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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 introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a \(K\)-theoretic deformation of the quasi-key basis and also a lift of the \(K\)-analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the \(K\)-analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)].
The second new basis is the kaon basis, a \(K\)-theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis.
Throughout, we explore how the relationships among these \(K\)-analogues mirror the relationships among their cohomological counterparts. We make several ``alternating sum'' conjectures that are suggestive of Euler characteristic calculations. Demazure character; Demazure atom; Lascoux polynomial; Lascoux atom; Grothendieck polynomial; quasi-Lascoux polynomial; kaon Symmetric functions and generalizations, Classical problems, Schubert calculus, Hopf algebras and their applications, Connections of Hopf algebras with combinatorics, Grassmannians, Schubert varieties, flag manifolds Polynomials from combinatorial \(K\)-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 This paper is a continuation of [Int. J. Math. 14, No. 6, 573-617 (2003; Zbl 1083.57037)]. In that paper, the author introduced the notion of \(G\)-Frobenius algebras together with graded and super versions, where \(G\) is a finite group. The motivation was an equivariant version of 1+1 dimensional topological field theory. In the present paper, the author reviews the notions and results from his previous article. Then ``Intersection'' and Jacobian \(G\)-Frobenius algebras are studied in detail. Special \(G\)-Frobenius algebras -- those with \(\dim A_g=1\) for all \(g\in G\) -- were described in the previous paper in terms of 2-cocycles, generalizing the well-known description of strongly graded algebras. A concrete presentation is given here for \(G={\mathbf S}_n\). The last section is devoted to ``second quantized Frobenius algebras'', again here \(G={\mathbf S}_n\). twisted Frobenius algebras; intersection categories; topological field theories R. M. Kaufmann. Second quantized Frobenius algebras. \textit{Comm. Math. Phys.}, 248(1):33-- 83, 2004. Associative rings and algebras arising under various constructions, Quasi-Frobenius rings, Topological quantum field theories (aspects of differential topology), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Projective and enumerative algebraic geometry, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Graded rings and modules (associative rings and algebras) Second quantized Frobenius algebras. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper studies an ODE arising geometrically in the study of the equivariant quantum cohomology of the Hilbert scheme of points on \(\mathbb C^2\) and the Gromov-Witten/Donaldson-Thomas theory of \(\mathbb P^1\times \mathbb C^2\).
Specifically, let \(\mathcal F\) denote the Fock space with creation operators \(\alpha_{-k}\) acting on the vacum vector \(|\emptyset\rangle\), and annihilation operators killing the vacuum. The commutation relations are
\[
[\alpha_k, \alpha_l]=k\delta_{k+l}.
\]
The differential equation studied in the paper takes the form
\[
q\frac{d}{dq}\Psi=\mathsf M_{D}\Psi,\,\,\,\Psi\in \mathcal F
\]
for the differential operator
\[
\mathsf M_{D}=\frac{t_1+t_2}{2}\sum_{k>0} \left(k\frac{(-q)^k+1}{(-q)^k-1}-\frac{(-q)+1}{(-q)-1}\right)\alpha_{-k}\alpha_k +\frac{1}{2}\sum_{k, l>0} (t_1t_2\alpha_{k+l}\alpha_{-k}\alpha_{-l}-\alpha_{-k-l}\alpha_k\alpha_l).
\]
This equation has regular singularities at \(q=0\), \(q=\infty\) and certain roots of unity.
Geometrically, the Fock space can be identified via the Nakajima basis with the equivariant cohomology of the Hilbert scheme of points on the plane,
\[
\mathcal F\otimes \mathbb C[t_1, t_2]\cong \bigoplus_n H^{\star}_{T}((\mathbb C^2)^{[n]}).
\]
A natural divisor \(D\) can be constructed from the first Chern class of the tautological quotient. Classical multiplication by \(D\) is related to the Hamiltonian of the Calogero-Sutherland system. The operator \(\mathsf M_D\) corresponds to the small quantum multiplication by \(D\) in the equivariant quantum cohomology of the Hilbert scheme.
For \(q=0\), the operator \(\mathsf M_D(0)\) has as eigenvalues the Jack symmetric functions \(\mathsf J^{\lambda}\) with eigenvalues
\[
c(\lambda, t_1, t_2)=\sum_{(i,j)\in \lambda}((j-1)t_1+(i-1)t_2),
\]
where \(\lambda\) is a partition. A solution of the ODE of the form
\[
\Psi=\mathsf Y^{\lambda}(q)q^{-c(\lambda, t_1, t_2)},\,\, \mathsf Y^{\lambda}(0)=\mathsf J^{\lambda}
\]
with \(\mathsf Y^{\lambda}(q)\in \mathbb C[[q]]\) can be constructed for \(|q|<1\). The value of \(\mathsf Y^{\lambda}(q)\) for \(q=-1\) is found and related to the Macdonald polynomials. Okounkov, A.; Pandharipande, R., The quantum differential equation of the Hilbert scheme of points in the plane, Transform. Groups, 15, 965, (2010) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Relationships between surfaces, higher-dimensional varieties, and physics The quantum differential equation of the Hilbert scheme of points in the 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 Let \(l\) be a prime and fix an isomorphism \(\overline{\mathbb{Q}}_l \rightarrow \mathbb{C}\). In a first part, the authors prove the following equivariant comparison theorem:
Let \(\mathcal{O}\) be the ring of integers of a number field \(K\) and let \(n\) be a positive integer divisible by \(l\). Assume that \(\overline{X}\) is a smooth, proper scheme over \(\mathcal{O}[1/n]\) and that \(X \subset \overline{X}\) is a dense open subscheme, such that \(\overline{X} \setminus X\) is a normal crossings divisor relative to \(\mathcal{O}[1/n]\). Fix a prime \(\mathfrak{p}\) of \(\mathcal{O}[1/n]\) and an embedding \(\mathcal{O}[1/n] \hookrightarrow \mathbb{C}\). If \(k(\mathfrak {p})\) denotes the residue field of \(\mathfrak{p}\) and \(\overline{k (\mathfrak{p})}\) denotes an algebraic closure of \(k(\mathfrak{p})\), then there are canonical isomorphisms
\[
H^i(X_{\overline {k({\mathfrak p})}},\overline{\mathbb{Q}}_l) \rightarrow H^i(X_{\mathbb{C}}^{\text{an}},\mathbb{C})
\]
and
\[
H^i_c(X_{\overline {k({\mathfrak p})}}, \overline {\mathbb{Q}}_l) \rightarrow H^i_c(X_{\mathbb{C}}^{\text{an}},\mathbb{C})
\]
where \(X_{\mathbb{C}}^{\text{an}}\) is the complex analytic space attached to \(X_{\mathbb{C}}\). In particular, if a group \(G\) acts on \(X\), then these isomorphisms are \(G\)-equivariant, without assuming that the action of \(G\) extends to \(\overline{X}\).
Then, the authors show how this can be used to convert the computation of the graded character of the induced action on cohomology into questions about numbers of rational points of varieties over finite fields. This is carried through in three applications: First, for the symmetric group acting on the moduli space of \(n\) points of a genus zero curve; second, for a unitary reflection group acting on the complement of its reflecting hyperplanes; and third for the symmetric group action on the space of configurations of points in any smooth variety which satisfies certain strong purity conditions. \(\ell\)-adic cohomology; smooth scheme; action of group; equivariant comparison; action on cohomology; numbers of rational points; Poincaré polynomials Kisin, M; Lehrer, GI, Equivariant Poincaré polynomials and counting points over finite fields, J. Algebra, 247, 435-451, (2002) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), \(p\)-adic cohomology, crystalline cohomology, Finite ground fields in algebraic geometry, Rational points, Group actions on varieties or schemes (quotients), Classical real and complex (co)homology in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Equivariant Poincaré polynomials and counting points over finite 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 In this paper, the author gives a construction of the \textit{elliptic double shuffle Lie algebra} \({{\mathfrak{ds}_\mathrm{ell}}}\) that generalizes the double shuffle Lie algebra \({{\mathfrak{ds}}}\) constructed by G.Racinet to the elliptic situation. The double shuffle, or dimorphic, relations satisfied by elements of the Lie algebra \({{\mathfrak{ds}}}\) express two families of algebraic relations between multiple zeta values that conjecturally generate all relations. In analogy with this, elements of the elliptic double shuffle Lie algebra \({{\mathfrak{ds}_\mathrm{ell}}}\) are Lie polynomials whose behaviors conjecturally describe the (dual of the) set of algebraic relations between elliptic multiple zeta values (constructed explicitly in [\textit{P. Lochak} et al., Int. Math. Res. Not. 2021, No. 1, 698--756 (2021; Zbl 1486.11107)]). The major construction relies on the notion of \textit{moulds} due to J.Écalle whose basic properties are concisely summarized in Appendix of this article, where one finds that various combinatorial behaviors of Lie polynomials can be controlled in terms of certain symmetric properties of associated (sequence of) rational functions in \(\prod_r\mathbb{Q}(u_1,\dots,u_r)\). The counterparts of \({{\mathfrak{ds}}}\), \({{\mathfrak{ds}_\mathrm{ell}}}\) in the Grothendieck-Teichmüller theory are the Lie algebra \({{\mathfrak{grt}}}\) of Drinfeld and its elliptic version \({{\mathfrak{grt}_\mathrm{ell}}}\) by \textit{B. Enriquez} [Sel. Math., New Ser. 20, No. 2, 491--584 (2014; Zbl 1294.17012)] respectively, and a conjecturally isomorphic injection from \({{\mathfrak{grt}}}\) to \({{\mathfrak{ds}}}\) is established by \textit{H. Furusho} [Ann. Math. (2) 174, No. 1, 341--360 (2011; Zbl 1321.11088)]. It is shown that the tangential sections \({{\mathfrak{grt}}}\hookrightarrow{{\mathfrak{grt}_\mathrm{ell}}}\) (Enriquez) and \({{\mathfrak{ds}}}\hookrightarrow{{\mathfrak{ds}_\mathrm{ell}}}\) (Écalle) arising from the Tate elliptic curve compatibly terminate into the common target derivation algebra \(\mathrm{Der}\mathrm{Lie}[a,b]\), while constructing a compatible morphism between \({{\mathfrak{grt}_\mathrm{ell}}}\) and \({{\mathfrak{ds}_\mathrm{ell}}}\) is posed as an important open problem. double shuffle relations; multiple zeta values; elliptic associators Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Multiple Dirichlet series and zeta functions and multizeta values, Lie algebras and Lie superalgebras Elliptic double shuffle, Grothendieck-Teichmüller and mould theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper gives a very good account of the algebraic theory of Gromov-Witten invariants, developed by the author [Invent. Math. 127, 601-617 (1997; Zbl 0909.14007)] together with \textit{Y. Manin} [Duke Math. J. 85, 1-60 (1996; Zbl 0872.14019)] and \textit{B. Fantechi} [Invent. Math. 128, 45-88 (1997; Zbl 0909.14006)]. The author manages to explain very clearly sophisticated and technical issues in a highly readable style, yet keeping rigor and precision and pointing out where serious difficulties arise or why abstract geometric objects and methods are unavoidable at some places. From the expository viewpoint, the author uses only particular instances of the graphs used to define the moduli stacks, with the nice effect of making the theory simpler. The price one has to pay is that some invariants are missing but all the original Gromov-Witten invariants are still contained in the theory. The milestones of the algebraic theory of Gromov-Witten invariants, namely the virtual class of the Deligne-Mumford moduli stack (moduli stacks are necessary because they carry universal families) and the intrinsic normal cone of a Deligne-Mumford stack are defined and their geometrical meaning and usefulness are nicely explained. Gromov-Witten invariants; virtual class; intrinsic normal cone; moduli stacks of curves Behrend, K., Localization and Gromov-Witten invariants, (1999) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Algebraic 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 The Betti cohomologies \(H^\bullet_c(X)\) of compact support of a complex affine variety \(X\) come equipped with a mixed Hodge structure, therefore one can define the mixed Hodge numbers \(h^{p,q;j}_c(X) := \dim_{\mathbb{C}} \mathrm{Gr}^F_p \mathrm{Gr}_{p+q}^{W \otimes \mathbb{C}} H^j_c(X)\), and consequently the mixed Hodge polynomial \(H_c(X; x,y,t) := \sum_{p,q,j} h^{p,q;j}_c(X) x^p y^q t^j\). The \(E\)-polynomial of \(X\) is then defined as \(E(X; x,y) := H_c(X; x,y,-1)\). By spreading-out \(X\) to a scheme \(\mathcal{X}\) over a \(\mathbb{Z}\)-algebra \(R\) in a reasonable way, one says that \(X\) has polynomial count if there exists a polynomial \(P_X \in \mathbb{Z}[X]\) such that for each \(\phi: R \to \mathbb{F}_q\), where \(q\) is a power of \(p\) for all but finitely many primes \(p\), we have \(|\mathcal{X}(\mathbb{F}_q)| = P_X(q)\). It is shown by Katz (appendix to [\textit{T. Hausel} and \textit{F. Rodriguez-Villegas}, Invent. Math. 174, No. 3, 555--624 (2008; Zbl 1213.14020)]) that having polynomial count implies \(E(X; x,y) = P_X(xy)\).
In this paper, the authors consider the character varieties \(\mathfrak{X}_\Gamma(G) = \mathrm{Hom}(\Gamma, G) /\!/ G\) (the GIT quotient, \(G\) acts by conjugation on \(\mathrm{Hom}(\Gamma, G)\)) in the case where \(G = \mathrm{SL}_2(\mathbb{C})\) and \(\Gamma\) is the free group of rank \(r\). Such varieties play a prominent role in geometry. Given Katz's result alluded to above, the main idea is to study its \(E\)-polynomial by counting points over various \(\mathbb{F}_q\). Since the conjugation action of \(G\) on \(\mathrm{Hom}(\Gamma, G)\) is not free, one proceeds by stratifying \(\mathrm{Hom}(\Gamma, G)\), a careful yet elementary analysis is required. An explicit form of \(E\) is given in the Theorem B. free group; conjugacy class; character variety; finite field; \(E\)-polynomial 3. S. Cavazos and S. Lawton, E-polynomial of SL2(\mathbb{C})-character varieties of free groups, Internat. J. Math.25(6) (2014), Article ID:1450058, 27pp, arXiv:1401.0228 [arXiv] . [Abstract] genRefLink(128, 'S0129167X15501001BIB3', '000343050000007'); Group actions on varieties or schemes (quotients), Algebraic moduli problems, moduli of vector bundles, Rational points, Finite ground fields in algebraic geometry \(E\)-polynomial of \(SL_{2}(\mathbb{C})\)-character varieties of free 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 Within the last few decades, a number of well established mathematical schools focused their efforts to investigate the amazing properties of the cohomology of complex homogeneous varieties, at the point that the subject may be considered among the most developed in mathematics, once one agrees to measure achievements in terms of amount of knowledge, production of literature and rapid refinements of techniques. The subject is not easy and highly non trivial, especially because of its rich connections with many mathematical branches, like representation theory, combinatorics, algebra, mathematical physics, and algebraic geometry.
As a matter of the paper under review, this is concerned with the (small) quantum equivariant cohomology \(QK_T(X)\) of a cominuscule (generalized) flag variety \(X:=G/P\) acted on by an algebraic torus \(T\): the main result is a type-uniform Chevalley formula for multiplication of divisor classes by arbitrary classes of \(QK_T(X)\). Recall that a subgroup \(P\) of a semisimple complex algebraic group is said to be \textsl{ parabolic} if the quotient \(G/P\) is a projective variety. A (generalized) \textsl{flag manifold} is any homogenoeus projective variety of the form \(X:=G/P\), which is clearly smooth. For the convenience of those readers not belonging (yet?) to the experts' audience, recall that any parabolic subgroup contains a Borel subgroup \(B\), which in turn contains a maximal torus \(T\) of \(G\). The pair \((G,T)\) defines \textsl{ weights} (which are characters of \(T\) satisfying certain non triviality conditions) and a \textsl{ root system}. A fundamental weight \(\omega\) is said to be \textsl{ co-minuscule} if \(<\omega, \alpha^\vee>=1\), where \(\alpha\) denotes the highest root. A parabolic subgroup \(P_\omega\) of \(G\) can be thence attached to \(\omega\). A \textsl{cominuscule homogeneous space} is then a homogeneous projective variety of the form \(G/P_\omega\). Examples of cominuscule flag varieties are orthogonal grassmannians, even dimensional quadrics and exceptional varieties such as the Cayley plane or the Freudenthal variety.
The Grothendieck group \(K^0(X)\) of a smooth algebraic variety is the free module generated by vector bundles on \(X\) modulo the usual relations \(E-E'-E''\), whenever \(E',E,E''\) fit into the exact sequence \(0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0\). Its group structure is induced by the tensor product. The small quantum equivariant cohomology \(QK_T(X)\) has been introduced by \textit{A. Givental} [Mich. Math. J. 48, 295--304 (2000; Zbl 1081.14523)], and can be considered, like the authors themselves put it, as the ultimate generalization of all the cohomology theories ruled by (generalized) Schubert calculus. The equivariant quantum Grothendieck group \(QK_T(X)\) of \(T\)-equivariant vector bundles is a ring as well, and natural bases of classes of vector bundles are parametrized by elements of the Weil group of \(P\) (in case \(X\) were a Grassmannian, the basis would be parametrized by partitions).
