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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 our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This, in particular, implies Grothendieck's conjecture on the perfectness of his pairing between the Néron component groups of an abelian variety and its dual. The point is that our formulation is well suited to Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, \(1\)-motives. abelian varieties; duality; Grothendieck topologies Abelian varieties of dimension \(> 1\), Galois cohomology, Étale and other Grothendieck topologies and (co)homologies Grothendieck's pairing on Néron component groups: Galois descent from the semistable 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 There are two constructions of the level-\((0,1)\) irreducible representation of the quantum toroidal algebra of type \(A\). In the first (due to Nakajima and Varagnolo-Vasserot) the representation is constructed on the direct sum of the equivariant \(K\)-groups of the quiver varieties of type \(\widehat{A}\). In the second (due to Saito-Takemura-Uglov and Varagnolo-Vasserot) the representation is constructed on the \(q\)-deformed Fock space introduced by Kashiwara-Miwa-Stern.
In the paper an~explicit isomorphism between these two constructions is given. To describe this isomorphism the author construct simultaneous eigenvectors on the \(q\)-Fock space using the nonsymmetric Macdonald polynomials. The isomorphism is given by attaching these vectors to the torus fixed points on the quiver varieties. representations of quantum toroidal algebras; q-Fock spaces; nonsymmetric Macdonald polynomials; K-theory of quiver representations Nagao, K.: \(K\)-theory of quiver varieties, \(q\)-Fock space and nonsymmetric Macdonald polynomials. Osaka J. Math. \textbf{46}(3), 877-907 (2009). http://projecteuclid.org/getRecord?id=euclid.ojm/1256564211. arXiv:0709.1767 Quantum groups (quantized enveloping algebras) and related deformations, Orthogonal polynomials and functions associated with root systems, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Representations of quivers and partially ordered sets \(K\)-theory of quiver varieties, \(q\)-Fock space and nonsymmetric Macdonald polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of the paper under review is to give a new proof of a recent result of [\textit{W. Fulton} and \textit{C. Woodward} [J. Algebr. Geom. 13, 641--661 (2004; Zbl 1081.14076)] related to the smallest degree that appears in the expansion of the product of two Schubert cycles in the small quantum cohomology ring of a Grassmann variety. The author's approach is combinatorial, and this method also yields an alternative characterization of this smallest degree in terms of the rim hook formula for the quantum product. quantum cohomology ring; Grassmann varieties; Schubert cycles; Gromov-Witten invariants A. Yong. ''Degree bounds in quantum Schubert calculus''. Proc. Amer. Math. Soc. 131 (2003), pp. 2649--2655.DOI. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Degree bounds in quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author proves in this very interesting paper the Hertling conjecture on the variance of the spectrum of a plane curve singularity and the reviewer's conjecture (involving an opposite inequality) for the spectrum at infinity of a polynomial function \(f:\mathbb C^2\to\mathbb C\) having just one branch at infinity. The two conjectures follow from precise formulas for the variance of the spectrum in the two situations, based on Eisenbud-Neumann diagrams. spectrum; singularity; polynomial; monodromy; variance Structure of families (Picard-Lefschetz, monodromy, etc.), Global theory of complex singularities; cohomological properties, Singularities in algebraic geometry On the spectral pairs of polynomials of two variables | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Denote by \(\Sigma\) a Riemann surface, endowed with a fixed finite set \(\mathcal{S}\subset\Sigma\) of marked points. Let \(\mathcal{M}\) denote the moduli space of rank \(n\) local systems on \(\Sigma\setminus\mathcal{S}\), with semisimple local monodromy around the marked points. These moduli spaces are known as character varieties.
\textit{T. Hausel} et al. [Duke Math. J. 160, No. 2, 323--400 (2011; Zbl 1246.14063)] computed the \(E\)-polynomial of \(\mathcal{M}\), allowing them to guess a conjectural formula for the corresponding full mixed Hodge polynomial in terms of modified Macdonald polynomials. In particular, their formula predicted the associated Poincaré polynomial of the character variety.
In this paper, the author proves that indeed the explicit formula predicted by Hausel, Letellier and Rodriguez-Villegas is the Poincaré polynomial of \(\mathcal{M}\). This is carried out by counting points over finite fields, in the spirit of \textit{O. Schiffmann}'s work [Ann. Math. (2) 183, No. 1, 297--362 (2016; Zbl 1342.14076)], where he computed the number of stable Higgs bundles on a curve over a finite field -- cf. also the work [\textit{S. Mozgovoy} and \textit{O. Schiffmann}, Compos. Math. 156, No. 4, 744--769 (2020; Zbl 07178692)] with Mozgovoy -- and thus obtained a recipe to compute the Poincaré polynomials of character varieties in the case where \(\mathcal{S}\) is empty.
However, the case with marked points appeared to be more difficult since, in particular, there was no known way of obtaining Macdonald polynomials by counting bundles of some sort. In the process of computing the Poincaré polynomial of \(\mathcal{M}\), the author is able to find such an interpretation of Macdonald polynomials. This is based on the fact that the number of partial flags over a finite field preserved by a given nilpotent matrix is obtained by the Hall-Littlewood polynomial. Poincaré polynomials; Macdonald polynomials; character varieties; Higgs bundles; Hall algebras Vector bundles on curves and their moduli, Character varieties Poincaré polynomials of character varieties, Macdonald polynomials and affine Springer fibers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \({\mathcal G}\) be a Grothendieck category. The notion of a tilting object of \({\mathcal G}\) is studied, and some basic facts of tilting theory are proved in this general setting. The results apply to categories of modules over arbitrary rings, as well as to the theory of sheaves in algebraic geometry. Grothendieck category; abelian category; tilting object; cotilting object; module; comodule; Grothendieck group; sheaves R. Colpi, \emph{Tilting in {G}rothendieck categories}, Forum Math. \textbf{11} (1999), no.~6, 735--759. \MR{1725595 (2000h:18018)} Grothendieck categories, Grothendieck groups, \(K\)-theory, etc., Torsion theories, radicals, Module categories in associative algebras, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Homological dimension in associative algebras Tilting in Grothendieck categories | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Recent developments in mathematics have provided powerful tools for comparing the eigenvalues of matrices related to each other via a moment map. In this paper, we survey some of the more concrete aspects of the approach with a particular focus on applications to quantum information theory. After discussing the connection between Horn's Problem and Nielsen's Theorem, we move on to characterizing the eigenvalues of the partial trace of a matrix. Entanglement; Majorization; Moment map; Schubert calculus S. Daftuar and P. Hayden, \textit{Quantum state transformations and the Schubert calculus}, \textit{Ann. Phys.}\textbf{315} (2005) 80 [quant-ph/0410052]. Quantum computation, Grassmannians, Schubert varieties, flag manifolds, Eigenvalues, singular values, and eigenvectors Quantum state transformations and the Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author discusses three groups of examples of noncommutative spaces over non-archimedean fields. First he considers quantum affinoid algebras, admissible Noetherian quotients of the algebra of power series (with coefficients from a non-archimedean field) with noncommutative variables. The next class is that of quantum Calabi-Yau varieties, topological spaces equipped with sheafs of noncommutative affinoid algebras, which are quantum affinoid outside of ``small'' subspaces. Finally, he introduces quantum deformations of the algebras of locally analytic functions and locally analytic distributions on a \(p\)-adic group [in the sense of \textit{P. Schneider} and \textit{J. Teitelbaum}, Invent. Math. 153, No. 1, 145--196 (2003; Zbl 1028.11070)].
In his treatment of rigid analytic spaces, the author follows the approach by \textit{V. Berkovich} [Spectral theory and analytic geometry over non-archimedean fields. Mathematical Surveys and Monographs, 33. Providence, RI: American Mathematical Society (1990; Zbl 0715.14013)]. The paper contains a lot of conjectures and unsolved problems. In the author's words, ``the paper does not present a piece of developed theory'', which ``explains its sketchy character''. quantum affinoid algebra; Calabi-Yau variety; \(p\)-adic quantum group; locally analytic function; locally analytic distribution Y. Soibelman, ''Quantum p-adic spaces and quantum p-adic groups,'' Geometry and Dynamics of Groups and Spaces, Progr. Math. 265, 697--719 (Birkhauser, Basel, 2008). Non-Archimedean valued fields, Calabi-Yau manifolds (algebro-geometric aspects), Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Calabi-Yau theory (complex-analytic aspects) Quantum \(p\)-adic spaces and quantum \(p\)-adic groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected, simply connected, simple, complex Lie group, and \(B\subset G\) a Borel subgroup. The presentation of the quantum cohomology ring of the flag variety \(G/B\) has been first computed by \textit{B.~Kim} [Ann. Math. 149, 129--148 (1999; Zbl 1054.14533)], where he points out its relation with the Toda lattice.
In this note, the author gives a shorter proof of this fact. His argument relies on the work of \textit{R.~Goodman} and \textit{N. R.~Wallach} [Commun. Math. Phys. 83, 355--386 (1982; Zbl 0503.22013)] concerning the Toda lattices, and then he applies \textit{B.~Siebert} and \textit{G.~Tian}'s result [Asian J. Math. 1, 679--695 (1997; Zbl 0974.14040)] about the presentation of the quantum cohomology ring of Fano varieties.
