloping algebras) and related deformations Quantum coinvariant theory for the quantum special linear group and quantum Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Second degree polynomial Heisenberg algebras are realized through the harmonic oscillator Hamiltonian, together with two deformed ladder operators chosen as the third powers of the standard annihilation and creation operators. The corresponding solutions to the Painlevé IV equation are easily found. Moreover, three different sets of eigenstates of the deformed annihilation operator are constructed, called the good, the bad and the ugly coherent states. Some physical properties of such states will be studied as well. Coherent states, Selfadjoint operator theory in quantum theory, including spectral analysis, Operator algebra methods applied to problems in quantum theory, Applications of Lie (super)algebras to physics, etc., Formal methods and deformations in algebraic geometry The good, the bad and the ugly coherent states through polynomial Heisenberg 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 See the review of the entire collection in [Zbl 1303.01004]. History of algebraic geometry, History of \(K\)-theory The influence of Alexandre Grothendieck in \(K\)-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 obtain a universal method to compute the Gromov-Witten type invariants using the localization technique. This method can be applied to any natural cohomology class on the moduli space of curves \(\overline{\mathcal M}_{g,n}\). As applications, we illustrate a new proof of Witten's conjecture as well as the proof of Mariño-Vafa formula. cohomology class; localization; Mariño-Vafa formula Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic) A new approach to deriving recursion relations for the Gromov-Witten 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 consider the smallest possible ramification. The corresponding pairs are represented by only finite set of points in the individual Hurwitz space, but the set of Riemann surfaces admitting the meromorphic functions with the smallest possible number of critical values is dense in the moduli space. Riemann surface; algebraic curves; Hurwitz space Visualizing algebraic curves: from Riemann to 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 In this paper we give a note on the relation of the positivity of polynomial matrices and their homogenizations on basic closed semi-algebraic sets. Base on this relation, we extend Putinar-Vasilescu's Positivstellensatz, in particular, Reznick's Positivstellensatz, and Dickinson-Povh's Positivstellensatz to (not necessarily homogeneous) polynomial matrices. This is a continuation of the work by \textit{C.-T. Lê} [Positivity 19, No. 3, 513--528 (2015; Zbl 1331.14056)]. Pólya's theorem; Putinar-Vasilescu's theorem; Reznick's theorem; Dickinson-Povh's theorem; polynomials matrix; Positivstellensätze Real algebraic and real-analytic geometry, Real algebra, Matrices over special rings (quaternions, finite fields, etc.), Positive matrices and their generalizations; cones of matrices A note on Positivstellsätze for matrix 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 simple method for computing the genus-zero two-point descendant Gromov-Witten invariants of a smooth proper Deligne-Mumford stack with projective coarse moduli space. Gromov-Witten invariants; Deligne-Mumford stacks A. Gholampour, H.-H. Tseng, On computations of genus 0 two-point descendant Gromov-Witten invariants. \textit{Michigan Math. J}. \textbf{62} (2013), 753-768. MR3160540 Zbl 1306.53077 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) On computations of genus zero two-point descendant Gromov-Witten 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 In this note, based on archival documents, we supplement and clarify information about the father of Alexander Grothendieck, most of which is in the biography of Grothendieck, written by W. Scharlau. We managed to establish the real name, given at birth to Alexander Petrovich Shapiro and establish with certainty that he had at least three brothers. In addition, we found out the fate of A. P. Shapiro's eldest son, David Alexandrovich, and established contact with his daughter. In the mid 90s, D. A. Shapiro dictated to her several pages of memoirs about his parents, where he told the real name of his father, grandmother, the history of acquaintance of parents and other information that he knew from the words of his mother. History of algebraic geometry, History of mathematics in the 20th century Russian trace in the family history and work of Alexander Grothendieck | 0 |
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