The Chevalley formula holding in \(QK_T(X)\), the main result of the paper, is proved in Section 4. The prerequisites about cominuscule homogeneous varieties are exposed in the pretty useful Section 3, while Section 5, instead, is devoted to the appropriate emphasis of a striking fact: the Chevalley formula, i.e. multiplication with \(K\)-theoretic classes of divisors, is enough to determine the whole ring structure of \(QK_T(X)\) for an arbitrary cominuscule \(X\). The importance of this implication is that it provides the key to prove a conjecture by \textit{V. Gorbounov} and \textit{C. Korff} [Adv. Math. 313, 282--356 (2017; Zbl 1386.14181)], concerning the equivariant quantum \(K\)-theory of Grassmannians of Lie type A. The paper concludes as usual with a reference list which, in this case, is very rich, detailed and comprehensive. It is almost sufficient, upon coupling it with the very useful and well written introduction, to give, even to the readers not planning to enter into details, a rather precise idea of what is the paper about, what is going on in terms of literature framework, motivations and necessary background. quantum \(K\)-theory; Chevalley formula; Gromov-Witten invariants; Schubert structure constants; cominuscule flag varieties; Molev-Sagan equations Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Equivariant \(K\)-theory A Chevalley formula for the equivariant quantum \(K\)-theory of cominuscule 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 compute Stokes matrices and monodromy of the quantum cohomology of projective spaces. This problem can be formulated in a ``classical'' framework, as the problem of computation of Stokes matrices and monodromy of differential equations with regular and irregular singularities. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves. We also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups. Stokes matrix; monodromy; quantum cohomology; projective space D. Guzzetti, Commun. Math. Phys. 207(2), 341--383 (1999). Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) Stokes matrices and monodromy of the quantum cohomology of projective spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a complex semisimple group of adjoint type with Lie algebra \(\mathfrak{g}\) and let \(q\) be a primitive root of unity of odd order \(l\) greater than the Coxeter number of \(\mathfrak{g}\) and prime to \(3\) if \(\mathfrak{g}\) has a factor of type \(G_2\). Then \textit{V. Ginzburg} and \textit{S. Kumar} [Duke Math. J. 69, No. 1, 179--198 (1993; Zbl 0774.17013)] proved that the cohomology algebra of the small quantum group \(u_q\) corresponding to \(\mathfrak{g}\) is isomorphic to the coordinate algebra \(\mathcal{O}(\mathcal{N})\) of the nilpotent cone \(\mathcal{N}\) of \(\mathfrak{g}\). Note that every irreducible object of the category \(\mathcal{P}Coh\) of \(G\)-equivariant coherent sheaves on \(\mathcal{N}\) with respect to the middle perversity is determined by a certain pair \((O,\mathcal{L})\) where \(O\) is some \(G\)-orbit in \(\mathcal{N}\) and \(\mathcal{L}\) is some irreducible \(G\)-equivariant vector bundle on \(O\). The main result of the paper under review is a description of the \(\mathcal{O}(\mathcal{N})\)-module \(H^\bullet(u_q,T)\) for a tilting object \(T\) in the principal block \({\mathfrak U}_q \mathrm{-mod}^0\) of the category of finite-dimensional graded modules over Lusztig's quantum group \({\mathfrak U}_q\) which goes as follows. If \(T\) is an indecomposable tilting module of \({\mathfrak U}_q\mathrm{-mod}^0\), then either \(H^\bullet(u_q,T)=0\) or \(H^\bullet (u_q,T)\) is isomorphic to the total cohomology of the irreducible object of \(\mathcal{P} Coh\) corresponding to the pair \((O_T,\mathcal{L}_T)\) uniquely determined by \(T\). As a consequence the author proves a sheaf-theoretic version of a conjecture of \textit{J. E. Humphreys} [in AMS/IP Stud. Adv. Math. 4, 69--80 (1997; Zbl 0919.17013)] (verified by \textit{V. V. Ostrik} [Funct. Anal. Appl. 32, No. 4, 237--246 (1998; Zbl 0981.17010)] for type \(A\)) which states that the support of the cohomology of an indecomposable tilting module of \({\mathfrak U}_q\mathrm{-mod}^0\) as a coherent sheaf on \(\mathcal{N}\) is the closure of the nilpotent orbit corresponding to the two-sided cell in the affine Weyl group given via Lusztig's bijection. Complex semisimple group; quantized enveloping algebra at a root of unity; cohomology; tilting module; principal block; support; two-sided cell; affine Weyl group; nilpotent cone; Springer resolution; equivariant coherent sheaf; orbit; equivariant vector bundle Bezrukavnikov, Roman, Cohomology of tilting modules over quantum groups and \(t\)-structures on derived categories of coherent sheaves, Invent. Math., 166, 2, 327-357, (2006) Quantum groups (quantized enveloping algebras) and related deformations, Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, Derived categories, triangulated categories, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Cohomology of tilting modules over quantum groups and \(t\)-structures on derived categories of coherent sheaves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 reconstruction method of big quantum \(K\)-ring based on the \(q\)-difference module structure in quantum \(K\)-theory [\textit{A. Givental} and \textit{Y.-P. Lee}, Invent. Math. 151, No. 1, 193--219 (2003; Zbl 1051.14063); \textit{A. Givental} and \textit{V. Tonita}, Math. Sci. Res. Inst. Publ. 62, 43--91 (2014; Zbl 1335.19002)]. The \(q\)-difference structure yields commuting linear operators \(A_{i,com}\) on the \(K\)-group as many as the Picard number of the target manifold. The genus-zero quantum \(K\)-theory can be reconstructed from the \(q\)-difference structure at the origin \(t=0\) if the \(K\)-group is generated by a single element under the actions of \(A_{i,com}\). This method allows us to prove the convergence of the big quantum \(K\)-rings of certain manifolds, including the projective spaces and the complete flag manifold \(\mathrm{Fl}_{3}\). Iritani, H.; Milanov, T.; Tonita, V., Reconstruction and convergence in quantum \textit{K}-theory via difference equations, Int. Math. Res. Not. IMRN, 11, 2887-2937, (2015) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory in geometry, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Grassmannians, Schubert varieties, flag manifolds Reconstruction and convergence in quantum \(K\)-theory via 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 Let \(R\) be a commutative Noetherian ring. In this article, we introduce the notion of local cohomology functor \(\gamma_W\) with support in a general subset \(W\) of Spec\(\,R\), which is a natural generalization of ordinary local cohomology functors supported in specialization-closed subsets. We propose a general principle behind the local duality theorem, and report that the vanishing theorem of Grothendieck type holds for \(\gamma_W\). Local cohomology and commutative rings, Derived categories and commutative rings, Vanishing theorems in algebraic geometry Local duality principle and Grothendieck's vanishing theorem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Fix a Weil cohomology theory for smooth proper varieties over a field. Grothendieck's ``conjecture C-plus'' asserts the Künneth projectors are algebraic. His ``conjecture D'' asserts that in the group of algebraic cycles numerical equivalence agrees with homological equivalence.
Differential graded categories are regarded as ``noncommutative schemes'' by the specialists; the differential graded category associated with an ordinary scheme being the category of perfect complexes. In the noncommutative world, noncommutative analogues of Grothendieck's conjectures have been formulated.
Let \(k\) be a perfect field of characteristic \(p\).
In the article under review, the author proves that these noncommutative conjectures (for ``smooth proper \(k\)-linear dg categories'') are equivalent to the corresponding commutative conjectures (for smooth proper \(k\)-varieties). Here, one uses the rational crystalline cohomology in the conjectures C-plus and D, and the topological periodic cyclic homology in the noncommutative versions of these
conjectures.
The author gives some applications. From the abstract: ``As a first application, we prove Grothendieck's original conjectures in the new cases of linear sections of determinantal varieties. As a second application, we prove Grothendieck's (generalized) conjectures in the new cases of `low-dimensional' orbifolds. Finally, as a third application, we establish a far-reaching noncommutative generalization of Berthelot's
cohomological interpretation of the classical zeta function and of Grothendieck's conditional approach to 'half' of the Riemann hypothesis.''
Reviewer's remark: There is a citation with broken hyperlink ``[?book]'' on p. 5052 in the published version of the article. It should point to the author's book [Noncommutative motives. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1333.14002)]. noncommutative algebraic geometry; motive; crystalline cohomology Noncommutative algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology, \(K\)-theory and homology; cyclic homology and cohomology, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Chain complexes (category-theoretic aspects), dg categories A note on Grothendieck's standard conjectures of type \(\mathrm{C}^+\) and \(\mathrm{D}\) in positive characteristic | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Quantum cohomology theory can be described in general terms as intersection theory in spaces of holomorphic curves in a given Kähler or almost Kähler manifold \(X\). By quantum \(K\)-theory we may similarly understand the study of complex vector bundles over the spaces of holomorphic curves in \(X\). In these notes, we will introduce a \(K\)-theoretic version of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation which expresses the associativity constraint of the ``quantum multiplication'' operation on \(K^*(X)\).
Intersection indices of cohomology theory,
\[
\int_{[\text{space of curves}]} \omega_1\wedge\cdots\wedge \omega_k
\]
obtained by evaluation on the fundamental cycle of cup products of cohomology classes are to be replaced in \(K\)-theory by Euler characteristics
\[
\chi\;(\text{space of curves};\, V_1\otimes\cdots\otimes V_k)
\]
of tensor products of vector bundles. The hypotheses needed in the definitions of the intersection indices and Euler characteristics -- that the spaces of curves are compact and nonsingular, or that the bundles are holomorphic -- are rarely satisfied. We handle this foundational problem by restricting ourselves throughout the notes to the setting where the problem disappears. Namely, we will deal with the so-called moduli spaces \(X_{n,d}\) of degree-\(d\) genus-\(0\) stable maps to \(X\) with \(n\) marked points assuming that \(X\) is a homogeneous Kähler space. Under this hypothesis, the moduli spaces \(X_{n,d}\) are known to be compact complex orbifolds. We use their fundamental cycle \([X_{n,d}]\), well-defined over \(\mathbb{Q}\), in the definition of intersection indices, and we use sheaf cohomology in the definition of the Euler characteristic of a holomorphic orbi-bundle \(V\):
\[
\chi(X_{n,d};\, V):= \sum(-1)^k\dim H^k(X_{n,d};{\mathcal O}(V)).
\]
Alexander Givental, On the WDVV equation in quantum \?-theory, Michigan Math. J. 48 (2000), 295 -- 304. Dedicated to William Fulton on the occasion of his 60th birthday. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), \(K\)-theory of schemes, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On the WDVV equation in quantum \(K\)-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 formulate quantum field theories on an algebraic curve and outline a `paradigm' interpreting Ward identities as reciprocity laws. algebraic curves; Clifford algebra; Heisenberg algebra; quantum field theories; Ward identities; reciprocity laws L. A. Takhtajan, ``Quantum field theories on an algebraic curve'', Lett. Math. Phys., 52:1 (2000), 79 -- 91 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Algebraic functions and function fields in algebraic geometry, Relationships between algebraic curves and physics Quantum field theories on 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 We are interested in applying quantum field theory (QFT) to arithmetics, i.e., in developing QFT methods for algebraic number fields and fields of algebraic functions. On several occasions, I discussed these topics with Moshé Flato, who had deep thoughts about possible relations between QFT and arithmetics. Thus, Moshé came up with a beautiful idea to use factoring of polynomials in several variables for the spacial quantization of the Nambu bracket (called Zariski quantization), developed in our paper [\textit{G. Dito}, \textit{M. Flato}, \textit{D. Sternheimer} and \textit{L. Takhtajan}, Deformation quantization and Nambu mechanics, Comm. Math. Phys. 183, No. 1, 1-22 (1997; Zbl 0877.70012)].
See also the article version of the author [Lett. Math. Phys. 52, No. 1, 79--91 (2000; Zbl 1024.81016)]. QFT methods for algebraic number fields and fields of algebraic functions; factoring of polynomials; Nambu bracket; Zariski quantization Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Algebraic functions and function fields in algebraic geometry, Quantum field theory; related classical field theories, Relationships between algebraic curves and physics, Applications of Lie (super)algebras to physics, etc., Hamiltonian and Lagrangian mechanics, Geometry and quantization, symplectic methods Quantum field theories on 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 A 2-category was introduced in the author's paper [Adv. Math. 225, No. 6, 3327--3424 (2010; Zbl 1219.17012)] that categorifies Lusztig's integral version of quantum \(\mathrm{sl}(2)\). Here we construct for each positive integer \(N\) a representation of this 2-category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible \((N+1)\)-dimensional representation of quantum \(\mathrm{sl}(2)\). categorification; iterated flag varieties; 2-representation Lauda, Aaron D., Categorified quantum \(\mathrm sl(2)\) and equivariant cohomology of iterated flag varieties, Algebr. Represent. Theory, 14, 2, 253-282, (2011) Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Equivariant homology and cohomology in algebraic topology Categorified quantum \(\mathrm{sl}(2)\) and equivariant cohomology of iterated 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 For a \(\mathbb{P}^1\)-orbifold \(\mathcal{C}\), we prove that its big quantum cohomology is generically semisimple. As a corollary, we verify a conjecture of Dubrovin for orbi-curves. We also show that the small quantum cohomology of \(\mathcal{C}\) is generically semisimple iff \(\mathcal{C}\) is Fano, i.e. it has positive orbifold Euler characteristic. quantum cohomology; orbi-curve; Dubrovin's conjecture Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On semisimplicity of quantum cohomology of \(\mathbb{P}^1\)-orbifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors give a proof of the genus 0 Virasoro conjecture. The Virasoro conjecture (due to \textit{T. Eguchi, K. Hori} and \textit{C.-S. Xiong} [Phys. Lett. B 402, No. 1-2, 71-80 (1997; Zbl 0933.81050)] and Katz [unpublished]) is an infinite series of conjectural relations among the Gromov-Witten invariants (and their gravitational descendants) of a projective manifold \(V\). The relations are given in an appealing form. One first packages all the Gromov-Witten invariants and their descendants into a generating function \(Z\) (the ``partition function'') and then defines a sequence of differential operators \(L_{-1}\), \(L_{0}\), \(L_{1}\), \(L_{2}\),\dots which conjecturally annihilate \(Z\). The coefficients of \(L_{i}Z=0\) are relations among the invariants, for example \(L_{-1}Z=0\) is equivalent to the string equation while \(L_{0}Z=0\) is a combination of the selection rule, the divisor equation, and the dilaton equation. Moreover, the \(L_{i}\)'s define a representation of the Virasoro algebra with central charge \(\chi (V)\), the Euler characteristic of the manifold \(V\). In other words, they satisfy the relation
\[
[L_{m},L_{n}]= (m-n)L_{m+n} + \delta _{m,-n}\frac{m (m^{2}-1)}{12}\chi (V).
\]
If one writes \((L_{n}Z)/Z\) as a Laurent series in \(\lambda \) (the formal variable indexing genus), then one finds that the \(\lambda ^{-2}\) term only involves the genus 0 invariants. Thus this term gives a series of conjectural relations among the genus 0 invariants (and their descendants) that is collectively referred to as the ``genus 0 Virasoro conjecture'' and is proved in this paper. The author's proof of this conjecture does not require \(V\) to be Fano (as Eguchi-Hori-Xiong [loc. cit.] assume) but only that \(V\) has only even cohomology (and they point out that even this assumption is not essential).
For a general exposition of the Virasoro conjectures and an independent proof of the genus 0 Virasoro conjecture, see the paper by \textit{E. Getzler} [Contemp. Math. 241, 147-176 (1999)]. quantum cohomology; Virasoro conjecture; gravitational descendants Liu, X; Tian, G, Virasoro constraints for quantum cohomology, J. Differential Geom., 50, 537-590, (1998) Spaces and manifolds of mappings (including nonlinear versions of 46Exx), Projective and enumerative algebraic geometry, Deformations of analytic structures Virasoro constraints for 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 No review copy delivered. Jin, L.; Xing, C., Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes, IEEE Trans. Inf. Theory, 58, 5484-5489, (2012) Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry, Quantum coding (general) Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(X\subset\mathbb{P}^N\) be a complex projective variety of dimension \(n\), and let \(Y=\mathbb{P}^{N-1}\cap X\) a hyperplane section such that \(U= X\setminus Y\) is smooth and \(n\)-dimensional. Then the classical Lefschetz hyperplane theorem asserts that the restriction morphism \(H^k(X,\mathbb{Z})\to H^k(Y,\mathbb{Z})\) is an isomorphism for \(k< n\) and is still injective for \(k= n-1\).
In the framework of quantum cohomology, an analoguous principle has been suggested by \textit{A. B. Givental} [Int. Math. Res. Not. 1996, No. 13, 613--663 (1996; Zbl 0881.55006)] and mathematically formalized by \textit{B. Kim} [Acta Math. 183, No. 1, 71--99 (1999; Zbl 1023.14028)], where the so-called quantum Lefschetz hyperplane section principle was formulated for complete intersections in a generalized flag variety. In this setting, the quantum Lefschetz hyperplane theorem is basically equivalent to the mirror theorem for this class of varieties, and a generalization of the quantum Lefschetz hyperplane theorem to a wider class of varieties would therefore imply a generalization of the mirror theorem to that class, too.
In the paper under review, the author provides a generalization of the quantum Lefschetz hyperplane theorem to arbitrary complete intersections \(Y\) in a smooth complex projective variety \(X\) and therefore a generalized mirror theorem as well.
More precisely, let \(X\) be a smooth projective variety embedded in \(\prod^N_{i=1} \mathbb{P}^{r_i}= P\), and let \(E\) be a vector bundle on \(X\) which is decomposed into a direct sum of pull-backs of convex and concave line bundles on \(P\) such that some nonnegativity condition for its first Chern class holds. Then there are two formal functions \(J^E_X\) and \(I^E_X\) with values in the cohomology ring of \(X\) and the author's first main theorem states that these two functions are equivalent up to mirror transformation. The second main theorem asserts under which conditions on the bundle \(E\) these two generating genus zero functions of Gromov-Witten invariants, \(J^E_X\) and \(I^E_X\), actually coincide.
From these two main results, whose proof is based on equivariant Gromov-Witten theory and virtual localization techniques, the authors deduce a fundamental relationship between the Gromov-Witten invariants of the variety \(X\) and those of a complete intersection \(Y\) in \(X\). This relationship shows that enumerative information on \(Y\) can be derived from the one on \(X\), and that in a way generalizing the quantum Lefschetz hyperplane theorem à la Givental-Kim.
The significance of the author's generalized approach is demonstrated by the fact that some of the already established versions of the mirror theorem can be rediscovered in this context.
As to the reconstruction theorems for Gromov-Witten invariants used in the course of the paper, the authors rely on separate recent results by \textit{Y.-P. Lee} and \textit{R. Pandharipande} [Am. J. Math. 126, No. 6, 1367--1379 (2004; Zbl 1080.14065)]. Gromov-Witten invariants; quantum cohomology; mirror symmetry; mirror conjecture; complete intersections; Lefschetz principle Y.-P. Lee, Quantum Lefschetz hyperplane theorem. \textit{Invent. Math}. \textbf{145} (2001), 121-149. MR1839288 Zbl 1082.14056 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Complete intersections, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Grassmannians, Schubert varieties, flag manifolds, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Quantum Lefschetz hyperplane theorem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The calculations of Betti numbers is the first step to understand the Chow or the cohomology rings of spaces. \textit{E. Getzler} and \textit{R. Pandharipande} [J. Algebr. Geom. 15, No. 4, 709--732 (2006; Zbl 1114.14032)] computed the Betti numbers of the moduli spaces of genus 0 stable maps to projective spaces. They used the equivariant Serre characteristics and the stratification of the moduli spaces indexed by trees weighted by automorphisms. Their strata are described as quotients of fibered products of vertex moduli spaces by the group of tree automorphisms. \textit{K. Behrend} and \textit{A. O'Halloran} [Invent. Math. 154, No. 2, 385--450 (2003; Zbl 1092.14019)] used the Betti number calculations for the calculation of the cohomology rings of these moduli spaces.
Bialynicki-Birula developed a powerful method in 70's, which enables one to compute Betti numbers of a space with torus action by means of the information from the fixed loci. \textit{D. Oprea} [Adv. Math. 207, No. 2, 661--690 (2006; Zbl 1117.14056)] used this idea to find a ``cell-decomposition'' of the moduli stack of the stable maps to projective spaces. The paper under review uses this to compute the Betti numbers \(\overline{M}_{0,0}(G(k,n),d)\) the moduli stack of zero-pointed genus 0 stable maps to the Grassmannian \(G(k,n)\) for \(d=2\) and \(3\). stable maps; Poincaré polynomial; torus action; cell decomposition Martín, A L, Poincaré polynomials of stable map spaces to Grassmannians, 193-208, (2014) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Poincaré polynomials of stable map spaces to 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 The authors show that when \(X\) is a facet symmetric smooth symplectic toric Fano manifold, then for any toric symplectic form \(\omega\), the small quantum cohomology \(QH^{0}(X,\omega)\) is semisimple over the field \(\mathbb{K}^{\uparrow}\) of generalized Laurent series which is defined as
\[
\mathbb{K}^{\uparrow}=\{ \sum_{\lambda\in \mathbb{R}} a_{\lambda}s^{\lambda} | a_{\lambda}\in \mathbb{C} \mathrm{ and } \{ \lambda | a_{\lambda}\neq 0 \} \mathrm{ is discrete and bounded below in } \mathbb{R} \}.