Using this approach, the author extends in [Adv. Math. 185, 347--369 (2004; Zbl 1137.53348)] the result to the infinite dimensional setting, corresponding to loop groups. flag varieties; Toda lattices A.-L. Mare ''Relations in the quantum cohomology ring of G/B,'' Math. Res. Lett. 11 (2004), 35--48. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Relations in the quantum cohomology ring of \(G/B\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 gave two forms of his ``main conjecture of anabelian geometry'', namely the section conjecture and the hom conjecture. He stated that these two forms are equivalent and that if they hold for hyperbolic curves, then they hold for elementary anabelian varieties too. We state a stronger form of Grothendieck's conjecture (equivalent in the case of curves) and prove that Grothendieck's statements hold for our form of the conjecture. We work with DM stacks, rather than schemes. If \(X\) is a DM stack over \(k\subseteq\mathbb{C}\), we prove that whether \(X\) satisfies the conjecture or not depends only on \(X_{\mathbb{C}}\). We prove that the section conjecture for hyperbolic orbicurves stated by \textit{N. Borne} and \textit{M. Emsalem} [Bull. Soc. Math. Fr. 142, No. 3, 465--487 (2014; Zbl 1327.14103)] follows from the conjecture for hyperbolic curves. section conjecture; anabelian geometry Homotopy theory and fundamental groups in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology, Varieties over global fields, Generalizations (algebraic spaces, stacks) Some implications between Grothendieck's anabelian conjectures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials 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.
In connection with proofs of the above conjectures for some special cases, \textit{B. Dwork} [Duke Math. J. 96, No. 2, 225--239 (1999; Zbl 0983.12004)] posed the problem of describing the set of those \(p\), for which the corresponding \(p\)-curvatures vanish. The author formulates analogs of the above problems in a more general geometric setting. He studies the interplay between properties of an integrable algebraic connection in characteristic zero, and properties of its reductions modulo \(p\) for large primes \(p\), with special emphasis on the case of connections of geometric origin. This covers some cases where the above problems are resolved. \(p\)-curvature; reduction modulo \(p\); Gauss-Manin connection Y. André, ''Sur la conjecture des \(p\)-courbures de Grothendieck--Katz et un problème de Dwork'' in Geometric Aspects of Dwork Theory, Vol. I, II, de Gruyter, Berlin, 2004, 55--112. \(p\)-adic differential equations, Arithmetic problems in algebraic geometry; Diophantine geometry, Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain On the Grothendieck-Katz \(p\)-curvature conjecture and a problem of Dwork | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 presentation for the Grothendieck group of Deligne-Mumford stacks over a field of characteristic zero. The generators for this presentation are smooth, proper Deligne-Mumford stacks and the relations are expressed in terms of stacky blow-ups. In the process we prove a version of the weak factorization theorem for Deligne-Mumford stacks. Grothendieck group of varieties; Deligne-Mumford stack; destackification; weak factorization Generalizations (algebraic spaces, stacks), Rational and birational maps, Motivic cohomology; motivic homotopy theory Weak factorization and the Grothendieck group of Deligne-Mumford stacks | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The understanding of the topology of the spectra of quantum Schubert cell algebras hinges on the description of their prime factors by ideals invariant under the maximal torus of the ambient Kac-Moody group. We give an explicit description of these prime quotients by expressing their Cauchon generators in terms of sequences of normal elements in chains of subalgebras. Based on this, we construct large families of quantum clusters for all of these algebras and the quantum Richardson varieties associated to arbitrary symmetrizable Kac-Moody algebras and all pairs of Weyl group elements. Along the way we develop a quantum version of the Fomin-Zelevinsky twist map for all quantum Richardson varieties. Furthermore, we establish an explicit relationship between the Goodearl-Letzter and Cauchon approaches to the descriptions of the spectra of symmetric CGL extensions. Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Kac-Moody groups, Simple and semisimple modules, primitive rings and ideals in associative algebras, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson 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 remarkable role of Schur functions and Schubert polynomials is known in the group representation theory; [\textit{I. N. Bernstein}, \textit{I. M. Gelfand}, \textit{S. I. Gelfand}, Usp. Mat. Nauk 28, No.3, 3-26 (1973; Zbl 0286.57025)] and [\textit{W. Kraskiewicz}, \textit{P. Pragacz}, C. R. Acad. Sci., Paris, Sér. I 304, 209-211 (1987; Zbl 0642.13011)]. This important paper is a concise and readable summary (the proofs are not given) of a ``noncommutative lifting'' of these notions and the corresponding basic facts. In this lifting the notions `tableau' and `nilplaxtic relation' are central.
Let us fix some (ordered) alphabet \(X=\{x_ 1<x_ 2<...\}\). In the free monoid \(X^*\) a word \(w=x_{i_ 1}x_{i_ 2}...x_{i_ k}\) with \(x_{i_ 1}>x_{i_ 2}>...>x_{i_ k}\) is called column; its length \(| w|\) is k and its contents C(w) is the set \(\{x_{i_ 1},...,x_{i_ k}\}\). If for two given columns v and w there exists a nonincreasing injection C(v)\(\to C(w)\), then it is said that w majorizes v (denoted by \(w\gg v)\). A finite product \(t=v_ 1v_ 2..\). of columns such that \(v_ 1\gg v_ 2\gg..\). is called a tableau; its shape is the partition \((| v_ 1|,| v_ 2|,...)\) of \(| t|\). A tableau t is called standard if its shape is a permutation on the set \(\{\) 1,2,...,\(| t| \}\); t is called key if for all k, \(v_{k+1}\) is a subword of \(v_ k\). On \(X^*\) there exist two important equivalences: the plaxtic (\(\equiv)\) and nilplaxtic (\(\cong)\) relation. The first is generated by elementary relations (PL1) \(a_ ka_ ia_ j\equiv a_ ia_ ka_ j\), \(a_ ja_ ia_ k\equiv a_ ja_ ka_ i\) and (PL2) \(a_ ja_ ia_ j\equiv a_ ja_ ja_ i\), \(a_ ja_ ia_ i\equiv a_ ia_ ja_ i\), (PL1) and (PL2) both in the case \(i<j<k\). The second is given also by (PL1) and (PL2) in all cases excluding that of i, j with \(j=i+1\)- then (PL2) is replaced by (Nil PL): \(a_ ia_{i+1}a_ i\cong a_{i+1}a_ ia_{i+1}\), \(a_ ia_ i=0\). The Schensted construction is primarily concerned with the plaxtic relation, it extends to the nilplaxtic case (Th. 1). The nilplaxtic relations contain Coxeter relations for transpositions. So the set of reduced decompositions of any permutation \(\mu\) is a disjoint union of nilplaxtic classes. Any such class (not containing 0) contains exactly one tableau t and its elements are in 1-1 correspondence with standard tableaux of the same shape as t. Using these remarkable facts the paper describes the way how general (noncommutative) Schubert polynomials \(X_{\mu}\) can be introduced as elements of \({\mathbb{Z}}<X>\) and how they can be found as certain sums of tableaux connected with the reduced decompositions of \(\mu\). Schur functions; Schubert polynomials; nilplaxtic relation; alphabet; monoid; word; Schensted construction; Coxeter relations; transpositions; nilplaxtic classes; standard tableaux; sums of tableaux; reduced decompositions Lascoux, A.; Schützenberger, M. P., Tableaux and noncommutative Schubert polynomials, Funct. Anal. Appl., 23, 223-225, (1990) Representations of finite symmetric groups, Free semigroups, generators and relations, word problems, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Combinatorial identities, bijective combinatorics, Symmetric functions and generalizations Noncommutative 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 book is intended to be a friendly introduction to quantum cohomology. It makes the reader acquainted with the notions of stable curves and stable maps, and their moduli spaces. These notions are central in the field. The authors' main reference are the notes of \textit{W.~Fulton} and \textit{R.~Pandharipande} [in: Algebraic geometry. Proc. Summer Res. Inst., Santa Cruz, CA, USA, 1995. Proc. Symp. Pure Math. 62 (pt.2), 45--96 (1997; Zbl 0898.14018)].
The main goal of the book is to prove Kontsevich's formula for the number \(N_d\) of rational plane curves of degree \(d\) passing through \(3d-1\) points. The result is viewed from three different perspectives: that of enumerative geometry, as a reconstruction result for the Gromov-Witten invariants, and as an instance of the associativity of the quantum multiplication.
The book contains five chapters. The first chapter introduces the notion of stable pointed curves of genus zero, with emphasis on \(\bar M_{0,4}\) and \(\bar M_{0,5}\). The authors describe the main properties of the moduli spaces \(\bar M_{0,n}\): the stabilization process, the forgetful morphism, contraction of unstable components, the glueing morphisms, and description of the boundary of \(\bar M_{0,n}\).
