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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 In this paper, the notion of the Gröbner cell for the Hilbert scheme of points in the plane, as well as that of the punctual Hilbert scheme is comprehensively defined. An explicit parametrization of the Gröbner cells in terms of minors of a matrix is recalled. The main core of this paper shows that the decomposition of the Punctual Hilbert scheme into Grönber cells induces that of the compactified Jacobians of plane curve singularities. As an important application of this decomposition, the topological invariance of an analog of the compactified Jacobian and the corresponding motivic superpolynomial for families of singularities is concluded. Hilbert schemes; affine plane; Grothendieck-Deligne map; Gröbner cells; zeta functions; plane curve singularities Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Plane and space curves, Exact enumeration problems, generating functions, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Jacobians, Prym varieties, Hecke algebras and their representations, Combinatorial aspects of representation theory, Braid groups; Artin groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) Gröbner cells of punctual Hilbert schemes in dimension two | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider a weak Fano projective toric orbifold \(\mathcal X\). The author introduces a \(\widehat \varGamma\)-integral structure on the quantum \(D\)-module of \(\mathcal X\), that is an integral structure on the space of flat sections of Dubrovin's connection for \(\mathcal X\) given by a class
\[
\widehat \varGamma(T\mathcal X)=\prod_{i=1}^{\dim \mathcal X}\varGamma(1+\delta_i),
\]
where \(\delta_i\)'s are Chern roots of \(\mathcal X\). The main theorem (Theorem 4.11) states that under some assumptions this integral structure corresponds, modulo Mirror Conjecture, to the natural integral local system on the mirror B-model \(D\)-module under the mirror isomorphism. In particular this holds for toric manifolds as assumptions are proven to hold. By assuming the existence of an integral structure, the author gives a natural explanation for the specialization to a root of unity in \textit{Y. Ruan}'s crepant resolution conjecture [in: AMS special session, San Francisco, CA, USA, May 3--4, 2003. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 403, 117--126 (2006; Zbl 1105.14078)]. quantum cohomology; variation of Hodge structures; semi-infinite variation of Hodge structures; mirror symmetry; Landau-Ginzburg model; toric Deligne-Mumford stack; orbifold; orbifold quantum cohomology; Crepant resolution conjecture; Ruan's conjecture; \(K\)-theory; McKay correspondence; oscillatory integral; hypergeometric function; GKZ-system; singularity theory; gamma class H. Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds. \textit{Adv. Math.}, 222(2009), No.3, 1016-1079. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds An integral structure in quantum cohomology and mirror symmetry for toric 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 superb paper under review is quite (and probably unavoidably) technical. That is why the reviewer will not put a special effort to simplify the description of its content. Otherwise this short summary may turn useless for both non experts and specialists: the former because in any case would lack background, and the latter because the essential mathematical content of the paper would be lost behind excess of simplification. In any case, to save a minimum of friendly shape, let us begin this short summary as if it were a tale.
Schubert calculus. At the very beginning was the intersection theory of the complex Grassmann manifolds \(G(k,n)\) parametrizing \(k\)-dimensional subspaces of \({\mathbb C}^n\). Grassmannians, however, are a special kind of flag varieties, which are in turn special kind of homogeneous projective varieties, i.e. quotient \(G/B\) of a complex connected semi-simple Lie group modulo the action of a Borel subgroup \(B\). Thus, nowadays, the location Schubert calculus has acquired a much broader meaning. Not only for Grassmannians, but on general flag varieties, and not only classical cohomology, but also quantum, equivariant or quantum-equivariant, up to its \(K\)-theory and its connective \(K\)-theory (a theory interpolating, in a suitable sense, the \(K\)-theory and the intersection theory of a homogeneous space).
The paper under review puts itself in this very general framework using in a creative manner a new algebraic tool, what the authors call \textsl{formal root polynomials}, with the purpose of studying the elliptic cohomology of the homogeneous space \(G/B\): this means a cohomology theory where all the odd parts vanish and there is an invertible element \(h\in H^2\) inducing a complex orientation with the same formal group law as that of an elliptic curve. There is a correspondence between generalized cohomology theories and formal group laws, and in particular the authors investigate the \textsl{hyperbolic group law} introduced in Section 2.2. The corresponding Schubert calculus is so called by the authors \textsl{hyperbolic Schubert calculus.} and enables to study the elliptic cohomology of homogeneous spaces, extending previous work by Billey and Graham-Willems letting it to work uniformly in all Lie type. The definition of formal root polynomial is quite technical and is not worth to be recalled in the present review. However the idea is that of replacing, or rather extend, the notion of root polynomials heavily used by \textit{S. C. Billey} [Duke Math. J. 96, No. 1, 205--224 (1999; Zbl 0980.22018)] and by \textit{M. Willems} [Bull. Soc. Math. Fr. 132, No. 4, 569--589 (2004; Zbl 1087.19004)].
After introducing the formal root polynomial, whose definition depends on a reduced word for a Weyl group element, the main Theorem 3.10 states that indeed it does not depend on such a word provided that the formal group law is the hyperbolic one. Section 4 is devoted to applications: in particular the authors show how their techniques provide an efficient method to compute the transition matrix between two natural bases of the formal \textsl{Demazure algebra}, another gadget introduced and explained in Section 2. Section 5 is concerned with localization formulas in cohomology and \(K\)-theory, while section 6 is not only devoted to show further applications of root polynomials to compute Bott-Samelson classes, but also to propose a couple of conjectures in the hyperbolic Schubert calculus, based mainly on analogies and experimental evidence. The paper ends with a comprehensive reference list: among the key ones, the paper by Goresky, Kottwitz and MacPherson [\textit{M. Goresky} et al., Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)], the important 1974 paper by \textit{M. Demazure} [Ann. Sci. Éc. Norm. Supér. (4) 7, 53--88 (1974; Zbl 0312.14009)] on desingularization of generalized Schubert varieties, a couple of papers by Graham and Graham-Kumar on equivariant \(K\)-theory, and the papers by Billey and Willems, that inspired the research developed in this amazing step forward a generalized cohomology Schubert calculus. Schubert calculus; equivariant oriented cohomology; flag variety; root polynomial; hyperbolic formal group law ] C. Lenart and K. Zainoulline, Towards generalized cohomology Schubert calculus via formal root polynomials, arXiv:1408.5952. Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Equivariant \(K\)-theory, Generalized (extraordinary) homology and cohomology theories in algebraic topology, Bordism and cobordism theories and formal group laws in algebraic topology, Algebraic combinatorics Towards generalized cohomology Schubert calculus via formal root 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 Fix a simple Lie algebra \(\mathfrak{g}\) over \(\mathbb{C}\), and let \(T\) be a maximal torus in the associated simply-connected complex algebraic group \(G\).
The nonsymmetric Macdonald polynomials \(\{E_\lambda(q,t)\}_{\lambda\in P}\) form a distinguished basis for \(R(T)\otimes\mathbb{Q}(q,t),\) where \(R(T)\) is the representation ring of \(T\), \(P=\Hom(T,\mathbb{C}^\times)\) is its weight lattice, and \(q\) and \(t\) are indeterminates. (After choosing a basis for \(P\), one can identify \(R(T)\) with a Laurent polynomial ring over \(\mathbb{Z}\).)
Guided by one of the original motivations for the study of Macdonald polynomials -- namely, to interpolate between the major families of orthogonal polynomials in representation theory -- many efforts have been devoted to understanding the images of \(E_\lambda(q,t)\) under various specializations of \(q\) and \(t\). In this paper, the authors study, from the viewpoints of representation theory and geometry, the polynomials \(E_\lambda(q^{-1},\infty)\) obtained by sending \(t\to\infty\).
By an earlier combinatorial result of [the reviewer and \textit{M. Shimozono}, J. Algebr. Comb. 47, No. 1, 91--127 (2018; Zbl 1381.05089)], one knows that \[E_\lambda(q^{-1},\infty)\in R(T)\otimes \mathbb{Z}_{\ge 0}[q],\] i.e., the specialized polynomials have positive integral coefficients. A marvelous result of [\textit{S. Kato}, Math. Ann. 371, No. 3--4, 1769--1801 (2018; Zbl 1398.14053)] explains this positivity by realizing these (and related) polynomials in terms of graded characters of spaces of sections of sheaves on the semi-infinite flag manifold associated to \(G\), parallel to the classical Demazure character formula.
The first main result of the present paper (Theorem 1.2; cf. Corollary 3.27) gives, by explicit presentation similar in spirit of [\textit{E. Feigin} and \textit{I. Makedonskyi}, Sel. Math., New Ser. 23, No. 4, 2863--2897 (2017; Zbl 1407.17028)], a cyclic module \(U_{\lambda}\) over the Iwahori subalgebra in \(\mathfrak{g}\otimes_{\mathbb{C}}\mathbb{C}[z]\) such that its graded character \(\mathrm{gch}\,U_\lambda\) coincides with \(E_\lambda(q^{-1},\infty)\), up to some twists by the long element \(w_0\). This ``local'' result is upgraded to a ``global'' geometric statement in Theorem 1.3 (cf. Theorem 4.7), which gives presentations, as cyclic Iwahori modules \(\mathbb{U}_\lambda\), for the spaces of the sections mentioned in the previous paragraph.
Finally, under the additional assumptions that \(\mathfrak{g}\) is simply-laced and not of type \(E_8\), the third main result of this paper (Theorem 1.5; cf. Theorem 5.12) asserts that the modules \(\{\mathbb{U}_{-\lambda}\}\) are dual under an Ext-pairing to the family of level-one affine Demazure modules associated to \(\mathfrak{g}\). (The need for an affine Dynkin diagram automorphism in the proof excludes type \(E_8\) from consideration; however, Theorem 1.5 is conjectured to hold in type \(E_8\)). In Appendix A, which is quite informative, it is explained how Theorem 1.5 lifts the well-known orthogonality between nonsymmetric Macdonald polynomials (at \(t=0\) and \(t=\infty)\) to the level of representation categories. Finally, we note that Theorem 1.5 is closely related to the work [\textit{V. Chari} and \textit{B. Ion}, Compos. Math. 151, No. 7, 1265--1287 (2015; Zbl 1337.17016)] pertaining to specialized \textit{symmetric} Macdonald polynomials. Macdonald polynomials; current algebra; semi-infinite flag manifold Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Representation theory for linear algebraic groups, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Representation theoretic realization of non-symmetric Macdonald polynomials at infinity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(n\) and \(g\) be positive integers, and \({\mathbb K}\) be a field that contains a primitive \(n\)th root of unity \(\zeta_n\). The \textit{twisted} \(\text{Sl}_n\)\textit{-character variety} \({\mathcal M}(\text{Sl}_n({\mathbb K}))\) is given by all the classes of \(2g\)-tuples \((A_1,B_1,\dots,A_g,B_g)\) in \(\text{Sl}_n({\mathbb K})^{2g}\) satisfying the matrix equation
\[
[A_1,B_1]\cdots [A_g,B_g] = \zeta_n\text{Id}
\]
modulo simultaneous \(\text{PGL}_n({\mathbb K})\)-conjugation.
The \(E\)-polynomial of a variety \(X\) is
\[
E(q;X) := H_c(\sqrt{q},\sqrt{q},-1;X)
\]
where
\[
H_c(x,y,t;X) := \sum h_c^{p,q;j}(X) x^py^qt^j
\]
and the \(h_c^{p,q;j}(X)\) are the mixed Hodge numbers with compact support of \(X\).
The aim of the paper is to compute the \(E\)-polynomials of the \({\mathcal M}(\text{Sl}_n({\mathbb C}))\) for any genus \(g\) and any dimension \(n\). This is achieved by counting the number \(N_n(q)\) of points that these varieties have over the finite fields \({\mathbb F}_q\) (where \(n|q-1\)), and by showing (this is the main theorem of the paper) that the character varieties \({\mathcal M}(\text{Sl}_n({\mathbb C}))\) have a polynomial count, that is, that there exists a polynomial \(P(q)\in{\mathbb Z}[q]\) such that \(\#{\mathcal M}(\text{Sl}_n({\mathbb F}_q))=P(q)\) for sufficiently many prime powers \(q\).
The author also finds explicit formulas for the counting functions and for the \(E\)-polynomials; as corollaries, he proofs that: {\parindent=6mm \begin{itemize}\item[(i)] the \(E\)-polynomial of \({\mathcal M}(\text{Sl}_n({\mathbb C}))\) is palyndromic and monic; \item [(ii)] the character varieties \({\mathcal M}(\text{Sl}_n({\mathbb C}))\) are connected; \item [(iii)] the Euler characteristic of \({\mathcal M}(\text{Sl}_n({\mathbb C}))\) is 1 for \(g=1\) and \(\mu(n)n^{4g-3}\) for \(n\geq 2\).
\end{itemize}} character varieties; polynomial count; Euler characteristic Mereb, M.: On the e-polynomials of a family of character varieties. Technical report Geometric invariant theory, Linear algebraic groups over finite fields On the \(E\)-polynomials of a family of \({\mathrm {Sl}}_n\)-character 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 See the reviews in Zbl 1046.14009 and Zbl 1046.14010. Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) The Grothendieck-Teichmüller group and the adelic beta function | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Geometric Algebra and Calculus are mathematical languages that encode fundamental geometric relations that theories of physics must respect, and eliminate from our vocabulary those they do not. We propose criteria given which statistics of expressions in geometric algebra are computable in quantum theory, in such a way that preserves their algebraic character. They are that one must be able to arbitrarily transform the basis of the Clifford algebra, via multiplication by elements of the algebra that act trivially on the state space; all such elements must be neighbored by operators corresponding to factors in the original expression and not the state vectors. We explore the consequences of these criteria for a physics of dynamical multivector fields. geometric algebra; quantum theory; Clifford bundle; electroweak Quantum state spaces, operational and probabilistic concepts, 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), Clifford algebras, spinors On computable geometric expressions in quantum 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 A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers-connections on the projective line with extra structure [\textit{M. Aganagic} et al., Trans. Mosc. Math. Soc. 2018, 1--83 (2018; Zbl 1422.22021); translation from Tr. Mosk. Mat. O.-va 79, No. 1, 1--95 (2018)]. In this paper, the authors describe a deformation of this correspondence for SL(N). They take another more geometric approach, involving \(q\)-connections, a difference equation version of flat G-bundles. It is shown that the quantum/ classical duality between the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model may be viewed as a special case of the \(q\)-Langlands correspondence. geometric Langlands correspondence; Bethe ansatz equations; XXZ spin chain Exactly solvable models; Bethe ansatz, Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, Geometric Langlands program (algebro-geometric aspects), Groups and algebras in quantum theory and relations with integrable systems, Geometric Langlands program: representation-theoretic aspects \((\text{SL}(N), q)\)-opers, the \(q\)-Langlands correspondence, and quantum/classical 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 Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichmüller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surface topology and moduli spaces of Riemann surfaces, Grothendieck calls there for a recasting of topology, in order to make it fit to the objects of semialgebraic and semianalytic geometry, and in particular to the study of the Mumford-Deligne compactifications of moduli spaces. A new conception of manifold, of submanifold and of maps between them is outlined. We review these ideas in the present chapter, because of their relation to the theory of moduli and Teichmüller spaces. We also mention briefly the relations between Grothendieck's ideas and earlier theories developed by Whitney, Lojasiewicz and Hironaka and especially Thom, and with the more recent theory of o-minimal structures. moduli spaces; Mumford-Deligne compactifications; o-minimal structures Complex-analytic moduli problems, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Teichmüller theory for Riemann surfaces, Families, moduli of curves (analytic) On Grothendieck's tame topology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Authors' abstract: We generalize the topological recursion of Eynard-Orantin ([\textit{L. Chekhov} et al., J. High Energy Phys. 2006, No. 12, 053, 31 p. (2006; Zbl 1226.81250); \textit{B. Eynard} and \textit{N. Orantin}, Commun. Number Theory Phys. 1, No. 2, 347--452 (2007; Zbl 1161.14026)]) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic curve in the cotangent bundle \(T^\ast C\) of an arbitrary smooth base curve \(C\). We then prove that these spectral curves are quantizable, using the new formalism. More precisely, we construct the canonical generators of the formal \(\hbar\)-deformation family of \(D\) modules over an arbitrary projective algebraic curve \(C\) of genus greater than 1, from the geometry of a prescribed family of smooth Hitchin spectral curves associated with the \(\mathrm{SL}(2,\mathbb C)\)-character variety of the fundamental group \(\pi_1(C)\). We show that the semi-classical limit through the WKB approximation of these \(\hbar\)-deformed \(D\) modules recovers the initial family of Hitchin spectral curves. quantum curve; Hitchin fibration; family of spectral curves; Higgs field; topological recursion; WKB approximation O. Dumitrescu and M. Mulase, \textit{Lectures on the topological recursion for Higgs bundles and quantum curves}, arXiv:1509.09007 [INSPIRE]. Families, moduli of curves (analytic), Vector bundles on curves and their moduli, Relationships between algebraic curves and physics, Singular perturbations, turning point theory, WKB methods for ordinary differential equations, Topological field theories in quantum mechanics Quantum curves for Hitchin fibrations and the Eynard-Orantin 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 Young's lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young's lattice, we introduce a new partial order on weak compositions that we call the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomials, a nonsymmetric polynomial generalization of the Schur basis. Generalizing skew Schur functions, we define skew key polynomials in terms of this new poset. Using weak dual equivalence, we give a nonnegative weak composition Littlewood-Richardson rule for the key expansion of skew key polynomials, generalizing the flagged Littlewood-Richardson rule of \textit{V. Reiner} and \textit{M. Shimozono} [J. Comb. Theory, Ser. A 70, No. 1, 107--143 (1995; Zbl 0819.05058)]. Young tableaux; Young lattice; skew Schur functions; key poset Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of partitions of integers Skew key polynomials and a generalized Littlewood-Richardson rule | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We compute the \(\mathbb C^{\ast} \)-equivariant quantum cohomology ring of \(Y\), the minimal resolution of the DuVal singularity \(\mathbb C^2/G\) where \(G\) is a finite subgroup of \(SU(2)\). The quantum product is expressed in terms of an ADE root system canonically associated to \(G\). We generalize the resulting Frobenius manifold to nonsimply laced root systems to obtain an \(n\) parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov-Witten potential of \([\mathbb C^2/G]\). quantum cohomology; root system; ADE Bryan, J.; Gholampour, A.: Root systems and the quantum cohomology of ADE resolutions, (2007) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Root systems and the quantum cohomology of ADE resolutions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with \(q\)-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi-Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology. quantum black holes; M-theory; D-brane; quantum partition function; AdS/CFT correspondence; supergravity; Calabi-Yau threefolds; elliptic cohomology Bytsenko, A. A.; Chaichian, M.; Szabo, R. J.; Tureanu, A., Quantum black holes, elliptic genera and spectral partition functions, Int. J. Geom. Methods Mod. Phys., 11, 145008, (2014) Black holes, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Quantization of the gravitational field, String and superstring theories in gravitational theory, Calabi-Yau manifolds (algebro-geometric aspects), Cohomological dimension of fields, Classical or axiomatic geometry and physics, Supergravity, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Quantum black holes, elliptic genera and spectral partition 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 The aim of the book is to make an attempt to expose a new unifying approach to the study of numerous quantized algebras. This approach is based on investigations of noncommutative schemes in categories. The point of departure in the first chapter of the book is the notion of a left spectrum \(\text{Spec}_l R\) of an associatve ring \(R\) with unity. It consists of all left ideals \(P\) in \(R\) with the following property. For any element \(x\in R\setminus P\) there exists a finitely generated additive subgroup \(H\) in \(R\) such that \(zHx\subseteq P\) where \(z\in R\) implies \(z\in P\). The set \(\text{Spec}_l R\) is nonempty since it contains the set \(\text{Max}_l R\) of all maximal left ideals and the set \(\text{Spec}_l' R\) of all completely prime left ideals. If \(P,P'\in \text{Spec}_l R\) then \(P\leq P'\) if there exists a finitely generated additive subgroup \(H\) in \(R\) such that \(zH\subseteq P\) implies \(z\in P'\). There exists a topology \(\tau_*\) on \(\text{Spec}_l\) with the base \(\{\bigcup_{P'\geq P} P'\mid P\in \text{Spec}_l R\}\). There exists a topology \(\tau^*\) on \(\text{Spec}_l R\) determined by the base of closed subsets of the form \(\{p\in \text{Spec}_l R\mid p\leq M\}\) where \(M\) runs through the set of all proper ideals in \(R\). Flat localizations of Abelian categories are discussed. A stability theorem with respect to localizations is proved for the left spectrum. Special categories of rings are selected. (Pre)images of morphisms in these categories preserve preorder \(\leq\). It is shown that the intersection of all elements of \(\text{Spec}_l R\) is the Levitzki radical of \(R\). In particular the topological space \(\text{Spec}_l R\) has a base of quasi-compact open sets. Structure presheaves of modules over rings, noncommutative quasi-affine schemes and projective spectra are introduced.