\]
quantum cohomology; Fano manifolds; superpotential function Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 outlines applications of completely integrable systems and inverse scattering to quantization problems from an algebraic point of view. The classical spectral curve method, Lax representation and methods of Hitchin are introduced and applications to the Gaudin model are sketched. The remainder of the article discusses applications in quantization, quantum inverse scattering, and quantum integrable systems. inverse scattering; Lax representation; Hitchin system Talalaev, Dmitry V., Quantum spectral curve method, Geometry and Quantization, Trav. Math., 19, 203-271, (2011), University Luxembourg, Luxembourg Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems, Relationships between algebraic curves and integrable systems, Deformation quantization, star products, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Quantum spectral curve 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 Topological strings on toric Calabi-Yau threefolds can be defined non-perturbatively in terms of a non-interacting Fermi gas of \(N\) particles. Using this approach, we propose a definition of quantum mirror curves as quantum distributions on phase space. The quantum distribution is obtained as the Wigner transform of the reduced density matrix of the Fermi gas. We show that the classical mirror geometry emerges in the strongly coupled, large \(N\) limit in which \(\hbar \sim N\). In this limit, the Fermi gas has effectively zero temperature, and the Wigner distribution becomes sharply supported on the interior of the classical mirror curve. The quantum fluctuations around the classical limit turn out to be captured by an improved version of the universal scaling form of \textit{N. L. Balazs} and \textit{G. G. Zipfel jun.}, ``Quantum oscillations in the semiclassical fermion \(\mu\)-space density'', [Ann. Phys. 77, No. 1--2, 139--156 (1973; \url{doi:10.1016/0003-4916(73)90412-0})]. matrix models; topological strings String and superstring theories in gravitational theory, Topological field theories in quantum mechanics, Calabi-Yau manifolds (algebro-geometric aspects) Quantum curves as quantum distributions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review deals with the quantum invariance of genus zero Gromov-Witten invariants, up to analytic continuation under ordinary flops \(f\) from \(X\) to \(X'\) over a non-trivial smooth base. Such flops form the building blocks to connect birational minimal models. This plays an important role in string theory, and also comparing various birational minimal models in higher-dimensional algebraic geometry. The local geometry is encoded in a triple \((S, F, F')\) where \(S\) is a smooth variety and \(F, F'\) are two rank \(r + 1\) vector bundles over \(S\). If \(Z \subset X\) is the \(f\)-exceptional loci, then \(Z\cong \mathbb P(F) \to S\) with fibers spanned by the flopped curves and \(N_{Z/X} = F'\otimes \mathcal O_Z(-1)\), and similar structure holds for the exceptional loci \(Z'\subset X'\). The most studied case of Atiyah flop corresponds to \(S=\) pt and \(r = 1\). An earlier work of the authors of the paper under review constructs a canonical correspondence between the quantum cohomologies of \(X, X'\) by means of the graph closure
\[
\mathcal F = [\overline{\Gamma}_f]_* : QH(X) \to QH(X')
\]
when \(S=\) pt, in which the crucial idea is to interpret \(\mathcal F\)-invariance in terms of analytic continuations in Gromov-Witten theory.
The main result of the paper under reivew is the generalization the quantum invariance above under ordinary flops over a smooth base. The main results of this paper determines the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. Various reductions to the local models are performed in the way by means of degeneration techniques, WDVV equations, topological recursion relations, and divisorial reconstructions. quantum cohomology; ordinary flops; analytic continuations; degeneration formula; reconstructions Lee, Yuan-Pin; Lin, Hui-Wen; Wang, Chin-Lung, Invariance of quantum rings under ordinary flops I: Quantum corrections and reduction to local models, Algebr. Geom., 3, 5, 578-614, (2016) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays) Invariance of quantum rings under ordinary flops. I: Quantum corrections and reduction to local models | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 study of modules over elliptic algebras, especially modules of Gelfand-Kirillov dimension 2. An elliptic algebra \(A\) is associated with a certain automorphism of a one-dimensional scheme \(E\), generally an elliptic curve, and every elliptic algebra defines a `non-commutative projective plane' proj-\(A\), sometimes called a quantum plane. Therefore, the study of modules translates into an interplay between the geometries of \(E\) and of quantum planes. The relation to geometry is studied by looking at `the points of a given module \(M\)' and corresponding `incidence relations' (a point \(p\) of \(E\) is said to be a point of \(M\) if there is a non-zero map from \(M\) to the corresponding point module \(N_p\)). non-commutative projective planes; point modules; line modules; Cohen-Macaulay modules; modules over elliptic algebras; Gelfand-Kirillov dimension; one-dimensional schemes; elliptic curves; quantum planes Ajitabh, K, Modules over elliptic algebras and quantum planes, Proc. London Math. Soc., 72, 567-587, (1996) Graded rings and modules (associative rings and algebras), Noncommutative algebraic geometry, Linear incidence geometry, Quantum groups (quantized enveloping algebras) and related deformations, Elliptic curves, Deformations of associative rings, Noncommutative topology, Noncommutative differential geometry Modules over elliptic algebras and quantum 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 See the review of the original hardback edition in [Zbl 1211.14003]. See the review of Volume I in [Zbl 1051.01015]. selected works; algebraic geometry; collections of reprinted articles; scientific correspondences; unpublished papers History of algebraic geometry, Collected or selected works; reprintings or translations of classics, Collections of reprinted articles, History of mathematics in the 20th century, Biographies, obituaries, personalia, bibliographies, Schools of mathematics Selected papers II. On algebraic geometry, including correspondence with Grothendieck. Edited by Ching-Li Chai, Amnon Neeman and Takahiro Shiota | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 establish explicit isomorphisms between three realizations of the quantum twisted affine algebra \(U_{q}(A_{2}^{(2)}\): the Drinfeld current realization, the Chevalley realization, and the so-called \(RLL\) realization proposed by Reshetikhin, Takhtajan, and Faddeev. quantum affine algebra; \(RLL\) realization Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Spinor and twistor methods applied to problems in quantum theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Three realizations of the quantum affine algebra \(U_{q}(A_{2}^{(2)}\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Quantum \(\mathbb P^n\)s are regular algebras of global dimension \(n+1\), and are considered to be noncommutative analogs of the polynomial ring in \(n+1\) variables. The quantum \(\mathbb P^2\)s have been classified by \textit{M. Artin} et al. [Prog. Math. 86, 33--85 (1990; Zbl 0744.14024)] by means of point schemes. However, the classification of quantum \(\mathbb P^3\)s is still unknown. Motivated by the classifcation of quantum \(\mathbb P^2\)s, the authors propose that the identification of the point schemes and line schemes that arise from quantum \(\mathbb P^3\)s should be the first step toward completing the classification of quantum \(\mathbb P^3\)s. Note that most regular algebras of global dimension \(4\) are quadratic, so attention is restricted to quadratic quantum \(\mathbb P^3\)s, ie. quadratic, Noetherian, AS-regular algebras with Hilbert series \((1-t)^{-4}\).
The current article computes the line schemes of a specific family of quadratic quantum \(\mathbb P^3\)s in order to supplement a lack of examples in the current literature and to validate certain predictions on the generic structure of quadratic quantum \(\mathbb P^3\)s. Van den Bergh has proved that generically quadratic quantum \(\mathbb P^3\)s have a point scheme consisting of twenty distinct points and a one-dimensional line scheme. Furthermore, explicit calculation of the line schemes of a certain family of quadratic quantum \(\mathbb P^3\)s lead \textit{R. G. Chandler} and the second author [J. Algebra 439, 316--333 (2015; Zbl 1348.14005)] to conjecture that the line scheme of a generic quadratic quantum \(\mathbb P^3\) consists of a union of two degree-four spatial elliptic curves and four planar elliptic curves. New in this article, the authors provide support for this conjecture by computing the line schemes of a \(1\)-parameter family of algebras \(\mathcal A(\alpha)\) which are quadratic quantum \(\mathbb P^3\)s. While the line schemes obtained from this family are not unions of elliptic curves as conjectured, they are in fact degenerations of the conjectured collections of planar and spatial elliptic curves. As such, the \(\mathcal A(\alpha)\) may reasonably be expected to be not generic but rather limits of families of generic curves. In this way, the conjecture is supported by the results of the paper. However, further calculation of certain line schemes of other families of quadratic quantum \(\mathbb P^3\)s have led the authors to modify the conjecture to include the possibility of four degree-four spatial elliptic curves and two nonsingular conics. The updated conjecture appears as Conjecture 4.3.
The algebras \(\mathcal A(\alpha)\) are specific cases of the generalized graded Clifford algebras constructed by Cassidy and Vancliff as possible generic quantum \(\mathbb P^3\)s. \(\mathcal A(\alpha)\) is presented explicitly as a quotient of the free algebra in four variables by six quadratic relations. The point scheme of \(\mathcal A(\alpha)\) is computed following the method of Artin et al. [loc. cit.] and shown to consist of twenty distinct points. The line scheme is computed following the method originally introduced by \textit{B. Shelton} and the second author [Commun. Algebra 30, No. 5, 2535--2552 (2002; Zbl 1056.14002)], which identifies it with the zero set of fourty-five quartic polynomials and one quadratic polynomial in six variables. Mathematica is used to compute these polynomials along with a Gröbner basis. Using this, the authors identify the line scheme with a union of eight irreducible curves in \(\mathbb P^5\): two lines, two nonsingular conics, two planar elliptic curves, one spatial elliptic curve, and one spatial rational curve with a singular point. The singular rational curve is viewed as a degeneration of a spatial elliptic curve, while each line + conic pair is viewed as a degeneration of a planar elliptic curve. Additionally, the authors investigate the intersection points of the irreducible components of the line scheme as well as the lines in the line scheme which contains points of the point scheme. In particular, they show that four points of the point scheme lie on infinitely many points of the line scheme, while the remaining \(16\) points of the point scheme lie on exactly six distinct lines of the line scheme, counting multiplicity. line scheme; point scheme; elliptic curve; regular algebra; Plücker coordinates Tomlin, D., Vancliff, M.: The one-dimensional line scheme of a family of quadratic quantum \({\mathbb{P}}^{3}\)s (2017) \textbf{(preprint)}. arXiv:1705.10426 Noncommutative algebraic geometry, Quadratic and Koszul algebras, Rings arising from noncommutative algebraic geometry The one-dimensional line scheme of a family of quadratic quantum \(\mathbb{P}^3\)s | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 how to associate to any polynomial \(P\), of degree \(d\), with non-negative integer coefficients and constant term equal to 1, a pair of elements \(y_P\) and \(w_P\) in the symmetric group \(S_n\) where \(n=d+P(1)+1\). Then he proves that \(P\) is indeed the Kazhdan-Lusztig polynomial of those two elements, by reducing the problem to the case when \(P-1\) is a monomial and by using intersection cohomology of Schubert varieties. Kazhdan-Lusztig polynomials; Schubert varieties; intersection cohomology P. Polo, Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups, \textit{Repre-} \textit{sent. Theory}, 3 (1999), 90--104.Zbl 0968.14029 MR 1698201 Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Representations of finite symmetric groups, Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric 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 Hall algebra \(\mathbf H_X\) of the category of coherent sheaves on an elliptic curve \(X\) defined over a finite field \(\mathbb F_l\) contains a natural `spherical' subalgebra \(\mathbf U_X^+\) which is a two-parameter deformation of the ring of diagonal invariants
\[
\mathbf R_n^+=\mathbb C[x_1,\dots,x_n,y_1^{\pm 1},\dots,y_n^{\pm 1}]^{\Sigma_n},
\]
where \(\Sigma_n\) acts simultaneously on the \(x\)-variables and the \(y\)-variables. For any \(n\geq 1\) the authors construct a surjective algebra homomorphism between the Drinfeld double \(\mathbf{DU}_X^+\) and the spherical subalgebra of Cherednik's double affine Hecke algebra of type \(\text{GL}_n\). This leads to a geometric construction of the Macdonald polynomials \(P_\lambda(q,t^{-1})\) in terms of certain Eisenstein series on the moduli space of semistable vector bundles on \(X\). elliptic curves; finite fields; Hall algebras; Cherednik Hecke algebras; Macdonald polynomials; double affine Hecke algebras Schiffmann (O.), and Vasserot (E.).â The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compos. Math. 147, p. 188-234 (2011). Hecke algebras and their representations, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Quantum groups (quantized enveloping algebras) and related deformations The elliptic Hall algebra, Cherednik Hecke algebras and 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 Let \(R\) be a discrete valuation ring with infinite residue field and \(X\) a smooth projective curve over \(R\). Let \(\mathbf{G}\) be a simple simply-connected group scheme over \(R\) and \(E\) a principal \(\mathbf{G}\)-bundle over \(X\). It is proved that \(E\) is trivial locally for the Zariski topology on \(X\) providing \(E\) is trivial over the generic point of \(X\). The main aim of the present paper is to develop a method rather than to get a very strong concrete result. Group schemes, Perfectoid spaces and mixed characteristic, Representation theory for linear algebraic groups, Linear algebraic groups over adèles and other rings and schemes Notes on a Grothendieck-Serre conjecture in mixed characteristic 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 The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of small quantum cohomology of complex Grassmannians are studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequence counting non-coalescing Grassmannians, and the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function. Grassmannians, Schubert varieties, flag manifolds, Distribution of primes, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), \(\zeta (s)\) and \(L(s, \chi)\) Coalescence phenomenon of quantum cohomology of Grassmannians and the distribution of prime numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{Z. Qin} and \textit{Y. Ruan} [Trans. Am. Math. Soc., 350, 3615--3638 (1998; Zbl 0932.14030)] introduced interesting techniques for the computation of the quantum ring of manifolds which are projectivized bundles over projective spaces. In this work we consider more examples of projectivised vector bundles on projective spaces and we give some applications. V. Ancona and M. Maggesi, ``On the quantum cohomology of Fano bundles over projective spaces'' in The Fano Conference (Turin, 2002) , University of Turin, Turin, 2004, 81-98. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fano varieties On the quantum cohomology of Fano bundles over projective 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 authors study certain questions on the polynomiality of orbifold Gromov-Witten invariants of the root stack \(X_{D,r}\). Among other things, they obtain a bound for the degree of this polynomial, and they show that if there are no maps with higher genus components of the source curve mapping into divisor \(D\), then the relative and orbifold invariants coincide for sufficiently large \(r\). Gromov-Witten theory; virtual localization; degeneration; moduli space Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On the polynomiality of orbifold Gromov-Witten theory of root stacks | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Review of [Zbl 1211.14003]. External book reviews, History of algebraic geometry, Collected or selected works; reprintings or translations of classics, Collections of reprinted articles, History of mathematics in the 20th century, Biographies, obituaries, personalia, bibliographies, Schools of mathematics Book review of: David Mumford, Selected papers, Volume II. On algebraic geometry, including correspondence with 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 Thom polynomials express invariants of singularities of a general map \(f:X\to Y\) between complex analytic manifolds in terms of invariants of \(X\) and \(Y\). Knowing the Thom polynomial of a singularity \(\eta\) one can compute the cohomology classes represented by \(\eta\)-points of \(f\). One of the most successful methods to compute Thom polynomials is the method of restriction equations developed mainly by Rimanyi which converts the problem to an algebraic one. Pragacz, in collaboration with Lascoux, combined the method with techniques of Schur functions and, with Weber, established that the coefficients of Schur function expansions of the Thom polynomials of stable singularities are nonnegative. In the paper under review, the author studies from this point of view the structure of the Thom polynomials for \(A_4(-)\) singularities. The Schur function expansions of these polynomials are analysed and it is shown that partitions indexing the Schur function expansions of Thom polynomials for \(A_4(-)\) singularities have at most four parts. The system of equations that determines these polynomials is simplified and a recursive description of Thom polynomials for \(A_4(-)\) singularities is given. Also, the author gives Thom polynomials for \(A_4(3)\) and \(A_4(4)\) singularities. Thom polynomials; singularities; global singularity theory; classes of degeneracy loci; Schur functions; resultants Symmetric functions and generalizations, Enumerative problems (combinatorial problems) in algebraic geometry, Singularities of differentiable mappings in differential topology On Thom polynomials for \(A_4(-)\) via Schur 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 We provide a group-theoretic realization of two-parameter quantum toroidal algebras using finite subgroups of \(\text{SL}_2(\mathbb{C})\) via McKay correspondence. In particular our construction contains the vertex representation of the two-parameter quantum affine algebras of ADE types as special subalgebras. two-parameter quantum affine algebra; finite groups; wreath products; McKay correspondence DOI: 10.1090/S0002-9947-2011-05284-0 Quantum groups (quantized enveloping algebras) and related deformations, McKay correspondence, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Two-parameter quantum vertex representations via finite groups and the McKay correspondence | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials On a symplectic manifold or a complex projective variety \(X\), the quantum cohomology ring is an extension of the ordinary cohomology ring \(H^*(X)\) which comes in small and big versions. The former is defined via 3-point genus 0 Gromov-Witten (GW) invariants. The latter is more complicated (it is a family of quantum cup products parametrized by \(H^*(X)\)) and contains all the genus 0 GW invariants. Alternatively, one can realize the big quantum cohomology as a flat connection on the tangent bundle of \(H^*(X)\). On toric varieties, the equivariant structure lifts to the (big) quantum cohomology ring and the result is called the equivariant quantum cohomology.
Following a (mirror symmetry) proposal of \textit{A. Givental} [Prog. Math. 160, 141--175 (1998; Zbl 0936.14031); in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 472--480 (1995; Zbl 0863.14021)] and \textit{K. Hori}-\textit{C. Vafa} [``Mirror symmetry'', preprint, \url{arXiv:hep-th/0002222}], this paper identifies the big equivariant quantum cohomology connection of a toric variety \(X\) with the Gauss-Manin connection of a so called ``universal mirror Landau-Ginzburg potential function'' on \((C^*)^D\), where \(D\) is the number of primitive generators of the toric fan \(\Sigma_X\) of \(X\). The former is known as the A-side and the latter is known the B-side.
If \(X\) is a compact ``semipositive'' toric variety, Givental's mirror theorem identifies the small equivariant quantum cohomology of \(X\) with the Jacoby ring of the function
\[
F_\lambda(x)=Q^{\beta_1} x^{b_1} + \ldots+ Q^{\beta_m} x^{b_m}+ \sum_{i=1}^D \lambda_i \log(x_i), \qquad x\in (\mathbb{C}^*)^D
\]
where \(b_1,\ldots,b_m \in \mathbb{Z}^D\) are the primitive generators of the toric fan \(\Sigma_X\), \(x^{b_i}\) are the associated monomials, \(\beta_i=\beta(b_i)\in H_2(X,\mathbb{Z})\) are certain curve classes, and \(\lambda_i\) are the equivariant parameters of the torus \(T\cong (\mathbb{C}^*)^D\). Alternatively, his theorem identifies the small equivariant quantum connection with the Gauss-Main connection of the twisted de Rham cohomology \(H^D(\Omega^\bullet_{(\mathbb{C}^*)^D}[z],zd+ dF_\lambda \wedge )\).
Theorem 1.1 of the present paper (which generalizes the non-equivariant mirror theorem of Barannikov-Douai-Sabbah over weighted projective spaces) establishes a similar duality between the big equivariant cohomology of a smooth semi-projective (so not necessarily compact) toric variety \(X\) (and its associated flat connection) and the associated B-model of the ``universal Landau-Ginzburg function''
\[
F_\lambda(x;y)=\sum_{k} y_k Q^{\beta(k)} x^k - \lambda \cdot \log(x),
\]
where the sum is taken over all the lattice points in the support of \(\Sigma_X\), \(\beta(k)\) are as before on \(b_i\) and satisfy \(\beta(k+l)=\beta(k)+\beta(l)\), and \(y\) is an infinite set of parameters that under mirror symmetry correspond to the parameter \(\tau\in H^*(X)\) of the big quantum cohomology. Theorem 1.2 characterizes the mirror map \(y \to y(\tau)\) as a solution to the PDE
\[
\frac{\partial \tau}{\partial y_k} = S_k (\tau),
\]
where \(S_k\) is the Seidel element associated to the \(\mathbb{C}^*\)-action of \(k\). An important question left to be answered is whether the inverse mirror map can be described as a generating function of certain open GW invariants as in the works of \textit{C.-H. Cho} and \textit{Y.-G. Oh} [Asian J. Math. 10, No. 4, 773--814 (2006; Zbl 1130.53055)] and Fukaya-Oh-Ohta-Ono [\textit{K. Fukaya} et al., Duke Math. J. 151, No. 1, 23--175 (2010; Zbl 1190.53078); Sel. Math., New Ser. 17, No. 3, 609--711 (2011; Zbl 1234.53023); Lagrangian Floer theory and mirror symmetry on compact toric manifolds. Paris: Société Mathématique de France (SMF) (2016; Zbl 1344.53001)]. H. Iritani, \textit{A mirror construction for the big equivariant quantum cohomology of toric manifolds}, arXiv:1503.02919 [INSPIRE]. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Mirror symmetry (algebro-geometric aspects), Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category A mirror construction for the big equivariant quantum cohomology of toric 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 In the paper under review the author uses the methods of Harder and Narasimhan from the theory of moduli of vector bundles to investigate the moduli of representations of a quiver.