The second chapter introduces the space of stable maps \(\bar M_{0,n}(\mathbb P^n,d)\), and sketches its construction. Again, the main features of this space are discussed: projectivity and normality, the evaluation maps at the marked points, the forgetful morphisms, and the description of the boundary divisors.
The third chapter contains the first proof of Kontsevich's formula. The problem is stated within the frame of enumerative geometry. Using transversality arguments, it is proved that counting stable maps and plane curves gives the same result. The cases of plane conics and rational cubics are discussed in detail.
The fourth chapter contains the definition of the Gromov-Witten invariants of \(\mathbb P^n\), and some of their properties: the so-called divisor and splitting axioms. Kontsevich's formula is recovered as a special case of the Reconstruction Theorem [see \textit{M.~Kontsevich} and \textit{Yu~Manin}, Commun. Math. Phys. 164, 525--562 (1994; Zbl 0853.14020)].
Finally, in the last chapter the authors introduce the Gromov-Witten potential, the quantum cup product, and prove its associativity. This gives a third method for computing the number \(N_d\).
Each chapter ends with references for further readings, and also with a set of exercices which help fixing the ideas introduced in that chapter. This makes the book especially useful for graduate courses, and for graduate students who wish to learn about quantum cohomology. stable curves; stable maps; enumerative problems; Kontsevich formula; Gromov-Witten invariants; quantum cohomology; quantum product; generating function Kock, J.; Vainsencher, I., An invitation to quantum cohomology. Kontsevich's formula for rational plane curves, Progress in Mathematics, vol. 249, (2007), Birkhäuser Boston, Inc. Boston, MA Research exposition (monographs, survey articles) pertaining to algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds An invitation to quantum cohomology. Kontsevich's formula for rational plane curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we will introduce \textit{Quantum Toric Varieties} which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we show that their category is equivalent to a certain category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure. toric geometry; non-commutative geometry; mirror symmetry; irrational fans and polytopes; quantum integrable systems; moduli spaces Toric varieties, Newton polyhedra, Okounkov bodies, Noncommutative algebraic geometry Quantum (non-commutative) toric geometry: foundations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Yomdin-Gromov algebraic lemma assets that a semialgebraic set \(X \subseteq [0,1]^n\) is covered by the images of finitely many semialgebraic \(\mathcal C^r\) maps \(\phi_1,\ldots, \phi_C:(0,1)^{\dim X} \to X\) whose maximum norms are not greater than one, and the number of the maps are bounded by the number determined of \(n\), \(r\), the dimension of \(X\) and the sum of degrees of polynomials defining \(X\).
This paper demonstrates a refined version of the Yomdin-Gromov algebraic lemma for general o-minimal structures using the notion of `cellular parametrization.' A definable set \(X \subseteq [0,1]^n\) is covered by the images of finitely many definable cellular \(\mathcal C^r\) maps \(\phi_\alpha:C_\alpha \to X\) defined on cells \(C_\alpha\) whose maximum norms are not greater than one. Here, a definable function \(f=(f_1,\ldots,f_l):C \to \mathbb R^l\) defined on a cell \(C\) is called cellular if the value of \(f_i(x)\) depends only on the first \(i\) coordinates of \(x\), and \(f_i(x_1,\ldots, x_i)\) is strictly increasing with respect to the last coordinate \(x_i\) when the first \(i-1\) coordinates \(x_1,\ldots, x_{i-1}\) are fixed. Gromov-Yomdin parametrization; o-minimality Semialgebraic sets and related spaces, Model theory of ordered structures; o-minimality, Topological entropy, Model theory (number-theoretic aspects) The Yomdin-Gromov algebraic lemma revisited | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author establishes a formula for the classes of certain tori in the Grothendieck ring of varieties, expressing them in terms of the natural lambda-structure on the Grothendieck ring. More explicitly, if \(L^\ast\) is the torus of invertible elements in the \(n\)-dimensional separable \(k\)-algebra \(L\), then the class of \(L^\ast\) can be expressed as an alternating sum of the images of the spectrum of \(L\) under the lambda-operations, multiplied by powers of the Lefschetz class. This formula is suggested from the cohomology of the torus, illustrating a heuristic method that can be used in other situations. To be able to perform these, the author introduces a homomorphism from the Burnside ring of the absolute Galois group of \(k\), to the Grothendieck ring of varieties over \(k\). In the process the author obtains some information about the structure of the subring generated by zero-dimensional varieties. torus; variety; Grothendieck group; Grothendieck ring DOI: 10.1353/ajm.2011.0026 Applications of methods of algebraic \(K\)-theory in algebraic geometry, (Equivariant) Chow groups and rings; motives The class of a torus in the Grothendieck ring of varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth complex projective variety and let \(H^i(X)\) denote its singular homology with rational coefficients. The \textit{coniveau} filtration on \(H^i(X)\) is a descending filtration defined by \(N^pH^i(X)=\sum_{\text{codim} S\geq p} \text{Ker}[H^i(X)\to H^i(X-S)]\). The generalized Hodge conjecture (GHC) predicts that \(N^pH^i(X)\) coincides with the \textit{level} filtration, which is the largest rational sub-Hodge structure contained in \(F^pH^i(X)\) [\textit{A. Grothendieck}, Topology 8, 299--303 (1969; Zbl 0177.49002)]. The main result of the paper is the following: if GHC holds for \(X\) than it holds for any variety defining the same class in a suitable completion of the Grothendieck group of varieties. Using motivic integration it is also proved that this is the case for birationally equivalent Calabi-Yau varieties or, more generally, K-equivalent varieties. Generalized Hodge conjecture; motivic integration; Calabi-Yau varieties DOI: 10.1307/mmj/1163789917 Transcendental methods, Hodge theory (algebro-geometric aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Relations of \(K\)-theory with cohomology theories Coniveau and the Grothendieck group of varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a dozen formulas concerning Schubert and Grothendieck polynomials, and their interrelations, half of them being new, and most of them interesting. In particular, we describe explicitly the decomposition of Schubert polynomials as positive sums of Grothendieck polynomials, and we show that non-commutative Schubert polynomials are obtained by reading the columns of a two-dimensional Cauchy kernel. A six pages summary in English has been added. Schubert polynomials; Grothendieck polynomials Lascoux, A., Schubert & Grothendieck: un bilan bidécennal, Sém. Lothar. Combin., 50, (2003/04) Symmetric functions and generalizations, Classical problems, Schubert calculus, Representations of finite symmetric groups Schubert and Grothendieck: a bidecennial balance | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 consider a certain \(1\)-parameter family \(A(q,\lambda)\) of four dimensional regular graded algebras in the sense of \textit{M. Artin} and \textit{W. F. Schelter} [Adv. Math. 66, 171-216 (1987; Zbl 0633.16001)], where \(\lambda\) is a nonzero parameter in an algebraically closed field of characteristic different from two and \(q^4=1\). The algebra \(A(q,\lambda)\) was defined by \textit{M. Vancliff}, \textit{K. Van Rompay} and \textit{L. Willaert} [Commun. Algebra 26, No. 4, 1193-1208 (1998; Zbl 0915.16035)] on four generators with six quadratic defining relations. The study of \(A(q,\lambda)\) sprouts in the geometric theory of regular algebras initiated by \textit{M. Artin}, \textit{J. Tate} and \textit{M. Van den Bergh} [Prog. Math. 86, 33-85 (1990; Zbl 0744.14024)], where the notion of quantum \(\mathbb{P}^2\) (quantum projective plane) was used to classify the regular algebras of dimension three generated by degree one elements. The authors suggest that an appropriate notion of quantum \(\mathbb{P}^3\) should help in the classification of regular algebras of global dimension four. The scheme-theoretic zero locus in \(\mathbb{P}^3\times\mathbb{P}^3\) of the defining relations of \(A(q,\lambda)\) is obtained by the multilinearization process explained by \textit{M. Artin}, \textit{J. Tate} and \textit{M. Van den Bergh} [loc. cit.]