In the second chapter all main `small' quantized rings are introduced: the quantum plane \(k_q[X, Y]\), the algebra of \(q\)-differential operators \(D_{q,h}\), quantum Heisenberg and Weyl algebra \(W_{q, 1}\), the quantum envelope \(U_q(sl(2))\), the coordinate ring \(M_q(2)\) of \(2\times 2\) matrices, the coordinate ring \(A(SL_q(2))\), twisted \(SL(2)\) group, \(W_v(sl(2))\) of Woronovicz. The goal of chapter 2 is to develop the representation theory of these algebras. It is shown that all these algebras are specializations of the hyperbolic ring \(A\{\theta, \zeta\}\) which is generated by a commutative ring \(A\) with fixed \(\theta\in \Aut A\), \(\zeta\in A\), and by elements \(x\), \(y\) with defining relations
\[
xa= \theta(a) x,\quad ay= y\theta(a),\quad a\in A;\quad xy= \zeta,\quad yx= \theta^{- 1}(\zeta).
\]
It is worth to mention that the theory of these and even more general classes of algebras under the name of generalized Weyl algebras is developed by \textit{V. V. Bavula} [Generalized Weyl algebras (Bielefeld, preprint, 1994)]. Unfortunately, it is not mentioned in the book.
In section 1 of chapter 2 the left spectrum of a skew polynomial ring \(A[X, \theta]\) is studied. These results are applied to the quantum plane. Section 3 contains an almost complete description of \(\text{Spec}_l R\) of a hyperbolic ring \(R\). This description is applied to the rings mentioned above.
Chapter 3 is devoted to the categorical point of view on geometrical objects. The author introduces the notion of the spectrum of an Abelian category, studies its behaviour with respect to localizations, Serre subcategories, Grothendieck categories, local Abelian categories, localizations at points, topologies on categories.
Chapter 4 contains generalization of the notion of a hyperbolic ring. Namely the author introduces the notion of a hyperbolic category over an Abelian category. The main result of the chapter is the description of a hyperbolic category which is naturally related to the spectrum of the underlying Abelian category. This result generalizes similar results from chapter 2.
Chapter 5 is concerned with skew PBW monads in a monoidal category \(A\). A skew PBW monad is a generalization to monads of the notion of a hyperbolic ring. Other examples of skew PBW monads are related to Kac-Moody and Virasoro Lie algebras. The author studies semigroup-graded monads \(F\) and their spectra. The main result of the chapter shows that all points of \(F\)-modules which `grow up' over a given point of \(\text{Spec } A\) can be represented by a graded module with the grading associated to this point. The chapter ends with considerations of quasi-holonomic modules, \(F\)-comodules, spectra of Weyl algebra and their quantizations, representations of Kac-Moody algebra, two-parameter deformations of the algebras \(M(2)\), \(GL(2)\).
In chapter 6 the author surveys major approaches to noncommutative spectral theory such as injective spectrum by Gabriel, Goldman's spectrum of a ring, affine scheme by F. Van Oystaeyen and A. Verschoren, Cohn's affine scheme. It is shown that these spectra can be deduced from the case considered in the book. The chapter ends with exposing properties of Gabriel-Krull dimension and a calculations of dimensions of some spectra.
The last chapter 7 is concerned with the projective spectrum of graded monads. Affine and projective fibres, blowing up and related topics are considered. flat localizations of Abelian categories; structure presheaves of modules; quantized algebras; noncommutative schemes in categories; left spectrum; maximal left ideals; completely prime left ideals; categories of rings; Levitzki radical; quasi-affine schemes; projective spectra; quantized rings; quantum planes; algebra of \(q\)-differential operators; Weyl algebras; quantum envelopes; coordinate rings; generalized Weyl algebras; skew polynomial rings; Serre subcategories; Grothendieck categories; hyperbolic rings; skew PBW monads; monoidal category; Kac-Moody and Virasoro Lie algebras; semigroup-graded monads; Gabriel-Krull dimension Rosenberg, A.L.: Algebraic Geometry Representations of Quantized Algebras. Kluwer Academic Publishers, Dordrecht, Boston London (1995) Research exposition (monographs, survey articles) pertaining to associative rings and algebras, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Quantum groups (quantized enveloping algebras) and related deformations, Noncommutative algebraic geometry, Torsion theories; radicals on module categories (associative algebraic aspects), Rings of differential operators (associative algebraic aspects), Local categories and functors, Abelian categories, Grothendieck categories, Graded rings and modules (associative rings and algebras), Associative rings of functions, subdirect products, sheaves of rings, ``Super'' (or ``skew'') structure, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Abstract manifolds and fiber bundles (category-theoretic aspects) Noncommutative algebraic geometry and representations of quantized 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 \(U\) be a regular connected affine semilocal scheme over a field \(k\). Let \(\boldsymbol{G}\) be a reductive group scheme over \(U\). Assuming that \(\boldsymbol{G}\) has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine \(k\)-scheme \(W\), a principal \(\boldsymbol{G}\)-bundle over \(W\times_kU\) is trivial if it is trivial over the generic fiber of the projection \(W\times_kU\to U\).
We also simplify the proof of the Grothendieck-Serre conjecture: let \(U\) be a regular connected affine semilocal scheme over a field \(k\). Let \(\boldsymbol{G}\) be a reductive group scheme over \(U\). A principal \(\boldsymbol{G}\)-bundle over \(U\) is trivial if it is trivial over the generic point of \(U\).
We generalize some other related results from the simple simply connected case to the case of arbitrary reductive group schemes. algebraic groups; principal bundles; local schemes; Grothendieck-Serre conjecture; affine Grassmannians Group schemes On the Grothendieck-Serre conjecture about principal bundles and its generalizations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(p\) be a multilinear polynomial in several noncommuting variables with coefficients in an arbitrary field \(K\). Kaplansky conjectured that for any \(n\), the image of \(p\) evaluated on the set \(M_n(K)\) of \(n\) by \(n\) matrices is a vector space. In this paper, we settle the analogous conjecture for a quaternion algebra. noncommutative polynomials; Kaplansky conjecture; quaternion algebra Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Noncommutative algebraic geometry, Ordinary and skew polynomial rings and semigroup rings The images of noncommutative polynomials evaluated on the quaternion algebra | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Gröbner bases) and geometry (via quiver theory). Gröbner bases serve as effective models for computation in algebras of various types. Although the theory of Gröbner bases was developed in the second half of the 20-th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced -- with big impact -- in the 1990s.
Divided into two parts, the book first discusses the theory of Gröbner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Gröbner bases to analysis on systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line.
While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.
The articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to associative rings and algebras, Proceedings, conferences, collections, etc. pertaining to statistics, Collections of articles of miscellaneous specific interest, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Vector bundles on curves and their moduli, Geometric invariant theory, Group actions on varieties or schemes (quotients), Free, projective, and flat modules and ideals in associative algebras, Representations of quivers and partially ordered sets, Quadratic and Koszul algebras, Theories (e.g., algebraic theories), structure, and semantics, 2-categories, bicategories, double categories, Resolutions; derived functors (category-theoretic aspects), Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain, Other special methods applied to PDEs, Momentum maps; symplectic reduction, Exterior differential systems (Cartan theory), Grammars and rewriting systems Two algebraic byways from differential equations: Gröbner bases and quivers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 report on the calculation of some quantum invariants, including Gromov-Witten invariants and FJRW invariants, via tautological relations on the moduli space of stable pointed curves. quantum invariants; tautological relations Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Singularities in algebraic geometry Quantum invariants via tautological relations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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, via Newton polyhedra, we define and study symmetric matrix polynomials which are nondegenerate at infinity. From this, we construct a class of (not necessarily compact) semialgebraic sets in \(\mathbb{R}^n\) such that for each set \(K\) in the class, we have the following two statements: (i) the space of symmetric matrix polynomials, whose eigenvalues are bounded on \(K\), is described in terms of the Newton polyhedron corresponding to the generators of \(K\) (i.e., the matrix polynomials used to define \(K\)) and is generated by a finite set of matrix monomials; and (ii) a matrix version of Schmüdgen's Positivstellensätz holds: every matrix polynomial, whose eigenvalues are ``strictly'' positive and bounded on \(K\), is contained in the preordering generated by the generators of \(K\). matrix polynomials; positivstellensätze; Newton polyhedra; nondegeneracy Real algebraic sets, Sums of squares and representations by other particular quadratic forms, Real algebra, Convex sets and cones of operators, Free algebras, Approximation by polynomials, Semialgebraic sets and related spaces A note on nondegenerate matrix polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Orlik-Solomon algebra of a matriod \(\mathcal{M}\) on the ground set \(\{1,2,\dots, n\}\) is the quotient of the exterior algebra on the points by the ideal \(\mathfrak{I}(\mathcal{M})\) generated by the circuits of \(\mathcal{M}\). The isomorphism between the Orlik-Solomon algebra of a complex matroid and the complement of a complex arrangement of hyperplanes was established by \textit{P. Orlik} and \textit{L. Solomon} [Math. Ann. 261, 339--357 (1982; Zbl 0491.51018)].
In the article under review a generalization of Orlik-Solomon algebras, called \(\chi\)-algebra, is considered. Examples of \(\chi\)-algebras also include Orlik-Solomon-Terao algebras [\textit{P. Orlik} and \textit{H. Terao}, Nagoya Math. J. 134, 65--73 (1994; Zbl 0801.05019)] and Cordovil algebras [\textit{R. Cordovil}, Discrete Comput. Geom. 27, No. 1, 73--84 (2001; Zbl 1016.52014)]. After the definition of \(\chi\)-algebras and the introduction of examples, special bases of the \(\chi\)-algebra are obtained from the ``no broken circuit'' sets of \(\mathcal{M}\) and corresponding basis for the ideal \(\mathfrak{I}_{\chi} (\mathcal{M})\). Furthermore, the reduced Gröbner basis for \(\mathfrak{I}_{\chi} (\mathcal{M})\) is constructed. Finally, following \textit{A. Szenes} [Int. Math. Res. Not. 1998, No. 18, 937--956 (1998; Zbl 0968.11015)], the so called diagonal basis for the \(\chi\)-algebra is defined and constructed. The results are illustrated by a small six point example. arrangement of hyperplanes; broken circuit; cohomology algebra; matroid; oriented matroid; Orlik-Solomon algebra Combinatorial aspects of matroids and geometric lattices, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), de Rham cohomology and algebraic geometry Gröbner and diagonal bases in Orlik-Solomon type 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 Motivated by the recent surge of interest in matrix-valued polynomials in optimization, the author of the paper under review considers positivity of matrix-valued polynomials in commuting variables. His main result is a generalization of Krivine's Positivstellensatz from real algebraic geometry and the proof is a slick application of Schur complements. positive polynomial; matrix polynomial; real algebraic geometry Cimprič, J., Strict positivstellensätze for matrix polynomials with scalar constraints, Linear algebra appl., 434, 8, 1879-1883, (2011) Real algebra, Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), Semialgebraic sets and related spaces Strict positivstellensätze for matrix polynomials with scalar constraints | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The goal of paper is to give an explicit description of the universal factorization algebra of the generic polynomial of degree \(n\) into product of two monic polynomials, one of degree \(r\),as a presentation of Lie algebra of \(n\times n\) matrices with polynomial entries. This is related to bosonic vortex presentation of infinite Lie algebra due to Date and Jimbo. Hasse-Schmidt derivation; exterior algebra; universal factorization algebra; Giambelli's formula; Bosonic and Fermionic representations by Date, Jimbo, Kashiwara and Miwa Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Universal factorization algebras of polynomials represent Lie algebras of endomorphisms | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is primarily concerned with questions about complete and non-projective toric varieties. The authors note early on that many examples of complete and non-projective varieties are ad-hoc variations of the classical example of Oda, and highlight a general lack of understanding as to how non-projective varieties arise among complete toric varieties. The authors highlight the need for more examples to better understand this aspect of the theory, and given the difficulty of these constructions point to the need to adopt a computer-aided approach. To this end, two algorithms are prposed for computing complete \(\mathbb{Q}\)-factorial fans over a set of vectors.
The first algorithm proposed is inefficient but of theoretical interest, leading to a conjecture and several possible applications. The second algorithm relies on the theory of Gröbner bases, where the authors deal with toric ideals arising from complete configurations, which cannot be homogeneous. The results here lead to an efficient algorithm for computing all projective complete simplicial fans with a given 1-skeleton that is based on the computation of the Gröbner fan of the associated toric ideal.
The paper concludes by presenting several nice concrete examples and outlining many opportunities and directions for future work. toric varieties; Gale duality; Gröbner fan; secondary fan; initial ideals; toric ideals Toric varieties, Newton polyhedra, Okounkov bodies, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Software, source code, etc. for problems pertaining to algebraic geometry Toric varieties and Gröbner bases: the complete \(\mathbb{Q}\)-factorial case | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathcal O_q(M_{m,n}(k))\) be a generic coordinate \(k\)-algebra on rectangular matrices of size \(m\times n\). It is assumed that \(m\leqslant n\). If \(I=\{i_1<\cdots<i_u\}\), \(K=\{k_1<\cdots<k_v\}\subset M=\{1,\dots,m\}\) and \(J=\{j_1<\cdots<j_u\}\), \(L=\{l_1<\cdots<l_v\}\subset\{1,\dots,n\}\), then \((I,J)\leqslant_{st}(K,L)\) if and only if \(u\leqslant v\) and \(i_s\leqslant k_s\), \(j_s\leqslant l_s\) for all possible indices \(s\). Denote by \(\Pi_{m,n}\) the set of all pairs of indices \((I,J)\) in which \(u=m\).
The quantum Grassmannian \(\mathcal O_q(G_{m,n}(k))\) is the subalgebra in \(\mathcal O_q(M_{m,n}(k))\) generated by all \(m\times m\) quantum minors. In the authors' previous paper [J. Algebra 301, No. 2, 670-702 (2006; Zbl 1108.16026)], it is shown that there is a vector basis in \(\mathcal O_q(G_{m,n}(k))\) consisting of monomials \([I_1\mid M]\cdots [I_t\mid M]\) such that \((I_1,M)\leqslant_{st}\cdots\leqslant_{st}(I_t,M)\).
Take \(\gamma\in\Pi_{m,n}\) and denote by \(\Pi_{m,n}^\gamma\) the set of all \(\alpha\in\Pi_{m,n}\) such that \(\alpha\ngeqslant_{st}\gamma\). The quantum Schubert variety \(\mathcal O_q(G_{m,n}(k))_\gamma\) associated with \(\gamma\) is the algebra \(\mathcal O_q(G_{m,n}(k))\) factorized by the ideal generated by \(\Pi_{m,n}^\gamma\). It is shown that
\[
\mathcal O_q(G_{m,n}(k))[Y^{\pm 1};\varphi]\simeq\mathcal O_q(G_{m,n}(k))_\gamma[\gamma^{-1}].
\]
It is given a criterion under which a quantum Schubert variety has left and right finite injective dimensions. Let \(I_t\) be the ideal in \(\mathcal O_q(M_{m,n}(k))\) generated by all minors of a fixed size \(t\). It is shown that \(\mathcal O_q(M_{m,n}(k))/I_t\) is a normal domain. rings arising in quantum group theory; generic coordinate algebras; quantum Grassmannians; quantum minors; quantum Schubert varieties; normal domains T. H. Lenagan and L. Rigal, Quantum analogues of Schubert varieties in the Grassmannian, Glasg. Math. J. 50 (2008), no. 1, 55 -- 70. , Grassmannians, Schubert varieties, flag manifolds, Divisibility, noncommutative UFDs, Rings arising from noncommutative algebraic geometry, Noetherian rings and modules (associative rings and algebras) Quantum analogues of Schubert varieties in the Grassmannian. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The categorification program was initiated by I. Frenkel with the aim of extending 3-dimensional topological field theories to dimensions 4 and higher [\textit{L. Crane} and \textit{I. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)]. This program was extended by the first author in his work on categorified tangle invariants [\textit{M. Khovanov}, Algebr. Geom. Topol. 2, 665--741 (2002; Zbl 1002.57006)].
The present paper is a part of ongoing research by the authors on categorification of quantum groups and their representations. A categorification of quantum \(sl(2)\) obtained previously by the second author is generalised to \(sl(n)\). More precisely the authors construct a linear 2-category whose Grothendieck category coincides with the idempotent form of quantum \(sl(n)\). Note that the interpretation of elements of Lustig's canonical basis of the idempotent form as classes of indecomposible ob jects established for \(sl(2)\) is still an open problem for \(sl(n)\). This category has potential applications in representation theory. The authors expect this category to manifest itself as a symmetry of various categories of interest in representation theory, ranging from derived categories of coherent sheaves on quiver varieties to categories of modules over cyclotomic and degenerate affine Hecke algebras. categorification; quantum group; quantum \(sl(n)\); iterated flag variety; 2-representation; 2-category Khovanov, M.; Lauda, A., A categorification of quantum \(\mathfrak{sl}_n\), Quantum Topol., 1, 1, 1-92, (2010) Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Grassmannians, Schubert varieties, flag manifolds, Ring-theoretic aspects of quantum groups A categorification of quantum \(\text{sl}(n)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Denote by \({\mathfrak M}_n\) the moduli space of semi-stable coherent sheaves \(F\) over the complex projective plane \(\mathbb{P}^2 (\mathbb{C})\) with \(\text{rank} (F)=2\), \(c_1 (F)=0\) and \(c_2(F) =n\). It is well-known that \({\mathfrak M}_n\) is a normal, irreducible projective variety of dimension \(4n-3\). In this particular case of the projective plane as base variety, the differential-geometrically defined Donaldson polynomial \(q_d(u)\) of degree \(d= 4n-3\), evaluated at the positive generator \(u\in H_2 (\mathbb{P}^2, \mathbb{Z})\), gives the so-called Donaldson number \(q_d\) which can be explicitly computed in terms of the ample generator \(D_n\) of \(\text{Pic} ({\mathfrak M}_n)\).
In the present paper, the author derives numerical relations between the Donaldson numbers \(q_d= q_{4n-3}\) and certain intersection numbers for suitable finite subschemes of length \(n+\ell^2\) in the Hilbert scheme \(\text{Hilb}^{n+ \ell^2} (\mathbb{P}^2 (\mathbb{C}))\), where the positive integers \(\ell\) are to satisfy the inequality \(n\leq (\ell+1) \cdot (\ell+2)\).
The author's strategy consists in using the moduli spaces for coherent systems (of coherent algebraic sheaves), established by himself recently [cf. \textit{J. Le Potier}, in: Vector bundles in algebraic geometry, Proc. Symp. Durham, Lond. Math. Soc. Lect. Note Ser. 208, 179-239 (1995; Zbl 0847.14005)], and in interpreting the Donaldson number \(q_d\) by means of certain integrals over the moduli spaces of coherent systems of a specific type depending on \(\ell\in \mathbb{N}\) with \((\ell+1) \cdot (\ell+2) \geq n\). This method provides the explicit values of the Donaldson numbers \(q_{4n-3}\) for \(2\leq n\leq 6\), thus re-establishing the already known results for \(2\leq n\leq 4\) and adding the new formulae for \(n=5\) and \(n=6\).