This is done by applying the notions of stability and of the Harder-Narasimhan filtration to the representation varieties of quivers. In this way, he constructs the Harder-Narasimhan stratification of the spaces of representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is obtained inside the quantized enveloping algebra of a Kac-Moody algebra. This yields a resolution of the Harder-Narasimhan system.
As an application, the cohomology of quiver moduli is explicitly computed. This generalizes some results on the cohomology of quotients in the literature. moduli space; quantum group; Harder-Narasimhan system; Hall algebra Reineke, M., The Harder-narasimham system in quantum groups and cohomology of quiver moduli, \textit{Invent. Math.}, 152, 349-368, (2003) Quantum groups (quantized enveloping algebras) and related deformations, Representations of quivers and partially ordered sets, Geometric invariant theory The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is the text of a talk given by the author in the ``automorphic semester'' held at the Centre Émile Borel at the Institut Henri Poincaré in which the author described his work (some of it with A.J. de Jong) on deformations of \(p\)-divisible groups and their Newton polygons. These results appeared in [J. Am. Math. Soc. 13, 209--241 (2000; Zbl 0954.14007)], [Ann. Math. (2) 152, 183--206, (2000; Zbl 0991.14016)] and [Prog. Math. 195, 417--440 (2001; Zbl 1086.14037)].
The main result, originally conjectured by A. Grothendieck in 1970, is that if \(G_0\) is a \(p\)-divisible group over a field \(K\) of characteristic \(p, \beta = {\mathcal N} (G_0)\) is the Newton polygon of \(G_0\) and \(\gamma\) is a Newton polygon below \(\beta\) (in the sense that every point of \(\gamma\) is on or below \(\beta\)) then there is a deformation \(G_\eta\) of \(G_0\) for which \({\mathcal N}(G_\eta) = \gamma\). There is also an analog of this result for principally polarized \(p\)-divisible groups of abelian varieties which has as a consequence a conjecture of Manin on realizing symmetric Newton polygons by abelian varieties. The author sketches some of the techniques needed to prove these results and concludes with several conjectures. An addendum (Nov. 2004) indicates that several of these are now theorems. abelian varieties; Barsotti-Tate groups; moduli spaces; Grothendieck conjecture Oort, F.: Newton polygons and p-divisible groups: a conjecture by Grothendieck. Automorphic forms I, Astérisque, No. 298, pp. 255--269. Société Mathématique de France, Paris (2005) Formal groups, \(p\)-divisible groups, Finite ground fields in algebraic geometry, Arithmetic ground fields for abelian varieties Newton polygons and \(p\)-divisible groups: a conjecture by 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 We construct geometric lifts of the Bost-Connes algebra to Grothendieck rings and to the associated assembler categories and spectra, as well as to certain categories of Nori motives. These categorifications are related to the integral Bost-Connes algebra via suitable Euler characteristic type maps and zeta functions, and in the motivic case via fiber functors. We also discuss aspects of \(\mathbb{F}_1\)-geometry, in the framework of torifications, that fit into this general setting. Geometry over the field with one element, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Algebraic cycles Bost-Connes systems and \(\mathbb{F}_1\)-structures in Grothendieck rings, spectra, and Nori motives | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(R\) be a discrete valuation ring and \(K\) its fraction field. Given an Abelian variety \(A_K\), we denote by \(A_K'\) its dual, by \(A,A'\) the associated Néron models and by \(\varphi,\varphi'\) their component groups. \textit{A. Grothendieck} defines [in: Sémin. Géométrie Algébrique, SGA 7 I, Exp. 9, Lect. Notes Math. 288, 313-523 (1972; Zbl 0248.14006), 1.2] a pairing \(\langle\cdot, \cdot\rangle: \varphi \times\varphi' \to\mathbb{Q}/ \mathbb{Z}\) which measures the obstruction to extending the Poincaré bundle on \(A_K\times A_K'\) as a biextension of \(A\times A'\) by \(\mathbb{G}_m\). Grothendieck's pairing is always a perfect duality for semistable Abelian varieties [\textit{A. Werner}, J. Reine Angew. Math. 486, 207-217 (1997; Zbl 0872.14037)]. For more general Abelian varieties, it is conjecturally a perfect duality if the residue field is perfect and indeed almost all has been proved in that direction, except for the case of equal positive characteristic and infinite residue field. If the residue field is not perfect, counterexamples to the perfectness of the pairing can be found [cf. \textit{A. Bertapelle} and \textit{S. Bosch}, J. Algebr. Geom. 9, 155-164 (2000; Zbl 0978.14044)].
In the present paper we prove the perfectness of Grothendieck's pairing on the \(l\)-parts of component groups when \(l\) is prime to the residue characteristic. This is sketched by \textit{A. Grothendieck} (loc. cit.; 11.3), but, as far as we know, no complete proof is to be found in the literature. Grotendieck establishes the perfectness of a similar pairing leaving it up to the reader to relate the two pairings. We show that they are equivalent up to sign. component groups of an abelian variety; discrete valuation ring; Grothendieck's pairing Bertapelle, A.: On perfectness of Grothendieck's pairing for \(l\)-parts of component groups. J. Reine Angew. Math. \textbf{538}, 223-236 (2001) Arithmetic ground fields for abelian varieties, Algebraic theory of abelian varieties, Abelian varieties of dimension \(> 1\), Local ground fields in algebraic geometry On perfectness of Grothendieck's pairing for the \(l\)-parts of component 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 articles of this volume will be reviewed individually. Quantum cohomology; Summer school; Lectures; CIME; Cetraro (Italy) Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to differential geometry, Proceedings, conferences, collections, etc. pertaining to quantum theory, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Quantization in field theory; cohomological methods, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology. Lectures given at the CIME summer school, Cetraro, Italy, June 30--July 8, 1997 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 specialize Möller's algorithm to lattices by using an additive monoid structure for the lattice and by considering the application of Möller's algorithm to linear codes. The complexity issue is discussed. Gröbner bases; Möller algorithm; lattices; label codes; group codes Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Applications to coding theory and cryptography of arithmetic geometry, Linear codes (general theory) Computing Gröbner bases associated with 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 \({\mathcal M}_{0,n}\) denote the moduli space of Riemann spheres with \(n\) ordered marked points. In this article we define the group \(\text{Out}_n^\#\) of quasi-special symmetric outer automorphisms of the algebraic fundamental group \(\widehat\pi_1({\mathcal M}_{0,n})\) for all \(n\geq 4\) to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of \(\widehat\pi_1 ({\mathcal M}_{0, n})\) and commuting with the group of outer automorphisms of \(\widehat\pi_1({\mathcal M}_{0,n})\) obtained by permuting the marked points. Our main result states that \(\text{Out}_n^\#\) is isomorphic to the Grothendieck-Teichmüller group \(\widehat {GT}\) for all \(n\geq 5\). braid group; moduli space of Riemann spheres; outer automorphisms; algebraic fundamental group; Grothendieck-Teichmüller group D. Harbater and L. Schneps: Fundamental groups of moduli and the Grothendieck--Teichmüller group , Trans. Amer. Math. Soc. 352 (2000), 3117--3148. JSTOR: Homotopy theory and fundamental groups in algebraic geometry, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Families, moduli of curves (algebraic), Fundamental groups and their automorphisms (group-theoretic aspects) Fundamental groups of moduli 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 We give a formula for the number of rational points of projective algebraic curves defined over a finite field, and a bound ``à la Weil'' for connected ones. More precisely, we give the characteristic polynomials of the Frobenius endomorphism on the étale \(\ell\)-adic cohomology groups of the curve. Finally, as an analogue of Artin's holomorphy conjecture, we prove that, if \(Y \to X\) is a finite flat morphism between two varieties over a finite field, then the characteristic polynomial of the Frobenius morphism on \(H_c^i(X,\mathbb Q_{\ell})\) divides that of \(H_c^i(Y,\mathbb Q_{\ell})\) for any \(i\). We are then enable to give an estimate for the number of rational points in a flat covering of curves. algebraic curve; finite field; rational point; zeta function Y. Aubry and M. Perret, \textit{On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields}, Finite Fields Appl., 10 (2004), pp. 412--431. Finite ground fields in algebraic geometry, Curves over finite and local fields, Arithmetic ground fields for curves, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite 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 One problem with quantum algebras is that one does not have a nice ``geometric object'' associated to them. The author proposes a definition of quantum algebras at roots of unity and---using this definition---constructs an associated geometric object, namely a (non-classical) \textit{Zariski geometry}. Zariski geometries were introduced by \textit{E. Hrushovski} and \textit{B. Zilber} [J. Am. Math. Soc. 9, No. 1, 1--56 (1996; Zbl 0843.03020)]. A Zariski geometry is a set \(X\) together with a topology on each cartesian power \(X^n\) satisfying some axioms inspired by the usual Zariski topologies; in particular, for any algebraically closed field \(K\) and any algebraic set \(V\) over \(K\), \(V(K)\) is a Zariski geometry. However, there are also Zariski geometries not coming from any algebraic set (called ``non-classical'').
Let me quickly give an idea of how a quantum algebra yields a Zariski geometry. If \(A\) is a commutative algebra, then the corresponding space \(\operatorname{Max} A\) parametrizes the one-dimensional representations of \(A\) (\(m \in \operatorname{Max} A\) yields the representation \(A/m\)). If \(A\) is non-commutative, we start with \(V := \operatorname{Max} Z_0\), where \(Z_0\) is the center of \(A\) and we consider a suitable family of representations \(M_m\) of \(A\) (of fixed dimension), parametrized by \(m \in V\). As in the commutative case, we would like the union \(\tilde{V} := \coprod_{m \in V} M_m\) to be a vector bundle; however, it is only a vector bundle ``up to automorphisms of \(M_m\)''. This structure can adequately be described by topologies on cartesian powers of \(\tilde{V}\). The main part of the article then consists in verifying that these topologies satisfy the axioms required by Zariski structures.
One question which arises is: is the used definition of a quantum algebra general enough? The author shows that several classical examples of quantum algebras satisfy his definition (including \(U_\epsilon(\mathfrak{sl}_2)\)); however, for more complicated quantum algebras, the question is left open.
Other results of the article are that Zariski structures obtained from quantum algebras have some additional nice model theoretical properties (they are \(\aleph_1\)-categorical and model complete), and that all Zariski geometries coming from the examples treated in the article are non-classical. quantum algebra at roots of unity; non-commutative geometry; Zariski geometry Zilber, A class of quantum zariski geometries, in: Model theory with applications to algebra and analysis I (2008) Models of other mathematical theories, Model-theoretic algebra, Noncommutative algebraic geometry, Quantum groups and related algebraic methods applied to problems in quantum theory A class of quantum Zariski geometries | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe the monodromy representation of a sum \(f+ g\) of two polynomials \(f\) and \(g\) in disjoint sets of variables in terms of the monodromy representations of \(f\) and \(g\). Complete results are obtained under the assumption that the bifurcation set of \(g\) is a one-point set. Thom-Sebastiani; monodromy representation A. Dimca and A. Némethi, Thom-Sebastiani construction and monodromy of polynomials, Tr. Mat. Inst. Steklova 238 (2002), no. Monodromiya v Zadachakh Algebr. Geom. i Differ. Uravn., 106 -- 123; English transl., Proc. Steklov Inst. Math. 3(238) (2002), 97 -- 114. Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Fibrations, degenerations in algebraic geometry Thom-Sebastiani construction and monodromy of 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 Quantum cohomology and \(K\)-theory algebras of geometrically relevant spaces, for example Nakajima quiver varieties, are important objects of equivariant geometry. The paper under review defines a version of such quantum \(K\)-theory algebra, not based on the customary approach of counting stable maps, but based on counting quasi-maps. With this approach the authors define some distinguished classes (``quantum tautological classes''), and a deformed product operation.
Motivated by relations to quantum integrable systems the authors consider certain functions, called partition functions. In fact these partition functions come in two flavors, and an operator mapping one to the other (capping operator) satisfies quantum difference equations of the style of qKZ equations.
The first main result of the paper is that -- for quiver varieties with discrete fixed point set -- the eigenvalues of the quantum product by quantum tautological classes can be expressed via a certain asymptotic of the vertex function.
The second main result concern the special case of the cotangent bundle of a partial flag variety of type A. In this case the \(K\)-theory algebra can be identified with the Hilbert space of a quantum integrable system called XXZ model. Using this correspondence the authors calculate the vertex functions, and -- through the correspondence mentioned above -- obtain formulas for the eigenvalues of the quantum multiplication operators. The interesting degeneration to the compact 0-section of the cotangent bundle is given in detail.
The third main topic of the paper concerns the further special case of cotangent bundle over the full flag variety. The paper presents theorems similar in spirit to Giventhal-Kim theorems in quantum cohomology. Namely, the quantum \(K\)-theory algebra is presented as the algebra of functions on a Lagrangian subvariety of the phase space of an integrable model, called trigonometric Ruijsenaars-Schneider model. Again, the interesting degeneration to the 0-section is spelled out in detail, where now the Lagrangian variety lives in the phase space of the relativistic Toda lattice. quantum \(K\)-theory of quiver varieties; quantum tautological classes; spectrum of the quantum multiplication; asymptotic of vertex functions Equivariant \(K\)-theory, Classical groups (algebro-geometric aspects), Representations of quivers and partially ordered sets, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Quantum groups (quantized enveloping algebras) and related deformations, Exactly solvable models; Bethe ansatz, Supersymmetric field theories in quantum mechanics Quantum \(K\)-theory of quiver varieties and many-body systems | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The modular invariant \(j^{\mathrm{qt}}\) of quantum tori is defined as a discontinuous, \(\mathrm{PGL}_{2}(\mathbb Z)\)-invariant multi-valued map of \(\mathbb R\). For \(\theta \in \mathbb Q\) it is shown that \(j^{\mathrm qt}(\theta) = \infty\). For quadratic irrationalities, experiments conducted with the PARI/GP computer algebra system suggest that \(j^{\mathrm{qt}}(\theta)\) is a finite set. In the case of the golden mean \(\varphi\), we produce explicit formulas for the experimental supremum and infimum of \(j^{\mathrm{qt}}(\varphi)\) involving weighted versions of the Rogers-Ramanujan functions. Finally, we define a universal modular invariant as a continuous and single-valued map of ultrasolenoids from which \(j^{\mathrm{qt}}\) as well as the classical modular invariant of elliptic curves may be recovered as subquotients. Castaño Bernard, C.; Gendron, T.M., Modular invariant of quantum tori, proc, Lond. Math. Soc., 109, 1014-1049, (2014) Complex multiplication and moduli of abelian varieties, Elliptic curves, Foliations in differential topology; geometric theory Modular invariant of quantum tori | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book contains lecture notes from the special year on enumerative geometry and its interaction with theoretical physics held at the Mittag-Leffler Institute in 1996. Part I of the book contains introductory lectures on the moduli space of stable maps, Gromov-Witten invariants, the definition of quantum cohomology and applications to enumerative geometry. Parts II and III of the book contain lectures on special topics related to quantum cohomology. The first part of this book follows the paper [\textit{W. Fulton} and \textit{R. Pandharipande}, Notes on stable maps and quantum cohomology; Proc. Symp. Pure Math. 62, Part 2, 45-96 (1997; Zbl 0898.14018), see also alg-geom/9608011] closely. The lectures in this book approach the subject from the point of view of algebraic geometry. Thus stable maps into projective algebra schemes are used in place of pseudo-holomorphic curves into symplectic manifolds and the Chow ring is used in place of cohomology. For an introduction to quantum cohomology based on the later notions one should confer to [\textit{D. McDuff} and \textit{D. Salamon}, \(J\)-holomorphic curves and quantum cohomology, University Lecture Series 6, Am. Math. Soc. (1994; Zbl 0809.53002)]. Quantum cohomology; quantum homology; enumerative geometry; theoretical physics; moduli space of stable maps; Gromov-Witten invariants; Chow ring P. Aluffi ed., \textit{Quantum cohomology at the Mittag-Leffler institute}, Edizioni della Normale, Italy, (1998). Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Proceedings, conferences, collections, etc. pertaining to quantum theory, Collections of articles of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Relationships between surfaces, higher-dimensional varieties, and physics, Quantization in field theory; cohomological methods Quantum cohomology at the Mittag-Leffler Institute | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(R\) be a discrete valuation ring with field of fractions \(K\). Let \(A_K\) be an abelian variety over \(K\) with dual variety \(A_K'\). Then, with regard to their respective Néron models \(A\) and \(A'\), there is a bilinear pairing of the component groups of these Néron models
\[
\langle,\rangle: \Phi_A \times \Phi_{A'} \to\mathbb{Q}/ \mathbb{Z},
\]