. If \(q^3\neq 1\), then \(\Gamma(q,\lambda)\) is the graph of an automorphism of a subscheme \(P(q,\lambda)\) of \(\mathbb{P}^3\) (Lemma 1.7) which consists of one point of multiplicity 20 (Lemma 1.5). This scheme is referred to as the point scheme of \(A(q,\lambda)\), and it parametrises the point modules over \(A(q,\lambda)\). The stalk \({\mathcal O}_{P(q,\lambda)}\) at the only point is explicitly computed and it becomes a Frobenius algebra (Theorem 2.4). Moreover, the zero locus \(\Gamma(q,\lambda)\) completely determines the relations of the algebra \(A(q,\lambda)\). regular algebras; quadratic algebras; point modules; point schemes; generators; relations; quantum projective planes; global dimensions Shelton, Brad; Vancliff, Michaela, Some quantum \(\mathbf{P}^3\)s with one point, Comm. Algebra, 27, 3, 1429-1443, (1999) Graded rings and modules (associative rings and algebras), Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, Finite rings and finite-dimensional associative algebras, Associative rings of functions, subdirect products, sheaves of rings, Quantum groups (quantized enveloping algebras) and related deformations, , Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting), Homological dimension in associative algebras Some quantum \(\mathbb{P}^3\)s with one point | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This short communication considers when a prevariety of locally convex spaces is a Grothendieck ideal space. It is shown that the prevariety P\({\mathfrak M}\) is Groth(\({\mathfrak A})\) for an injective ideal \({\mathfrak A}\) iff, for any cardinal \({\mathfrak m}\) \((={\mathfrak m^{c}})\), the universal generator F(\({\mathfrak m})\) has a ``stabilized base'' of unit neighbourhoods. No proofs are included. prevariety of locally convex spaces; Grothendieck ideal space; universal generator; stabilized base Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.), General theory of locally convex spaces, Foundations of algebraic geometry Grothendieck prevarieties of locally convex spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the relation between the scaling 1-hook property of the coloured Alexander polynomial $\mathcal {A}_R^{\mathcal{K}}\left( q \right)$ and the KP hierarchy. The Alexander polynomial arises as a special case of the HOMFLY polynomial $\mathcal {H}_R^{\mathcal{K}}\left(q,a \right)$ of the knot $\mathcal{K}$ coloured with representation $R$ [\textit{E. Witten}, Commun. Math. Phys. 121, No. 3, 351--399 (1989; Zbl 0667.57005); \textit{ D. Bar-Natan}, J. Knot Theory Ramifications 4, No. 4, 503--547 (1995; Zbl 0861.57009)]:
\[{\mathcal {H}}_R^{\mathcal {K}}\left( {q,a} \right) = \frac{1} {Z}\int {DA{e^{ - \frac{i} {\hbar }{S_{CS}}\left[ A \right]}}{W_R}\left( {K,A} \right)}, \]
where the Wilson loop is
\[{W_R}\left( {K,A} \right) = {\text{t}}{{\text{r}}_R}P \exp \left( {\oint {A_\mu ^a\left( {\text{x}} \right){T^a}d{{\text{x}}^\mu }} } \right),\]
and the Chern-Simons action is
\[{S_{CS}}\left[ A \right] = \frac{\kappa } {{4\pi }}\int\limits_M {\text{Tr} \left( {A \wedge dA + \frac{2} {3}A \wedge A \wedge A} \right)}, \]
with $q = {e^\hbar },a = N\hbar $ and $\hbar = \frac{{2\pi i}} {{\kappa + N}}.$
The limiting case $\hbar \to 0,N \to \infty $ such that $N\hbar$ remains fixed, i.e., $q=1$, of the HOMFLY polynomials gives the special polynomials ${\mathcal {H}}_R^{\mathcal {K}}\left( {q,a} \right) = \sigma _R^{\mathcal {K}}\left( a \right)$, whose $R$ dependence makes them expressible in the form $\sigma _R^{\mathcal {K}}\left( a \right) = {\left( {\sigma _{\left[ 1 \right]}^{\mathcal {K}}\left( a \right)} \right)^{\left| R \right|}}$ for the Young diagram $R = \left\{ {{R_i}} \right\},{R_1} \geqslant {R_2} \geqslant \ldots \geqslant {R_{l\left( R \right)}},\left| R \right|: = \sum\nolimits_i {{R_i}} $. This provides the construction of a KP $\tau$-function (see for example [\textit{P. Dunin-Barkowski} et al., J. High Energy Phys. 2013, No. 3, Paper No. 021, 85 p. (2013; Zbl 1342.57004)]).
\par The dual limit as $a \to 1$ of the HOMFLY polynomials, i.e., ${\mathcal {H}}_R^{\mathcal {K}}\left( {q,1} \right)$, for the fundamental representation $R$ gives the Alexander polynomial, the coloured version of which exhibits a dual property with respect to $R$, viz., ${\mathcal {A}}_R^{\mathcal {K}}\left( q \right) = {\mathcal {A}}_{\left[ 1 \right]}^{\mathcal {K}}\left( {{q^{\left| R \right|}}} \right)$, which holds only for the representations corresponding to 1-hook Young diagrams ${R = \left[ {r,{1^L}} \right]}.$
\par In this paper the authors study this property perturbatively and claim that while the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials induce the equations of the KP hierarchy.
\par The main result of the paper is stated in Section 5, where, by considering the generating function of the KP hierarchy, replacing the Hirota operators with the Casimir eigenvalues and symmetrizing the identity, the authors find that Hirota KP bilinear equations are satisfied if and only if
\({\mathcal {A}}_R^{\mathcal {K}}\left( q \right) = {\mathcal {A}}_{\left[ 1 \right]}^{\mathcal {K}}\left( {{q^{\left| R \right|}}} \right)\).
The authors give only the first half of the proof of this result. \par The paper explores interesting
connections between the KP hierarchy and the coloured Alexander polynomials. Chern-Simons theory; knot invariant; Kontsevich integral; Vassiliev invariants; Hirota bilinear identities; KP hierarchy; Young diagrams; Gromov-Witten theory; Schur polynomial Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Knots and links in the 3-sphere, Invariants of knots and \(3\)-manifolds, Knots and links (in high dimensions) [For the low-dimensional case, see 57M25], Eta-invariants, Chern-Simons invariants, Casimir effect in quantum field theory, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Coloured Alexander polynomials and KP hierarchy | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m)KdV equations discretized on a space lattice. This braided Yang-Baxter equation still ensures that the transfer matrix generates operators in involution which form the Cartan sub-algebra of the braided quantum group. Representations diagonalizing these operators are described through relying on an easy generalization of algebraic Bethe ansatz techniques. The conjecture that this monodromy matrix algebra leads, in the cylinder continuum limit, to a perturbed minimal conformal field theory description is analysed and supported. D. Fioravanti and M. Rossi, \textit{A braided Yang-Baxter algebra in a theory of two coupled lattice quantum KdV: algebraic properties and ABA representations}, \textit{J. Phys.}\textbf{A 35} (2002) 3647 [hep-th/0104002] [INSPIRE]. Quantum groups and related algebraic methods applied to problems in quantum theory, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Quantum field theory on lattices, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, KdV equations (Korteweg-de Vries equations), Structure of families (Picard-Lefschetz, monodromy, etc.), Quantum groups (quantized enveloping algebras) and related deformations A braided Yang-Baxter algebra in a theory of two coupled lattice quantum KdV: algebraic properties and ABA representations. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A geometric categorification is given for arbitrary-large-finite- dimensional quotients of quantum \( \mathfrak{osp}(1| 2)\) and tensor products of its simple modules. The modified quantum \( \mathfrak{osp}(1| 2)\) of Clark-Wang, a new version in this paper and the modified quantum \( \mathfrak{sl}(2)\) are shown to be isomorphic to each other over a field containing \( \mathbb{Q}(v)\) and \( \sqrt {-1}\). quantum \(\mathfrak{osp}(1\|2)\); quantum modified algebra; tensor product module; categorification; perverse sheaf Fan, Z., Li, Y.: A geometric setting for quantum \({osp(1|2)}\). Trans. Am. Math. Soc. \textbf{367}, 7895-7916 (2015). arXiv:1305.0710 Quantum groups (quantized enveloping algebras) and related deformations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Schur and \(q\)-Schur algebras A geometric setting for quantum \(\mathfrak{osp}(1|2)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Grothendieck introduced the notion of a ``motif'' in a letter to \textit{J.-P. Serre} in 1964. Later he wrote that, among the objects he had been privileged to discover, they were the most charged with mystery and formed perhaps the most powerful instrument of discovery.