The paper is written in a very careful, lucid, detailed and nearly self-contained manner. Section 1 recalls the basics on coherent systems and their moduli spaces, while section 2 summarizes some new results in the infinitesimal study of the moduli spaces of coherent systems, basically due to \textit{Min He} [Thèse d'État, Univ. Paris VII (in preparation)]. Section 3 provides a (theoretic) method for computing the codimension of the Harder-Narasimhan strata for complete families of coherent systems. All this is applied in section 4, which is devoted to the intersection theory of certain moduli spaces of coherent systems within an ambient Hilbert scheme of \(\mathbb{P}^2 (\mathbb{C})\), and the results obtained here are then applied, in the concluding section 5, to the explicit computation of the Donaldson numbers \(q_{4n-3}\), for \(2\leq n\leq 6\), as integrals over moduli spaces of certain coherent systems.
In the course of the paper, the author gives several hints to the related work by others, in particular to the recent results of \textit{G. Ellingsrud} and \textit{S. A. Strømme} [cf. J. Am. Math. Soc. 9, No. 1, 175-193 (1996; Zbl 0856.14019)]. moduli space of semi-stable coherent sheaves; Donaldson numbers; intersection numbers J. Le Potier, ``Systèmes cohérents et polynômes de Donaldson'' in Moduli of Vector Bundles (Sanda, Japan, 1994; Kyoto, Japan, 1994) , Lecture Notes in Pure and Appl. Math. 179 , Dekker, New York, 1996, 103--128. Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Étale and other Grothendieck topologies and (co)homologies Coherent systems and Donaldson polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Grothendieck conjecture states that a linear differential equation with coefficients from \(\mathbb Q(x)\) admits a basis of algebraic solutions, if and only if, for almost all primes \(p\), its reduction \(\mod p\) has the same property. If an equation with coefficients from \(\mathbb F_p(x)\) is considered, then such a basis exists if an associated \(p\)-curvature vanishes. A more general conjecture by \textit{N. Katz} [Invent. Math. 18, 1--118 (1972; Zbl 0278.14004)] is reduced to the Grothendieck conjecture if almost all the \(p\)-curvatures vanish.
The first result of the present paper proves the general conjecture of Katz for locally symmetric varieties associated to a connected, reductive, adjoint algebraic group \(G\) over \(\mathbb Q\) with \(G=G_1\times\ldots\times G_m\) such that \(G_i, 1\leq i\leq m,\) has \(\mathbb R\)-rank \(\geq 2\) and is of type \(A,B,C,\) or \(D\). Note that this result in particular applies to Hilbert-Blumenthal modular varieties and the moduli spaces \(\mathcal{A}_g,\, g\geq 1\) of pricipally polarized abelian varieties.
In case all the factors \(G_i\) have \(\mathbb Q\)-rank \(\geq 1\) as well, the conjecture of Katz-Grothendieck is also proved for groups of exceptional type.
These results are proved by combining different results: On the one hand, the authors prove new properties of the structure of vector bundles on locally symmetric varieties, which in turn are based on Margulis' rigidity results on lattices in semisimple Lie groups from [\textit{G. A. Margulis}, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 17. (Berlin) etc.: Springer-Verlag (1991; Zbl 0732.22008)], namely the superrigidity theorem and the normal subgroups theorem. On the other hand, the authors use earlier results of Katz and André [\textit{Y. André}, Adolphson, Alan (ed.) et al., Geometric aspects of Dwork theory. Vol. I, II. Berlin: Walter de Gruyter. 55-112 (2004; Zbl 1102.12004)] for the Grothendieck conjecture concerning Picard-Fuchs equations. Grothendieck-Katz conjecture; \(p\)-curvature; Hilbert-Blumenthal modular varieties; moduli spaces Modular and Shimura varieties, \(p\)-adic differential equations Rigidity, locally symmetric varieties, and the Grothendieck-Katz conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Noncommutative tori associated with \(C^*\)-algebras are studied in the article. Bridge seminorms for \(C^*\)-algebras are considered. The author proves that curved noncommutative tori are Leibniz quantum compact metric spaces. Moreover, it is shown that they form a continuous family over the group of invertible matrices. The latter is taken with entries in the image of the quantum tori supplied with a conjugation operator in the regular representation. The considered group is endowed with a length function. noncommutative torus; Leibniz quantum space; compact metric space; regular representation; \(C^*\)-algebra Latrémolière, F., Curved noncommutative tori as Leibniz compact quantum metric spaces, J. Math. Phys., 56, 12, (2015) Noncommutative algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Compact (locally compact) metric spaces, Noncommutative geometry in quantum theory, Theory of matrix inversion and generalized inverses Curved noncommutative tori as Leibniz quantum compact metric spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a Grothendieck-Lefschetz theorem for equivariant Picard groups of non-singular varieties with finite group actions. Picard groups, Group actions on varieties or schemes (quotients), Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants A Grothendieck-Lefschetz theorem for equivariant Picard groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be an \(r\)-dimensional smooth complex quasiprojective variety and denote the Hilbert scheme parametrizing zero-dimensional subschemes of length \(n\) of \(X\) by \(\text{Hilb}^nX\). A nested Hilbert scheme on \(X\) is defined to be a scheme of the form
\[
Z_{\mathbf n}(X):=\{(Z_1, Z_2,\dots, Z_m) : Z_i \in \text{Hilb}^{n_i}X
\]
and \(Z_i\) is a subscheme of \(Z_j\) if \(i < j\},\) where the symbol \(\mathbf n\) is used as a shorthand for the \(m\)-tuple \((n_1, n_2,\dots,n_m)\). If \(({\mathcal U}_1, {\mathcal U}_2,\dots, {\mathcal U}_m)\) is the universal element over the nested Hilbert scheme \(Z_{\mathbf n}(X)\), we call the scheme \({\mathcal U}_1\times_{Z_{\mathbf n}(X)}{\mathcal U}_2 \times_{Z_{\mathbf n}(X)}\cdots \times_{Z_{\mathbf n}(X)} {\mathcal U}_m\) the universal family over \(Z_{\mathbf n}(X)\). \textit{J. Cheah} [J. Algebr. Geom. 5, No. 3, 479-511 (1996)], expressed the virtual Hodge polynomials of the smooth Hilbert schemes \(\text{Hilb}^n X\) in terms of that of \(X\). In the paper under review we indicate how the arguments of the cited paper can be modified to express the virtual Hodge polynomials of all the smooth nested Hilbert schemes (and those of their universal families when \(r\geq 2\)) in terms of that of \(X\). More generally, we obtain the virtual Hodge polynomials of the schemes
\[
\begin{cases} \text{Hilb}^nX,\\ Z_{n-1,n}(X),\\ {\mathcal F}_n(X), \\ {\mathcal F}_{n-1,n}(X),\\ \{(P,Z_1,Z_2)\in X\times \text{Hilb}^{n-1}X\times \text{Hilb}^n X: \\ \qquad P \text{ lies in the support of }Z_2, Z_1 \text{ is a subscheme of }Z_2\} \\ \{(P, Q, Z) \in X \times X\times \text{Hilb}^nX: P \text{ and }Q\text{ lie in the support of }Z\}\end{cases}\tag{1}
\]
in terms of the virtual Hodge polynomial of \(X\) and those of the reduced schemes
\[
\text{Hilb}^k (\mathbb{A}^r,0) = \{Z \in \text{Hilb}^k\mathbb{A}^r : Z\text{ is supported at the origin\}}
\]
and
\[
{\mathcal Z}_{k-1,k}(\mathbb{A}^r, 0)=\{(Z_1, Z_2) \in \text{Hilb}^{k-1}(\mathbb{A}^r,0)\times \text{Hilb}^k(\mathbb{A}^r,0): Z_1\text{ is a subscheme of }Z_2\}.
\]
When \(r= 2\) or \(n\geq 3\), the virtual Hodge polynomials of the spaces listed in (1) can be given purely in terms of that of \(X\). Note that when \(r\geq 2\), this includes all the smooth nested Hilbert schemes \(Z_{\mathbf n}(X)\) and their universal families. If \(r = 1\), the schemes \(Z_{\mathbf n}(X)\) are products of symmetric powers of \(X\) and their virtual Hodge polynomials are easily determined using the formula for the virtual Hodge polynomials of symmetric powers given in the paper cited above.
From the equations of virtual Hodge polynomials, we obtain for free analogous equations of virtual Poincaré polynomials and Euler characteristics. In fact, since the Euler characteristics of the spaces \(\text{Hilb}^k(\mathbb{A}^r,0)\) and \({\mathcal Z}_{k-1,k}(\mathbb{A}^r,0)\) can be expressed in terms of the numbers of certain higher dimensional partitions, the Euler characteristics of the schemes listed in (1) are expressible in terms of these numbers and the Euler characteristic of \(X\). If \(X\) is projective, we also obtain formulae giving the Hodge (resp. Poincaré) polynomials of the smooth nested Hilbert schemes in terms of that of \(X\). virtual Hodge polynomials; nested Hilbert schemes; Poincaré polynomials; Euler characteristic J. Cheah, ''The Virtual Hodge Polynomials of Nested Hilbert Schemes and Related Varieties,'' Math. Z. 227, 479--504 (1998). Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The virtual Hodge polynomials of nested Hilbert schemes and related 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 obtain an interesting realization of the de Rham cohomological operators of differential geometry in terms of the noncommutative \(q\)-superoscillators for the supersymmetric quantum group \(GL_{qp}(1|1)\). In particular, we show that a unique quantum superalgebra, obeyed by the bilinears of fermionic and bosonic noncommutative \(q\)-(super)oscillators of \(GL_{qp}(1|1)\), is exactly identical to that obeyed by the de Rham cohomological operators. A set of discrete symmetry transformations for a set of \(GL_{qp}(1|1)\) covariant quantum superalgebras turns out to be the analogue of the Hodge duality * operation of differential geometry. A connection with an extended Becchi-Rouet-Stora-Tyutin (BRST) algebra obeyed by the conserved and nilpotent (anti-)BRST and (anti-)co-BRST charges, the conserved ghost charge and a conserved bosonic charge (which is equal to the anticommutator of (anti-)BRST and (anti-)co-BRST charges) is also established. Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Quantization in field theory; cohomological methods, Noncommutative geometry methods in quantum field theory, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Supersymmetric field theories in quantum mechanics, Transcendental methods, Hodge theory (algebro-geometric aspects), Superalgebras Cohomological operators and covariant quantum superalgebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review develops a theory of polynomial representations of the super general linear group \(\mathrm{GL}(m|n,A)\), defined over an arbitrary commutative superalgebra \(A\). The methods used adapt and parallel Green's approach to the usual Schur algebra via comodules, as presented in \S 2 of [\textit{J. A. Green}, Polynomial representations of \(\mathrm{GL}_n\). With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green and M. Schocker. 2nd corrected and augmented edition. Berlin: Springer (2007; Zbl 1108.20044)]. Thus, rather than work directly with super modules for \(\mathrm{GL}(m|n,A)\), the author first defines, for each suitable sub-super coalgebra \(D\) of the super algebra of finitary functions \(\mathrm{GL}(m|n,A)\), a module category \(\mathrm{mod}_D(A\mathrm{GL}(m|n,A))\) of representations of the super group algebra \(A\mathrm{GL}(m|n,A)\) such that the coefficient functions of the representing matrices lie in \(D\). There is an equivalence of categories
\[
\mathrm{mod}_D(A\mathrm{GL}(m|n,A)) \simeq \mathrm{com}(D)
\]
between this category and the category of super \(D\)-comodules. Taking \(D\) to be the super coalgebra of polynomial functions \(\mathrm{GL}(m|n,A)\) of degree \(r\) and dualizing, the author obtains a superalgebra \(S_A(m|n,r)\) that is the super analogue of the usual Schur algebra. A key result on the Schur algebra is that if \(F\) is an infinite field then category of polynomial representations of \(\mathrm{GL}(n,F)\) of polynomial degree \(r\) is equivalent to the category of representations of the Schur algebra \(S_F(n,r)\), defined over the field \(F\). The author proves the analogous result for representations of \(S_A(m|n,r)\) in Proposition 4.2.
The main result of this paper, described as `super Schur duality' is Theorem 5.1. In it \(A\) is a commutative superalgebra over an infinite field \(F\) and \(E_A = A^{m|n}\) is a free rank \(m|n\) \(A\)-module, generated by \(m\) even elements and \(n\) odd elements. The free \(A\)-module \(E_A^{\otimes r}\) is acted on by \(\mathrm{GL}(m|n,A)\) with coefficient functions of polynomial degree \(r\). It may therefore be regarded as a representation of \(S_A(m|n,r)\). The symmetric group \(S_r\) acts on \(E_A^{\otimes r}\) by permuting the factors (with signs coming from the super structure). Theorem 5.1 states that the natural action map
\[
S_A(m|n,r) \rightarrow \mathrm{End}_A(E_A^{\otimes r}) \tag{\(\star\)}
\]
is injective, and its image is precisely those \(A\)-endomorphisms of \(E_A^{\otimes r}\) which commute with the action of the symmetric group \(S_r\). This is the super version of Schur's theorem (see [loc. cit., Theorem 2.6c]) that \(S_F(n,r) \cong \mathrm{End}_{S_r}(V^{\otimes r})\), where \(V\) is the natural \(\mathrm{GL}_n(F)\)-module. As is the case with Schur's theorem, the main force of the result is that \emph{every} \(S_r\) endomorphism of the tensor algebra comes from an element of the super Schur algebra, and so is induced by the action of a suitable linear combination of elements in the super group algebra \(A\mathrm{GL}(m|n,A)\). An important corollary is that if \(F\) has infinite characteristic or prime characteristic \(p > r\) then \(S_A(m|n,r)\) is semisimple.
The author begins with a brief but useful survey of other approaches to Schur-Weyl duality, emphasising that the main novel feature in his paper is to work with the supergroup \(\mathrm{GL}(m|n)\) rather than its super Lie algebra \(\mathfrak{gl}(m|n)\). In this connection we mention that modules for \(S_A(m|n,r)\) are direct sums of the special class of covariant \(\mathfrak{gl}(m|n)\)-modules: see Chapter 3 of Moens' Ph.D.~thesis [Supersymmetric Schur functions and Lie superalgebra representations. Universiteit Gent (2007)] for an excellent introduction. In general, and in contrast to the case for \(S_A(m|n,r)\), modules for \(\mathfrak{gl}(m|n)\) are not completely reducible. Another reference one might add to the author's list is [\textit{D. Benson} and \textit{S. Doty}, Arch. Math. 93, No. 5, 425--435 (2009; Zbl 1210.20039)], which shows (amongst other results) that the Schur algebra analogue of (\(\star\)) holds over any field \(F\) such that \(|F| > r\).
The paper under review includes all the results needed to perform \(p\)-modular reduction on the category of representations of the Super Schur algebra \(S(m|n,r)\). The author remarks that `It seems to us such a modular theory is needed for a geometric theory'. modular representations; supergroups; \(\mathrm{GL}(m|n)\); Schur's duality; super Schur algebra; finitary maps; comodules Modular Lie (super)algebras, Superalgebras, Group actions on varieties or schemes (quotients), Representations of finite symmetric groups, Modular representations and characters, Lie bialgebras; Lie coalgebras, Graded Lie (super)algebras, Vector and tensor algebra, theory of invariants Polynomial representations of \(\mathrm{GL}(m|n)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is concerned with the quantum \(K\)-theory of cominuscule homogeneous spaces. To help the reader, who is not specialist in the subject, to catch the flavor of the paper under review, some basic vocabulary is needed. Recall that if \(G\) is an algebraic group, an algebraic subgroup \(P\) of \(G\) is said to be \textsl{parabolic} if the quotient \(G/P\) is a complete algebraic variety. Any parabolic subgroup contains a Borel subgroup \(B\) which contains a maximal torus \(T\) of \(G\). The pair \((G,T)\) defines \textsl{weights} (which are characters of \(T\) satisfying certain non triviality conditions) and a \textsl{root system}. A fundamental weight \(\omega\) is said to be \textsl{minuscule} if and only if \(|<\omega, \alpha>|\leq 0\), for each positive root \(\alpha\), where \(<,>\) is the pairing induced by the natural duality between the characters and the co-characters of \(T\). A fundamental weigth \(\omega\) is said to be \textsl{co-minuscule} if and only if \(<\omega, \alpha^\vee>=1\), where \(\alpha\) denotes the highest root. To each such weight a parabolic subgroup \(P_\omega\) of \(G\) can be associated, and the corresponding quotient \(G/P_\omega\) is said to be a (co)minuscule homogeneous space.
Example of cominuscules homogeneous varieties are type A Grassmannians, Lagrangian grassmannians \(LG(n,2n)\) and, more exotically, the two exceptional homogeneous spaces: the Cayley Plane and the Freudenthal variety.
The paper under review deals with the finiteness of the cominuscule quantum \(K\)-theory. The product of two Schubert classes in the quantum \(K\)-theory of a homogeneous space \(X=G/P\) is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on \(X\). The remarkable result proven by the authors is that if \(X\) is cominuscule then the power series expansion of the product has only finitely many non zero terms (Theorem 1 stated in the Introduction and proven in Section 5). The proof consists in a fine analysis of the geometry of the Schubert varieties of cominuscule homogeneous spaces. They all have at most rational singularities. Furthermore boundary Gromov-Witten varietes defined by two Schubert varieties are either empty or unirational. General details on the nature of the theorems and their proofs are provided in the comprehensive introduction to the paper. In order to prove their main results the authors are led to use an adaptation of a result by \textit{M. Brion} [J. Algebra 258, No. 1, 137--159 (2002; Zbl 1052.14054)] about a Kleiman-Bertini's like theorem regarding rational singularities (Theorem 2.5), which is interesting in its own. Section 3 is devoted to the geometry of the Gromov-Witten varieties. Section 4 is very important as it supplies the list of the Gromov-Witten varieties of cominuscule spaces. Indeed, this is what the authors need to reach the climax of the paper in the last section, where the main theorem about the finiteness of the expansion of the product in quantum \(K\)-theory, stated in the introduction, is finally proven. quantum \(K\)-theory; Gromov-Witten varieties; rational singularities; rational connectedness; quantum Schubert calculus; cominuscule homogeneous spaces Buch, A. S.; Chaput, P. E.; Mihalcea, L. C.; Perrin, N., Finiteness of cominuscule quantum \textit{K}-theory, Ann. Sci. Éc. Norm. Supér. (4), 46, 3, 477-494, (2013) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Finiteness of cominuscule 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 For a 3-dimensional quantum polynomial algebra \(A=A(E,\sigma)\), Artin-Tate-Van den Bergh showed that \(A\) is finite over its center if and only if \(|\sigma|<\infty\). Moreover, Artin showed that if \(A\) is finite over its center and \(E\ne\mathbb{P}^2\), then \(A\) has a fat point module, which plays an important role in noncommutative algebraic geometry, however, the converse is not true in general. In this paper, we show that, if \(E\ne\mathbb{P}^2\), then \(A\) has a fat point module if and only if the quantum projective plane Proj\(_{\text{nc}}A\) is finite over its center if and only if \(|\nu^*\sigma^3|<\infty\) where \(\nu\) is the Nakayama automorphism of \(A\). As a byproduct, we show that \(|\nu^*\sigma^3|= 1\) or \(\infty\) if and only if the isomorphism classes of simple 2-regular modules over \(\nabla A\) are parameterized by \(E\subset\mathbb{P}^2\). quantum polynomial algebra; quantum projective planes Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry, Derived categories and associative algebras, Abelian categories, Grothendieck categories Characterization of the quantum projective planes finite 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 Topological recursion of \textit{B. Eynard} and \textit{N. Orantin} [Commun. Number Theory Phys. 1, No. 2, 347--452 (2007; Zbl 1161.14026)] is an active research area during the past decade. It governs a variety of problems in enumerative geometry. In this paper, the authors study the question about the enumeration of a-hypermaps which may be regarded as the generalizations of Ribbon graphs.