which has been introduced by \textit{A. Grothendieck} [SGA 7, I, Lect. Notes Math., 288, Springer Verlag (1972; Zbl 0237.00013)] thirty years ago. Grothendieck himself conjectured that this pairing is perfect. In the meantime, Grothendieck's conjecture has been proved in various special cases, but also series of counter-examples in other cases have been found. In the paper under review, the authors investigate the case of the Jacobian \(J_K\) of a smooth proper curve \(X_K\) admitting a \(K\)-rational point. Their main result is an explicit formula for the pairing \(\langle,\rangle\), which is then used to prove Grothendieck's conjecture for certain types of such Jacobians, on the one hand, and disprove it for particular Jacobians in the case where the residue field of the ground ring \(R\) is imperfect. The method of proof is purely geometric and based on the invention of a certain pairing attached to a symmetric matrix, which leads to a more practical description of Grothendieck's pairing. Many explicit examples, at the end of the paper, demonstrate the power of the authors' approach. abelian varieties over arithmetic ground fields; Néron models; Jacobians over arithmetic ground fields Bosch, S., Lorenzini, D.: Grothendieck's pairing on component groups of Jacobians. Invent. Math. \textbf{148}(2), 353-396 (2002) Arithmetic ground fields for abelian varieties, Abelian varieties of dimension \(> 1\), Jacobians, Prym varieties Grothendieck's pairing on component groups of Jacobians. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \((R, \mathfrak {m})\) be a noetherian local ring with maximal ideal \(\mathfrak{m}, G = \bigoplus_{n\geq 0}G_n\) a standard graded algebra with \(D := \dim(\text{Proj}(G))\) and \(I\) a \(R\)-submodule of \(G_1\). We use the degree of Rees polynomial of \(I\) to show that coefficient modules exist in general case for standard graded algebras and for \(R\)-submodules \(E\) of \(R^p\) over a noetherian local ring \(R\). coefficient modules; Rees polynomials Multiplicity theory and related topics, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Dimension theory, depth, related commutative rings (catenary, etc.) Coefficient modules and Rees polynomials of arbitrary modules | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\Delta\) be the set of all quadruples \((\nu, \mu, d, e)\) of integers that satisfy \[0 \leq \frac{D - d}{2} \leq \nu \leq \mu \leq D - d \leq D,\] \[e + d + D \text{ is even }, \quad |e| \leq 2 \nu - D + d,\] \[d \in \{e + D - 2 \nu, \min \{D - \mu, e + D - 2 \nu + 2(N - 2 D) \} \}.\] In [\textit{P. Terwilliger}, J. Algebr. Comb. 2, No. 2, 177--210 (1993; Zbl 0785.05091)], it is shown for the Terwilliger algebra \(T\) of the Grassmann scheme \(J_q(N, D)\), \(N \geq 2 D\), that the isomorphism classes of irreducible \(T\)-modules \(W\) are determined by their endpoint \(\nu\), dual endpoint \(\mu\), diameter \(d\), and auxiliary parameter \(e\), which come from the Leonard system attached to \(W\), and it is claimed without proof that the quadruples \((\nu, \mu, d, e)\) belong to \(\Delta\), if \(d \geq 1\). Let \(\Lambda\) be the set of triples \((\alpha, \beta, \rho)\) of non-negative integers that satisfy \[0 \leq \alpha \leq \frac{D - \rho}{2},\] \[0 \leq \beta \leq \frac{N - D - \rho}{2},\] \[0 \leq \alpha + \beta \leq D - \rho.\] We construct a mapping from \(\Lambda\) to \(\Delta\) which is bijective if \(N > 2 D\) and \(2 : 1\) if \(N = 2 D\). We show that the set \(\Lambda\) naturally parameterizes the isomorphism classes of irreducible \(T\)-modules, by embedding the standard module of \(J_q(N, D)\) in a bigger space that allows a \(U_{\sqrt{q}}(\hat{\mathfrak{sl}}_2)\)-module structure. As a byproduct we have the following: for a fixed \(\rho\), \(0 \leq \rho \leq D\), set \(N^\prime = N - 2 \rho\), \(D^\prime = D - \rho\), and \(\Lambda_\rho = \{(\alpha, \beta) |(\alpha, \beta, \rho) \in \Lambda\} \). Then \(\Lambda_\rho\) is precisely the set that parameterizes the isomorphism classes of irreducible \(T\)-modules for the Johnson scheme \(J(N^\prime, D^\prime)\). Grassmann scheme; Terwilliger algebra; Leonard system; quantum affine algebra Association schemes, strongly regular graphs, Combinatorial aspects of representation theory, Quantum groups (quantized function algebras) and their representations, Connections of hypergeometric functions with groups and algebras, and related topics, Grassmannians, Schubert varieties, flag manifolds The Terwilliger algebra of the Grassmann scheme \(J_q(N,D)\) revisited from the viewpoint of the quantum affine algebra \(U_q(\hat{\mathfrak{sl}}_2)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors exhibit a relation between the chiral complex of a lattice vertex algebra and the equivariant signature of the loop space, related to the Ochanine-Witten elliptic genus. chiral complex; lattice vertex algebra; Ochanine-Witten elliptic genus Malikov F. and Schechtman V. (2003). Deformations of vertex algebras, quantum cohomology of toric varieties and elliptic genus. Commun. Math. Phys. 234(1): 77--100 Elliptic genera, Vertex operators; vertex operator algebras and related structures, Toric varieties, Newton polyhedra, Okounkov bodies, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects) Deformations of vertex algebras, quantum cohomology of toric varieties, and elliptic genus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Given \(\mathcal{Y}\) a non-compact manifold or orbifold, we define a natural subspace of the cohomology of \(\mathcal{Y}\) called the narrow cohomology. We show that despite \(\mathcal{Y}\) being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The narrow cohomology proves useful for the study of genus zero Gromov-Witten theory. When \(\mathcal{Y}\) is a smooth complex variety or Deligne-Mumford stack, one can define a quantum \(D\)-module on the narrow cohomology of \(\mathcal{Y}\). This yields a new formulation of quantum Serre duality. Gromov-Witten theory; quantum \(D\)-modules; quantum Serre duality; mirror symmetry Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Narrow quantum \(D\)-modules and quantum Serre duality | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We obtain Weyl type asymptotics for the quantised derivative \(\textit{\text{đ}} f\) of a function \(f\) from the homgeneous Sobolev space \(\dot{W}^1_d(\mathbb{R}^d)\) on \(\mathbb{R}^d.\) The asymptotic coefficient \(\|\nabla f\|_{L_d(\mathbb R^d)}\) is equivalent to the norm of \(\textit{\text{đ}} f\) in the principal ideal \(\mathcal{L}_{d,\infty },\) thus, providing a non-asymptotic, uniform bound on the spectrum of \(\textit{\text{đ}} f.\) Our methods are based on the \(C^{\ast } \)-algebraic notion of the principal symbol mapping on \(\mathbb{R}^d\), as developed recently by the last two authors and collaborators. Sobolev space; quantized derivative; spectral triples Noncommutative topology, Noncommutative function spaces, Operator ideals, Parametrization (Chow and Hilbert schemes) Asymptotics of singular values for quantum derivatives | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 will demonstrate how calculations in toric geometry can be used to compute quantum corrections to the relations in the chiral ring for certain gauge theories. We focus on the gauge theory of the del Pezzo 2, and derive the chiral ring relations and quantum deformations to the vacuum moduli space using Affleck-Dine-Seiberg superpotential arguments. Then we calculate the versal deformation to the corresponding toric geometry using a method due to Altmann, and show that the result is equivalent to the deformation calculated using gauge theory. In an appendix we will apply this technique to a few other examples. This is a new method for understanding the infrared dynamics of certain quiver gauge theories. S. Pinansky. Quantum deformations from toric geometry. \textit{Journal of High Energy Physics} (2006), 055, 26~pp. (electronic). String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Toric varieties, Newton polyhedra, Okounkov bodies, Supersymmetric field theories in quantum mechanics, Rational and ruled surfaces Quantum deformations from toric 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 In this paper, several branches of expository accounts are given on nonabelian Galois action, Grothendieck-Teich{\-}müller theory and anabelian geomety from the activities of GTEM -- European Network ``\textbf{G}alois \textbf{T}heory and \textbf{E}xplicit \textbf{M}ethod''. Researches within this network are listed as (1) (Explicit) finite Galois groups over \(\mathbb Q\). (2) The Inverse Galois Problem (IGP). (3) Dessins d'enfants. (4) Grothendieck-Teichmüller Theory: \(\roman{Gal}_{\mathbb Q}\) and \(\roman{GT}\). (5) Arithmetic of elliptic curves over number fields. (6) Algorithms in number theory, in particular class field theory. (7) Differential Galois Theory (8) Arithmetic of covers, arithmetic fundamental groups. (9) (Birational) anabelian geometry. (10) Miscellaneous: Iwasawa Theory, Invariant Theory, explicit implementations,... The author selected several topics mainly concerned with the ground field of the rational numbers, giving overviews on those subjects, positioning new results, and supplying bibliographical informations. arithmetic fundamental group; moduli space of curves; Galois group over Q Galois theory, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Curves over finite and local fields, Inverse Galois theory, Coverings of curves, fundamental group The Grothendieck-Teichmüller group and Galois theory of the rational numbers -- European network GTEM | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(\mathcal O_q(M(n))\) be the coordinate ring of quantum \(n\times n\) matrices and let \(\mathcal O_q(\text{GL}(n))\) be the coordinate ring of quantum general linear group which is obtained from \(\mathcal O_q(M(n))\) by inverting the quantum determinant. There is an adjoint coaction of the Hopf algebra \(\mathcal O_q(M(n))\) on \(\mathcal O_q(M(n))\). Denote by \(C\) the subalgebra of coinvariants. The main result of the article under review is that \(\mathcal O_q(M(n))\) is a free graded left \(C\)-module. A similar result is also obtained for another quantization of the coordinate ring of \(n\times n\) matrices, namely the so-called reflection equation algebra. These results can be seen as quantum analogues of a classical theorem of \textit{B. Kostant} [Am. J. Math. 85, 327-404 (1963; Zbl 0124.26802)]. quantum groups; coordinate rings of quantum matrices; algebras of coinvariants; invariant theory; reflection equation algebras; harmonic polynomials Aizenbud, A.; Yacobi, O., A quantum analogue of kostant's theorem for the general linear group, J. Algebra, 343, 183-194, (2011) Ring-theoretic aspects of quantum groups, Quantum groups (quantized enveloping algebras) and related deformations, Geometric invariant theory, Quantum groups and related algebraic methods applied to problems in quantum theory A quantum analogue of Kostant's theorem for the general linear 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 In his Ph. D. thesis ``A Formula for the Gromov-Witten Invariants of Toric Varieties'' (Université Louis Pasteur, Strasbourg, 1999; Zbl 0964.14045), the author of the present paper has recently proven an explicit combinatorial formula for the genus-zero Gromov-Witten invariants of a smooth projective toric variety \(X\) with respect to the distinguished cohomology class \(\beta = 1\) in \(H^0(\overline{\mathcal M}_{0,m},\mathbb{Q})\), where \(\overline{\mathcal M}_{0,m}\) denotes the Deligne-Mumford compactification of the moduli space of genus-zero curves with \(m\) marked points.
In the paper under review, the author continues this study by deriving a similar formula for the case where the class \(\beta\) is the maximal product of Chern classes of cotangent lines to the marked points, that is for those classes that are Poincaré dual to a finite number of points in the moduli space \(\overline{M}_{0,m}\). At the end of the paper, this new formula is illustrated by explicitely exhibiting it for the special toric variety \(\mathbb{P}_{\mathbb{P}^1}({\mathcal O} \oplus(r-2)\oplus {\mathcal O}(1)\oplus {\mathcal O}(1))\). This is then used to compute the quantum cohomology ring of that special manifold, which provides an alternative approach to the one carried out earlier by \textit{V. V. Batyrev} in [Astérisque 218, 9-34 (1993; Zbl 0806.14041)]. Gromov-Witten invariants; quantum cohomology; toric varieties; symplectic manifolds; moduli spaces of stable curves; cohomology classes Spielberg, H.: Multiple quantum products in toric varieties, Int. J. Math. math. Sci. 31, No. 11, 675-686 (2002) Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Multiple quantum products 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 Since the appearance of quantum groups, there were several attempts to provide ``geometric'' or ``topological'' constructions of these mathematical objects. The book under review belongs to this trend; it can be considered as a continuation of [\textit{V. Schechtman} and \textit{A. Varchenko}, Quantum groups and homology of local systems, in: Algebraic geometry and analytic geometry, ICM-90 Satell. Conf. Proc., 182-197 (1991; Zbl 0760.17014)].
Let \(A\) be a finite Cartan matrix, let \(l\) be a positive integer satisfying suitable technical hypothesis with respect to \(A\); let \({\mathbf k}\) be a field such that char \({\mathbf k}\) is not divisible by \(l\), and contains a primitive \(l\)-th root of 1 \(\zeta\). Let \({\mathfrak u}\) be the small quantum group (also called Frobenius-Lusztig kernel by some authors) attached to \(A\), \({\mathbf k}\), \(\zeta\). There is a triangular decomposition \({\mathfrak u}= {\mathfrak u}^+ {\mathfrak u}^0{\mathfrak u}^-\); \({\mathfrak u}^0\) is the group algebra of the group-like elements of \({\mathfrak u}\). Let \(\mathcal C\) be the category of finite dimensional \({\mathfrak u}\) where \({\mathfrak u}^0\) ``acts by powers of \(\zeta\)''. By results of Kazhdan, Lusztig and the authors, \(\mathcal C\) is a rigid balanced tensor category, or ribbon category in the terminology of Turaev.
The main result of this book is the identification of the ribbon category \(\mathcal C\) with a category arising from the topology of configuration spaces. Specifically, the authors introduce the notion of ``factorizable sheaf''; this is a compatible collection of perverse sheaves over configuration spaces, in a suitable sense. The category \(\mathcal{FS}\) of factorizable sheaves has a structure of tensor category and the authors show that \(\mathcal{FS}\) is isomorphic to \(\mathcal{C}\) as tensor category. Then they offer global versions of this result, placing sheaves into points of an arbitrary algebraic curve. braided categories; quantum groups; configuration spaces R. Bezrukavnikov, M. Finkelberg, V. Schechtman, Factorizable Sheaves and Quantum Groups, Lecture Notes in Math., Vol. \textbf{1691} (1998). Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Research exposition (monographs, survey articles) pertaining to quantum theory, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Intersection homology and cohomology in algebraic topology, Families, moduli of curves (algebraic), Universal enveloping (super)algebras, Categories of spans/cospans, relations, or partial maps, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Factorizable sheaves and quantum groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A quasischeme is, by definition, a Grothendieck category, and the objects of such a category are called modules. A subscheme \(Y\) of a scheme \(X\) is then a full subcategory, and \(Y\) is said to be closed when it is closed under subquotients and the inclusion functor \(i_*\colon X\subseteq Y\) has a left adjoint \(i^*\). If, in addition, \(i_*\) has a right adjoint \(i^!\), then the subscheme is said to be biclosed. The author considers a biclosed subscheme \(Y\) of a quasischeme \(X\) which is a regularly embedded hypersurface in the sense that there is an automorphism \(\sigma\) of \(Y\) such that the derived functor \(R^1i^!\) is naturally isomorphic to \(\sigma^{-1}\circ i^*\). A pure curve module is then defined as a module in \(X\) of Krull dimension 1 such that \(i^!C=0\). When every proper quotient of \(C\) is of finite length, it is said to be irreducible.
This general framework was considered by \textit{S. P. Smith} and \textit{J. J. Zhang} [Algebr. Represent. Theory 1, No. 4, 311-351 (1998; Zbl 0947.16029)]. A pure curve module \(C\) is called well positioned (Definition 3.0.2) if (1) \(i^*C\) is of finite length, (2) all simple quotients of \(C\) are objects of \(Y\), and (3) given two elements \(p,q\) in the set \([i^*C]\) of simple subquotients of \(i^*C\) and \(n\in \mathbb{Z}\), the integers, then \(p\cong q^{\sigma^n}\) if and only if both \(n=0\) and \(p\neq q\). An \(Y\)-multistrand (Definition 3.2.4) is a pure curve module \(C\) such that the module \(C/C_3\) is a direct sum of woven modules \(B^1,\dots,B^n\) such that for \(p\in[B^j]\) and \(q\in[B^k]\) with \(j\neq k\), then \(\text{Ext}_Y^1(\sigma^m(p),\sigma^l(q))=0\) for all integers \(m,l\in\mathbb{Z}\). Here, \(C_3\) is a term of the sequence recursively defined as \(C_0=C\), and \(C_n\) as the kernel of the counit map \(C_{n-1}\to i^*C_{n-1}\).
The notion of a woven module is introduced in this Definition 3.2.1: they are the finite length modules \(A\) with pairwise nonisomorphic simple subquotients such that every subquotient is either indecomposable or semisimple. Woven modules are characterized in terms of the pattern of existence of certain nonsplit short exact sequences between the simples appearing in each two consecutive levels in the Loewy series of \(A\). The precise statement (Definition 3.1.3 and Remark 3.2.2) is given in terms of a binary relation defined in \([A]\) called pointing, and its associated graph, called skeleton.
The first substantial results appear in Section 5, where the smallest set \({\mathbf H}_C\) of modules in \(X\) containing a well-positioned, irreducible \(Y\)-multistrand module \(C\) and closed under submodules and nonsplit extensions by simple modules in \(Y\) is indexed by a subset \(\Lambda\) of \(\mathbb{Z}^l\) (\(l\) denotes the number of subquotients of \(i^*C\)). The set \(\Lambda\) is completely determined by the skeleton of \(C/C_3\) (Definition 5.1.1). This indexing maps each \(\alpha\in\Lambda_C\) onto an irreducible \(Y\)-multistrand module \(C(\alpha)\) in such a way that the skeleton of \(i^*C(\alpha)\) is completely determined by that of \(i^*C\) together with the automorphism \(\sigma\) and \(\alpha\) itself (Proposition 5.2.1). Some other nice properties of this indexing are also proved.
The author also considers the curve category defined by \(C\), namely, the full subcategory \({\mathcal C}_C\) of \(X\) whose objects are all subquotients of direct sums of modules in \({\mathbf H}_C\). It is a closed subscheme of \(X\), and, as a Grothendieck category, \({\mathcal C}_C\) is locally Noetherian. In fact, Section 6 investigates an appropriate subset of \({\mathbf H}_C\) which turns out to be a set of Noetherian generators of \({\mathcal C}_C\) of projective dimension at most 1 (Lemmas 6.2.2 and 6.3.5, Corollary 6.3.6).
Finally, the author connects the abstract theory with the research program that conceives noncommutative projective algebraic geometry as the study of (noncommutative) graded rings by replacing the projective variety by a suitable Grothendieck category. His main theorem in this setting is Theorem 7.3.4, which asserts that if the irreducible multistrand module \(C\) is such that most nontrivial quotients have \(n\) nonzero woven summands, then \({\mathcal C}_C\) is equivalent to a quotient category of the category of graded modules over an appropriate \(I\)-algebra. The localizing subcategory (in the sense of \textit{P. Gabriel} [Bull. Soc. Math. Fr. 90, 323-448 (1962; Zbl 0201.35602)]) is that of graded modules of Krull dimension less or equal than \(n-2\). All these results have a concrete start point, coming from the line modules over a Sklyanin-type algebra.
From the abstract: ``\dots this extension allows examination of the category created from a line module over more general AS-regular rings than those considered by Smith and Zhang. For instance, suppose that \(C\) is a generic line module over \(R_d\), Stafford's Sklyanin-like algebra. [\dots] Then \({\mathcal C}_C\) is equivalent to the category of graded \(k[x,y]/(x^2-y^2)\) modules under the \(\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\)-grading where \(\deg(x)=(-1,0)\) and \(\deg(y)=(-1,1)\).'' Sklyanin algebras; Grothendieck categories; noncommutative curves; noncommutative projective geometry; graded rings; full subcategories; categories of graded modules; Krull dimension; non-commutative schemes; quasi-schemes; quasi-coherent sheaves Rings arising from noncommutative algebraic geometry, Grothendieck categories, Module categories in associative algebras, Noncommutative algebraic geometry, Homological dimension in associative algebras, Graded rings and modules (associative rings and algebras) Curves in Grothendieck 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 The article is devoted to weighted projective lines. For them mutations of derived categories are studied. For this purpose bounded derived categories of coherent sheaves are investigated. A review of previous works in this area is given. Applications to Lie and quantum algebras are described. Among them Ringel-Hall Lie algebras, the Kac-Moody algebra, Tits' automorphisms, Lustig's symmetries are scrutinized. derived category; weighted projective line; Kac-Moody algebra; quantum enveloping algebra Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Representations of quivers and partially ordered sets, Quantum groups (quantized enveloping algebras) and related deformations, Derived categories, triangulated categories, Graded Lie (super)algebras, Special properties of functors (faithful, full, etc.) Applications of mutations in the derived categories of weighted projective lines to Lie and 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 Let \(A=\mathbb{C} [[x_ 1,\dots,x_ n]]\) be the formal power series ring in \(n\) variables over the complex numbers, and \(G\) a finite abelian group acting linearly and faithfully on \(A\). The aim of this paper is to study the Grothendieck group \(K_ 0\pmod R\) of the category of finitely generated modules over the invariant ring \(R=A^ G\). \textit{M. Auslander} and \textit{I. Reiten} [J. Pure Appl. Algebra 39, 1-51 (1986; Zbl 0576.18008)] showed that \(K_ 0\pmod R\) is finitely generated by at most \(c(G)\) elements, where \(c(G)\) denotes the class number of \(G\). In particular, \(K_ 0\pmod R\) is a factor group of \(\mathbb{Z} [G^*]\) where \(G^*\) denotes the character group of \(G\). The authors prove that \(K_ 0\pmod R \simeq \mathbb{Z} [G^*]/K\), where \(K\) is defined as in the paper of \textit{J. Herzog} and \textit{H. Sanders} [Singularities, representation of algebras, and vector bundles, Proc. Symp., Lambrecht 1985, Lect. Notes Math. 1273, 134-149 (1987; Zbl 0652.14016)]. Grothendieck group; quotient singularity J. Herzog, E. Marcos and R. Waldi, On the Grothendieck group of a quotient singularity defined by a finite abelian group,J. Algebra 149 (1992), 122--138 \(K_0\) of group rings and orders, Grothendieck groups (category-theoretic aspects), Actions of groups on commutative rings; invariant theory, Singularities in algebraic geometry, Formal power series rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry On the Grothendieck group of a quotient singularity defined by a finite abelian 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 When given a quantized analytic cycle $(X, \sigma)$ in $Y$, \textit{Sh. Yu} [``Todd class via homotopy perturbation theory'', Preprint, \url{arXiv:1510.07936}] discovered a geometric condition by computing the quantized cycle of a closed embedding of complex manifolds using homotopy perturbation theory. Damien Calaque and Julien Grivaux give a categorical Lie-theoretic interpretation of this condition, which involves the second formal neighborhood of $X$ in $Y$.