In this article, I shall explain what motives are, and why Grothendieck valued them so highly. motives; Weil conjectures; standard conjectures Milne, J.S.: Motives--Grothendieck's dream. In: Ji, L., Poon, Y.-S., Yau, S.-T. (eds.) Open Problems and Surveys of Contemporary Mathematics, vol. 6 of Surv. Mod. Math., pp. 325--342. Int. Press, Somerville, MA (2013) (Equivariant) Chow groups and rings; motives, Drinfel'd modules; higher-dimensional motives, etc., Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Motives -- Grothendieck's dream | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace [18]. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in [18] and with excellent parameters. Namely, we are able to provide new binary records according to [21] and non-binary codes improving the ones available in the literature. quantum codes; trace; subfield-subcodes; cyclotomic cosets Quantum coding (general), Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry Stabilizer quantum codes defined by trace-depending polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct quantum theta functions over noncommutative \({{\mathbb T}}^d \) with general embeddings. Manin has constructed quantum theta functions from the lattice embedding into vector space \(\times \) finite group. We extend Manin's construction of quantum thetas to the case of general embedding of vector space \(\times \) lattice \(\times \) torus. It turns out that only for the vector space part of the embedding there exists the holomorphic theta vector, while for the lattice part there does not. Furthermore, the so-called quantum translations from embedding into the lattice part become non-additive, while those from the vector space part are additive. DOI: 10.1088/1751-8113/41/10/105201 Noncommutative geometry in quantum theory, Theta functions and abelian varieties Quantum thetas on noncommutative \({\mathbb T}^d \) with general embeddings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum cohomology ring of a Kähler manifold is a deformation of the usual cohomology ring introduced by \textit{E. Witten} [Commun. Math. Phys. 118, No. 3, 411-449 (1988; Zbl 0674.58047)]. The computation of a quantum cohomology ring is generally a difficult task. Givental' and Kim proposed a presentation of the quantum cohomology rings in the case of flag varieties. In the paper under review a method for computing these rings is described and the presentation proposed by Givental' and Kim is established. It is observed that this same method should also apply to more general homogeneous spaces. quantum cohomology ring; flag varieties Ionuţ Ciocan-Fontanine, Quantum cohomology of flag varieties, Internat. Math. Res. Notices 6 (1995), 263 -- 277. (Co)homology theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Model quantum field theories, Kähler manifolds Quantum cohomology of flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We identify, in the Grothendieck group of complex varieties \(K_0(\mathrm{Var}_\mathbb{C})\), the classes of \(\mathbb{Q}\)-homological planes. Precisely, we prove that a connected smooth affine complex algebraic surface \(X\) is a \(\mathbb{Q}\)-homological plane if and only if \([X]=[\mathbb{A}^2_\mathbb{C}]\) in the ring \(K_0(\mathrm{Var}_\mathbb{C})\) and \(\mathrm{Pic}(X)_\mathbb{Q}:=\mathrm{Pic}(X)\otimes_\mathbb{Z}\mathbb{Q}=0\). motivic nearby cycles; motivic Milnor fiber; nearby motives Rational and birational maps, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Homological planes in the Grothendieck ring of varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is mainly an introduction of expository nature to quiver varieties and their applications, however also new results relating quiver varieties to quantum enveloping algebras of affine Lie algebras and their finite-dimensional representations are presented. First the author explains the construction of algebras via convolution products and geometric constructions of quantized enveloping algebras due to Ringel and Lusztig. Then the definition of quiver varieties is given and their relations to (non-affine) quantum enveloping algebras is explained.
See also the author's paper [J. Am. Math. Soc. 14, No. 1, 145--238 (2001; Zbl 0981.17016)]. H. Nakajima, Quiver varieties and quantum affine algebras, Sugaku Expositions, 19 (2006), no. 1, 53-78. Quantum groups (quantized enveloping algebras) and related deformations, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets, Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics Quiver varieties and quantum affine algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the study of a finite dimensional hereditary algebra of infinite representation type, understanding regular modules is essential. Recently, \textit{M. Herschend} et al. [Adv. Math. 252, 292--342 (2014; Zbl 1339.16020)] introduced the notions of \(d\)-representation infinite algebra and \(d\)-regular module, extending the above notions to finite dimensional algebras of global dimension \(d\geq 1\). Since the Beilinson algebras of AS-regular algebras of dimension \(d+1\) are typical examples of \(d\)-representation infinite algebras, the purpose of this paper is to study the behavior of \(d\)-regular modules over such algebras. In particular, we will show that the isomorphism classes of simple 2-regular modules over a 2-representation tame quantum Beilinson algebra of Type \(S\) are parameterized by \(\mathbb P^2\). regular modules; Beilinson algebras; AS-regular algebras; representation infinite algebras; preprojective algebras Representation type (finite, tame, wild, etc.) of associative algebras, Rings arising from noncommutative algebraic geometry, Ordinary and skew polynomial rings and semigroup rings, Graded rings and modules (associative rings and algebras), Noncommutative algebraic geometry Regular modules over 2-dimensional quantum Beilinson algebras of type \(S\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this note, we report on our work on the formalism of the Grothendieck six operations on o-minimal sheaves. As an application to the theory of definable groups, we see that the cohomology of a definably compact group with coefficients in a field is a connected, bounded, Hopf algebra of finite type. Edmundo, M.; Prelli, L.: The six Grothendieck operations on o-minimal sheaves Model theory of ordered structures; o-minimality, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Compact groups, Hyperspaces in general topology The six Grothendieck operations on o-minimal sheaves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) universal coefficient formula; (iii) derived base change formula; (iv) Künneth formula; (v) local and global Verdier duality. o-minimal structures; proper direct image; sheaves; cohomology; semi-algebraic; globally sub-analytic Model theory of ordered structures; o-minimality, Sheaves in algebraic geometry, Compact groups, Hyperspaces in general topology, Sheaf cohomology in algebraic topology The six Grothendieck operations on o-minimal sheaves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Double Kostka polynomials \(K_{\lambda,\mu}(t)\) are polynomials in \(t\), indexed by double partitions \({\lambda,\mu}\). As in the ordinary case, \(K_{\lambda,\mu}(t)\) is defined in terms of Schur functions \(s_\lambda(x)\) and Hall-Littlewood functions \(P_\mu(x;t)\). In this paper, we study combinatorial properties of \(K_{\lambda,\mu}(t)\) and \(P_\mu(x;t)\). In particular, we show that the Lascoux-Schützenberger type formula holds for \(K_{\lambda,\mu}(t)\) in the case where \(\mu = (-,\mu^{\prime\prime})\). Moreover, we show that the Hall bimodule \(\mathscr{M}\) introduced by \textit{M. Finkelberg} et al. [Sel. Math., New Ser. 14, No. 3--4, 607--628 (2009; Zbl 1215.20041)] is isomorphic to the ring of symmetric functions (with two types of variables) and the natural basis \(\mathfrak{u}_\lambda\) of \(\mathscr{M}\) is sent to \(P_\lambda(x;t)\) (up to scalar) under this isomorphism. This gives an alternate approach for their result. Schur functions; Hall-Littlewood functions S. Liu and T. Shoji, Double Kostka polynomials and Hall bimodule, \doihref10.3836/tjm/1475723088Tokyo J. Math., 39 (2017), 743--776. Symmetric functions and generalizations, Combinatorial aspects of representation theory, Representation theory for linear algebraic groups, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds Double Kostka polynomials and Hall bimodule | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 matrix polynomial of the form
\[
W(z) = B^0 z^m + \cdots + B^m, \quad B^i \in \mathrm{Mat}_n(\mathbb{C}).
\]
Let \(C \to \mathbb{CP}^1\) be the associated spectral curve whose affine part is given by
\[
\{(z,w)\in \mathbb{C}^2 ~|~\det\left(w\cdot \mathbf{1}_n - W(z)\right) = 0\}.
\]
If \(W\) is generic, then \(C\) is smooth and irreducible with \(n\) pairwise distinct points \(P_1, \dots, P_n\) over \(\infty\). Note that the spectral curve \(C\) comes with a natural line bundle \(\mathcal{L}\) defined by the eigenvectors of \(W\). Finally, we denote by \(\theta\) the Riemann theta function of \(C\) associated with a fixed canonical basis of \(H_1(C,\mathbb{Z})\).
The present article studies \(N\)-differentials \(\nu_{N, \mathbb{Q}}\) on \(C\) for \(N\geq 2\) and \(N\)-tuples \(\mathbb{Q}\) of points on \(C\). The \(\nu_{N, \mathbb{Q}}\) are defined by the \(N\)-th logarithmic differential of \(\theta\) at the point in the Jacobian corresponding to \(\mathcal{L}\) (after tensoring with an appropriate divisor) and the \(N\)-tuples \(\mathbb{Q}\). The surprising main result of this articles expresses each \(\nu_{N, \mathbb{Q}}\) explicitly in terms of \(W\) and \(\mathbb{Q}\) only. If \(W\) has rational coefficients, i.e. \(B^i \in \mathrm{Mat}_n(\mathbb{Q})\), then it is further shown as a corollary that \(\nu_{N, \mathbb{Q}}\) has only rational coefficients when expanded around \(\infty\).
The key of proving these statements (for \(N \geq 3\); for \(N=2\) the methods are related but more direct) is the relationship to the \(n\)-wave (or AKNS-D) hierarchy defined by
\[
[L_{a, k}, L_{b,l}] = 0,\quad L_{a, k} = \frac{\partial}{\partial t^a_k} - U_{a, k}(\mathbf{t}; z).
\]
Here \(U_{a, k}(\mathbf{t}; z)\) are any \(n \times n\)-matrix-valued polynomials in \(z\) of degree \(k+1\) of a special form. Now given a spectral curve \(C\) as above and \(N\geq 0\), a solution to the \(n\)-wave hierarchy is constructed (Proposition 2.20) following Krichever's approach and using a vector-valued Baker-Akhiezer function. We note that the \(U_{a,k}\) of the hierarchy are in fact constructed from the spectral curve \(C\). A key result (Proposition 2.24) is that the tau-function attached to such a solution (reviewed in Appendix A) can be expressed as
\[
\tau(\mathbf{t}) = a(\mathbf{t})\cdot \theta(V(\mathbf{t}) - \mathbf{u}_0).
\]
Here \(\theta\) is the Riemann theta function of \(C\) as above and \(a(\mathbf{t})\), \(V(\mathbf{t})\) are certain scalar-/vector-valued functions.