From the generating function of the enumeration of a-hypermaps, the authors derive its quantum curve following the philosophy of \textit{S. Gukov} and \textit{P. Sulkowski} [J. High Energy Phys. 2012, No. 2, Paper No. 070, 57 p. (2012; Zbl 1309.81220)].
Then they propose the topological recursion conjecture for the enumeration of a-hypermaps from its spectral curve which is the semi-classical limit of the quantum curve, and also provide some evidences. topological recursons; quantum curves; hypermap enumeration Do, N; Manescu, D, Quantum curves for the enumeration of ribbon graphs and hypermaps, Commun. Number Theory Phys., 8, 677-701, (2013) Relationships between algebraic curves and physics, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Calabi-Yau manifolds (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Quantum curves for the enumeration of ribbon graphs and hypermaps | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider mirror symmetry (A-side \textit{vs} B-side, namely singularity side) in the framework of quantum differential systems. We focus on the logarithmic non-resonant case, which describes the geometric situation and we show that such systems provide a good framework in order to generalize the construction of the rational structure given by Katzarkov, Kontsevich and Pantev for the complex projective space. As an application, we give a closed formula for the rational structure defined by the Lefschetz thimbles on the flat sections of the Gauss-Manin connection associated with the Landau-Ginzburg models of weighted projective spaces (a class of Laurent polynomials). As a by-product, using a mirror theorem, we get a rational structure on the orbifold cohomology of weighted projective spaces. The formula on the B-side is more complicated than the one on the A-side (the latter agrees with one of Iritani's results), depending on numerous combinatorial data which are rearranged after the mirror transformation. mirrow symmetry; quantum differential systems Douai, A, Quantum differential systems and rational structures, Manuscr. Math., 145, 285-317, (2014) Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Mirror symmetry (algebro-geometric aspects), Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms Quantum differential systems and construction of rational 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 A theorem of Witten asserts that the fusion ring \(\mathcal{F}(\widehat{\mathfrak{gl}}(n))_k\) of integrable highest weight \(\widehat{\mathfrak{gl}}(n)\)-modules at level \(k\) is isomorphic to the specialization at \(q=1\) of the quantum cohomology ring \(qH^{\bullet}(\mathrm{Gr}_{k,n+k})_{q=1}\) for the Grassmannian \(\mathrm{Gr}_{k,n+k}\). The main aim of the paper under review is to give a mathematically rigorous realization of \(\mathcal{F}(\widehat{\mathfrak{sl}}(n))_k\) as a quotient of \(qH^{\bullet}(\mathrm{Gr}_{k,n+k})\) with the defining relations explained via Bethe Ansatz equations of a quantum integrable system.
Both rings are also described combinatorially in terms of symmetric polynomials (Schur polynomials) in pairwise noncommuting variables. The starting point of authors' analysis is an observation that generating function of the elementary symmetric polynomials turns out to be the transfer matrix for a quantum integrable system. As a consequence, the authors obtain a simple particle formulation of both rings leading to new recursion formulae for the structure constants of the fusion ring as well as for Gromov-Witten invariants. The proposed approach also leads naturally to the Verlinde formula and equips the combinatorial fusion algebra with a \(\mathrm{PSL}(2,\mathbb{Z})\)-action. quantum cohomology; integrable module; plactic algebra; Hall algebra; Bethe Ansatz; fusion ring; Verlinde algebra; symmetric function Korff, C.; Stroppel, C., The \(s l(n)\)-WZNW fusion ring: a combinatorial construction and a realisation as quotient of quantum cohomology, Adv. Math., 225, 200, (2010) Quantum groups (quantized enveloping algebras) and related deformations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Symmetric functions and generalizations, Exactly solvable models; Bethe ansatz, Inverse scattering problems in quantum theory The \(\widehat {\mathfrak {sl}}(n)_k\)-WZNW fusion ring: a combinatorial construction and a realisation as quotient of 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 This article deals with the quantum cohomology of the classical flag manifold \(\text{GL}(n,\mathbb C)/B\), where \(B\) is the subgroup of upper triangular nonsingular matrices. The cohomology ring can be obtained as a factor ring of the polynomials in \(n\) variables modulo the ideal generated by the elementary symmetric polynomials. A distinguished basis for the classical cohomology ring is given by the Schubert polynomials. The quantum cohomology ring again is a factor ring of the polynomial ring but now with \(n-1\) additional variables, the deformation parameters, modulo the ideal generated by the ``quantum elementary polynomials''. The author relates the quantum Schubert polynomial and some other related basis with problems in algebraic combinatorics. For the proofs and further details he mainly refers to the following publications of the author [Adv. Math. 136, No. 2, 224-250 (1998; Zbl 0920.05069) and ``On algebraic and combinatorial properties of Schur and Schubert polynomials'' (Bayreuther Math. Schr. 59) (2000; Zbl 0958.05001)]. quantum cohomology; flag manifold; Schubert polynomial; elementary symmetric polynomial; standard elementary monomial Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry From quantum cohomology to algebraic combinatorics: The example of flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is a survey on a joint work with \textit{K. Sorlin} [Transform. Groups 18, No. 3, 877--929 (2013; Zbl 1308.14050); ibid. 19, No. 3, 887--926 (2014; Zbl 1319.14056)] concerning the theory of character sheaves on the exotic symmetric space. After explaining the historical background, we introduce character sheaves on this variety.
By using those character sheaves, we show that modified Kostka polynomials, indexed by a pair of double partitions, can be interpreted in terms of the intersection cohomology of the orbits in the exotic nilpotent cone, as conjectured by \textit{P. N. Achar} and \textit{A. Henderson} [Adv. Math. 228, No. 5, 2984--2988 (2011; Zbl 1225.14040)]. Classical groups (algebro-geometric aspects), Symmetric functions and generalizations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Positive characteristic ground fields in algebraic geometry, Reflection and Coxeter groups (group-theoretic aspects) Character sheaves on exotic symmetric spaces and Kostka polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper studies the classical and quantum cohomology of a class of quasi-homogeneous, but not homogeneous, spaces known as Mihai's odd symplectic Grassmannian of lines. The classical cohomology ring can be described in terms of Pieri and Giambelli-type formulas by exploiting relations to the even symplectic and the ordinary Grassmannians. The quantum deformation is obtained by a careful study of enumerativity of Gromov-Witten invariants and a transversality lemma.
The quantum cohomology ring being semi-simple motivates the check of a conjecture of Dubrovin's, according to which semi-simplicity should be equivalent to the existence of a full exceptional collection. Pech constructs such an exceptional collection for the odd symplectic Grassmannian by modifying Kuznetsov's exceptional collection for the even symplectic Grassmannian.
The paper is a pleasant read and can be understood with familiarity with cohomology of homogeneous spaces and some background knowledge of Gromov-Witten theory. quantum cohomology; quasi-homogeneous spaces; Grassmannians; Pieri and Giambelli formulas; exceptional collections in derived categories Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum cohomology of the odd symplectic Grassmannian of lines | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that quantum curves arise in infinite families and have the structure of singular vectors of a relevant symmetry algebra. We analyze in detail the case of the Hermitian one-matrix model with the underlying Virasoro algebra, and the super-eigenvalue model with the underlying super-Virasoro algebra. In the Virasoro case we relate singular vector structure of quantum curves to the topological recursion, and in the super-Virasoro case we introduce the notion of super-quantum curves. We also discuss the double quantum structure of the quantum curves and analyze specific examples of Gaussian and multi-Penner models. Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Supersymmetric field theories in quantum mechanics, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Relationships between algebraic curves and physics, Virasoro and related algebras Singular vector structure of 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 The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type \(A\) by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual \(k\)-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's \(r\)-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual \(k\)-Schur functions given by studying the affine grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual \(k\)-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially the positivity of the multiplication of a dual \(k\)-Schur function by a Schur function. Bruhat order; Schubert polynomials; \(k\)-Schur functions; Hopf algebras Symmetric functions and generalizations, Hopf algebras and their applications, Classical problems, Schubert calculus Schubert polynomials and \(k\)-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 show in some detail how to implement Shor's efficient quantum algorithm for discrete logarithms for the particular case of elliptic curve groups. It turns out that for this problem a smaller quantum computer can solve problems further beyond current computing than for integer factorization. A 160 bit elliptic curve cryptographic key could be broken on a quantum computer using around 1000 qubits while factoring the security-wise equivalent 1024 bit RSA modulus would require about 2000 qubits. In this paper we only consider elliptic curves over \(\mathrm{GF}(p)\) and not yet the equally important ones over \(\mathrm{GF}(2^{n})\) or other finite fields. The main technical difficulty is to implement Euclid's gcd algorithm to compute multiplicative inverses modulo \(p\). As the runtime of Euclid's algorithm depends on the input, one difficulty encountered is the ``quantum halting problem''. Proos, J., Zalka, C.: Shor's discrete logarithm quantum algorithm for elliptic curves. Quantum Inf. Comput. \textbf{3}, 317-344 (2003) Quantum computation, Quantum cryptography (quantum-theoretic aspects), Cryptography, Applications to coding theory and cryptography of arithmetic geometry Shor's discrete logarithm quantum algorithm for elliptic curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(p(x)=p(x_1,\dots,x_g)\) be a noncommutative polynomial in \(g\) variables with real coefficients. Such a polynomial \(p\) is called symmetric if \(p^T=p\), where the involution \(^T\) is first defined on monomials (words in the letters \(x_1,\dots,x_g\)) by sending a word to a word with the same letters written in the reverse order, e.g., \((x_jx_l)^T=x_lx_j\), and then naturally extended to polynomials (which are linear combinations of words). A noncommutative polynomial \(p\) can be evaluated on \(g\)-tuples \(X=(X_1,\dots,X_g)\) of real symmetric \(n\times n\) matrices for any \(n\), so that \(p(X)\) is a symmetric matrix if \(p\) is symmetric. The positivity domain \(\mathcal{D}_p^n\) of \(p\) in dimension \(n\) is the closure of the component of \(0\) of the set of \(g\)-tuples \(X\) of symmetric \(n\times n\) matrices such that \(p(X)\) is positive definite. The positivity domain \(\mathcal{D}_p\) of \(p\) is a disjoint union of the domains \(\mathcal{D}_p^n\), \(n=1,2,\dots\).
The main result of the present paper is that if the symmetric noncommutative polynomial \(p\) satisfies some natural conditions (\(p\) is decreasing near the boundary of \(\mathcal{D}_p\), \(p\) is a minimum degree defining polynomial for \(\mathcal{D}_p\), \(p\) has the generic full rank boundary property), which are defined and discussed in detail in the paper, and \(\mathcal{D}_p\) is convex, then \(p\) has degree four or less. In addition, a finer structure of \(p\) is discussed in detail. linear matrix inequalities; convex sets of matrices; noncommutative semialgebraic geometry H. Dym, W. Helton, and S. McCullough, ''Irreducible noncommutative defining polynomials for convex sets have degree four or less,'' Indiana Univ. Math. J., vol. 56, iss. 3, pp. 1189-1231, 2007. Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), Miscellaneous inequalities involving matrices, Semialgebraic sets and related spaces, Linear operator inequalities, Dilations, extensions, compressions of linear operators, Convex sets and cones of operators, Convexity of real functions of several variables, generalizations Irreducible noncommutative defining polynomials for convex sets have degree four or less | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We define a Grothendieck ring of varieties with actions of finite groups and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural \(\lambda\)-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized (`motivic') Euler characteristics of higher orders can be defined as homomorphisms from this ring to the Grothendieck ring of varieties extended by powers of the class of the complex affine line. We give an analogue of the Macdonald type formula for the generating series of the generalized higher-order Euler characteristics of wreath products. finite group actions; complex quasi-projective varieties; Grothendieck rings; lambda-structure; power structure Applications of methods of algebraic \(K\)-theory in algebraic geometry, Group actions on varieties or schemes (quotients), Grothendieck groups (category-theoretic aspects), Finite automorphism groups of algebraic, geometric, or combinatorial structures Grothendieck ring of varieties with actions of finite 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 If \(X\) is a complex algebraic variety, the virtual Poincaré polynomial \(P_X(t)\) is uniquely determined by the properties:
(1) \(P_X(t)= P_{X-Y} (t)+P_Y(t)\), for every closed subvariety \(Y\).
(2) If \(X\) is smooth and complete, then \(P_X(t)\) is the usual Poincaré polynomial of \(X\).
The authors consider the case \(X=G/H\), where \(G\) is a complex connected linear algebraic group and \(H\) is its closed subgroup.
The main result: \(P_{G/H}(t) =t^{2u} (t^2-1)^rQ_{G/H} (t^2)\), where \(Q_{G/H} (t^2)\) is a polynomial with non-negative integer coefficients. For a regular embedding \(X\) of \(G/H\) [\textit{E. Bifet}, \textit{C. De Concini} and \textit{C. Procesi}, Adv. Math. 82, 1-34 (1990; Zbl 0743.14018)] it is proved (provided \(G\) is complete and \(H\) is connected) that \(P_X(t)= Q_{G/H}(t^2) R_X(t^2)\), where \(R_X(t^2)\) is a polynomial with non-negative integer coefficients.
Reviewer's remark. The authors' assertion that ``the usual Poincaré polynomials for homogeneous spaces are generally unknown'' is properly not quite correct. As soon as the cohomology of homogeneous spaces can be described by the H. Cartan algebra, the Poincaré polynomial can be written using its algebraic characteristics (Gröbner bases, for instance) [see \textit{I. Z. Rozenknop}, Usp. Mat. Nauk 25, No. 5, (155), 245-246 (1970; Zbl 0204.06001)]. virtual Poincaré polynomial Brion, Michel; Peyre, Emmanuel, The virtual Poincaré polynomials of homogeneous spaces, Compositio Math., 0010-437X, 134, 3, 319-335, (2002) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Homogeneous spaces and generalizations The virtual Poincaré polynomials of homogeneous spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let A be a one-dimensional Cohen-Macaulay local ring, (b,e,\(\rho\)) a triplet of integers. Let \(\rho_{0,b,e}=(r+1)e-\left( \begin{matrix} r+b\\ r\end{matrix} \right)\), where r is the integer such that \(\left( \begin{matrix} b+r-1\\ r\end{matrix} \right)\leq e<\left( \begin{matrix} b+r\\ r+1\end{matrix} \right)\) and \(\rho_{1,b,e}=e(e-1)/2-(b-1)(b-2)/2\). The main result of the paper is that there exists a one-dimensional Cohen-Macaulay local ring A with embedding dimension b, multiplicity e and reduction number \(\rho\) iff \(b=e=1\) and \(\rho =0\) or \(2\leq b\leq e\) and \(\rho_{0,b,e}\leq \rho \leq \rho_{1,h,e}\). In this last case, one can choose A to be the quotient of a power series ring over an algebraically closed field of characteristic zero. Using this, the Hilbert-Samuel function of one- dimensional Cohen-Macaulay rings of small multiplicity is computed. Lastly, conditions for the Cohen-Macaulayness of gr(A) are established in terms of the reduction number. curve singularities; one-dimensional Cohen-Macaulay local ring; embedding dimension; multiplicity; reduction number; Hilbert-Samuel function J. Elias, Characterization of the Hilbert-Samuel polynomial of curve singularities, Compositio Math. 74 (1990), 135--155 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Singularities of curves, local rings, Cohen-Macaulay modules Characterization of the Hilbert-Samuel polynomials of curve singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Author's abstract: This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set \(D_L\), called a free Hilbert spectrahedron, of the linear operator inequality (LOI) \(L(X) = A_0 \otimes I + \sum_{j=1}^g A_j \otimes X_j \geq 0\), where \(A_j\) are self-adjoint linear operators on a separable Hilbert space, \(X_j\) matrices and \(I\) is an identity matrix. If \(A_j\) are matrices, then \(L(X) \geq 0\) is called a linear matrix inequality (LMI) and \(D_L\) a free spectrahedron. For monic LMIs, i.e., \(A_0 = I\), and nc matrix-valued polynomials the certificates of positivity were established by Helton, Klep and McCullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry. Since the full strength of the theory of complete positivity is not restricted to finite dimensions, but works well also in the infinite-dimensional setting, we use it to tackle our problems. First we extend the characterization of the inclusion \(D_{L_1} \subseteq D_{L_2}\) from monic LMIs to monic LOIs \(L_1\) and \(L_2\).
As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron \(D_L\) and its projection, called a free Hilbert spectrahedrop. Further on, using this characterization in a separation argument, we obtain a certificate for multivariate matrix-valued nc polynomials F positive semidefinite on a free Hilbert spectrahedron defined by a monic LOI. Replacing the separation argument by an operator Fejér-Riesz theorem enables us to extend this certificate, in the univariate case, to operator-valued polynomials F. Finally, focusing on the algebraic description of the equality \(D_{L_1} = D_{L_2}\), we remove the assumption of boundedness from the description in the LMIs case by an extended analysis. However, the description does not extend to LOIs case by counterexamples. free convexity; linear matrix inequality (LMI); spectrahedron; spectrahedrop; completely positive; Positivstellensatz; polar dual; Gleichstellensatz; quadratic module; free real algebraic geometry; noncommutative polynomial; free positivity Zalar, A, Operator positivstellensätze for noncommutative polynomials positive on matrix convex sets, J. Math. Anal. Appl., 445, 32-80, (2017) Semialgebraic sets and related spaces, Matrix pencils, Operator spaces and completely bounded maps, Linear operator inequalities, Sums of squares and representations by other particular quadratic forms, Real algebra, Other ``noncommutative'' mathematics based on \(C^*\)-algebra theory, Applications of functional analysis in optimization, convex analysis, mathematical programming, economics, Operator spaces (= matricially normed spaces) Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 semi-simple simply connected algebraic group with Lie algebra \(\mathfrak{g}\), and \(\mathcal{B}\) be its (full) flag variety. Then \(T^*\mathcal{B}\) is a symplectic resolution of the nilpotent cone \(\mathcal{N}\subseteq\mathfrak{g}^*\), called the Springer resolution. Consider the action of \(\mathbb{G}:=G\times\mathbb{C}^*\) on \(T^*\mathcal{B}\), where \(\mathbb{C}^*\) acts by multiplication on the cotangent fibers. The authors study \(H^*_{\mathbb{G}}(T^*\mathcal{B})\), the \(\mathbb{G}\)-equivariant quantum cohomology ring of \(T^*\mathcal{B}\). The paper is a part of a broader project to demonstrate that mirror symmetry structures on certain symplectic resolutions generalize classical notions of geometric representation theory. Many auxilliary constructions are carried out in greater generality with an eye on this larger picture. The main result is a formula for the quantum multiplication by divisors, which for Springer resolutions generate the quantum cohomology ring. A limiting procedure recovers Kim's description of the quantum cohomology of the base \(\mathcal{B}\) in terms of quantum Toda lattices [\textit{B. Kim}, Ann. Math. (2) 149, No. 1, 129--148 (1999; Zbl 1054.14533)]. A generalization to Slodowy slices of \(\mathcal{N}\) is also given.