\par
A quantized cycle $(X, \sigma)$ is tame if $\sigma^* \mathrm{N}_{X/Y}$ extends to a locally free sheaf at the second order. If this condition is satisfied, the authors prove that the derived Ext algebra $\mathcal{RH}om_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)$ is isomorphic to the universal enveloping algebra $U(\mathrm{N}_{X/Y}[-1])$ of the shifted normal bundle $\mathrm{N}_{X/Y}[-1]$, endowed with a specific Lie structure (Theorem A, page 34): assuming that $(X, \sigma)$ is a tame quantized cycle in $Y$, the class $\alpha$ defines a Lie coalgebra structure on the shifted conormal bundle $\mathrm{N}^*_{X/Y}[1]$, which gives a Lie algebra structure on $\mathrm{N}_{X/Y}[-1]$. Moreover, $\mathcal{RH}om^{\ell}_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)$ and $\mathcal{RH}om^{r}_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)$ are algebra objects in the derived category $\mathrm{D}^{\mathrm{b}}(X)$, and the following diagrams
\[
\begin{tikzcd}
\sigma_\ast\mathcal{RH}om_{\mathcal O_S}(\mathcal O_X, \mathcal O_X) \ar[r] \ar[dd, "\simeq", "\mathrm{HKR}" '] & \mathcal{RH}om^\ell_{\mathcal O_Y}(\mathcal O_X, \mathcal O_X) \ar [d, "\simeq" ', "\mathrm{HKR}"] \\
& \mathrm{S(N}_{X/Y}[-1]) \ar[d, "\simeq" ', "\mathrm{PBW}"]\\
\mathrm{T(N}_{X/Y}[-1]) \ar [r] & \mathrm{U(N}_{X/Y}[-1])
\end{tikzcd}
\]
and
\[
\begin{tikzcd}
\sigma_\ast\mathcal{RH}om_{\mathcal O_S}(\mathcal O_X, \mathcal O_X) \ar[r] \ar[dd, "\simeq", "\mathrm{dual\ HKR}" '] & \mathcal{RH}om^r_{\mathcal O_Y}(\mathcal O_X, \mathcal O_X) \ar [d, "\simeq" ', "\mathrm{dual\ HKR}"] \\
& \mathrm{S(N}_{X/Y}[-1]) \ar[d, "\simeq" ', "\mathrm{PBW}"]\\
\mathrm{T(N}_{X/Y}[-1]) \ar [r] & \mathrm{U(N}_{X/Y}[-1])
\end{tikzcd}
\] commute, where all horizontal arrows are algebra morphisms.
The authors also give a new Lie-theoretic proof of S. Yu's result for the tame quantized cycle class (Theorem B, page 35): letting $(X, \sigma)$ to be a tame quantized cycle in $Y$, the quantized cycle class of $(X, \sigma)$ defined by \textit{J. Grivaux} [Int. Math. Res. Not. 2014, No. 4, 865--913 (2014; Zbl 1312.14027)] is the Duflo element of the Lie algebra object $\mathrm{N}_{X/Y}[-1]$. closed embeddings; formal neighborhoods; Todd class; Ext algebra; derived categories; Lie algebras; enveloping algebras; Duflo element; PBW isomorphism Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Formal neighborhoods in algebraic geometry, Universal enveloping (super)algebras, Universal enveloping algebras of Lie algebras, Cycles and subschemes The Ext algebra of a quantized cycle | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is an introductory paper about the theory of determinantal ideals \(I_t (X)\) of a matrix \(X= (X_{ij})\) of indeterminates. The algebraic properties of these ideals are examined from a number of view points.
(1) First, the authors use the straightening law of \textit{P. Doubilet}, \textit{G.-C. Rota} and \textit{J. Stein} (see [On the foundations of combinatorial theory IX: Combinatorial methods in invariant theory, Stud. Appl. Math. 53, 185--216 (1975; Zbl 0426.05009)]) to show that the rings \(K[X]/I_t\) are normal domains, to compute the symbolic powers of \(I_t\), and to describe the primary decomposition of products \(I_{t_1}\cdots I_{t_n}\).
(2) Second, they introduce the theory of initial ideals and initial algebras. They use this theory to prove that the rings \(K[X]/I_t\) are Cohen-Macaulay and to determine their Hilbert functions and their multiplicities.
(3) Third, the Knuth-Robinson-Schensted correspondence is applied to translate statement about monomials in \(K[X]\) to statements about standard bitableaux. This yields a Gröbner basis and the initial ideal of \(I^k_t\) for \(k\geq 1\) as well as a minimal system of generators.
(4) Finally, the theory of initial algebras is employed to prove the normality and Cohen-Macaulay property of the Rees algebras \({\mathcal R}(I_{t_1},\cdots I_{t_n})\), the symbolic Rees algebras \({\mathcal R}(I_t)\) and the algebra \(K[M_t]\) generated by the minors of size \(t\times t\) of \(X\). For the last ring, the authors also determine the canonical module and characterize when it is Gorenstein.
The presentation is very clear and concise. This overview article can be recommended to everyone who wants to study the algebra of determinantal ideals from a modern point of view. determinantal ideal; Cohen-Macaulay ring; Gröbner basis; straightening law Bruns, W.; Conca, A., Gröbner bases and determinantal ideals, \textit{Commutative Algebra, Singularities and Computer Algebra}, 9-66, (2003), Kluwer Academic Publishers, Dordrecht Rings with straightening laws, Hodge algebras, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Determinantal varieties, Combinatorial aspects of representation theory Gröbner bases and determinantal ideals | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials If \(G\) is a polyhedral group, then Nakamura's \(G\)-Hilbert scheme \(Y=G\)-Hilb\((\mathbb{C}^3)\) gives a natural Calabi-Yau resolution of the quotient singularity \(\mathbb{C}^3/G\). The authors describe the quantum geometry of \(Y\) in terms of \(R\), an ADE root system associated to \(G\). They give an explicit formula for the Gromov-Witten partition function of \(Y\) and a prediction for the orbifold Gromov-Witten invariants of the quotient singularity \(\mathbb{C}^3/G\), via the Crepant Resolution Conjecture. polyhedral group; \(G\)-Hilbert scheme; Calabi-Yau resolution; Gromov-Witten invariants Jim Bryan and Amin Gholampour, The quantum McKay correspondence for polyhedral singularities, Invent. Math. 178 (2009), no. 3, 655-681. Global theory and resolution of singularities (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) The quantum McKay correspondence for polyhedral singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The mirror map and the quantum coupling are known to have many interesting number theoretic properties. For example, the Fourier coefficients of the mirror map are integral in all known cases. In this paper, the authors prove that some K3 mirror maps are algebraic over \(\mathbb{Q} (J)\) by solving the Schwarzian differential equation in terms of the \(J\)-function, which leads to a uniform proof that these mirror maps have integral Fourier coefficients. The authors also give another proof that those mirror maps are integral by using the fact that such maps are genus zero functions as Riemann mappings, and conjecture a connection between K3 mirror maps and the Thompson series. Finally, they discuss \(\text{mod } p\) congruences for the cases of threefolds and the quintics. mirror map; quantum coupling; K3 mirror maps; Schwarzian differential equation Lian B., H.; Yau, S.-T., Arithmetic properties of mirror map and quantum coupling, Commun. Math. Phys, 176, 163-191, (1996) \(K3\) surfaces and Enriques surfaces, Supervarieties, \(3\)-folds Arithmetic properties of mirror map and quantum coupling | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 version of the quantum geometric Langlands conjecture in characteristic \(p\). Namely, we construct an equivalence of certain localizations of derived categories of twisted crystalline \(\mathcal{D}\)-modules on the stack of rank \(N\) vector bundles on an algebraic curve \(C\) in characteristic \(p\). The twisting parameters are related in the way predicted by the conjecture and are assumed to be irrational (i.e., not in \(\mathbb{F}_{p}\)). We thus extend some previous results Braverman and Bezrukavnikov concerning a similar problem for the usual (nonquantum) geometric Langlands. In the course of the proof, we introduce a generalization of \(p\)-curvature for line bundles with nonflat connections, define quantum analogues of Hecke functors in characteristic \(p\), and construct a Liouville vector field on the space of de Rham local systems on \(C\). geometric Langlands; \(D\)-modules; characteristic \(p\); Azumaya algebra; quantization; quantum Hecke functors; \(p\)-curvature Geometric Langlands program (algebro-geometric aspects), Positive characteristic ground fields in algebraic geometry Quantum geometric Langlands correspondence in positive characteristic: the \(\mathrm{GL}_N\) 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 The paper under review presents a new algebraic approach to quantum cluster algebras based on noncommutative ring theory. The paper proposes a general construction of quantum cluster algebra structures on a broad class of algebras. Initial clusters and mutations are constructed in a uniform and intrinsic way, in particular, avoiding any ad hoc constructions with quantum minors.
The main theorem of the paper asserts that every algebra in a very large, axiomatically defined class of quantum nilpotent algebras admits a quantum cluster algebra structure. Furthermore, for all such algebras, the latter equals the corresponding upper quantum cluster algebra.
This theorem has a broad range of applications and the required axioms are easy to verify. Many classical families of algebras fall within this axiomatic class. In particular, an application of this theorem gives an explicit quantum cluster algebra structures on the quantum Schubert cell algebras for all finite dimensional simple Lie algebras. quantum cluster algebras; quantum nilpotent algebras; iterated Ore extensions; noncommutative unique factorization domain K. R. Goodearl and M. T. Yakimov, \textit{Quantum cluster algebra structures on quantum nilpotent algebras}, Memoirs of the American Mathematical Society \textbf{247} (2017). Ring-theoretic aspects of quantum groups, Cluster algebras, Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Quantum cluster algebra structures on quantum nilpotent algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors propose Gamma conjectures for Fano manifolds which can be thought of as a square root of the index theorem. The quantum connection of a Fano manifold \(F\) has a regular singularity at \(z=\infty\) and an irregular singularity at \(z=0\). Flat sections near \(z=\infty\) are constructed by the Frobenius method and can be put into correspondence with cohomology classes in a natural way. Flat sections near \(z=0\) are classified by their exponential growth order (along a sector). Under the assumption of Property \(\mathcal O\), Gamma conjecture I claims that the flat section with the smallest asymptotics as \(z\to 0\) will transport to the flat section near \(z=\infty\) corresponding to the Gamma class \(\hat{\Gamma}_F\). Under further semi-simplicity assumption, Gamma conjecture II says that cohomology classes \(A_i\) that correspond to flat sections asymptotic to \(e^{-u_i/z}\) as \(z\to 0\), where \(u_i\) are eigenvalues of \(C_1(F)\star_0\), can be written as \(A_i=\hat{\Gamma}_F\mathrm{Ch}(E_i)\), where \(\{E_i\}\) form an exceptional collection of the bounded derived category of coherent sheaves on \(F\). Gamma conjecture II refines a part of Dubrovin's conjecture which says that the Stokes matrix of the quantum connection equals the matrix formed by Euler pairings among the objects in an exceptional collection. The authors then prove the Gamma conjectures for projective spaces by elementary methods (without using mirror symmetry, instead they directly look at solutions to the quantum differential equation and its Laplace transform). They also prove Gamma conjectures for Grassmannians, which follow from the case of projective spaces and the quantum Satake principle, or abelian/non-abelian correspondence, which says that the quantum connection of \(\mathrm{Gr}(r, N)\) is the \(r\)-th wedge of the quantum connection of \(\mathbb P^{N-1}\). Fano varieties; quantum cohomology; Frobenius manifolds; Dubrovin's conjecture; gamma class; Apery limit; derived category of coherent sheaves; exceptional collection; Grassmannians; abelian/nonabelian correspondence; quantum Satake principle; mirror symmetry Galkin, Sergey and Golyshev, Vasily and Iritani, Hiroshi, Gamma classes and quantum cohomology of {F}ano manifolds: gamma conjectures, Duke Mathematical Journal, 165, 11, 2005-2077, (2016) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Arithmetic mirror symmetry, Fano varieties, Mirror symmetry (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Gamma classes and quantum cohomology of Fano manifolds: gamma conjectures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be a field of characteristic zero. Denote by \(A\) the associative \(k\)-algebra generated by \(x,y,z,w\) subject to the defining relations
\[
\begin{aligned} yx=-xy, &\quad wx=-xw+y^2+ayz+bz^2,\\ zx=xz, &\quad wy=-yw+x^2+cxz+dz^2,\\ zy=yz, &\quad wz=zw+xy,\end{aligned}
\]
where \(a,b,c,d\in k\). Then (1) \(A\) is a Noetherian Artin-Schelter regular domain of global dimension 4; (2) \(A\) is an infinite module over its center; (3) there exists a finite point scheme parameterizing graded cyclic \(A\)-modules \(M\) with Hilbert series \((1-t)^{-1}\) if \((b,d)\neq (0,0)\); in this case there are at most 5 closed points; (4) if \((a,b,c,d)\neq (0,0,0,0)\) then there exists a one-parameter family of graded \(A\)-modules \(M\) with Hilbert series \((1-t)^{-2}\).
This example shows that the result of \textit{M. Artin, J. Tate} and \textit{M. Van den Bergh} [Invent. Math. 106, No. 2, 335-388 (1991; Zbl 0763.14001)] cannot be extended for algebras of global dimension 4. noncommutative algebraic geometry; regular algebras; quantum spaces; point modules; point schemes; Hilbert series Stephenson, Darin R.; Vancliff, Michaela, Some finite quantum \(\mathbb{P}^3\)s that are infinite modules over their centers, J. Algebra, 297, 1, 208-215, (2006) Rings arising from noncommutative algebraic geometry, Quadratic and Koszul algebras, Noncommutative algebraic geometry, Graded rings and modules (associative rings and algebras) Some finite quantum \(\mathbb{P}^3\)s that are infinite modules over their centers. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the series `Geometry of \(G/P\)', \textit{V. Lakshmibai}, \textit{C. Musili} and \textit{C. S. Seshadri} [see Bull. Am. Math. Soc., New. Ser. 1, 432-435 (1979; Zbl 0466.14020)] developed a standard monomial theory for semisimple algebraic groups as a generalization of the Hodge-Young standard monomial theory for \(GL(n)\). Standard monomial theory consists in constructing explicit bases for spaces of sections of effective line bundles on the generalized flag variety. Standard monomial theory has led to very many important geometric and representation-theoretic consequences.
In this article, the author gives a different approach to standard monomial theory (which avoids the case by case consideration) through path models of representations and their associated bases. The bases are constructed using the theory of quantum groups at a root of unity. Using these bases, geometric and representation-theoretic consequences -- vanishing theorems for higher cohomology for effetive line bundles on Schubert varieties, a proof of the Demazure character formula, projective normality of Schubert varieties, good filtration property -- are deduced.
It should be added that the path model theory has also led to a Littlewood-Richardson rule for symmetrizable Kac-Moody algebras proved by the author [cf. \textit{P. Littelmann}, Invent. Math. 116, No. 1-3, 329-346 (1994; Zbl 0805.17019)]. flag variety; standard monomial theory; path models of representations; quantum groups; vanishing theorems; Schubert varieties; Demazure character formula; projective normality; good filtration property Peter Littelmann, The path model, the quantum Frobenius map and standard monomial theory, Algebraic groups and their representations (Cambridge, 1997) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 517, Kluwer Acad. Publ., Dordrecht, 1998, pp. 175 -- 212. Homogeneous spaces and generalizations, Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds The path model, the quantum Frobenius map and standard monomial theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Suppose that \(b=(\varepsilon_1,\varepsilon_2,\ldots ,\varepsilon_n)\) is a sequence of plus and minus signs. Then there are two well-defined operations that can be carried out on \(b\). We cross out all pairs \((+,-)\) occurring in \(b\) and repeat this process until there are no such pairs left; call the sequence with signs crossed out \(\widetilde{b}\). Let \(r\) be the position of the rightmost minus sign in \(\widetilde{b}\) and let \(s\) be the position of the leftmost plus sign in \(\widetilde{b}\) (keeping the numbering of \(b\)). Define \(e(b)\) to be \(b\) with the sign of \(\varepsilon_r\) changed to plus, and define \(f(b)\) to be \(b\) with the sign of \(\varepsilon_s\) changed to minus.
This combinatorial device appears in two seemingly different contexts: in Kleshchev's modular branching rule for the symmetric group, in which case the sequence \(b\) appears as a sequence of addable and removable nodes of a Young tableau associated with a residue [see \textit{A. S. Kleshchev}, J. Algebra 178, 493-511 (1995; Zbl 0854.20013), J. Reine Angew. Math. 459, 163-212 (1995; Zbl 0817.20009), J. Lond. Math. Soc. (2) 54, 25-38 (1996; Zbl 0854.20014) and J. Algebra 201, 547-572 (1998; Zbl 0931.20014)] and in the action of Kashiwara's operators on the crystal graph [\textit{M. Kashiwara}, Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)] for the module of an affine Lie algebra of type \(A_{r-1}^{(1)}\) of highest weight given by the first fundamental weight \(\Lambda_0\). (See also \textit{T. Nakashima} and \textit{M. Kashiwara} [J. Algebra 165, 295-345 (1994; Zbl 0808.17005)] for explict examples of this combinatorics.)
This book can be seen as an explanation of this coincidence, which of course is not a coincidence after all, and relies on deep connections between the representation theory of symmetric groups and Hecke algebras on the one side, and the crystal graphs of modules for affine quantized enveloping algebras on the other.
The book has two main parts. The first part, chapters 1-9, is an introduction to the theory of quantum groups (the quantum algebras of the title). Some background in the theory of simple Lie algebras and their classification by Dynkin diagrams [see e.g. \textit{H. Samelson}, Notes on Lie algebras, Springer-Verlag, New York (1990; Zbl 0708.17005)], and Kac-Moody Lie algebras, would be useful to the reader, but a good explanation is given also in these areas in order to understand the correspondence mentioned above. The author gives a good introduction to the algebraic aspects of this fast-developing field, including the theory of the global crystal, or canonical basis, for a quantized enveloping algebra associated to a symmetrizable Kac-Moody Lie algebra as developed by \textit{G. Lusztig} [J. Am. Math. Soc. 3, 447-498 (1990; Zbl 0703.17008)] and M. Kashiwara (reference as above). The example \(A_{r-1}^{(1)}\) (affine type) is used throughout, which works well -- the reader is able to see how things work first hand, as well as obtaining the explicit case needed for the correspondence with modular representation theory.
In the second part, the author discusses the connection between the modular representation theory of the symmetric group (and the representation theory of Hecke algebras) and the canonical and crystal bases mentioned above. Let \(U\) denote the quantized enveloping algebra of type \(A_{r-1}^{(1)}\). The author describes the Hayashi realisation [\textit{T. Hayashi}, Commun. Math. Phys. 127, 129-144 (1992; Zbl 0701.17008) and \textit{K. C. Misra} and \textit{T. Miwa}, Commun. Math. Phys. 134, 79-88 (1990; Zbl 0724.17010)] of the module \(V(\Lambda_0)\) of \(U\) of highest weight \(\Lambda_0\) (the first fundamental weight) in terms of Young tableaux, in fact giving a new proof. He shows how this can be used to obtain a realisation of an arbitrary highest weight \(U\)-module via multipartitions. He goes on to show how this can be used in the \(\Lambda_0\) case to obtain a description of the crystal basis of \(V(\Lambda_0)\) in terms of the combinatorics of good nodes appearing in the modular representation theory of symmetric groups (a result of K. C. Misra and T. Miwa) as well as discussing the general case.