From the previous formula it follows that the \(N\)-differentials \(\nu_{N, \mathbb{Q}}\) are equivalently defined as certain \(N\)-th logarithmic differentials of the tau-function \(\tau(\mathbf{t})\). Applying results on such differentials for more general solutions of the \(n\)-wave hierarchy (proven in Appendix A), it is possible to express them in terms of \(W(z)\) for the solutions attached to \(C\). Thereby the main theorem follows (see end of Section 2.3).
Despite being technical, this article is well written and organized. It contains several interesting results along the way. For example, results on the divisor of normalized eigenvectors of \(W\) (see Section 2) and its relation to the relative Jacobian \(J(C; P_1, \dots, P_n)\) (which can be considered as the Jacobi variety of the singular curve obtained from \(C\) by identifying the points \(P_i\) over \(\infty\)). Finally, explicit examples of the main results are provided as well as an appendix on tau-functions of solutions to the \(n\)-wave hierarchy. Riemann surfaces; Riemann theta function; tau-functions; integrable hierarchies Relationships between algebraic curves and integrable systems, Theta functions and abelian varieties, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Algebraic spectral curves over \(\mathbb{Q}\) and their tau-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 In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix-like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes-Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general \(q\)-polynomials. asymptotics; matrix models; partition function; Stieltjes-Wigert polynomials; theta function Model quantum field theories, Applications of basic hypergeometric functions, Asymptotic approximations, asymptotic expansions (steepest descent, etc.), Theta functions and abelian varieties, Random matrices (probabilistic aspects), Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) Asymptotics of partition functions in a fermionic matrix model and of related \(q\)-polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as \(\hslash\to 0\), and becomes non-commutative or ``quantum'' away from this limit. For a classical curve defined by the zero locus of a polynomial \(A(x,y)\), we provide a construction of its non-commutative counterpart \(\hat A(\hat x,\hat y)\) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing \(\hat A\) that, surprisingly, turns out to be much simpler than any of the existent methods. In particular, as a bonus feature of our approach comes a curious observation that, for all curves that ``come from geometry,'' their non-commutative counterparts can be determined just from the first few steps of the topological recursion. We also propose a K-theory criterion for a curve to be ``quantizable,'' and then apply our construction to many examples that come from applications to knots, strings, instantons, and random matrices. The material contained in this chapter was presented at the conference \textit{Mirror Symmetry and Tropical Geometry} in Cetraro (July 2011) and is based on the work: [\textit{S. Gukov} and \textit{P. Sulkowski}, J. High Energy Phys. 2012, No. 2, Paper No. 070, 57 p. (2012; Zbl 1309.81220)]. S. Gukov and P. Sulkowski, \textit{A-polynomial, B-model, and quantization}, in \textit{Homological mirror symmetry and tropical geometry}, \textit{Lect. Notes Unione Mat. Ital.}\textbf{15}, Springer, Cham Switzerland (2014), pg. 87. String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Relationships between surfaces, higher-dimensional varieties, and physics, Noncommutative algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Applications of methods of algebraic \(K\)-theory in algebraic geometry A-polynomial, B-model, and Quantization | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as \(\hslash \to 0\), and becomes non-commutative or ``quantum'' away from this limit. For a classical curve defined by the zero locus of a polynomial \(A(x, y)\), we provide a construction of its non-commutative counterpart \(\widehat{A}\left({\widehat{x},\widehat{y}} \right)\) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing \(\widehat{A}\) that, surprisingly, turns out to be much simpler than any of the existent methods. In particular, as a bonus feature of our approach comes a curious observation that, for all curves that come from knots or topological strings, their non-commutative counterparts can be determined just from the first few steps of the topological recursion. We also propose a K-theory criterion for a curve to be ``quantizable,'' and then apply our construction to many examples that come from applications to knots, strings, instantons, and random matrices. matrix models; non-commutative geometry; Chern-Simons theories; topological strings Gukov, S.; Sułkowski, P., A-polynomial, B-model, and quantization, J. High Energy Phys., 1202, (2012) String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Topological field theories in quantum mechanics, Noncommutative geometry methods in quantum field theory, Eta-invariants, Chern-Simons invariants, Quantization in field theory; cohomological methods, Relationships between algebraic curves and physics A-polynomial, B-model, and quantization | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An unpublished result of D. Peterson states that the complexified quantum cohomology of a homogeneous manifold \(G/P\) (where \(P\) is an parabolic subgroup in \(G\)) is isomorphic to the coordinate ring of a subvariety in \(G^\vee / B^\vee\) (where \(G^\vee\) is the Langlands dual group and \(B^\vee\) is a Borel subgroup in \(G^\vee\)). This paper verifies Peterson's result for the Lagrangian and orthogonal Grassmannians, using explicit presentations of their quantum cohomology rings given by \textit{A. Kresch} and \textit{H. Tamvakis} [Algebr. Geom. 12, No. 4, 777--810 (2003; Zbl 1051.53070)] and [Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083)]. In \S 3 and \S 4, explicit construction of the corresponding subvarieties and descriptions of their elements are given. \S 5 applies these explicit formulas to prove a Vafa-Intriligator type formula for the \(m\)-pointed Gromov-Witten invariants of these Grassmannians. It also verifies that the quantum Euler classes [\textit{L. Abrams}, Isr. J. Math. 117, 335--352 (2000; Zbl 0954.53048)] of these Grassmannians are invertible, which implies that the quantum cohomology rings are semisimple when restricted some \(q \in \mathbb C^*\). Vafa-intriligator; quantum cohomology; quantum Euler class Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations Vafa-intriligator type formulas and quantum Euler classes for Lagrangian and orthogonal Grassmannianns | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 effective algorithm of determining Gromov-Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound) from Gromov-Witten invariants of the ambient space is proposed. Gromov-Witten; quantum Lefschetz hyperplane theorem Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Towards a quantum Lefschetz hyperplane theorem in all genera | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 and \(\Pi\) is a partially ordered homogeneous subset in a graded \(k\)-algebra \(A\) generating the algebra \(A\). It is assumed that each element from \(\Pi\) has positive degree. A standard monomial on \(\Pi\) is a product \(a_1\cdots a_s\) where \(a_1\leqslant\cdots\leqslant a_s\). The algebra \(A\) is an \(ASL\)-algebra if standard monomials are linearly independent and the following properties are satisfied: 1) if \(a,b\in\Pi\) are not comparable then \(ab\) belongs to the span of elements of \(\Pi\) and the standard monomial \(lm\), where \(l,m\in\Pi\) and \(l<a\), \(l<b\). 2) if \(a,b\in\Pi\) then \(ab-c_{a,b}ba\) belongs to the same span as in 1) for some \(c_{a,b}\in k^*\).
Standard monomials form a base of \(A\). A quantum graded \(ASL\)-algebra is Noetherian and of polynomial growth. A subset \(\Omega\) of \(\Pi\) is a \(\Pi\)-ideal if it is an ideal with respect to the order in \(\Pi\). The \(k\)-algebra \(A/\langle\Omega\rangle\) is again an \(ASL\)-algebra on the image of \(\Pi\setminus\Omega\).
Under some minor restriction it is shown that an \(ASL\)-algebra is \(AS\)-Cohen-Macaulay. Let \(R\) be a commutative integral domain with \(k\) as the field of fractions. Main examples of \(ASL\)-algebras are the coordinate algebra \(\mathcal O_u(M_{m,n}(A))\) on matrices of size \(m\times n\) and quantum Grassmannian \(\mathcal O_u(G_{m,n}(A))\). So the notion of \(ASL\)-algebra is unifying both examples and allows to clarify some proofs making them more explicit. This approach allows also to consider quantum analogs of Schubert varieties. quantum matrices; quantum Grassmannians; quantum Schubert varieties; quantum determinantal rings; straightening laws Lenagan, T. H.; Rigal, L., Quantum graded algebras with a straightening law and the \textit{AS}-Cohen-Macaulay property for quantum determinantal rings and quantum Grassmannians, J. Algebra, 301, 2, 670-702, (2006) Rings arising from noncommutative algebraic geometry, Graded rings and modules (associative rings and algebras), Quadratic and Koszul algebras, Grassmannians, Schubert varieties, flag manifolds Quantum graded algebras with a straightening law and the AS-Cohen-Macaulay property for quantum determinantal rings and 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 In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory and Lurie proposed a conjectural substitute, later termed the \textit{fundamental local equivalence}. With a few exceptions, we prove this conjecture and its extension to the affine flag variety by using what amount to Soergel module techniques. highest-weight modules; Whittaker sheaves; geometric Langlands Geometric Langlands program (algebro-geometric aspects), Geometric Langlands program: representation-theoretic aspects Fundamental local equivalences in quantum geometric Langlands | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 aims to extend the quantum Riemann-Roch theorem given by Coates and Givental in Gromov-Witten theory to smooth Deligne-Mumford stacks. The orbifold GW invariants of a smooth Deligne-Mumford stack is defined. A quantum Lefschetz hyperplane theorem is also derived. In this paper, one can also find the relations between genus 0 GW invariants of Deligne-Mumford stacks and that of a complete intersection, with additional assumptions. quantum Rieman-Roch; quantum Lefschetz; quantum Serre; Deligne-Mumford stack; Gromov-Witten theory H.-H. Tseng, Orbifold quantum Riemann-Roch, Lefschetz and Serre. \textit{Geom. Topol}. \textbf{14} (2010), 1-81. MR2578300 Zbl 1178.14058 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Riemann-Roch theorems, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Stacks and moduli problems Orbifold quantum Riemann-Roch, Lefschetz and Serre | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the present paper, we study the geometrically pro-\(p\) fundamental groups of hyperbolic polycurves, i.e., successive extensions of families of hyperbolic curves. Among other results, we show that the isomorphism class of a hyperbolic polycurve of dimension \(\leq 4\) over a sub-\(p\)-adic field satisfying a certain group-theoretic condition is completely determined by the geometrically pro-\(p\) fundamental group equipped with surjection onto the absolute Galois group of the base field. pro-\(p\) Grothendieck conjecture; hyperbolic polycurve Coverings of curves, fundamental group, Families, moduli of curves (algebraic), Arithmetic ground fields for curves Pro-\(p\) Grothendieck conjecture for hyperbolic polycurves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 simple finite dimensional Lie algebra \(\widehat{g}\) of type ADE, let \(g\) be the corresponding (untwisted) affine Lie algebra and \(U_q(\widehat g)\) its quantum affine algebra. In this paper, the author studies finite dimensional representations of \(U_q(\widehat g)\) using geometry of quiver varieties. His purpose is to solve the following conjecture affirmatively, that is, an equivariant \(K\)-homology group of the quiver variety gives the quantum affine algebra \(U_q(\widehat g)\), and to derive results whose analogues are known for \(H_q\).