Divisors of \(T^*\mathcal{B}\) are in one-to-one correspondence with weights \(\lambda\) of the Borel subgroup of \(G\), and the group algebra of the Weyl group acts on \(H^*_{\mathbb{G}}(T^*\mathcal{B})\) as described by \textit{G. Lusztig} [Publ. Math., Inst. Hautes Étud. Sci. 67, 145--202 (1988; Zbl 0699.22026)]. With this in mind, and denoting the classical multiplication by a divisor as \(x_\lambda\) the authors' formula for the quantum multiplication is
\[
D_\lambda=x_\lambda+t\sum_{\alpha\in R_+}(\lambda,\alpha^\vee)\frac{q^{\alpha^\vee}}{1-q^{\alpha^\vee}}(s_\alpha-1). \,\eqno(1)
\]
The sum is over positive simple roots, \(\alpha^\vee\) are the coroots and \(s_\alpha\) are the corresponding Weyl reflections, parameter \(t\) can be identified with the equivariant parameter of the \(\mathbb{C}^*\) factor of \(\mathbb{G}\). If \(d_\lambda\) is the derivative in the direction of \(\lambda\) then \(\nabla_\lambda:=d_\lambda-D_\lambda\) defines a flat connection on a trivial bundle over the dual maximal torus with fiber \(H^*_{\mathbb{G}}(T^*\mathcal{B})\). This connection is shown to be the affine Knizhnik-Zamolodchikov connection. Its quantum \(D\)-module coincides with the quantum Calogero-Moser module of the Langlands dual to \(G\).
As a consequence, the authors are able to describe \(H^*_{\mathbb{G}}(T^*\mathcal{B})\) in terms of the classical Calogero-Moser integrable system, identifying in particular the shift operators of \textit{E. M. Opdam} [Invent. Math. 98, No. 1, 1--18 (1989; Zbl 0696.33006)]. The limit \(t\to\infty\), appropriately interpreted, gives Kim's description of \(H^*_{\mathbb{G}}(\mathcal{B})\) in terms of quantum Toda lattices. This description requires the use of another action \(\overline{s}_\alpha\) by the Weyl reflections, which is nilpotent, and restricts the sum in (1) to a subset \(R'_+\) of simple positive roots
\[
D_\lambda^{\mathcal{B}}=x_\lambda+\sum_{\alpha\in R_+}(\lambda,\alpha^\vee)\,q^{\alpha^\vee}\overline{s}_\alpha\,.
\]
If \(G\) is simply laced \(R'_+=R_+\).
Finally, the authors generalize their results to so-called Slodowy slices \(\mathcal{S}_n\subseteq\mathcal{N}\) transversal to the \(G\)-orbit of \(n\) in \(\mathcal{N}\). The lift \(\widetilde{\mathcal{S}}_n\) to the total space of \(T^*\mathcal{B}\to\mathcal{N}\) is a symplectic resolution of \(\mathcal{S}_n\), for \(n=0\) we recover \(\mathcal{S}_n=\mathcal{N}\) and \(\widetilde{\mathcal{S}}_n=T^*\mathcal{B}\). The analog of \(H^*_{\mathbb{G}}(T^*\mathcal{B})\) is \(H^*_{Z_n\times\mathbb{C}^*}(T^*\mathcal{B})\), where \(Z_n\) is the centralizer of \(n\) in \(G\). The authors show that if the restriction \(H^2(T^*\mathcal{B},\mathbb{Z})\to H^2(\widetilde{\mathcal{S}}_n,\mathbb{Z})\) is onto, e.g., if \(G\) is simply laced, then quantum multiplication by divisors of \(\widetilde{\mathcal{S}}_n\) is still given by (1). equivariant quantum cohomology; Springer resolution; nilpotent cone; mirror symmetry; KZ connection; symplectic resolution; Calogero-Moser integrable system; Opdam operators; Slodowy slices D. Maulik and A. Okounkov, \textit{Quantum Groups and Quantum Cohomology}, arXiv:1211.1287 [INSPIRE]. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Group actions on varieties or schemes (quotients) Quantum cohomology of the Springer resolution | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using \textit{F. S. Macaulay}'s correspondence [Proc. Lond. Math. Soc. (2) 26, 531--555 (1927; JFM 53.0104.01)] we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for the dimension of cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur. cactus rank; Artinian Gorenstein local algebra Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Computational aspects of higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On polynomials with given Hilbert function and applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review the authors show that the quantized coordinate ring \(\mathbb{K}_q[\mathrm{Gr}(2,n)]\), \(n\geq 3\), of the Grassmannian is a quantum cluster algebra of type \(A_{n-3}\). Using computer-aided calculation the authors also show that \(\mathbb{K}_q[\mathrm{Gr}(3,n)]\), \(n=6,7,8\), are quantum cluster algebras of types \(D_4\), \(E_6\) and \(E_7\), respectively. Using this, the authors obtain quantum cluster algebra structures on quantum Schubert cells of the \(k=2\) Grassmannians. It turns out that the quantum Schubert cells associated to the partition \((t,s)\), where \(t\geq s\) and \(t,s\leq n-2\), is of type \(A_{s-1}\). These cases are precisely those where the quantum cluster algebra is of finite type and the structures the authors describe quantize the structures known in the classical case. cluster algebra; quantum Grassmannian; Schubert cell; finite type; seed Jan E. Grabowski, Stéphane Launois, Lifting quantum cluster algebra structures, 2015, in preparation. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Cluster algebras, Representations of quivers and partially ordered sets Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe a pre-\(\lambda \)-structure on the Grothendieck ring of stacks (originally studied by Torsten Ekedahl) and the corresponding power structures over it, discuss some of their properties and give some explicit formulae for the Kapranov zeta-function for some stacks. In particular, we show that the \(n\)-th symmetric power of the class of the classifying stack \(\mathrm{BGL}(1)\) of the group \(\mathrm{GL}(1)\) coincides, up to a power of the class \(\mathbb L\) of the affine line, with the class of the classifying stack \(\mathrm{BGL}(n)\). Grothendieck ring; Hilbert scheme Gusein-Zade, SM; Luengo, I; Melle-Hernández, A, On the pre-\(\lambda \)-ring structure on the Grothendieck ring of stacks and the power structures over it, Bull. Lond. Math. Soc., 45, 520-528, (2013) Parametrization (Chow and Hilbert schemes), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) On the pre-\(\lambda \)-ring structure on the Grothendieck ring of stacks and the power structures over it | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 part I of this paper the author studied the beta function \(B_\sigma(s,t)\) \((s,t\in\mathbb{Q}/ \mathbb{Z}\); which takes values in a certain arithmetic ring) associated with each element \(\sigma\) of the Grothendieck-Teichmüller group \(GT\). He has shown there, by using the 5-cycle relation defining \(GT\), that \(B_\sigma\) splits into the product of the ``hyperadelic gamma functions'' \(\Gamma_\sigma(s)\), as
\[
B_\sigma(s,t)=\frac {\Gamma_\sigma(s) \Gamma_\sigma(t)} {\Gamma_\sigma (s+t)},
\]
and also that the logarithmic derivative \(D \log\Gamma_\sigma(s)- D\log\Gamma_\sigma(0)\) of \(\Gamma_\sigma (s)\) can be expressed explicitly in terms of the generalized logarithmic Kummer quasi 1-cocycles \(\Psi_n^{(j)} (\sigma)\) \((n\in\mathbb{N}\), \(j\in \mathbb{Z}/n\), \(\sigma\in GT)\). These are generalizations of results of \textit{G. Anderson} [Invent. Math. 95, No. 1, 63--131 (1989; Zbl 0682.14011)] from the case \(\sigma\in G_\mathbb{Q}=\text{Gal} (\overline\mathbb{Q}/ \mathbb{Q})\) to \(\sigma\in GT\), with essentially different methods of proofs. We still do not know whether \(G_\mathbb{Q}\subsetneqq GT\), but the author suspects so, and in particular that Anderson's multiplication formula [loc. cit.] Corollary 8.6.3 for \(\Gamma_\sigma (\sigma\in G_\mathbb{Q})\):
\[
\left(\prod_{nc=0} \Gamma_\sigma(s+c) \Gamma_\sigma (c)^{-1}\right) \Gamma_\sigma (ns)^{-1}=1\otimes \exp \bigl(2\pi ns\Psi_n^{(0)} (s)\bigr)
\]
\((s\in \mathbb{Q}/ \mathbb{Z},n\in\mathbb{N})\) could be a key test condition for an element \(\sigma\) of \(GT\) to belong to \(G_\mathbb{Q}\).
The main purpose of this article under review is to present two other necessary conditions (K) and (A') for \(\sigma\in GT\) to belong to \(G_\mathbb{Q}\), and discuss logical relations among (K), (A), (A'). One of them, (K), arises from the compatibility of the \(G_\mathbb{Q}\)-actions on the algebraic fundamental groups over \(\overline\mathbb{Q}\) associated with the following morphisms of curves:
\[
\begin{matrix} \mathbb{P}^1-\{0,1, \infty\} & \leftarrow &\mathbb{P}^1-\{0, \mu_n,\infty\} & \hookrightarrow & \mathbb{P}^1-\{0,1,\infty\},\\ t^n & \mapsto & t & \mapsto & t\end{matrix}
\]
\((n\in \mathbb{N})\). This compatibility gives rise to defining a certain subgroup, which the author calls \(GTK\), of \(GT\) containing \(G_\mathbb{Q}\). Thus, our first task is to define \(GTK\) explicitly (\S1). Then in \S2, there is discussed its relations with the above (A), and with another natural condition on our quasi 1-cocycles \(\Psi_n^{(j)} (\sigma)\)'s, called (A'), each defining some subgroups \(GTA\), \(GTA'\) of \(GT\) containing \(G_\mathbb{Q}\). The main purpose of \S2 is to show that \(GTK\subseteqq GTA'\) and also that (A') is equivalent to ``\(D\log A\)''.
The author claims that he found some unfortunate sign errors in part I of this paper [loc. cit.]. In view of their close relations with the present paper, the indications for corrections are listed in the supplement (page 10). , On beta and gamma functions associated with the Grothendieck-Teichmüller group. II, J. Reine Angew. Math., 527 (2000), 1-11. Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), \(q\)-gamma functions, \(q\)-beta functions and integrals On beta and gamma functions associated with the Grothendieck-Teichmüller group. II. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), à la Mallios, in a topos-theoretic environment, and hence, the extension of the ``mechanism of differentials'' in the quantum regime. The process of gluing information, within diagrams of commutative algebraic localizations, generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection. Zafiris E.: Quantum observables algebras and abstract differential geometry: The topos-theoretic dynamics of diagrams of commutative algebraic localizations. Int. J. Theor. Phys. 46, 319--382 (2007) Operator algebra methods applied to problems in quantum theory, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Fibered categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Applications of local differential geometry to the sciences, Topos-theoretic approach to differentiable manifolds Quantum observables algebras and abstract differential geometry: the topos-theoretic dynamics of diagrams of commutative algebraic localizations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Two hierarchies of quantum principal bundles over quantum real projective spaces are constructed. One hierarchy contains bundles with \(U(1)\) as a structure group, the other has the quantum group \(SU_{q}(2)\) as a fibre. Both hierarchies are obtained by the process of prolongation from bundles with the cyclic group of order 2 as a fibre. The triviality or otherwise of these bundles is determined by using a general criterion for a prolongation of a comodule algebra to be a cleft Hopf-Galois extension. quantum real projective space; quantum sphere; principal comodule algebra; prolongation Brzeziński, T.; Zieliński, B., Quantum principal bundles over quantum real projective spaces, J. geom. phys., 62, 1097-1107, (2012) Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Quantum principal bundles over quantum real 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 quantum torus is an algebra generated by two elements \(x\) and \(y\) subject to relation \(yx=qxy\) where \(q\) is a complex number. In this article an action of the group \(\mathrm{SL}(2,\mathbb{Z})\) on quantum torus is defined and invariants of finite sub-groups of this group are studied. In particular, \(0\)-degree Hochschild homology of these sub-algebras of invariants are computed. quantum torus; invariants; Hochschild homology Baudry, J., Invariants du tore quantique, Bull. sci. math., 134, 531-547, (2010) Noncommutative algebraic geometry, Ring-theoretic aspects of quantum groups, Quantum groups and related algebraic methods applied to problems in quantum theory Invariants of the quantum torus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let a representation of a reductive group \(G\) in a linear space \(V\) be given. Let \(X\) be an irreducible \(G\)-invariant subvariety of the projective space \(\mathbb{P}(V)\). The variety \(X\) is said to be spherical if a Borel subgroup \(B\) of the group \(G\) has an open orbit on \(X\). We denote by \(F[X]\) the homogeneous coordinate ring of the variety \(X\); this ring is graded by the degree of polynomials, \(F[X]= \bigoplus^\infty_{n=0} F[X]_n\). We note that, for any classical group \(G\) and any spherical \(G\)-variety \(X\), we can define, in the real space \(\mathbb{R}^{\dim B}\) of dimension \(\dim B\), a polytope \(\Delta(X)\) and a lattice \(\widetilde P\) with the following properties.
Proposition. If the variety \(X\) is normal, then \(\dim F[X]_n=\text{card}\{n\Delta(X) \cap \widetilde P\}\), i.e., the Hilbert polynomial of the variety \(X\) coincides with the Ehrhart polynomial of the polytope \(\Delta(X)\). For every variety \(X\) we have \(\deg X=(\dim X) !\text{ vol} \Delta(X)\), where the volume of the cell of \(\widetilde P\) is normalized to unity. spherical variety; Hilbert polynomial Okounkov A.\ Y., Note on the Hilbert polynomial of a spherical variety, Funct. Anal. Appl. 31 (1997), no. 2, 138-140. Geometric invariant theory, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Note on the Hilbert polynomial of a spherical variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials From the abstract: ``We define analogues of homogeneous coordinate algebras for noncommutative two-tori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some non-commutative two-torus with real multiplication. These results are based on the techniques of \textit{A.~Polishchuk} and \textit{A.~Schwarz} [Commun. Math. Phys. 236, 135--159 (2003; Zbl 1033.58009)] allowing to interpret all the data in terms of autoequivalences of the derived categories of coherent sheaves on elliptic curves.'' Koszul dual algebra; Morita autoequivalence; derived category of coherent sheaves Polishchuk, A., Noncommutative two-tori with real multiplication as noncommutative projective varieties, J. Geom. Phys., 50, 162-187, (2004) Noncommutative algebraic geometry, Elliptic curves, Derived categories, triangulated categories, Rings arising from noncommutative algebraic geometry, Noncommutative geometry (à la Connes) Noncommutative two-tori with real multiplication as noncommutative projective varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The localization theorem of Beilinson-Bernstein used Hodge theory in the representation theory of complex semisimple Lie algebras. This has lead to a theory called \textit{microlocalization}, which is localization on an open subset of the quantized cotangent bundle. This theory can be applied to other important algebras, such as the Cherednik-type algebras.
The authors establish the theory of derived microlocalization for a lot of quantized symplectic resolutions, as well as more general quantized birational symplectic morphisms.
At first, algebras obtained by quantum Hamiltonian reduction from quantizations of smooth affine symplectic varieties with group action, is considered. In the cases considered, Hamiltonian reductions are rings of global twisted differential operators on algebraic stacks, and under mildt assumptions, it is proved that the representation categories of such algebras are derived equivalent to microlocal derived categories on open sets in the symplectic quotient stack.
Because of such equivalences, the article results in new proofs of known derived equivalences for enveloping algebras, independent of the original Beilinson-Bernstein approach. Also, it gives the derived version of microlocalization of rational Cherednik algebras of type A, and hypertoric enveloping algebras, and gives a new derived microlocalization theorem relating cyclotomic rational Cherednik algebras with quantizations of the Hilbert schemes \(\widetilde{\mathbb C^2/\Gamma}^{[n]}\) of points on minimal resolutions \(\widetilde{\mathbb C^2/\Gamma}\) of cyclic quotient singularities.
The new derived equivalences require the establishment of derived microlocal categories. This induces a lot of basic properties, including compact generation, indecomposibility, and an equivalence between the two natural definitions of the derived categories in the case of quantizations of cotangent bundles.
More specific about the microlocalization theorem: Let \(\mathsf W\) be a smooth affine complex symplectic variety, with a Hamiltonian action of a connected redugtive group \(G\) and moment map \(\mu:\mathsf W\rightarrow\mathfrak g^\ast.\) Then one can associate a usually singular, affine symplectic quotient \(X=\mu^{-1}(0)//G=\text{Spec}\mathbb C[\mu^{-1}(0)]^G\). Or, choosing a character \(\chi:G\rightarrow\mathbb G_m=\mathbb C^\ast\), a better quotient is defined as \(\mathfrak X=\mu^{-1}(0)//_{\chi}G\) by geometric invariant theory. There is a natural projective morphism \(f:\mathfrak X\rightarrow X\), and \(\chi\) can be chosen so that \(\mathfrak X\) gives a symplectic resolution of the singular symplectic variety \(X\).
It is possible to quantize the situation by replacing the functions on \(\mathsf{W}\) by a filtered noncommutative algebra \(\mathsf{A}\) whose associated graded algebra is \(\mathbb C[\mathsf{W}]\). The method of Hamiltonian reduction yields an algebra \(U_c\) for a Lie algebra character \(c:\mathfrak g\rightarrow\mathbb C\) that works as a quantum analogue of the algebra \(\mathbb C[X]\) of functions on \(X\).
In the same way, there is a natural quantum analogue \(D(\mathcal E_{\mathfrak X}(c))\) of the derived category of \(\mathfrak X\), and there are quantizations \(\mathbb L\mathsf{f}^\ast\),\(\mathbb R\mathsf{f}_\ast\) of the inverse and direct images \(\mathbb L f^\ast\), \(\mathbb R f_\ast\) associated with \(f:\mathfrak X\rightarrow X\). Letting \(\mathsf{W}=T^\ast Z\) and assuming that \(G\) acts freely on \(\mu^{-1}(0)^{\text{ss}}\), \(\mathfrak X\) is a smooth symplectic variety, and the category \(D(\mathcal E_{\mathfrak X}(c))\) can be described explicitly as a derived category of modules over a noncommutative deformation of the sheaf \(\mathcal O_{\mathfrak X}\) of functions on \(\mathfrak X\). In general, the authors define \(D(\mathcal E_{\mathfrak X}(c))\) via a categorical quotient, but the resulting category still have the invariants such as characteristic cycles that relate the representation theory of \(U_c\) to the geometry of \(\mathfrak X\). There is an added value that it has good properties even if the action of \(G\) on \(\mu^{-1}(0)^{\text{ss}}\) is not free.
The main result in the article is a criterion, with a mild set of assumptions, for proving that \(\mathbb L\mathsf{f}^\ast\) and \(\mathbb R\mathsf{f}_\ast\) are mutually quasi-inverse equivalences of derived categories. As one of several corollaries to this result, the authors prove that if \(U_c\) has finite global dimension, then \(\mathbb L\mathsf{f}^\ast\) is an exact equivalence of categories. This, together with the other corollaries provides derived equivalences between the algebras \(U_c\) even when the natural shift functors cross walls.
An interesting class of examples are given, the spherical Cherednik algebras associated with wreath products of cyclic groups. When \(\mu_l\subset\text{SL}(2)\) denotes a cyclic subgroup, then for the spherical Cherednik algebra \(U_{k,c}\) associated with the wreath product \(S_n\wr\mu_l\), the functor \(D(U_{k,c})\rightarrow D(\mathcal E_{\mathfrak X}(k,c))\) is an exact equivalence of triangulated categories away from a finite collection of hyperplanes. These hyperplanes are explicitly described.
The article's main result also generalizes to non-affine situations in which a good quotient exists. Using étale charts on an algebraic curve \(C\), one can reduce to \(C=\mathbb A^1\): Let \(C\) be a smooth algebraic curve. Let \(U_c\) denote the sheaf (on \(\text{Sym}^n(C)\)) of sperical subalgebras of the type A Cherednik algebra of \(C\). Let \(D(\mathcal E_{(T^\ast C)^{[n]}(c)})\) denote the associated microlocal category. Then \(D(\mathcal E_{(T^\ast C)^{[n]}(c)})\simeq D(U_c)\) if \(c\notin\{-\frac{p}{q}\in\mathbb Q|1\leq p<q\leq n\}.\)
Also, the author mention another interesting class of examples: The Mori dream spaces. This includes flag varieties, spherical varieties, and toric varieties. The constructions of the article can be extended to (micro)localization for Mori Dream spaces, and thus the authors reprove derived localization for the examples above, independently of the classical methods of Beilinson-Bernstein. The authors' mild assumptions are easier to prove than that the corresponding central reductions of the enveloping algebra has finite global dimension.