Next, the author describes the LLT algorithm [\textit{A. Lascoux, B. Leclerc} and \textit{J.-Y. Thibon}, Commun. Math. Phys. 181, 205-263 (1996; Zbl 0874.17009)], for obtaining a canonical basis element of \(V(\Lambda_0)\) and their conjecture that the coefficients obtained in a decomposition of such an element in terms of the natural tableau basis arising from the Hayashi realisation are decomposition numbers for the Hecke algebra with parameter \(q\) given by a primitive \(r\)th root of unity. The key theorem (12.5 in the book) is the author's generalisation of this conjecture to the Hecke algebras \(H_n\) of type \(G(m,1,n)\). The remainder of the book is devoted to the author's proof of this result and its important applications. Specifically, he shows that, for an appropriate choice of the \(m\) parameters (powers of \(q\)), the direct sum of the complexified Groethendieck groups of the \(H_n\) affords a structure as module for \(U\) where the Chevalley generators act as certain restriction and induction operators. Moreover, the module so obtained is a highest weight module (with highest weight obtained from the exponents in the parameters), and if the characteristic of the field of definition of \(H_n\) is zero, the canonical basis of this module specialised at \(q=1\) coincides with the basis given by the indecomposable projective modules. The proof involves perverse sheaves and the Hall algebra of the cyclic quiver.
Remark: For an alternative approach to induction and restriction operators on these Grothendieck rings and the link with quantum groups, see also \textit{I. Grojnowksi}, preprint arXiv:math.RT/ 9907129).
Overall, this is a well-written and clear exposition of the theory needed to understand the latest advances in the theory of the canonical/global crystal basis and the links with the representation theory of symmetric groups and Hecke algebras.
The book finishes with an extensive bibliography of papers, which is well organised into different areas of the theory for easy reference. quantized enveloping algebra; affine Lie algebra; Young tableau; crystal basis; canonical basis; Hayashi realisation; Hecke algebra; LLT algorithm; Kleshchev branching rules; modular representation theory; symmetric group; combinatorial representation theory Ariki, S., Representations of quantum algebras and combinatorics of Young tableaux, Univ. Lecture Ser., vol. 26, (2002), American Mathematical Society Providence, RI, Translated from the 2000 Japanese edition and revised by the author Quantum groups (quantized enveloping algebras) and related deformations, Combinatorial aspects of representation theory, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Hecke algebras and their representations, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Representations of finite groups of Lie type, Module categories in associative algebras, Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Quantum groups and related algebraic methods applied to problems in quantum theory Representations of quantum algebras and combinatorics of Young tableaux. Transl. from the Japanese and revised from the author | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is the first half of the author's work devoted to the systematic treatment of canonical spin polynomial invariants of algebraic surfaces. The notion of a spin polynomial of a smooth simple connected compact complex algebraic surface \(S\) was introduced in the author's recent works [see, e.g., Russ. Acad. Sci., Izv., Math. 42, No. 2, 333-369 (1994); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 2, 125-164 (1993; Zbl 0823.14031)] in the differential-geometric setting. In the present paper the spin polynomials are defined in the algebro-geometric setting. The paper consists of two chapters.
In chapter 1 a treatment of Jacobians \(J_k^i(S)\) and theta-loci \(\Theta_k^i(S)\) is given. Here \(i\) indexes the spin chambers of the Kähler cone of \(S\), \(J_k^i(S)\) is the Gieseker-Maruyama moduli space \(M^i(2, c_1=K_S, c_2=c_2(S)+k)\) with respect to the almost-canonical polarization lying in the \(i\)-th chamber, and \(\Theta_k^i(S)= \{[F]\in J_k^i(s)\mid h^0(F)\geq 1\}\). Then the spin polynomials \(s\gamma(k,n,i)\in S^dH^2(S,\mathbb{Z})\), \(d=3c_2(S)- K_S^2+3k-p_a(S)- 2n+1\), \(n\geq 0\), are defined by the cohomological correspondence using the discriminant of the universal quasifamily of sheaves on \(S\times \Theta_k^i(S)\). -- In chapter 2 the differential-geometric construction of spin polynomials is discussed and its coincidence with the algebro-geometric one is proved; the spin polynomials \(s\gamma(k,n,i)\) are then interpreted as ``fundamental'' algebraic classes of intermediate dimension \(2d\) in the cohomology ring of the Hilbert scheme \(\text{Hilb}^dS\). Jacobian; canonical spin polynomial invariants; theta-loci; Gieseker-Maruyama moduli space; almost-canonical polarization; Hilbert scheme Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Differentiable structures in differential topology, Moduli, classification: analytic theory; relations with modular forms, Parametrization (Chow and Hilbert schemes), Theta functions and abelian varieties Canonical spin polynomials of an algebraic surface. I | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 well known that a quantum loop algebra was defined as a generalization of the Drinfeld's new realization of the quantum affine algebra to the loop algebra of any Kac-Moody algebra. It has been shown by Schiffmann that the Hall algebra of the category of coherent sheaves on a weighted projective line is closely related to the quantum loop algebra of some Kac-Moody algebra with a star-shaped Dykin diagram. In this paper the authors study Drinfeld's presentation of the quantum loop algebra in the double Hall algebra setting and find out a collection of generators of the double composition algebra and verify that they satisfy all the Drinfeld relations. quantum loop algebra; Drinfeld's presentation; Hall algebra; weighted projective line; coherent sheaf Dou, R; Jiang, Y; Xiao, J, The Hall algebra approach to drinfeld's presentation of quantum loop algebras, Adv. Math., 231, 2593-2625, (2012) Vector bundles on curves and their moduli, Quantum groups (quantized enveloping algebras) and related deformations, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) The Hall algebra approach to Drinfeld's presentation of quantum loop 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 point out an explicit connection between graphs drawn on compact Riemann surfaces defined over the field \(\overline{\mathbb Q}\) of algebraic numbers -- the so-called Grothendieck's dessins d'enfants -- and a wealth of distinguished point-line configurations. These include simplices, cross-polytopes, several notable projective configurations, a number of multipartite graphs and some ''exotic'' geometries. Among them, remarkably, we find not only those underlying Mermin's magic square and magic pentagram, but also those related to the geometry of two- and three-qubit Pauli groups. Of particular interest is the occurrence of all the three types of slim generalized quadrangles, namely GQ(2,1), GQ(2,2) and GQ(2,4), and a couple of closely related graphs, namely the Schläfli and Clebsch ones. These findings seem to indicate that dessins d'enfants may provide us with a new powerful tool for gaining deeper insight into the nature of finite-dimensional Hilbert spaces and their associated groups, with a special emphasis on contextuality. Grothendieck's dessins d'enfants; quantum contextuality; finite geometries M. Planat, A. Giorgetti, F. Holweck and M. Saniga, Quantum contextual finite geometries from dessins d'enfants, Int. J. Geom. Methods Mod. Phys. 12 (2015), 1550067, 10.1142/ s021988781550067x. Arithmetic aspects of dessins d'enfants, Belyĭ theory, Contextuality in quantum theory, Quantum information, communication, networks (quantum-theoretic aspects), Dessins d'enfants theory, Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices Quantum contextual finite geometries from dessins d'enfants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 tropical vertex is an extension of the results of \textit{P. Bousseau} [Invent. Math. 215, No. 1, 1--79 (2019; Zbl 07015696)] to the setting of the classical tropical vertex group introduced by M. Gross, R. Pandharipande and B. Siebert. This proves the coincidence of two previously introduced refinements for scattering diagrams and some higher genera log Gromov-Witten invariants of some log Calabi-Yau surfaces. Additionally to this interpretation the author proves an integrality statement, known as Ooguri-Vafa/open BPS, about the generating series of the invariants and provide a refined conjecture about a coarser integrality.
Two dimensional scattering diagrams are combinatorial objects. They consist in a family of half-lines \((D)\) in the plane, and to each half-line \(D\) is attached an element \(\theta_D\) in some group. In the setting of the Gross-Pandharipande-Siebert, this group is the tropical vertex group. A scattering diagram is called balanced if for each loop \(\gamma\) in the plane, the product of the elements \(g_D\) for the half-lines met by \(\gamma\) is equal to the identity. Given an unbalanced scattering diagram, there is a way to \textit{balance} it by adding some half-lines. As proven by Gross-Pandharipande-Siebert, the group elements attached to the half-lines added to balance the scattering diagram are related to log Gromov-Witten invariants. This relation passes through the intermediate step of tropical invariants and thus uses a correspondence theorem, proven by G. Mikhalkin for the planar case, and T. Nishinou and B. Siebert for the higher dimensional setting.
Two refinements of the previous setting have been provided. The first one is due to M. Kontsevich and Y. Soibelman. It is obtained by considering a natural \(q\)-deformation of the tropical vertex group used in the scattering diagrams. As in the classical case, this deformation is related to a tropical count of rational curves using the refined multiplicity of Block and Göttsche, as worked out by S.A. Filippini and J. Stoppa. The second refinement is provided by Gross-Pandharipande-Siebert as they noticed that the log Gromov-Witten invariants appearing in the scattering diagram setting have a natural generalization to higher genera by the insertion of a top dimensional \(\lambda\)-class, which is in fact the setting of \textit{P. Bousseau} [Invent. Math. 215, No. 1, 1--79 (2019; Zbl 07015696)]. The values of the newly considered invariants are encoded in a generating series using a new indeterminate \(\hbar\) whose power is \(2g-2\), with \(g\) the genus. The content of the first theorem is then to prove using the results of [loc. cit.] that both refinement agree, using the change of variable \(q=e^{i\hbar}\). Concretely, this means that the refined \(q\)-invariant and the change \(q=e^{i\hbar}\) give the generating series of log Gromov-Witten invariants with the insertion of a top dimensional \(\lambda\)-class.
The second result of the paper is a combination of the first theorem and an enhancement of results of Kontsevich-Soibelman. It consists in some integrality statement about the log Gromov-Witten invariants, known as Ooguri-Vafa/open BPS. Briefly, the generating series of these invariants become some specific rational function in \(q\) after the change of variable \(q=e^{i\hbar}\). scattering diagrams; quantum tori; Gromov-Witten invariants Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Combinatorial aspects of tropical varieties The quantum tropical vertex | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Given a $D$-module $M$ generated by a single element, and a polynomial $f$, one can construct several $D$-modules attached to $M$ and $f$ and can define the notion of the (generalized) $b$-function following M. Kashiwara. These modules are closely related to the localization and the local cohomology of $M$.\par We show that the $b$-function, if it exists, controls these modules and present general algorithms for computing these modules and the $b$-function if it exists without any further assumptions. We also give some examples of multiplicity computation of such $D$-modules including a possibly well-known explicit formula for the localization of the polynomial ring by a hyperplane arrangement. \(D\)-module; Gröbner basis; localization; local cohomology; \(b\)-function; hyperplane; arrangement; multiplicity Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Local cohomology and commutative rings, Local cohomology and algebraic geometry, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Localization, local cohomology, and the \(b\)-function of a \(D\)-module with respect to a polynomial | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 of Chow motives admits two natural opposite \(\lambda\)-ring structures, one of which is a special structure allowing the definition of Adams operations on the ring. In this work I present algorithms which allow an effective simplification of expressions that involve both \(\lambda\)-ring structures, as well as Adams operations. In particular, these algorithms allow the symbolic simplification of algebraic expressions in the sub-\(\lambda\)-ring of motives generated by a finite set of curves into polynomial expressions in a small set of motivic generators. As a consequence, the explicit computation of motives of some moduli spaces is performed, allowing the computational verification of some conjectural formulas for these spaces. lambda-rings; symbolic computations of motives; Chow motives; moduli spaces; Higgs bundles moduli space Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli, Computational aspects of higher-dimensional varieties, \(K\)-theory of schemes, Symbolic computation and algebraic computation Simplification of \(\lambda\)-ring expressions in the Grothendieck ring of Chow motives | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The complex projective space \(\mathbb P^n\) has no deformations as a classical algebraic variety, but several noncommutative ones which the author calls quantum deformations.
The article concentrates on the case \(n=3\), and relations to problems in Poisson geometry and foliation theory. Noncommutative projective geometry is modelled by the homogeneous coordinate rings, noncommutative analogues of polynomial rings satisfying the Artin-Schelter (AS) regularity criterion. From the fact that a Poisson structure on a surface is determined by a section of the anticanonical bundle, there is a link between the noncommutative algebras and the geometry of anticanonical divisors in the classical surfaces. The case of \(\mathbb P^3\) in the form of AS regular algebras of global dimension 4 has been studied extensively, and the full classification remains one of the central questions in noncommutative projective geometry.
From the subclass of AS regular algebras consisting of the Calabi-Yau algebras, it is possible to obtain other AS regular algebras by twisting procedures. Although a given quantum deformation of \(\mathbb P^n\) may be representable by many different AS regular algebras, there will be a unique one that is Calabi-Yau.
The main result of the article is a classification of the quantum deformations of \(\mathbb P^3\): The flat deformations the graded Calabi-Yau algebra \(\mathbb C[x_0,x_1,x_2,x_3]\) is contained in six irreducible families, each realized as closures of the \(\mathrm{GL}(4,\mathbb C)\)-orbits of specified normal forms.
The families intersect nontrivially, and it is not quite clear how to describe all possible intersections. Thus the focus is on suitable generic elements of each family.
To prove the main result, the problem of classification is reduced to the study of the semi-classical limits using Kontsevich's results on deformation quantization. Such deformation quantization of a given Poisson structure will be a Calabi-Yau algebra if and only if the Poisson structure is unimodular. Then the result can be reduced to proving that the variety parametrizing unimodular quadratic Poisson structures in four dimensions has exactly six irreducible components.
The author gives explicit formulas for the generic Poisson brackets in each component. This is possible by using work of \textit{D. Cerveau} and \textit{A. Lins Neto} [Ann. Math. (2) 143, No. 3, 577--612 (1996; Zbl 0855.32015)] regarding codimension-one foliations on \(\mathbb P^3\) and recent refinements of Loray, Pereira, and Touzet [\textit{F. Loray} et al., Math. Nachr. 286, No. 8--9, 921--940 (2013; Zbl 1301.37032)] and the well-known correspondence between homogeneous quadratic Poisson structures on projective space.
The author consider quantization as equivariant geometry, and finds that the corresponding quantum \(\mathbb P^3\) contains 3 commutative rational curves; a line, a plane conic, and a twisted cubic. These all correspond to various embeddings of \(\mathbb P^1\) in its symmetric power. There is also a pencil of noncommutative sextic surfaces whose classical limits are the level sets of the \(j\)-invariant, and the Schwarzenberger bundles which served as early examples of indecomposable vector bundles on projective space quantize to give graded bimodules over the \(\mathrm E(3)\) algebra.
The article reviews basic facts about deformations of quadratic algebras and their semiclassical limits in noncommutative projective geometry. Calabi-Yau algebras are introduced, superpotentials, their role as unique homogeneous coordinate rings for quantum \(\mathbb P^n\)s, and the connection with the unimodularity condition for Poisson structures. To complete the proof of the main result, the author gives normal forms for the appearing algebras and compare their semiclassical limits with the classification of foliations.
A quantum projective space is described by a noncommutative graded algebra \(A=\bigoplus_{k\geq 0}A_k\) which arises as a deformation of the usual product on the polynomial ring. Invariantly, consider a vector space \(V\), and present the symmetric algebra as a quotient \(S^\bullet V=T^\bullet V/(\bigwedge V)\) of the tensor algebra by the ideal generated by the skew- symmetric two-tensors. Quantizations of the homogeneous coordinate ring are constructed by deforming the subspace \(\bigwedge^2 V\subset V\otimes V\) to a new element \(R\in\mathrm{Gr}(r,V\otimes V)\) of the relevant Grassmannian. This means that \(A\) is deformed as a quadratic algebra. The deformations are supposed to have the same same invariants as the polynomial ring in the sense that the Hilbert series is constant.
This article uses the flat, noncommutative, deformations of the ``commutation ideal'' as the quantum deformations, and these are the main objects of investigation. Define a \textit{superpotential} on a vector space \(V\) of dimension \(d\) as an element \(\Phi\in V^{\otimes d}\) such that \(\mathrm{cyc}(\Phi)=(-1)^d\Phi\), where \(\mathrm{cyc}:V^{\otimes d}\rightarrow V^{\otimes d}\) is the linear automorphism that cyclically permutes the tensor factors. From a (possibly) twisted superpotential, one obtains a quadratic algebra by differentiating, and the link to Calabi-Yau algebras is given by a result stating that A Koszul algebra \(A\) is Calabi-Yau (resp. twisted Calabi-Yau) if and only if it is derived from a superpotential (respectively, twisted superpotential) whose associated complex is exact.
The article illustrates the deformation theory in a nice way, containing nice and explicit examples. Its results is important as it illustrates the link between different methods in noncommutative projective geometry. noncommutative projective geometry; Calabi-Yau algebra; Possin structure; holomorphic foliation; noncommutative deformation; foliations; deformation quantization; quadratic algebra; skew polynomial ring; quantum deformations; Kozul algbebra; superpotensial Pym, B., Quantum deformations of projective three-space, Adv. Math., 281, 1216-1241, (2015) Noncommutative algebraic geometry, Poisson manifolds; Poisson groupoids and algebroids, Dynamical aspects of holomorphic foliations and vector fields Quantum deformations of projective three-space | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Generalized subrings of the ring of integers that give birational models for the field of rationals are studied. A homogeneous strengthening of Evdokimov's theorem is proved. An approach to calculation of homotopy groups with the help of generalized rings is proposed. Relevant commutative algebra Nonclassical birational models for \(\operatorname{Spec}\mathbb{Q}\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(G_q\) denote the Manin quantization of \(GL_n\) and denote by \(B_q\) a Borel subgroup. The author's main result says that induction from \(B_q\) to \(G_q\) may be obtained by iterating rank 1 inductions corresponding to a reduced expression of the longest word in the Weyl group. The same statement is true for an arbitrary reductive algebraic group, see the paper by \textit{E. Cline, B. Parshall} and \textit{L. Scott} [Invent. Math. 47, 41-51 (1978; Zbl 0399.20039)], and for quantized enveloping algebras, see the joint work of the reviewer, \textit{P. Polo} and \textit{K. Wen} [ibid. 104, No. 1, 1-59 (1991; Zbl 0724.17012)].
It is also proved that for a dominant weight \(\lambda\) the natural homomorphisms from \(\text{Ind}_{B_q}^{G_q}\lambda\) to certain iterated partial subsequences of rank 1 inductions are surjective. This corresponds to the fact that for reductive groups the restrictions to (any) Schubert variety gives a surjection on the sections of effective line bundles, see the reviewer's paper [ibid. 79, 611-618 (1985; Zbl 0591.14036)]. (Compare also the above mentioned joint work for the quantized enveloping algebras case.) As consequences he gets many of the important cohomology vanishing results for quantized \(GL_n\).
The approach used in this paper involves rather explicit computations with bideterminants. In particular, it is proved that there exists a basis for the above iterated rank 1 induced modules consisting of bideterminants. Manin quantization; Borel subgroups; induction; reductive algebraic groups; quantized enveloping algebras; dominant weights; line bundles; cohomology vanishing results; bideterminants; induced modules J. Hu, ''A combinatorial approach to representations of quantum linear groups,'' Comm. Algebra 26(8) (1998), 2591--2621. Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations, Homogeneous spaces and generalizations, Cohomology theory for linear algebraic groups A combinatorial approach to representations of quantum linear 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 article strengthens the connections between Cherednik algebras and geometry by showing that they can be regarded as noncommutative deformations of Hilbert schemes of points in the plane. The article explicitly defines the rational Cherednik algebra \(H_c\) of type \(A_{n-1}\), and its spherical subalgebra \(U_c\). Let \(W=\mathfrak S_n\) be the symmetry group on \(n\) letters, regarded as the Weyl group of type \(A_{n-1}\) acting on its \((n-1)\)-dimensional representation \(\mathfrak h\in\mathbb{C}^n\) by permutations. Then \(H_c\) may be regarded as a deformation of the twisted group ring \(D(\mathfrak h)\ast W\), where \(D(\mathfrak h)\) is the ring of differential operators on \(\mathfrak h \) with the natural action of the symmetric group \(W=\mathfrak S_n.\) The algebra \(U_c\) is then the corresponding deformation of the fixed ring \(D(\mathfrak h)^W.\)
\(U_c\) and \(H_c\) both have a natural filtration by order of differential operators, with associated graded rings \(\operatorname{gr}U_c\cong\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W\) and \(\operatorname{gr}H_c\cong\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]\ast W\), and so \(U_c\) may be regarded as a deformation of \(\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W.\) The article is mostly concerned about \(U_c\), but the authors prove that \(U_c\) and \(H_c\) are Morita equivalent.