In \S 1, the author recalls a new realization of \(U_q(\widehat g)\), called Drinfeld realization and introduces the quantum loop algebra \(U_q(Lg)\) as a subquotient of \(U_q(\widehat g)\), which will be studied rather than \(U_q(\widehat g)\). The basic results are recalled on finite dimensional representations of \(U_\varepsilon(Lg)\). And, several useful concepts are introduced.
In \S 2, the author introduces two types of quiver varieties \({\mathcal M}(w)\) and \({\mathcal M}_0(\infty, w)\) as analogues of \(T^*{\mathcal B}\) and the nilpotent cone \(\mathcal N\) respectively. Their elementary properties are given.
In \S 3--\S 8, the author prepares some results on quiver varieties and \(K\)-theory which will be used in later sections.
In \S 9--\S 11, the author considers an analogue of the Steinberg variety
\[
Z(w) = {\mathcal M}(w)\times _{{\mathcal M}_0(\infty,w)}{\mathcal M}(w)
\]
and its equivariant \(K\)-homology \(K^{G_w\times \mathbb{C}^*}(Z(w))\). An algebra homomorphism is constructed from \(U_q(Lg)\) to \(K^{G_w\times \mathbb{C}^*}(Z(W)) \otimes_{\mathbb{Z}[q,q^{-1}]}\mathbb{Q}(q)\).
In \S 12, the author shows that the above homomorphism induces a homomorphism from \(U^\mathbb{Z}_q(Lg)\) to \(K^{G_w\times \mathbb{C}^*}(Z(w))/\text{torsion}\).
In \S 13, the author introduces a standard module \(M_{x,a}\). Thanks to a result in \S 7, it is proved to be isomorphic to \(H_*({\mathcal M}(w)^A_x,\mathbb{C})\) via the Chern character homomorphism. Also, it is shown that \(M_{x,a}\) is a finite dimensional \(l\)-highest weight module. It is conjectured that \(M_{x,a}\) is a tensor product of \(l\)-fundamental representations in some order, which is proved when the parameter is generic in \S 14.1.
In \S 14, it is verified that the standard modules \(M_{x,a}\) and \(M_{y,a}\) are isomorphic if and only if \(x\) and \(y\) are contained in the same stratum. Furthermore, the author shows that the index set \(\{\rho\}\) of the stratum coincides with the set \({\mathcal P} =\{P\}\) of \(l\)-dominant \(l\)-weights of \(M_{0,a}\), the standard module corresponding to the central fiber \(\pi^{-1}(0)\). And, the multiplicity formula \([M(P) : L(Q)] =\dim H^*(i^!_x IC({\mathcal M}^{\text{reg}}_0(\rho_Q)))\) is proved. The result here is simpler than the case of the affine Hecke algebra: nonconstant local systems never appear.
Let \(\text{Res }M(P)\) be the restriction of \(M(P)\) to a \(U_\varepsilon(g)\)-module. In \S\ 15, the author shows the multiplicity formula \([\text{Res }M(P) : L(w - v)] \dim H^*(i_x^! IC({\mathcal M}^{\text{reg}}_0(v, w)))\). This result is compatible with the conjecture that \(M(P)\) is a tensor product of \(l\)-fundamental representations since the restiction of an \(l\)-fundamental representation is simple for type \(A\), and Kostka polynomials give tensor product decompositions.
Two examples are given where \({\mathcal M}^{\text{reg}}_0(v, w)\) can be described explicitly.
As mentioned in the Introduction of this paper, \(U_q(\widehat{g})\) has another realization, called the Drinfeld new realization, which can be applied to any symmetrizable Kac-Moody algebra \(g\), not necessarily a finite dimensional one. This generalization also fits the result in this paper, since quiver varieties can be defined for arbitrary finite graphs. If finite dimensional representations are replaced by \(l\)-integrable representations, parts of the result in this paper can be generalized to a Kac-Moody algebra \(g\), at least when it is symmetric.
If equivariant \(K\)-homology is replaced by equivariant homology, one should get the Yangian \(Y(g)\) instead of \(U_q(\widehat{g})\). The conjecture is motivated again by the analogy of quiver varieties with \(T^*\mathcal B\). As an application, the affirmative solution of the conjecture implies that the representation theory of \(U_q(\widehat g)\) and that of the Yangian are the same. quantum affine algebra; quiver variety; equivariant \(K\)-theory; finite dimensional representation Nakajima, H., Quiver varieties and finite-dimensional representations of quantum affine algebras, \textit{J. Amer. Math. Soc.}, 14, 1, 145-238, (2001) Quantum groups (quantized enveloping algebras) and related deformations, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets, Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics Quiver varieties and finite dimensional representations of quantum affine algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X=(X_{ij})\) be an \(m\times n\) matrix of indeterminates over a field \(K\). In the book: ``Enumerative combinatorics of Young tableaux'' (1988; Zbl 0643.05001), \textit{S. S. Abhvankar} considers ideals \(I[p,a]\) in the polynomial ring \(K[X_{ij}| i=1,\ldots,m, j=1,\ldots,n]\) whose systems of generators consist of some combinatorially described minors of \(X\). [For an alternative description of the same ideals see the book by \textit{W. Bruns} and the reviewer ``Determinantal rings'' (1988; Zbl 0673.13006).] He shows that the Hilbert polynomial of these ideals is completely determined by certain integer valued functions. The author proves ``some important properties of these integer valued functions''. He especially gives an affirmative answer to two questions asked in Abhyankar's book. determinantal ideals; Hilbert polynomial Udpikar S G, On Hilbert polynomial of certain determinantal ideals,Int. Math. Math. Sci. 14 (1991) 155--162 Linkage, complete intersections and determinantal ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Determinantal varieties On Hilbert polynomial of certain determinantal ideals | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is devoted to the study of categories of vector bundles over ringed spaces \((X,\mathcal A)\), where \(X\) is an irreducible projective variety over a finite field \(k\). In particular, the authors analyze the case in which \(X\) is a curve and the sheaf \(\mathcal A\) of rings is given by \(\mathcal A(U)=\mathcal O(U)[G]\), where \(\mathcal O\) denotes the structure sheaf of \(X\) and \(G\) is a finite group.
The authors start by proving basic facts on sheaves of algebras, radicals, the Wedderburn decomposition, and isotriviality. Next given a coherent sheaf \(\mathcal A\) of algebras on \(X\), various categories of \(\mathcal A\)-bundles on \(X\) are introduced, and the class group \(\text{Cl}^{lf}(\mathcal A)\) of locally free \(\mathcal A\)-bundles is studied. If \(X\) is a curve, then for commutative \(\mathcal A\) it is shown that \(\text{Cl}^{lf}(\mathcal A)\) is isomorphic to the Picard group, while for a sheaf \(\mathcal A\) of orders a version of Quillen localization is proved.
Assuming that \(\mathcal A\) is a sheaf of orders, the authors next study a natural filtration on the Grothendieck group of locally free \(\mathcal A\)-bundles over \(X\). They next derive and study an idelic description of \(\text{Cl}^{lf}(\mathcal A)\) in the case that \(X\) is a curve. The theory developed so far is then applied to study the case \(\mathcal A=\mathcal O[G]\) for a finite group \(G\) whose order is not divisible by the characteristic \(p\) of \(k\).