The article finally discuss how to extend the main results to deformation quantization of arbitrary symplectic resolutions.
The methods used in the article shifts from the classical methods of Beilinson-Bernstein which depends on the Azumaya property, i.e. the large centers of the noncommutative rings appearing, to more homotopical methods by establishing the existence of right adjoints. The Čheck methods comes to consideration.
The article is advanced, and assume basic knowledge of derived categories. When that is in place though, the authors explain their particular idea and methods in a very stringent and nice way. Most important, the authors put their theory in to the big picture and gives a good overview of the field of localized derived categories and the philosophy behind it. This article gives an important contribution to the theory. derived equivalences; quantum Hamiltonian reduction; microlocalization; Beilinson-Bernstein theory; hodge theory; localized derived category; quantum Hamiltonian reduction; chomological boundedness; wreath product; Cherednik algebras; right adjoint; left adjoin; microlocal derived categories; microlocal abelian categories; Serre duality of derived categories McGerty, K., Nevins, T.: Derived equivalence for quantum symplectic resolutions. Selecta Math. (N.S.) \textbf{20}(2), 675-717 (2014) Vector bundles on curves and their moduli, Derived categories, triangulated categories Derived equivalence for quantum symplectic resolutions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 polynomial \(p(X)\) in the polynomial algebra \(K[X]=K[x_1,\ldots,x_n]\) over a field \(K\) is called a test polynomial if any endomorphism fixing \(p(X)\) is an automorphism. The main purpose of the paper under review is to construct new classes of test polynomials and to give in explicit form all automorphisms fixing these test polynomials. In particular, the authors solve some problems raised in the recent paper by the reviewer and one of the authors [\textit{V. Drensky} and \textit{J.-T. Yu}, J. Algebra 207, No. 2, 491-510 (1998)]. polynomial algebras; automorphisms; test polynomials; endomorphism; Jacobian problem Feng, K. -Q.; Yu, J. -T.: New classes of test polynomials of polynomial algebras. Sci. China ser. A 42, 712-719 (1999) Polynomial rings and ideals; rings of integer-valued polynomials, Jacobian problem, Actions of groups on commutative rings; invariant theory, Morphisms of commutative rings New classes of test polynomials of polynomial algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It has been recently proved that the arc-analytic type of a singular Brieskorn polynomial determines its exponents. This last result may be seen as a real analogue of a theorem by Yoshinaga and Suzuki concerning the topological type of complex Brieskorn polynomials. In the real setting it is natural to investigate further by asking how the signs of the coefficients of a Brieskorn polynomial change its arc-analytic type. The aim of the present paper is to answer this question by giving a complete classification of Brieskorn polynomials up to the arc-analytic equivalence. The proof relies on an invariant of this relation whose construction is similar to the one of Denef-Loeser motivic zeta functions. The classification obtained generalizes the one of Koike-Parusiński in the two variable case up to the blow-analytic equivalence and the one of Fichou in the three variable case up to the blow-Nash equivalence. singular Nash function germs; arc-analytic equivalence; Brieskorn polynomials; motivic zeta functions; virtual Poincaré polynomial Arcs and motivic integration, Singularities in algebraic geometry, Nash functions and manifolds, Topology of real algebraic varieties, Equisingularity (topological and analytic) Complete classification of Brieskorn polynomials up to the arc-analytic equivalence | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 ring. A polynomial \(f_1\in R[X]:=R[X_1,\dots,X_n]\) is called a coordinate if there exist \(f_2,\dots,f_n\in R[X]\) such that \(R[X]:=R[f_1,\dots,f_n].\) The problem how to recognize coordinates is fundamental in the study of polynomial automorphisms (for instance in the jacobian conjecture). There are known solutions of this problem in the case \(R\) is a field. The authors study this problem in many aspects in the case \(R\) is a \(\mathbb Q\)-algebra and \(n=2.\) The main results concern the following aspects:
1. criteria for \(f \in R[X,Y]\) to be a coordinate (in terms of derivation \(f_y\partial _x-f_x\partial _y\) in \(R[X]\) and in terms of residual polynomial rings \(k_{\mathfrak p}[X,Y]\) over the fields \(k_{\mathfrak p}\) for \(\mathfrak p\in {\text{Spec}}(R)),\)
2. conditions (for a given \(f \in R[X_1,X_2]\)) for an equivalence between the property of being a coordinate in \(R[X_1,X_2]\) and in \(S[X_1,X_2,\dots,X_n]\), where \(S\) is a ring extension of \(R\),
3. criteria for an \(R\)-endomorphism of \(R[X,Y]\) which sends linear coordinates to coordinates to be an \(R\)-automorphism,
4. generalizations of the famous Abhyankar-Moh theorem on the embedding of the line into the plane to the case when the field of coefficients is replaced by a commutative \(\mathbb Q\)-algebra,
5. partial results concerning coordinates in more than two variables. polynomial in two variables; coordinate; polynomial automorphism; derivation; Jacobian conjecture; Abhyankar-Moh theorem; embedding of the line into the plane A. van den Essen and P. van Rossum, Coordinates in two variables over a \(\mathbb{Q}\)-algebra, Trans. Amer. Math. Soc. 356 (2004), 1691--1703. Polynomials over commutative rings, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Coordinates in two variables over a \(\mathbb{Q} \)-algebra | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\Gamma \subset \mathrm{SL}_2(\mathbb{C})\) be a nontrivial finite subgroup and the surface \(S = \widehat{\mathbb{C}^2/\Gamma}\) be the minimal resolution of \(\mathbb{C}/\Gamma\). Associated to \(\Gamma\) is a Heisenberg algebra of affine type, \(\mathfrak{h}_\Gamma\), and the Hilbert schemes of points \(\mathrm{Hilb}^n(S)\). \textit{I. Grojnowski} [Math. Res. Lett. 3, No. 2, 275--291 (1996; Zbl 0879.17011)] and \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] construct a representation of the Heisenberg algebra (actually a slightly different version from the one considered in this paper) on the cohomology of the Hilbert schemes. Algebraically, \textit{I. Frenkel}, \textit{N. Jing} and \textit{W. Wang} [Int. Math. Res. Not. 2000, No. 4, 195--222 (2000; Zbl 1011.17020)] construct the basic representation of \(\mathfrak{h}_\Gamma\) on the Grothendieck group of the category of \(\mathbb{C}[\Gamma^n \rtimes S_n]\)-modules.
In this paper, the authors define a 2-category \(\mathcal{H}_\Gamma\) and their first main result (3.4) states that \(\mathcal{H}_\Gamma\) categorifies the Heisenberg algebra \(\mathfrak{h}_\Gamma\).
The second main result of the paper (4.4) is a categorical action of \(\mathcal{H}_\Gamma\) on a 2-category \(\bigoplus_{n\geq 0} D(A_n^\Gamma -\mathrm{gmod})\). Here \(D(A_n^\Gamma -\mathrm{gmod})\) denotes the bounded derived category of finite-dimensional, graded \(A_n^\Gamma\)-modules, where
\[
A_n^\Gamma = [(\mathrm{Sym}^*((\mathbb{C}^2)^\vee) \rtimes \Gamma) \otimes \ldots \otimes (\mathrm{Sym}^*((\mathbb{C}^2)^\vee) \rtimes \Gamma) ] \rtimes S_n.
\]
As explained in Section 8, \(D(A_n^\Gamma -\mathrm{gmod})\) is known to be equivalent to \(D\mathrm{Coh}(\mathrm{Hilb}^n(S))\) and thus the second main theorem categorifies a representation similar to that of Grojnowski [Zbl 0879.17011] and Nakajima [Zbl 0915.14001].
In Section 9, another 2-representation of \(\mathcal{H}_\Gamma\) is introduced that is related to the first by Koszul duality. In section 9.6, it is shown that this 2-representation categorifies an action similar to that constructed by Frenkel, Jing and Wang [Zbl 1011.17020].
For the most part geometry appears only in Section 8. The main definitions are algebraic and a number of the proofs are based on graphical calculus. Section 10 contains a description of various connections to other categorical actions and some open problems.
As the case \(\Gamma = \mathbb{Z}/2\) differs slightly, the necessary modifications are addressed separately in a short appendix. categorification; Heisenberg algebra; McKay correspondence; Hilbert scheme S. Cautis & A. Licata, ``Heisenberg categorification and Hilbert schemes'', Duke Math. J.161 (2012) no. 13, p. 2469-2547 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Quantum groups and related algebraic methods applied to problems in quantum theory, Frobenius induction, Burnside and representation rings, Lie algebras and Lie superalgebras, Parametrization (Chow and Hilbert schemes) Heisenberg categorification 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 \({\mathcal O}_q (M_n)\) denote the coordinate ring of quantum \(n \times n\) matrices. We show there exists a subvariety \({\mathcal P}_n\) of \(\mathbb{P} (M_n)\) and an automorphism \(\sigma_n\) of \({\mathcal P}_n\) such that \({\mathcal O}_q (M_n)\) determines, and is determined by, the geometric data \(\{{\mathcal P}_n, \sigma_n\}\); the linear span of the defining relations of \({\mathcal O}_q (M_n)\) is the set of all those elements of \(M^*_n \otimes M^*_n\) that vanish on the graph of \(\sigma_n\). Moreover, if \(q^2 \neq 1\), the variety \({\mathcal P}_n\) is independent of \(q\). Our main result is that there are two natural descriptions of \({\mathcal P}_n\). Firstly, if \(q \in k^\times\), there is a natural bijection between \({\mathcal P}_n\) and the point modules over \({\mathcal O}_q (M_n)\), and the automorphism \(\sigma_n\) is the shift functor on point modules. Secondly, since \({\mathcal O}_q (M_n)\) is a graded flat deformation of \({\mathcal O}_1 (M_n)\), the polynomial ring \({\mathcal O} (M_n)\), there is a homogeneous Poisson bracket on \({\mathcal O} (M_n)\), and an associated Poisson structure on \(\mathbb{P} (M_n)\). In this context, if \(q^2 \neq 1\), the variety \({\mathcal P}_n\) consists of those points of \(\mathbb{P} (M_n)\) which are the zero-dimensional symplectic leaves with respect to this Poisson structure. coordinate ring of quantum matrices; coordinate ring of quantum \(n \times n\) matrices; automorphisms; defining relations; variety; point modules; graded flat deformations; polynomial rings; homogeneous Poisson brackets; Poisson structures; symplectic leaves Vancliff, M.: The defining relations of quantum n\(\times n\) matrices. J. lond. Math. soc. (2) 52, No. 2, 255-262 (1995) Twisted and skew group rings, crossed products, Noncommutative algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations, Graded rings and modules (associative rings and algebras), Quantum groups and related algebraic methods applied to problems in quantum theory, Automorphisms of curves, Low codimension problems in algebraic geometry The defining relations of quantum \(n\times n\) matrices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a new description of the Pieri rule for \(k\)-Schur functions using the Bruhat order on the affine type-A Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and \(k\)-Schur functions. \(k\)-Schur functions; Pieri rule; Bruhat order; Macdonald polynomials; Hall-Littlewood polynomials; \(k\)-tableaux Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Combinatorial aspects of representation theory, Enumerative problems (combinatorial problems) in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds The ABC's of affine Grassmannians and Hall-Littlewood 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 Artin-Schelter regular algebras of global dimension 3 have been completely classified. Much has been written about the associated geometry in the case where \(A\) is a regular algebra generated by elements of degree 1. In this paper, the author continues a study of the geometry of \(\text{Tails }A=\text{GrMod }A/\text{Fdim }A\) when \(A\) is a regular algebra of global dimension 3 that is not generated by elements of degree 1. Here \(\text{GrMod }A\) is the category of graded right \(A\)-modules and \(\text{Fdim }A\) is the full subcategory consisting of direct limits of finite-dimensional modules.
This paper starts by sharpening the classification of regular algebras of weight \((1,1,n)\) with \(n>1\). \textit{D. R. Stephenson} [J. Algebra 183, No. 1, 55-73 (1996; Zbl 0868.16027)] proved that a regular algebra \(A\) of weight \((1,1,n)\) is isomorphic to an Ore extension \(A\cong R[z;\sigma,\delta]\) where \(R\) is regular of global dimensional 2, \(\sigma\) is a graded automorphism of \(R\), and \(\delta\) is a graded left \(\sigma\)-derivation on \(R\). In Section 1 of this paper, by direct computation, the author determines all possibilities for \(\sigma\) and \(\delta\) in each case. In Section 3, a point module over \(A\) is defined to be a graded, normalized right \(A\)-module which is 1-critical and has multiplicity at most 1. In Section 5, it is shown that the set of isomorphism classes of point modules over \(A\) has the structure of the graph of an automorphism \(\tau=\tau_A\) of a subscheme \(D_A\subseteq P(1,1,n)\). If \(R\cong k\{x,y\}/(yx-qxy)\) and \(A\cong R[z;\sigma,\delta]\) with \(\sigma\) generic, it is proved that \(D_A\) breaks down into three components: two lines and a nonsingular curve. The automorphism \(\tau\) stabilizes these components. Other possibilities exist for the decomposition of \(D_A\). Some examples are given in Section 5 and 6. noncommutative schemes; quantum planes; Artin-Schelter regular algebras; categories of graded right modules; finite-dimensional modules; Ore extensions; graded automorphisms; point modules Darin R. Stephenson, Quantum planes of weight (1,1,\?), J. Algebra 225 (2000), no. 1, 70 -- 92. , Graded rings and modules (associative rings and algebras), Quantum groups (quantized enveloping algebras) and related deformations, Elliptic curves, Homological dimension in associative algebras, Ordinary and skew polynomial rings and semigroup rings, Rings arising from noncommutative algebraic geometry Quantum planes of weight \((1,1,n)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author describes how to specify vector bundles of conformal blocks for $\mathfrak{sl}_{2M}$ with rectangular weights of ranks $0$, $1$, and larger than $1$, respectively. For rank one bundles, the author further shows that their first Chern classes determine a finitely generated subcone of the nef cone of the moduli space of stable $n$-pointed rational curves. In order to prove the results, the author uses Witten's dictionary and Kostka numbers for computing ranks by counting Young tableaux. vector bundle; moduli space; stable rational curve Families, moduli of curves (algebraic) Quantum Kostka and the rank one problem for \(\mathfrak{sl}_{2m}\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 stratification associated with the number of generators of the ideals of the punctual Hilbert scheme of points on the affine plane has been studied since the 1970s. In this paper, we present an elegant formula for the E-polynomials of these strata. Symmetric functions and generalizations, Polynomial rings and ideals; rings of integer-valued polynomials, Parametrization (Chow and Hilbert schemes) A note on the E-polynomials of a stratification of the Hilbert scheme of points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper starts with an introduction to some basic concepts of algebraic geometry and their relations with commutative algebra. In the second chapter the authors introduce standard bases (generalization of Gröbner bases to non-well-orderings), which are needed for applications to local algebraic geometry (singularity theory), and a method for computing syzygies and free resolutions. The last chapter describes a new algorithm for computing the normalizition of a reduced affine ring and gives an elementary introduction to singularity theory. Moreover algorithms are given, which use standard bases to compute infinitesimal deformations and obstructions. Gröbner bases; local algebraic geometry; singularity theory; free resolutions; computing syzygies; non-well-orderings Greuel, G. -M.; Pfister, G.: Gröbner bases and algebraic geometry. Gröbner bases and applications, 109-143 (1998) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Computational aspects in algebraic geometry, Syzygies, resolutions, complexes and commutative rings Gröbner bases and algebraic geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the relation between left and right adjoint functors to the precomposition functor. As a consequence, we obtain various dualities for the Ext-groups in the category of strict polynomial functors. strict polynomial functor; Poincaré duality; Ext-group Functor categories, comma categories, Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.), Ext and Tor, generalizations, Künneth formula (category-theoretic aspects), Linear algebraic groups over arbitrary fields, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Poincaré duality for Ext-groups between strict polynomial functors | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 field \(\mathbf{k}\) and a noetherian commutative algebra \(A\) over \(\mathbf{k}\), impose an ascending filtration on \(A\) to build the Rees ring of the filtration, which is a free \(\mathbf{k}[t]\)-algebra \(\mathcal R\) such that \(A \cong \mathcal R / (t - \lambda)\mathcal R\) for all \(\lambda \in \mathbf{k}^\times\), while \(\mathcal R / t \mathcal R\) is isomorphic to the associated graded ring. Geometrically, if \(\mathbf{k}\) is algebraically closed, then the variety associated to \(\mathcal R\) is a flat family over the affine line, whose generic fiber is isomorphic to \(\text{Spec}\: A\) and whose zero fiber is isomorphic to \(\text{Spec}~\textsf{gr} A\). In this context, the fiber over \(0\) is called a degeneration of \(\text{Spec}\: A\). A standard result from algebraic geometry states that if the fiber over \(0\) is regular (Gorenstein, Cohen-Macaulay, etc.), then all fibers are regular (Gorenstein, Cohen-Macaulay, etc.)
The idea of studying a ring by imposing a filtration and passing to the associated graded ring is a basic tool for algebraists, and can be applied outside of a geometric context. In fact, commutativity is not a necessary condition for filtered-to-graded methods to work. In the spirit of noncommutative algebraic geometry, one studies the case where \(A\) is noetherian, \(\mathbb{N}\)-graded and connected, i.e., \(A_0 = \mathbf{k}\). Although in this case there are no varieties associated to such algebras as in the commutative setting, there are suitable analogues of the notions of being regular, Gorenstein, or Cohen-Macaulay, defined in purely homological terms. Thus it makes sense to ask whether these properties are ``stable by flat deformation'', i.e., if \(\textsf{gr} A\) has any of these properties, then \(A\) also has that property. In [J. Algebra 372, 293--317 (2012; Zbl 1279.16029); Algebr. Represent. Theory 18, No. 5, 1155--1186 (2015; Zbl 1388.16030)], the authors develop these ideas in order to study natural classes of noncommutative algebraic varieties, and show that good geometric properties are stable by degeneration in this context.
In the commutative setting, the usual flag variety and its Schubert subvarieties are examples where the degeneration method is successful. On the other hand, the theory of quantum groups provides natural quantum analogues of flag and Schubert varieties, whose classical counterparts can be recovered as semi-classical limits when the deformation parameter tends to \(1\). This is the class of noncommutative varieties that the authors study.
In [Transform. Groups 7, No. 1, 51--60 (2002; Zbl 1050.14040)], \textit{P. Caldero} was the first to prove that any Schubert variety of an arbitrary flag variety degenerates to an affine toric variety. A degeneration to a toric variety is particularly convenient since the geometric properties of toric varieties are easily tractable, being encoded in the combinatorial properties of a semigroup naturally attached to them in the affine case. Caldero uses the theory of quantum groups (the global bases of Lusztig and Kashiwara) to produce bases of the coordinate rings of the classical objects by relying on Littelmann's string parametrization of the crystal basis of the negative part of the quantized enveloping algebra, which allows him to show that his bases have good multiplicative properties. Thus, one can build adequate filtrations of the coordinate rings, leading to a toric degeneration.