It is known that the map \(\tau:\text{Hilb}^n\mathbb{C}\rightarrow \mathbb{C}^{2n}/W\) is a resolution of singularities, and Haiman has described \(\text{Hilb}^n\) as ``Proj'' of Rees rings. This article proves that for \(\text{Hilb}(n)=\tau^{-1}(\mathfrak h\oplus\mathfrak h^\ast/W)\), \(\tau:\text{Hilb}(n)\rightarrow \mathfrak h\oplus\mathfrak h^\ast/W \) is a crepant resolution of singularities. The ring \(U_c\) has finite global homological dimension so one should expect the properties of a smooth deformation of \(\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W,\) that is, its properties should be more closely related to those of \(\text{Hilb}(n)\) than to \(\mathfrak h\oplus\mathfrak h^\ast/W\). The main ideal of this article is to formalize this idea by showing that there exists a second way of passing to associated graded objects that maps \(U_c\)-modules precisely to coherent modules on \(\text{Hilb}(n).\)
The main result of this article gives an affirmative answer to the question of whether the following diagram can be completed:
\[
\begin{tikzcd} {?} \ar[d,"\mathrm{gr}" '] & U_c \ar[l,"\sim" ']\ar[d,"\mathrm{gr}"] \\ \mathcal O_{\mathrm{Hilb}(n)} & \mathcal O(\mathfrak h\oplus\mathfrak h^\ast/W) \ar[l,"\tau" ']\rlap{\,.} \end{tikzcd}
\]
Given a graded ring \(R\), write \(R\)-qgr for the quotient category of noetherian graded \(R\)-modules modulo those of finite length. The main result then says that
There exists a graded ring \(B\), filtered by order of differential operators, such that
(1) there is an equivalence of categories \(U_c\text{-mod}\cong B\text{-qgr};\)
(2) there is an equivalence of categories \(\text{gr }B\text{-qgr}\cong\text{Coh}(\text{Hilb}(n))\).
The construction of \(B\) is not the same as Haiman's construction of the Rees ring, but a noncommutative deformation. This leads to the theory of \(\mathbb{Z}\)-algebras and Poincaré series in this setting. The authors then need results concerning Morita equivalence of the Cherednik algebras. The authors give explicit definitions and results about the Cherednik algebras and its representations. They study thoroughly the Morita equivalence of these algebras. Then Haiman's constructions of the Hilbert scheme and the resulting formulas for the Poincaré series are derived, and a geometric interpretation is given. Finally, \((\mathfrak h\oplus\mathfrak h^\ast)/W)\) is blown up.
In the noncommutative case, the Rees rings have to be replaced by noncommutative analogs. These are called \(\mathbb{Z}\)-algebras and a nice introduction is given here. The main theorem is then proved by introducing the noncommutative Poincaré series, and this is higly nontrivial.
This article is a very nice introduction to the theory and problems of noncommutative (graded) projective geometry. \(\mathbb Z\)-algebras; resolution of singularities Opdam, E.: Complex reflection groups and fake degrees (1998) arXiv:math/9808026\textbf{(preprint)} Parametrization (Chow and Hilbert schemes), Modifications; resolution of singularities (complex-analytic aspects), Deformations of associative rings, Module categories in associative algebras, Noncommutative algebraic geometry Rational Cherednik algebras and Hilbert schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth projective complex surface. The author introduces analogues of Donaldson polynomials, christened ``almost canonical spin polynomials'': They are defined by intersecting certain ``divisors'' on the moduli space of (stable) rank two vector bundles \(V\) on \(X\) with \(c_1(V)= c_1(K_X)\), \(c_2(V)= c_2(X)+k\), and \(h^0(V)>0\). The main result is the so called shape theorem, which asserts that these spin polynomials are expressible in terms of the intersection form, the canonical class, and the Poincaré duals of certain curves on \(X\). Donaldson polynomials; spin polynomials Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Special surfaces Canonical and almost canonical spin polynomials of an algebraic surface | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper offers an interesting contribution to Galois module theory in the setting of Hopf Galois theory. The governing question is the following: Given a finite extension \(L/K\) of local fields and a \(K\)-algebra \(A\) acting on \(L\), under which conditions will the ring \(O_L\) of integers be free over the so-called associated order \(\mathfrak A=\{\alpha\in A \mid \alpha O_L\subset O_L\}\)? The classical setting is a \(G\)-Galois extension \(L/K\), and \(A=K[G]\). If \(L/K\) is tame, then the associated order \(\mathfrak A\) is exactly \(O_K[G]\), and Noether's theorem says that \(O_K\) is free over it. Childs' monograph [\textit{L. N. Childs}, Taming wild extensions: Hopf algebras and local Galois module theory. Mathematical Surveys and Monographs. 80. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0944.11038)] proved much more generally that we have freeness if \(\mathfrak A\) is a Hopf order in \(A=K[G]\). Without any hypothesis one should not expect an affirmative answer, there are counterexamples.
The present paper generalises this result to Hopf Galois extensions \(L/K\) in the sense of Greither-Pareigis (see loc. cit. or the original paper in [\textit{C. Greither} and \textit{B. Pareigis}, J. Algebra 106, 239--258 (1987; Zbl 0615.12026)]). The Hopf algebra \(H\) which plays the role of \(A\) now is a form of a group ring. The extension \(L/K\) need not even be Galois in the classical sense. There are two cases that are captured: either \(L/K\) is unramified, or the degree of \(L/K\) is coprime to the residue characteristic and \(H\) is commutative. In both cases the author shows that the associated order is Hopf. Basically this is descent theory. In the latter case the associated order turns out to be the \textit{maximal} order. In the former case, the situation is just the opposite: if \(E/K\) is a Galois extension splitting \(H\) (so \(E\otimes_K H\) is a group ring \(E[N]\)), then \(\mathfrak A\) is gotten by taking the \(\text{Gal}(E/K)\)-fixed points of the \textit{minimal} order \(O_E[N]\).
In the final section, the author uses standard techniques to combine the local results into the following global result: Let \(L/K\) be a domestic extension of number fields, which is \(H\)-Galois for a finite commutative \(K\)-Hopf algebra \(H\). Then \(O_L\) is locally free over the associated order. (An extension \(L/K\) is called \textit{domestic} if no prime of \(K\) lying over a divisor \(p\) of the degree \([L:K]\) is ramified in \(L\). Note that this is considerably more restrictive than tameness.) Noether's theorem; integral normal basis; Hopf orders; associated orders; descent Truman, P. J., Towards a generalisation of noethers theorem to nonclassical Hopf-Galois structures, \textit{New York J. Math}, 17, 799-810, (2011) Integral representations related to algebraic numbers; Galois module structure of rings of integers, Group actions on varieties or schemes (quotients) Towards a generalisation of Noether's theorem to nonclassical Hopf-Galois 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 By classical work of Serre, Tate and Hartshorne, Grothendieck's trace map for a smooth, projective curve over a finite field can be expressed as a sum of residues over all closed points of the curve. Subsequently, this result was generalized in several ways, up to algebraic varieties of any dimension, using higher-dimensional adèles. In all these generalizations one only deals with varieties over a field.
The paper gives the first extension non-varieties, namely to arithmetic surfaces. Let \({\mathcal O}_k\) be a Dedekind domain of characteristic zero and with finite residue fields, and let \(K\) be its field of fractions. The extension given in Theorem 3.1 of the paper applies to a normal scheme, proper and flat over \(S=\text{Spec}{\mathcal O}_K\), whose generic fibre is a smooth, geometrically connected curve.
This extension requires three main steps: i) the definition of suitable local residue maps, either on spaces of differential forms or on local cohomology groups; ii) using the local residue maps, the definition of the dualizing sheaf; iii) patching together the local residue maps to define Grothendieck's trace map on the cohomology of the dualizing sheaf.
The first two steps are essentially already contained in a previous paper of the same author [New York J. Math. 16, 575--627 (2010; Zbl 1258.14031)], while the third step is achieved in the present paper.
The paper ends with applications to adelic duality for the arithmetic surface. reciprocity laws; higher adèles; arithmetic surfaces; Grothendieck duality; residues Morrow, M.: Grothendieck's trace map for arithmetic surfaces via residues and higher adèles, Algebra number theory 6, No. 7, 1503-1536 (2012) Arithmetic ground fields for curves, Local cohomology and algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Grothendieck's trace map for arithmetic surfaces via residues and higher adèles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials One says that a pair \((P,Q)\) of ordinary differential operators specify a quantum curve if \([P,Q]=\hslash\). If a pair of difference operators \((K,L)\) obey the relation \(KL=qLK\), where \(q =e^{\hslash}\), we say that they specify a discrete quantum curve.
This terminology is prompted by well known results about commuting differential and difference operators, relating pairs of such operators with pairs of meromorphic functions on algebraic curves obeying some conditions.
The goal of this paper is to study the moduli spaces of quantum curves. We will relate the moduli spaces for different \(\hslash\). We will show how to quantize a pair of commuting differential or difference operators (i.e., to construct the corresponding quantum curve or discrete quantum curve). Schwarz, A., Quantum curves, Commun. Math. Phys., 338, 483, (2015) Commutation relations and statistics as related to quantum mechanics (general), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Families, moduli of curves (algebraic), Relationships between algebraic curves and physics Quantum curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(U_q\) denote the quantum group for \(sl_3\) over \(\mathbb{Q}(q)\) and let \(U_A\) be the \(A=\mathbb{Z}[q,q^{-1}]\)-form of \(U_q\) defined via quantum divided powers. If \(V\) is a finite-dimensional simple module for \(U_q\) with highest weight vector \(v\), consider the \(A\)-form \(V_A=U_Av\). The author proves that \(V_A\) has an \(A\)-basis which is compatible with all quantum Demazure submodules. She also checks that the transition matrix from this basis to Lusztig's canonical basis for \(V_A\) is upper triangular (with respect to an appropriate ordering).
The formulations of these results generalize naturally to other semisimple Lie algebras. The author conjectures that they do in fact hold in general, and she has proved her conjecture in [\textit{V. Lakshmibai}, Proc. Symp. Pure Math. 56, Pt. 2, 149-168 (1994; Zbl 0848.17020)]. quantum Demazure modules; crystal basis; transition matrix Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Bases for quantum Demazure modules. I | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 deals with the quantum cohomology ring of \(\mathbb P(E)\) where \(E\) is a Fano bundle which is a direct sum of line bundles over \(\mathbb P^{n_1}\times \mathbb P^{n_2}\cdots \times \mathbb P^{n_s}\). Fano bundle means that \(\mathbb P(E)\) is a Fano manifold, i.e., the negative of its canonical line bundle is ample. The authors formulate a conjecture for the presentation of its quantum cohomology ring in terms of generators and relations. Under certain numerical conditions on the line bundles involved in defining \(E\) they are able to prove the conjecture to be true. The conjecture generalizes a conjecture of Batyrev (\(s=1\)). In this case it was proved by \textit{Z. Qin} and \textit{Y. Ruan} [Trans. Am. Math. Soc. 350, No. 9, 3615-3638 (1998; Zbl 0932.14030)], again under some numerical conditions on the line bundles involved.
The article under review starts with determining when \(\mathbb P(E)\) is a Fano manifold. This is followed by a calculation of extremal rays. Some Gromov-Witten invariants are computed. The assumed numerical conditions guarantee that the ``quantum corrections'' (defining the deformed product) come only from homology classes which generate the extremal rays. The article closes with additional examples supporting the conjecture. Fano manifolds; Gromov-Witten invariants; quantum cohomology; Fano bundle; quantum corrections Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fano varieties, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Projective techniques in algebraic geometry Quantum cohomology of projective bundles over \({\mathbb{P}}^{n_1}\times\cdots\times{\mathbb{P}}^{n_s}\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper studies the representation of a positive polynomial \(f(x)\) on a noncompact semialgebraic set \(S=\{x\in \mathbb{R}^n: g_1(x)\geq 0, \dots, g_s(x)\geq 0\}\) modulo its KKT (Karush-Kuhn-Tucker) ideal. Under the assumption that the minimum value of \(f(x)\) on \(S\) is attained at some KKT point, we show that \(f(x)\) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if \(f(x)>0\) on \(S\); furthermore, when the KKT ideal is radical, we argue that \(f(x)\) can be represented as a sum of squares (SOS) of polynomials modulo the KKT ideal if \(f(x)\geq 0\) on \(S\). This is a generalization of results of \textit{J. Nie, J. Demmel} and \textit{B. Sturmfels} [Math. Program. 106, No. 3 (A), 587--606 (2006; Zbl 1134.90032)], which discusses the SOS representations of nonnegative polynomials over gradient ideals. J. Demmel, J. Nie, and V. Powers, \textit{Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals}, J. Pure Appl. Algebra, 209 (2007), pp. 189--200, . Real algebra, Sums of squares and representations by other particular quadratic forms, Semialgebraic sets and related spaces, Semidefinite programming Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a global, intrinsic, and coordinate-free quantization formalism for Gromov-Witten invariants and their B-model counterparts, which simultaneously generalizes the quantization formalisms described by \textit{E. Witten} [in: Surveys in differential geometry. Vol. I: Proceedings of the conference on geometry and topology, held at Harvard University, Cambridge, MA, USA, April 27-29, 1990. Providence, RI: American Mathematical Society; Bethlehem, PA: Lehigh University. 243--310 (1991; Zbl 0757.53049)], \textit{A. B. Givental} [in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 472--480 (1995; Zbl 0863.14021)], and \textit{M. Aganagic} et al. [Commun. Math. Phys. 277, No. 3, 771--819 (2008; Zbl 1165.81037)]. Descendant potentials live in a Fock sheaf, consisting of local functions on Givental's Lagrangian cone that satisfy the \((3g-2)\)-jet condition of Eguchi-Xiong; they also satisfy a certain anomaly equation, which generalizes the holomorphic anomaly equation of \textit{M. Bershadsky} et al. [Nucl. Phys., B 405, No. 2--3, 279--304 (1993; Zbl 0908.58074)]. We interpret Givental's formula for the higher-genus potentials associated to a semisimple Frobenius manifold in this setting, showing that, in the semisimple case, there is a canonical global section of the Fock sheaf. This canonical section automatically has certain modularity properties. When \(X\) is a variety with semisimple quantum cohomology, a theorem of \textit{C. Teleman} [Invent. Math. 188, No. 3, 525--588 (2012; Zbl 1248.53074)] implies that the canonical section coincides with the geometric descendant potential defined by Gromov-Witten invariants of \(X\). We use our formalism to prove a higher-genus version of Ruan's crepant transformation conjecture for compact toric orbifolds. When combined with our earlier joint work with \textit{Y. Jiang} [Adv. Math. 329, 1002--1087 (2018; Zbl 1394.14036)], this shows that the total descendant potential for a compact toric orbifold \(X\) is a modular function for a certain group of autoequivalences of the derived category of \(X\). Gromov-Witten invariants; geometric quantization; modular forms; mirror symmetry; toric orbifold Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Geometric quantization A Fock sheaf for Givental quantization | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum cohomology of Grassmannians exhibits two symmetries related to the quantum product, namely a cyclic action and an involution related to complex conjugation. We construct a new ring by dividing out these symmetries in an ideal theoretic way and analyze its structure, which is shown to control the sum of all coefficients appearing in the product of cohomology classes. We derive a combinatorial formula for the sum of all Littlewood-Richardson coefficients appearing in the expansion of a product of two Schur polynomials. H. Hengelbrock, An involution on the quantum cohomology ring of the Grassmannian , Research exposition (monographs, survey articles) pertaining to algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Symmetries in 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 We show an explicit relation between the Chow form and the Grothendieck residue; and we clarify the role that the residue can play in the intersection theory besides its role in the division problem. Chow form; Grothendieck residue; intersection theory ELKADI (M) . - Résidu de Grothendieck et forme de Chow , Publ. Math., t. 38, 1994 , p. 381-393. MR 96e:32007 | Zbl 0837.32002 Analytic algebras and generalizations, preparation theorems, Residues for several complex variables, Relevant commutative algebra, Polynomial rings and ideals; rings of integer-valued polynomials Grothendieck residue and Chow form | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 topological group acting continuously on a topological space \(X\). In order to extend the formalism of Grothendieck's six operations on the derived category of sheaves (over a fixed field) to the equivariant setting, \textit{J. Bernstein} and \textit{V. Lunts} introduced the equivariant derived category \(D^b_G(X)\) [``Equivariant sheaves and functors'', Lect. Notes Math. 1578 (1994; Zbl 0808.14038)]. The category \(D^b_G(X)\) is a triangulated \(t\)-category and according to a theorem of Bernstein and Lunts the heart of the \(t\)-structure is isomorphic to the category \(Sh_G(X)\) of \(G\)-equivariant sheaves on \(X\).
The objective of the present paper is to give an alternative proof of this fact. Unlike the original argument, the present one does not rely on the results of Deligne about simplicial topological spaces. As an application, it is shown that the Grothendieck groups of the categories \(Sh_G(X)\) and \(D^b_G(X)\) are naturally isomorphic. equivariant derived category; Grothendieck group; triangulated category; t-structure Derived categories, triangulated categories, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Presheaves and sheaves in general topology, Equivariant homology and cohomology in algebraic topology, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Grothendieck group of an equivariant derived category | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 simply connected complex semisimple Lie group, \(\mathfrak g=\text{Lie}\;G\) and \(X^{\vee}\) the flag manifold associated to the Langlands dual to \(G\). Let \(CH(X^{\vee})\) be the complexified quantum cohomology of \(X^{\vee}\). Let \(f\in\mathfrak{g}\) be a principal nilpotent element, \(G^f_0\), the identity component of the centralizer of \(f\in\mathfrak g\), and \(\mathfrak g^f= \text{Lie}\;G^f_0\) Let \(F_f\) be the field of rational functions on \(G^f_0\). Using results of Dale Peterson and earlier results of ours, particularly on the Toda lattice, we show that \(CH(X_{\vee})\) naturally injects into \(F_f\). Among other things it follows that \(CH(X_{\vee}\) has the structure of \(U(\mathfrak{g}^f)\) module where \(U(\mathfrak g^f)\) is the universal enveloping algebra of \(\mathfrak g^f\). Furthermore generators of \(CH(X_{\vee})\) are explicitly given in terms of the matrix entries of \(\pi_{\rho}| G^f_0\) where \(\pi_{\rho}\) is the irreducible representation of \(G\) whose highest weight is one-half the sum of the positive roots. A particularly important role here is played by Newton's equation of motion for the Toda lattice. For example the quantum parameters \(q_i\) appear as ``forces''. Kostant, B.: Quantum cohomology of the flag manifold as an algebra of rational functions on a unipotent algebraic group. Deformation Theory and Symplectic Geometry (D. Sternheimer et al., ed.), Amsterdam: Kluwer, 1997, pp. 157--175 (Co)homology theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Universal enveloping (super)algebras Quantum cohomology of the flag manifold as an algebra of rational functions on a unipotent algebraic 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 We describe an evaluation/interpolation approach to compute modular polynomials on a Hilbert surface, which parametrizes abelian surfaces with maximal real multiplication. Under some heuristics we obtain a quasi-linear algorithm. The corresponding modular polynomials are much smaller than the ones on the Siegel threefold. We explain how to compute even smaller polynomials by using pullbacks of theta functions to the Hilbert surface. modular polynomials; cyclic isogeny; abelian surface; Humbert surface; moduli space of Hilbert and Siegel; theta constant Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Arithmetic aspects of modular and Shimura varieties, Algebraic number theory computations, Modular and Shimura varieties Modular polynomials on Hilbert surfaces | 0 |
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