The authors wish to produce degenerations of quantum Schubert varieties at the noncommutative level, leading to introduce noncommutative analogues of affine toric varieties as a suitable target for degeneration and their geometric properties, where these properties are encoded in the associated semigroup. Inspired by Caldero's work, the authors show that quantum Schubert varieties degenerate into quantum toric varieties (Theorem 27, page 1141), followed by exploring their consequences. quantum toric degeneration; quantum flag variety; quantum Schubert varieties; AS-Gorenstein; AS-Cohen-Macaulay Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Quantum groups (quantized enveloping algebras) and related deformations, Deformations of associative rings Quantum toric degeneration of quantum flag and Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper studies the representation of a non-negative polynomial \(f\) on a non-compact semi-algebraic set \(K\) modulo its KKT (Karush-Kuhn-Tucker) ideal. Under the assumption that \(f\) satisfies the boundary Hessian conditions (BHC) at each zero of \(f\) in \(K\), we show that \(f\) can be represented as a sum of squares (SOS) of real polynomials modulo its KKT ideal if \(f\geq 0\) on \(K\). non-negative polynomials; sum of squares (SOS); optimization of polynomials; semidefinite programming (SDP) Semialgebraic sets and related spaces, Sums of squares and representations by other particular quadratic forms, Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Semidefinite programming Representations of non-negative polynomials 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 The authors study connections between ideal lattices and multivariate polynomial rings over integers using the theory of Gröbner bases. Let us consider the vectors \(b_1,\ldots ,b_n\in \mathbb{R}^m\). Then, the set of all integral combinations of these vectors is called a lattice. Now, given a monic polynomial \(f\in \mathbb{Z}[x]\) of degree \(N\), an ideal lattice is an integer lattice in \(\mathbb{Z}^N\) isomorphoic to an ideal in \(\mathbb{Z}[x]/\langle f \rangle\). The authors introduce first the notion of multivariate cyclic lattices and show that ideal lattices are a generalization of these ideals in the multivariate case. Then, using this generalization, they construct hash functions by using the theory of Gröbner bases. Finally, shortest substitution problem with respect to an ideal in \(\mathbb{Z}[x_1,\ldots ,x_n]\) is defined and its computational hardness is used to discuss the collision resistance of the hash functions. Gröbner bases over rings; multivariate ideal lattices; hash functions Applications to coding theory and cryptography of arithmetic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) On ideal lattices, Gröbner bases and generalized hash 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 An abelian surface \(A\) is called a QM-abelian surface if its endomorphism ring includes an order of an indefinite quaternion algebra, and a curve \(C\) of genus two is called a QM-curve if its jacobian variety is a QM-abelian surface. We give a computational result about the distribution of the arguments of the eigenvalues of the Frobenius endomorphisms of QM-abelian surfaces modulo good primes, which supports an analogue of the Sato-Tate Conjecture for such abelian surfaces. We also make some remarks on the field of definition of QM-curves and their endomorphisms. Quaternionic multiplication; \(L\)-functions Hashimoto, Ki-ichiro; Tsunogai, Hiroshi, On the Sato-Tate conjecture for QM-curves of genus two, Math. Comp., 68, 228, 1649-1662, (1999) \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Complex multiplication and moduli of abelian varieties, Families, moduli of curves (algebraic), Arithmetic ground fields for abelian varieties On the Sato-Tate conjecture for QM-curves of genus two | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 states: ``The goal of this note is to describe the Grothendieck-Serre duality on an arithmetic surface after the fixing of a horizontal divisor on this surface. We apply this description to the generalization of $\theta$-invariants from the case of arithmetic curves to the case of arithmetic surfaces.'' Pontryagin dual group; ind-Euclidean lattice; pro-Euclidean lattice; Arakelov degree Arithmetic varieties and schemes; Arakelov theory; heights, Adèle rings and groups Grothendieck-Serre duality and theta-invariants on arithmetic surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials
``The intensive activity in four-dimensional Riemannian geometry in recent years has led to the interesting discovery that the general trichotomy of geometric objects, namely the elliptic, parabolic and hyperbolic cases, is determined in the four-dimensional case by smoothness...
Kodaira's trichotomy for simply-connected algebraic surfaces, namely into the cases of rational surfaces, surfaces with elliptic pencils, and surfaces of general type, determines, according to the latest conjectives and predictions, the smoothness class of the underlying oriented four- manifold. - Although this view is confirmed at the moment only in the case of three basic examples, it is based on a number of very promising results obtained recently, which clarify the differential geometric meaning of certain algebro-geometric concepts and constructions. Thus, these constructions enable one to produce exotic smooth structures on all kinds of simply connected algebraic surfaces. On the other hand, the smoothness invariants which arise in this way are, in the case of simple surfaces, of a very fine algebraic geometric nature: e.g. in the case of \({\mathbb{P}}^ 2\), they are the constants of the Schubert calculus, and one can hope to prove in this way the uniqueness of the smooth structure on \({\mathbb{P}}^ 2\). - The aim of this survey is to analyse the algebro- geometric aspects of this theme... We have preferred to concentrate on the non-standard algebraic geometric constructions which offer methods for solving the many interesting open problems which arose while the lecture course for which this survey was originally planned as notes, developed...
...For the analytic constructions needed, we refer to the monograph ``Instantons and four-manifolds'' by \textit{D. Freed} and \textit{K. Uhlenbeck} (1984; Zbl 0559.57001). Throughout the article, we consider only the simply-connected algebraic case...'' Donaldson polynomials; smooth structures of simply connected algebraic surfaces; underlying oriented four-manifold Tyurin, A.N.: Algebraic geometric aspects of smooth structures I. The Donaldson polynomials. Russ. Math. Surv.44, 113--178 (1990) Special surfaces, Differentiable structures in differential topology, Compact complex surfaces Algebro-geometric aspects of smooth structure. I: The Donaldson polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce new mathematical aspects of the Bell states using matrix factorizations, non-noetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial p consists of two matrices \(\phi_{1}\), \(\phi_{2}\) such that \(\phi_{1}\phi_{2} = \phi_{2}\phi_{1} = p\mathrm{id}\). Using this notion, we show how the Bell states emerge from the separable product of two mixtures, by defining pure states over complex matrices rather than just the complex numbers. We then show in an idealized algebraic setting that pure states are supported on non-noetherian singularities. Moreover, we find that the collapse of a Bell state is intimately related to the representation theory of the noncommutative blowup along its singular support. This presents an exchange in geometry: the \textit{nonlocal commutative} spacetime of the entangled state emerges from an underlying \textit{local noncommutative} spacetime. entanglement; Bell state; nonlocality; emergence; non-Noetherian ring; matrix factorization; noncommutative blowup; quantum foundations; quantum information; noncommutative algebraic geometry 5. C. Beil, The Bell states in noncommutative algebraic geometry, to appear in Int. J. Quantum Inform. Quantum coherence, entanglement, quantum correlations, Noncommutative geometry in quantum theory, Noncommutative algebraic geometry The Bell states in noncommutative algebraic geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{R. L. Graham, P. Diaconis} and \textit{B. Sturmfels} [Bolyai Soc. Math. Stud. 2, 173--192 (1996; Zbl 0852.05014)] give a combinatorial description of the Graver basis for any rational normal curve in terms of primitive partition identities. Their result is extended by the author to rational normal scrolls. The description of the Graver bases of scrolls is given in terms of colored partition identities. This leads to a sharp bound on the degree of Graver basis elements, which is always attained by a circuit.
The main result of the paper under review shows the following assertions hold:
(1) The degree of any binomial in the Graver basis (and the universal Gröbner basis) of any rational normal scroll is bounded above by the degree of the scroll.
(2) Let \(X\) be any toric variety that can be obtained from a scroll by a sequence of projections to some of the coordinate hyperplanes. Then the degree of an element of any reduced Gröbner basis of the toric ideal \(I_{X}\) is at most the degree of the toric variety \(X\). Petrović, S., On the universal Gröbner bases of varieties of minimal degree, Math. Res. Lett., 15, 6, 1211-1221, (2008) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Computational aspects in algebraic geometry, Rational and ruled surfaces, Varieties of low degree, Toric varieties, Newton polyhedra, Okounkov bodies, Projective techniques in algebraic geometry On the universal Gröbner bases of varieties of minimal degree | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 studies DAHA-Jones polynomials of torus knots for reduces root systems and in the case of \(C^\vee C_1\). The main results of the paper establish the polynomiality conjecture and the evaluation conjecture from a previous work by the author and extend the theory by studying the color exchange symmetry. The DAHA-Jones polynomials in the case \(C^\vee C_1\) are related to DAHA superpolynomials of type \(A\). A connection of DAHA superpolynomials to Kohavonov-Rozansky homology and superpolynomials defined via rational DAHA is discussed. Also, a connection to the absolute Galois group is investigated. double affine Hecke algebras; Jones polynomial; Khovanov-Rozansky homology; torus knot; Macdonald polynomial; Askey-Wilson polynomial; Verlinde algebra; Galois group; Chern-Simons theory Cherednik, I., DAHA-Jones polynomials of torus knots, Selecta Math., 22, 2, 1013-1053, (2016) Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Root systems, Lie algebras of linear algebraic groups, Hecke algebras and their representations, Braid groups; Artin groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Compact Riemann surfaces and uniformization, Knots and links in the 3-sphere, Floer homology DAHA-Jones polynomials of torus knots | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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, \(k\langle X_ 1,\ldots,X_ n\rangle\) the free associative algebra in the \(X_ i\) (each of degree 1 over \(k)\). Let \(g_ j\) be homogeneous in the \(X_ i\). Consider the graded algebra \(A=k\langle X_ 1,\ldots,X_ n\rangle/(g_ 1,\ldots,g_ r)\). The Hilbert series of the graded algebra \(A=\oplus^{\infty}_{i=0}A_ i\) is defined by \(A(z)=\sum_{i\geq 0}\dim_ kA_ i\cdot z^ i\).
The authors study properties that hold for commutative algebras, but do not for noncommutative algebras. Given \(r,d,d_ 1,\ldots,d_ r\), consider \(R=k[X_ 1,\ldots,X_ n]/(f_ 1,\ldots,f_ r)\) as a point in \(A=\mathbb{A}^ N_ k\), where \(\deg(f_ i)=d_ i\) and \(N=\sum^ r_{i=1}{n+d_ i-1\choose d_ i}\). \(R\) is a complete intersection of dimension \(n-r\) if \(f_ 1,\ldots,f_ r\) constitute a regular sequence. If \(r\leq n\), then most points in \(A\) correspond to complete intersections. The points in \(A\) only give rise to finitely many Hilbert series. The Hilbert series is constant on a Zariski-open set for any given sequence of integers \(n,r,d_ 1,\ldots,d_ r\). In 2-related algebras \(T=k\langle X_ 1,...\ldots,X_ n\rangle/(f_ 1,\ldots,f_ r)\), the \(f_ i\) are linear combinations of \(n^ 2\) monomials, thus \(T\) is a point in \(A'=\mathbb{A}_ k^{rn^ 2}\). It is shown that \(T(z)\geq(1- nz-rz^ 2)^{-1}\).
It is shown that the exceptional points do not constitute a closed set even in the Euclidean sense and that, for a given sequence of integers, there may be infinitely many Hilbert series and that there is no series \(A(z)\) for which the set of points with series \(A(z)\) is Zariski-open and nonempty. Hilbert series of a graded algebra; complete intersection Fröberg, R.; Löfwall, C., On Hilbert series for commutative and noncommutative graded algebras, J. Pure Appl. Algebra, 76, 33-38, (1990) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Graded rings and modules (associative rings and algebras), Graded rings, Complete intersections, Linkage, complete intersections and determinantal ideals On Hilbert series for commutative and noncommutative graded 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 \textit{S. Toda} [SIAM J. Comput. 20, No. 5, 865-877 (1991; Zbl 0733.68034)] proved that the (discrete) polynomial time hierarchy, \(\mathbf{PH}\), is contained in the class \(\mathbf P ^{\#P }\), namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class \(\# \mathbf P\). This result, which illustrates the power of counting, is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of \textit{L. Blum, M. Shub} and \textit{S. Smale} real machines [Bull. Am. Math. Soc., New Ser. 21, No.~1, 1--46 (1989; Zbl 0681.03020)]) has been missing so far. In this paper we formulate and prove a real analogue of Toda's theorem. Unlike Toda's proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature. As a consequence of our techniques, we are also able to relate the computational hardness of two extremely well-studied problems in algorithmic semi-algebraic geometry: the problem of deciding sentences in the first-order theory of the reals with a constant number of quantifier alternations, and that of computing Betti numbers of semi-algebraic sets. We obtain a polynomial time reduction of the compact version of the first problem to the second. This latter result may be of independent interest to researchers in algorithmic semi-algebraic geometry. polynomial hierarchy; Betti numbers; semi-algebraic sets; Toda's theorem S. Basu and T. Zell. Polynomial hierarchy, Betti numbers, and a real analogue of Toda's theorem. \textit{Found. Comput. Math}., 10(4):429-454, 2010. Semialgebraic sets and related spaces, Topology of real algebraic varieties, Symbolic computation and algebraic computation Polynomial hierarchy, Betti numbers, and a real analogue of Toda's 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 Author's abstract: Let \(X\) be a geometrically split, geometrically irreducible variety over a field \(F\) satisfying Rost nilpotence principle. Consider a field extension \(E/F\) and a finite field \(\mathbb F\). We provide in this note a motivic tool giving sufficient conditions for so-called outer motives of direct summands of the Chow motive of \(X_{E}\) with coefficients in \(\mathbb F\) to be lifted to the base field. This going down result has been used by \textit{S. Garibaldi, V. Petrov} and \textit{N. Semenov} [``Shells of twisted flag varieties and non-decomposibility of the Rost invariant'', \url{arXiv:1012.2451}] to give a complete classification of the motivic decompositions of projective homogeneous varieties of inner type \(E_{6}\), and to answer a conjecture of Rost and Springer. Chow groups; Grothendieck motives; projective homogeneous varieties C. De Clercq, A going down theorem for Grothendieck Chow motives , Q. J. Math. 64 (2013), 721-728. (Equivariant) Chow groups and rings; motives, Linear algebraic groups over arbitrary fields, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) A going down theorem for Grothendieck-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 For each finite group \(G\), we define the Grothendieck-Teichmüller group of \(G\), denoted \(\mathrm{{GT}}(G)\), and explore its properties. The theory of dessins d'enfants shows that the inverse limit of \(\mathrm{{GT}}(G)\) as \(G\) varies can be identified with a group defined by Drinfeld and containing \(\mathrm{{Gal}}(\overline{\mathbb Q}/ \mathbb {Q})\). We give, in particular, an identification of \(\mathrm{{GT}}(G)\), in the case when~\(G\) is simple and non-abelian, with a certain very explicit group of permutations that can be analyzed easily. With the help of a computer, we obtain precise information for~\(G= \mathrm{PSL}_2(\mathbb {F}_q)\) when \(q \in \{4, 7, 8, 9, 11, 13, 16, 17, 19\}\), and we treat \(A_7\), \(\mathrm{PSL}_3({\mathbb {F}}_3)\) and \(M_{11}\). In the rest of the paper we give a conceptual explanation for the technique which we use in our calculations. It turns out that the classical action of the Grothendieck-Teichmüller group on dessins d'enfants can be refined to an action on ``\(G\)-dessins'', which we define, and this elucidates much of the first part. P. Guillot, The Grothendieck-Teichmüller group of a finite group and \textit{G}-dessins d'enfants, Symmetries in Graphs, Maps, and Polytopes, Springer Proc. Math. Stat. 159, Springer, Cham (2016), 159-191. Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Arithmetic aspects of dessins d'enfants, Belyĭ theory The Grothendieck-Teichmüller group of a finite group and \(G\)-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 We compute \(t\)-analogs of \(q\)-characters of all \(l\)-fundamental representations of the quantum affine algebras of type \(E^{(1)}_6\), \(E^{(1)}_7\), \(E^{(1)}_8\) by a supercomputer. (Here \(l\) stands for the loop.) In particular, we prove the fermionic formula for Kirillov-Reshetikhin modules conjectured by [\textit{G. Hatayama, A. Kuniba, M. Okado, T. Takagi} and \textit{Y. Yamada}, Remarks on fermionic formula. Contemp. Math. 248, 243--291 (1999; Zbl 1032.81015)] for these classes of representations. We also give explicitly the monomial realization of the crystal of the corresponding fundamental representations of the quantum enveloping algebras associated with finite dimensional Lie algebras of types \(E_6\), \(E_7\), \(E_8\). These are computations of Betti numbers of graded quiver varieties, quiver varieties and determination of all irreducible components of the Lagrangian subvarieties of quiver varieties of types \(E_6\), \(E_7\), \(E_8\), respectively. H. Nakajima, \textit{t-analogs of q-characters of quantum affine algebras of type A}\_{}\{\(n\)\}\textit{, D}\_{}\{\(n\)\}, math/0204184. Quantum groups (quantized enveloping algebras) and related deformations, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Representations of quivers and partially ordered sets \(t\)-analogs of \(q\)-characters of quantum affine algebras of type \(E_6\), \(E_7\), \(E_8\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0642.00007.]
Let \(J_{\rho}\) denote the Jacobi quartic (of modulus \(\rho)\) defined inhomogeneously by the equation \(Y^ 2=1-2\rho X^ 2+X^ 4\) over the function field \(K={\mathbb{Q}}(\rho)\) where \(\rho\) is a transcendental variable over the field \({\mathbb{Q}}\) of rational numbers. There is associated to \(J_{\rho}\) the non-zero differential 1-form of the first kind
\[
\omega_{\rho}=\frac{dX}{Y}=\frac{dX}{\sqrt{1-2\rho X^ 2+X^ 4}}=\sum^{\infty}_{n=0}P_ n(\rho)X^{2n}dX
\]
where \(P_ n(\rho)\) denotes the n-th Legendre polynomial. The purpose of this paper is to study the Jacobi quartic \(J_{\rho}\), Legendre polynomials \(P_ n(\rho)\) (n\(\in {\mathbb{N}})\) and the formal groups associated to them in the framework of the Honda theory of commutative formal groups [cf. \textit{T. Honda}, J. Math. Soc. Japan 22, 213-246 (1970; Zbl 0202.031)]. Jacobi quartic over function field; Legendre polynomial Formal groups, \(p\)-divisible groups, Spherical harmonics Jacobi quartics, Legendre polynomials and formal groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article contains some algebraic tools that can be used to make computations in the cohomology ring of Lagrangian flag manifolds, and Lagrangian degeneracy loci. The main tool is the study of several operators on a certain basis, which is orthonormal under a scalar product. This basis is useful in studying Schubert classes in Lagrangian manifolds.
The article also contains some simple proofs of previously known results, for example of the Giambelli-type formula for maximal Lagrangian Schubert classes. Lagrangian flag manifolds; Lagrangian degeneracy loci; Schubert polynomials; Giambelli-type formula Lascoux, A; Pragacz, P, Operator calculus for \({\widetilde{Q}}\)-polynomials and Schubert polynomials, Adv. Math., 140, 1-43, (1998) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Determinantal varieties, Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Operator calculus for \(\widetilde{Q}\)-polynomials and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum duality principle (QDP) for homogeneous spaces gives four recipes to obtain, from a quantum homogeneous space, a dual one, in the sense of Poisson duality. One of these recipes fails (for lack of the initial ingredient) when the homogeneous space we start from is not a quasi-affine variety. In this work we solve this problem for the quantum Grassmannian, a key example of quantum projective homogeneous space, providing a suitable analogue of the QDP recipe. Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups (quantized function algebras) and their representations, Geometry of quantum groups Quantum duality principle for quantum Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(M(n,d)\) denote the moduli space of stable vector bundles of rank \(n\) and degree \(d\) on a smooth curve \(X\). The Poincaré and Hodge polynomials of \(M(n,d)\) for \(n\) and \(d\) coprime were determined by several authors. In case \(n\) and \(d\) are not coprime, a few partial results are known. In this paper, the author determines the virtual Poincaré polynomial of \(M(n,d)\) for arbitrary \(n, d\). The virtual Poincaré polynomial of a variety over a finite field \(k\) is defined in terms of the weight filtration on the compactly supported \(l\)-adic cohomology groups. To determine the virtual Poincaré polynomial of \(M(n,d)\), the author introduces a new \(\lambda\) ring called the ring of \(c\)-sequences. A Poincaré function is associated to each \(c\)-sequence. The virtual Poincaré polynomial of \(M(n,d)\) is the Poincaré function of a \(c\)-sequence \(a_{n,d}\), where \(a_{n,d}(k)\) denotes the number of stable sheaves over \(X\) which remain stable over all finite field extensions.
The author gives conjectures regarding virtual Hodge polynomials of \(M(n,d)\) and also virtual Poincaré and Hodge polynomials of the motive of \(M(n,d)\). virtual Poincaré polynomial; moduli of stable bundles; smooth curve Mozgovoy, S.: Invariants of moduli spaces of stable sheaves on ruled surfaces (2013). arXiv:1302.4134 Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Enumerative problems (combinatorial problems) in algebraic geometry Poincaré polynomials of moduli spaces of stable bundles over curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We correct a definition in ``Discriminants in the Grothendieck ring'' [ibid. 164, No. 6, 1139--1185 (2015; Zbl 1461.14020)]. discriminants; Grothendieck ring of varieties Arcs and motivic integration, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Fibrations, degenerations in algebraic geometry Erratum to: ``Discriminants in the Grothendieck ring'' | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We sketch a proof of a conjecture of [\textit{M. Finkelberg} and \textit{I. Mirković}, Transl., Ser. 2, Am. Math. Soc. 194(44), 81--112 (1999; Zbl 1076.14512)] that relates the geometric Eisenstein series sheaf with semi-infinite cohomology of the small quantum group with coefficients in the tilting module for the big quantum group. semi-infinite cohomology Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Ring-theoretic aspects of quantum groups, Geometric Langlands program: representation-theoretic aspects, Geometry of quantum groups Eisenstein series 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 This is an application of a new algebraic reformulation of the Picard group \(\text{pic} (G)\) of a quasi-compact subset \(G \subseteq X = \text{Spec} A\) for a commutative ring \(A\). A torsion theoretic (algebraic) proof is given of \(A\). Grothendieck's theorem that a complete intersection that is factorial in co-dimension 3 is factorial. Our proof is along the lines of Grothendieck's SGA2 proof, but eliminates the need for spectral sequences and (formal) sheaf theory.
All torsion theoretic details are given in a lengthy appendix, including the proof of the long standing conjecture that \(Q_G = \varinjlim Q_U\) for \(G\) quasicompact and \(U\) open, where \(Q_G\) is defined in the following way: Denote by \(F_G = \{I \subseteq A |I \nsubseteq\) any \(p \in G\}\) the torsion filter corresponding to \(G\). \(F_G\) is a directed set under reverse inclusion so we can form, for any \(A\)-module \(M\), the \(A\)-module \(P_G (M) : = \varinjlim_{I \in F_G} \Hom (I,M)\) and \(Q_G (M) : = P_G (P_G(M))\). This yields the interesting result that restriction of an \({\mathcal O}_X\)-Module to a quasi-compact, generically closed subset does not require the sheafification process. Picard group; complete intersection; factorial; torsion Call, F.: A theorem of Grothendieck using Picard groups for the algebraist. Math. scand. 74, 161-183 (1994) Torsion modules and ideals in commutative rings, Picard groups, Linkage, complete intersections and determinantal ideals A theorem of Grothendieck using Picard groups for the algebraist. -- Appendix | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 first observes that the classical Bernstein polynomials
\[
B_{N}(f)(x)=\sum_{j=0}^{N}\binom{N}{j}f\left( \frac{j}{N}\right) x^{j}(1-x)^{N-j}
\]
of a smooth function \(f\in C^{\infty}([0,1])\) and their asymptotic expansion
\[
B_{N}(f)(x)\sim\sum_{\mu=0}^{\infty}L_{\mu}f(x)N^{-\mu}
\]
(where \(L_{\mu}=L_{\mu}\left( x,\frac{\text{d}}{\text{d} x}\right) \) are certain polynomial differential operators), are closely related to the Bergman-Szegö kernels for the Fubini-Study metric on the \(N\)-th powers \(\mathcal{O}(N)\) of the hyperplane line bundle \(\mathcal{O} (1)\rightarrow\mathbb{CP}^{1}\). This holds also for functions \(f\) of \(m\) variables over the cube, with \(\mathbb{CP}^{1}\) replaced by \(\mathbb{CP}^{m}\).
The paper generalizes this picture to a toric projective Kähler variety \(M\) and a Delzant polytope \(P\). Generalized Bernstein polynomial approximations \(B_{h^{N}}\) are defined and studied for \(f\in C(P)\), with respect to a Hermitian metric \(h\) on a line bundle \(L\rightarrow M\). A generalization of the asymptotic expansion of Bernstein polynomials is proved, as well as an asymptotic expansion for Dedekind-Riemann sums. Bernstein polynomials; toric varieties; Delzant polytope; Dedekind-Riemann sums; asymptotic expansions; Bergman-Szegö kernels S. Zelditch. Bernstein polynomials, Bergman kernels, and toric Kähler varieties, J. Symplectic Geom. (to appear); arXiv: 0705.2879. Kähler manifolds, Toric varieties, Newton polyhedra, Okounkov bodies Bernstein polynomials, Bergman kernels and toric Kähler varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the connection between the geometry of Hilbert schemes and integrable hierarchies. The main goal of the paper is to establish a direct link between equivariant cohomology rings of Hilbert schemes \(X^{[n]}\) of \(n\)-points on a quasi-projective surface \(X\) and integrable hierarchies as well as the correspondence with stationary Gromov-Witten theory. The authors study the case of \(X={\mathbb C}^2\) -- the affine plane. The action of \(T={\mathbb C}^*\) given by the formula \(t(w,z)=(tw,t^{-1}z)\) induces an action on the Hilbert scheme \(X^{[n]}\) with finitely many fixed points parametrized by partitions of \(n\) [cf. \textit{G. Ellingsrud} and \textit{S. A. Strømme}, Invent. Math. 87, 343--352 (1987; Zbl 0625.14002)].
\textit{E. Vasserot} [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 1, 7--12 (2001; Zbl 0991.14001)] has shown that the construction of the Heisenberg algebra given in [\textit{H. Nakajima}, Lectures on Hilbert schemes of points on surfaces (1999; Zbl 0949.14001)] can be extended to the \(T\)-equivariant cohomology of Hilbert schemes. All information of the equivariant cohomology ring \(H^{*}_T(X^{[n]})\) is then encoded in \({\mathbb H}_n=H^{2n}_T(X^{[n]})\) and \({\mathbb H}_{X}=\bigoplus_{n\geq 0}{\mathbb H}_n\) becomes the bosonic Fock space of a Heisenberg algebra. The ring \({\mathbb H}_n\) was identified in [Zbl 0991.14001] with the class algebra of the symmetric group \(S_n\).
This allows the authors to establish the correspondence between the \(k\)-th equivariant Chern characters of the tautological rank \(n\) vector bundle and the \(k\)-th power sum of Jucys-Murphy elements. The authors introduce the moduli spaces \({\mathcal M}(m,n)\) , where \(m\in {\mathbb Z}, n\geq 0\). The equivariant cohomology ring of \({\mathcal M}(m,n)\) corresponds to a ring \({\mathbb H}_n^{(m)}\) and \(\bigoplus_{n,m}{\mathbb H}_n^{(m)}\) can be identified with the fermionic Fock space via the boson-fermion correspondence. The introduction of the spaces \({\mathcal M}(m,n)\) allows one to reduce the study of equivariant intersection theory on the Hilbert schemes to the study of intersection numbers of equivariant Chern characters in \({\mathcal M}(m,n)\). The intersection numbers of equivariant Chern characters are studied via the generating functions.
The authors consider three types of generating functions: the \(N\)-point function, the multipoint trace function and the \(\tau\)-function. The authors show that the first function is related to the \(N\)-point disconnected series of stationary Gromov-Witten invariants of \({\mathbb P}^1\), the second is related to the characters on the fermionic Fock space and the third is actually the \(\tau\)-function for the Toda hierarchy. W.-P. Li, Z. Qin, and W. Wang, ''Hilbert schemes, integrable hierarchies, and Gromov-Witten theory,'' Int. Math. Res. Not. 40 (2004), 2085--2104. Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Virasoro and related algebras Hilbert schemes, integrable hierarchies, and Gromov-Witten theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Gau-Wu number of a matrix \(A \in M_n(\mathbb{C})\), denoted by \(k(A)\), is defined by the maximum size of an orthonormal set \(\{x_1, \dots, x_k\} \subseteq \mathbb{C}^n\) such that the values \(\langle Ax_j, x_j\rangle\) lie on the boundary of the numerical range of \(A\). \textit{H.-W. Gau} and \textit{P. Y. Wu} [Oper. Matrices 7, No. 2, 465--476 (2013; Zbl 1278.15026)] showed that \(2 \leq k(A) \leq n\). If \(H_1\) and \(H_2\) are the real and imaginary parts of \(A\), respectively, then the homogeneous polynomial \(F_A(x : y : t) = \det(xH_1 + yH_2 + tI_n)\) is called the base polynomial and the curve \(F_A(x : y : t) = 0\) in the projective space \(\mathbb{C}\mathbb{P}^2\) is said to be the base curve of the matrix \(A\).
As a continuation of the work \textit{K.-Z. Wang} and \textit{P. Y. Wu} [Linear Algebra Appl. 438, No. 1, 514--532 (2013; Zbl 1261.15032)] on classifying the Gau-Wu numbers for \(3 \times 3\) matrices, the authors consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify \(k(A)\). They generalize the concept of flat portions on the boundary of the numerical range, and for \(n \times n\) unitarily irreducible matrices \(A\), give necessary conditions for \(k(A) = 2\), and characterize \(k(A) = n\) in terms of singularities. They use their results to consider the effect of singularities on \(k(A)\) for any \(4 \times 4\) matrix \(A\). In addition, they consider unitarily irreducible matrices \(A\) with reducible base polynomials, and thus, extend the characterization of the irreducible base curves of \(4 \times 4\) matrices by \textit{M.-T. Chien} and \textit{H. Nakazato} [Electron. J. Linear Algebra 23, 755--769 (2012; Zbl 1253.15030)]. Finally, they use the recently proved Lax conjecture to give a new proof of a theorem of \textit{J. W. Helton} and \textit{I. M. Spitkovsky} [Oper. Matrices 6, No. 3, 607--611 (2012; Zbl 1270.15014)]. Several nice examples of matrices with different singularities and values of \(k(A)\) are presented as well. numerical range; field of values; Gau-Wu number; boundary generating curve; irreducible; singularity Norms of matrices, numerical range, applications of functional analysis to matrix theory, Special algebraic curves and curves of low genus Singularities of base polynomials and Gau-Wu numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors of the paper under review explain how the standard Gröbner algorithms need to be modified to handle general valued fields. A key tool to overcome the issue that the standard normal form algorithm need not terminate is to replace it by a modification of Mora's tangent cone algorithm using an appropriate écart function. Complexity and implementation issues are discussed. The obtained results have applications in tropical geometry. Gröbner bases; valued field Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Foundations of tropical geometry and relations with algebra, Max-plus and related algebras Gröbner bases over fields with valuations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(X \hookrightarrow \Pi\) be the Plücker embedding of the manifold \(X\) of complete flags in \(\mathbb C^{r+1}\) into the product of projective spaces \(\Pi = \prod_{i=1}^{r} \mathbb C\mathbb P^{n_{i}-1}\). The space of degree \(d\) holomorphic maps \(\mathbb C\mathbb P^1 \to \mathbb C\mathbb P^n\) is compactified to a complex projective space of dimension \((N+1)(d+1)-1\), which is denoted by \(\mathbb C\mathbb P^N_d\). This construction defines the compactification \(\Pi_d = \prod_{i=1}^r \mathbb C\mathbb P_{d_i}^{n_i-1}\) of the space of degree \(d\) maps from \(\mathbb C\mathbb P^1\) to \(\Pi\). Composing degree \(d\) holomorphic maps from \(\mathbb C\mathbb P^1\) to \(X\) with the Plücker embedding, we can embed the space of such maps into \(\Pi_d\). The closure \(QM_d\) of the image of this embedding is called the Drinfel'd compactification of the space of degree \(d\) maps from \(\mathbb C\mathbb P^1\) to \(X\), or the space of quasi maps.
Since \(X\) is a homogeneous space and there is a natural action of \(S^1\) on \(\mathbb C\mathbb P^1\) we have an action of \(G=S^1 \times SU_{r+1}\) on \(QM_d\). We denote by \(P = (P_1, \dots , P_r)\) the \(G\)-equivariant line bundles obtained by pulling back the Hopf bundles over the factor of \(\Pi_d\), and by \(P^z = P^{z_1}_1 \otimes \cdots \otimes P^{z_r}_r\) its tensor product. Next, we define the function
\[
{\mathcal G}(Q, z, q, \Lambda) = \sum_d \; Q^d \chi_G (H^*\widetilde{QM_d}, P^z)),
\]
where \(q\) and \(\Lambda_0, \dots, \Lambda_r\) are multiplicative coordinates on \(S^1\) and on the maximal torus \(T^r\) of \(SU_{r+1}\), respectively, and \(Q = (Q_1, \dots, Q_r)\) are formal variables. Here, \(\widetilde{QM_d}\) is a \(G\)-equivariant desingularization of \(QM_d\).
The main result of the paper states that the function
\[
G(Q,Q') = {\mathcal G}(Q, \frac{\ln Q'- \ln Q}{\ln q}, q, \Lambda)
\]
is the eigenfunction of the Toda operator
\[
\widehat{H}_{Q', q}G = (\Lambda_0 + \dots + \Lambda_r)G,
\]
where
\[
\widehat{H}_{Q,q} = q ^{\frac{\partial}{\partial q_0}} + q ^{\frac{\partial}{\partial q_1}} (1-e^{t_0-t_1}) + \cdots + q ^{\frac{\partial}{\partial q_r}}(1-e^{t_{r-1}-t_r}).
\]
The authors also explore the possibility to extend the above theorem for a general case of flag manifolds \(X=G/B\) where \(G\) is an arbitrary semi-simple complex Lie group \(G\). quantum \(K\)-theory; flag manifold; quantum group; Toda lattice; Drinfel'd compactification A. Givental and Y.-P. Lee, \textit{Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups}, math.AG/0108105. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Quantum \(K\)-theory on flag manifolds, finite-difference Toda lattices 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 We propose a notion of functor of points for noncommutative spaces, valued in categories of bimodules, and endowed with an action functional determined by a notion of hermitian structures and height functions, modeled on an interpretation of the classical functor of points as a physical sigma model. We discuss different choices of such height functions, based on different notions of volumes and traces, including one based on the Hattori-Stallings rank. We show that the height function determines a dynamical time evolution on an algebra of observables associated to our functor of points. We focus in particular the case of noncommutative arithmetic curves, where the relevant algebras are sums of matrix algebras over division algebras over number fields, and we discuss a more general notion of noncommutative arithmetic spaces in higher dimensions, where our approach suggests an interpretation of the Jones index as a height function. noncommutative arithmetic spaces; functor of points; Arakelov height; arithmetic curves Arithmetic varieties and schemes; Arakelov theory; heights, Noncommutative algebraic geometry Functor of points and height functions for noncommutative Arakelov 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 [\textit{V. Tarasov} and \textit{A. Varchenko}, Eur. J. Math. 7, No. 2, 706--728 (2021; Zbl 1475.14111)] the equivariant quantum differential equation \((qDE)\) for a projective space was considered and a compatible system of difference \(qKZ\) equations was introduced; the space of solutions to the joint system of the \(qDE\) and \(qKZ\) equations was identified with the space of the equivariant \(K\)-theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant \(K\)-theory algebra. This paper is a continuation of the paper by Tarasov and Varchenko. \par We describe the relation between solutions to the joint system of the \(qDE\) and \(qKZ\) equations and the topological-enumerative solution to the \(qDE\) only. definitioned as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant \(K\)-theory algebra, the equivariant Gamma class of the projective space, and the equivariant first Chern class of the tangent bundle of the projective space. \par We consider a Stokes basis, the associated exceptional basis in the equivariant \(K\)-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincaré pairing wrt to the basis, which is the left dual to the associated exceptional basis. \par We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space. \par These statements are equivariant analogs of results of G. Cotti, B. Dubrovin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani. differential geometry; algebraic geometry Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain, Applications of Lie algebras and superalgebras to integrable systems, Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms Equivariant quantum differential equation and \(qKZ\) equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-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 It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting variety of some parabolic subalgebras of \(M_{n}(\Bbbk)\) and to the punctual nested Hilbert scheme. By this method, we obtain a lower bound on the dimension of these moduli spaces. We characterize the cases where they are irreducible. In some reducible cases, we describe the irreducible components and their dimensions. Hilbert scheme; commuting variety; GIT; parabolic algebra; nilpotent orbit [4] Michael Bulois &aLaurent Evain, &Nested punctual Hilbert schemes and commuting varieties of parabolic subalgebras&#xhttp://arxiv.org/abs/1306.4838 Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Geometric invariant theory, Coadjoint orbits; nilpotent varieties Nested punctual Hilbert schemes and commuting varieties of parabolic subalgebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 regular local ring, let \(K\) be the fraction field of \(R\) and let \(G\) be a reductive group scheme over \(R\). A famous conjecture of Grothendieck and Serre predicts that the restriction map \(H^1(R,G)\to H^1(K,G)\) between étale cohomology pointed sets has trivial kernel. This conjecture is also commonly considered in the more general case where \(R\) is a regular semilocal domain.
The main result of the paper proves the extended conjecture when \(R\) is of Krull dimension \(\leq 1\), i.e., when \(R\) is a semilocal Dedekind domain. In the course of proving this, the author also establishes a decomposition theorem: for every reductive group scheme \(G\) over a semilocal Dedekind domain \(R\), we have
\[
\prod_{v} G(K_v)=G(K)\cdot\prod_v G(R_v),
\]
where \(v\) ranges over the maximal ideals of \(R\), and \(R_v\) and \(K_v\) are the completions of \(R\) and \(K\) at the \(v\)-adic topology, respectively. Separately, the author shows that if the Grothendieck-Serre conjecture holds for a particular regular semilocal domain \(R\), then every reductive group scheme over \(K\) admits at most one reductive model over \(R\). In particular, the latter holds when \(R\) is a semilocal Dedekind domain.
The many cases where the Grothendieck-Serre conjecture was previously known are surveyed in the introduction of the paper.
Historical note: \textit{Y. A. Nisnevich} [Etale cohomology and Arithmetic of Semisimple Groups. Harvard University (PhD Thesis) (1983); C. R. Acad. Sci., Paris, Sér. I 299, 5--8 (1984; Zbl 0587.14033)] proved the Grothendieck-Serre conjecture when \(R\) is a discrete valuation ring (DVR). His work was based on an unpublished result of Tits which settled the complete DVR case, and in turn relied on Bruhat-Tits theory [\textit{F. Bruhat} and \textit{J. Tits}, Publ. Math., Inst. Hautes Étud. Sci. 60, 1--194 (1984; Zbl 0597.14041)]. However, as the author notes (p.~899), there are some unclear points in the Nisnevich-Tits argument, possibly due to the fact that [loc. cit.] appeared in print two years after Nisnevich's works. The proof of the author's main result follows the outline of the Nisnevich-Tits argument, but with some changes which circumvent the said issues. Thus, in proving the their main result, the author also provides a clear and self-contained proof of Nisnevich's result. reductive group; torsor; Dedekind domain; Grothendieck-Serre conjecture; étale cohomology; weak approximation Group schemes, Cohomology theory for linear algebraic groups, Étale and other Grothendieck topologies and (co)homologies, Linear algebraic groups over local fields and their integers The Grothendieck-Serre conjecture over semilocal Dedekind rings | 0 |
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