Finally, the authors study the case \(\mathcal A=\mathcal O[G]\) for a cyclic group \(G\) of order \(p^n\) for some \(n>0\). They describe the Picard group in terms of Witt vectors and characterize those elements of the class group which may be realized by unramified \(G\)-covers. Grothendieck group; realizable class of class group; ringed space; Witt vector; locally free \(\mathcal A\)-bundles; finite field 10.1023/A:1011843723810 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Finite ground fields in algebraic geometry, Vector bundles on curves and their moduli, Grothendieck topologies and Grothendieck topoi, \(K_0\) of group rings and orders, \(K\)-theory of schemes Grothendieck groups of bundles on varieties over finite fields | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Hamiltonian monodromy is known to be the first obstruction to the existence of global action coordinates in integrable systems. Its manifestation in quantum systems can be seen as characteristic defects of the regular lattice formed by the joint eigenvalues of mutually commuting quantum operators. The relation between topology of singular fibers of classical integrable fibrations and patterns formed by joint spectrum of corresponding quantum systems is discussed. The notion of the sign of `elementary monodromy defect' is introduced on the basis of `cut and glue' construction of the lattice defects. Special attention is paid to non-elementary defects which generically appear in phyllotaxis patterns and can be associated with plant morphology. Zhilinskií B., Quantum monodromy and pattern formation, 2010, J. Phys. A, 43, \#434033 Groups and algebras in quantum theory and relations with integrable systems, Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory, Monodromy on manifolds, Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics, Structure of families (Picard-Lefschetz, monodromy, etc.) Quantum monodromy and pattern formation | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 with field of fractions \(K\). Let \(\mathcal{A} \) be an Azumaya algebra over \(R\). From the paper: ``Our work was motivated by \dots{} a paper by \textit{I.A. Panin} and \textit{M. Ojanguren} [Math. Z. 237, No. 1, 181-198 (2000; Zbl 1042.11024)] on the Grothendieck conjecture for Hermitian spaces. The point was to offer a good axiomatization for the method used in that paper.''
The results of the paper are as follows:
If \(R\) contains a field, then \(R^{\ast }/\)Nrd\((\mathcal{A}^{\ast })\rightarrow K^{\ast }/\)Nrd\((\mathcal{A}_{K}^{\ast })\) is injective; and the canonical map \(H_{\text{ét}}^{1}(R,\text{SL}_{1,\mathcal{A} })\rightarrow H_{\text{ét}}^{1}(K,\text{SL}_{1,\mathcal{A}_{K}})\) is injective. These results were originally proved by \textit{I. Panin} and \textit{A.A. Suslin} [St. Petersbg. Math. J. 9, No. 4, 851-858 (1998); translation from Algebra Anal. 9, No. 4, 215-223 (1997; Zbl 0902.16019)]. Here ``Nrd'' is the reduced norm.
If \(\sigma \) is a unitary involution of \(\mathcal{A}\) and \(R\) contains an infinite field of characteristic not equal to \(2\), then \(U(C)/\text{Nrd}(U( \mathcal{A}))\rightarrow U(C_{K})/ \text{Nrd}(U(\mathcal{A}_{K}))\) is injective and the canonical map
\[
H_{\text{ét}}^{1}(R,\text{SU}_{1,\mathcal{A} })\rightarrow H_{\text{ét}}^{1}(K,SU_{1,\mathcal{A}_{K}})
\]
is trivial. Here \(U(C)\) is the unitary group of the center of \(\mathcal{A}.\)
For \(d\) a natural number, if \(R\) contains a field, then \(R^{\ast }/\)Nrd\(( \mathcal{A}^{\ast })(R^{\ast })^{d}\rightarrow K^{\ast }/\text{Nrd}(\mathcal{A} _{K}^{\ast })(K^{\ast })^{d}\) is injective.
If \(\sigma \) is a unitary involution of \(\mathcal{A}\) and \(R\) contains an infinite field of characteristic not equal to \(2\), then \(U(C)/\text{Nrd}(U( \mathcal{A}))U(C)^{d}\rightarrow U(C_{K})/\text{Nrd}(U(\mathcal{A} _{K}))U(C_{K})^{d}.\)
This paper originally appeared in Russian. The English translation can be difficult to follow. regular local ring; Azumaya algebra; Grothendieck conjecture for Hermitian spaces Homogeneous spaces and generalizations, Regular local rings On Grothendieck's conjecture about principal homogeneous spaces for some classical algebraic groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Crepant Resolution Conjecture predicts a relationship between the Gromov-Witten theory of an orbifold and the Gromov-Witten theory of any crepant resolution. The paper under review checks this conjecture for the orbifold \(\left[\mathbb {C}^{2}/\mathbb {Z}_{3}\right]\).
In particular, the authors compute the genus \(0\) Gromov-Witten potential for
(i) the crepant resolution \(Y\) of the quotient singularity \(\mathbb C^2/\mathbb Z_3\),
(ii) the orbifold quotient \(\mathcal X=\left[\mathbb C^2/\mathbb Z_3\right]\).
The two potentials are then matched after a linear change of variables. The computation in (i) is performed by embedding the resolution \(Y\) into a Calabi-Yau threefold whose Gromov-Witten invariants have been previously computed. The computation in (ii) is related to integrals of the Hodge class \(\lambda_{g-1}\) over any connected component of the Hurwitz scheme of curves in \({\overline M}_g\) which admit a cyclic triple cover of \(\mathbb P^1\). The determination of these integrals is carried out in the appendix of the paper.
Finally, the authors write down conjectural changes of variables relating the Gromov-Witten potentials of DuVal singularities \([\mathbb C^2/G]\) and their resolutions, for any finite subgroup \(G\) of \(\text{SU}(2)\). This conjecture is answered for the \(A_{2}\) surface singularity in \textit{J. Bryan} and \textit{T. Graber}'s preprint [The Crepant Resolution Conjecture, \url{arXiv:math. AG/ 0610129}]. Of course, the current article establishes the case of the \(A_{3}\) singularity. The case of \(A_{n}\) singularities is settled by \textit{T. Coates, A. Corti, H. Iritani} and \textit{H.-H. Tseng} [The Crepant Resolution Conjecture for Type A Surface Singularities, \url{arXiv:0704.2034}]. Finally, we would like to point out related work of \textit{J. Bryan} and \textit{A. Gholampour} [Hurwitz-Hodge Integrals, the \(E_6\) and \(D_4\) root systems, and the Crepant Resolution Conjecture, \url{arXiv:0708.4244}] for computations concerning finite subgroups of \(\text{SO}(3)\). Bryan, J.; Graber, T.; Pandharipande, R., The orbifold quantum cohomology of \(\mathbb{C}^2/Z_3\) and Hurwitz-Hodge integrals, J. Algebr. Geom., 17, 1-28, (2008) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Generalizations (algebraic spaces, stacks) The orbifold quantum cohomology of \(\mathbb{C}^{2}/\mathbb{Z}_3\) and Hurwitz-Hodge integrals | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 derive a recursion relation for loop-level scattering amplitudes of Lagrangian field theories that generalises the tree-level Berends-Giele recursion relation in Yang-Mills theory. The origin of this recursion relation is the homological perturbation lemma, which allows us to compute scattering amplitudes from minimal models of quantum homotopy algebras in a recursive way. As an application of our techniques, we give an alternative proof of the relation between non-planar and planar colour-stripped scattering amplitudes. scattering amplitudes; BRST quantization; gauge symmetry \(2\)-body potential quantum scattering theory, Quantum groups and related algebraic methods applied to problems in quantum theory, Yang-Mills and other gauge theories in mechanics of particles and systems, Homotopy theory and fundamental groups in algebraic geometry Loop amplitudes and quantum homotopy 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 Grothendieck conjectured in the sixties that the even Künneth projector (with respect to a Weil cohomology theory) is algebraic and that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note, we extend these celebrated conjectures from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck's original conjectures in the case of intersections of quadrics and linear sections of determinantal varieties. Along the way, we also prove the case of quadric fibrations and intersections of bilinear divisors. Noncommutative algebraic geometry, (Equivariant) Chow groups and rings; motives, Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Determinantal varieties, Enriched categories (over closed or monoidal categories), Derived categories, triangulated categories A note on Grothendieck's standard conjectures of type \(C^+\) and \(D\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a compact connected Lie group and let \(H\subset G\) be the centralizer of a one parameter subgroup in \(G\): the homogeneous space \(G/H\) is a flag variety. Let \(T\subset H\) be a maximal torus and \(W\) (resp \(W'\)) the Weyl group of \(G\) (resp of \(H\)).
The Grothendieck cohomology of coherent sheaves on \(G/H\) is canonically isomorphic to the \(K\)-theory of complex vector bundles on \(G/H\). The group \(K(G/H)\) is a free \(\mathbb{Z}\)-module with rank equal to the quotient of the order of \(W\) by the order of \(W'\). There are two distinguished additive bases of \(K(G/H)\): the first valid for the case \(H= T\), is indexed by elements from the Weyl group \(\{a_w\in K(G/T)\mid w\in W\}.\)
The second comes from identifying the set of left cosets of \(W'\) in \(W\) with
\[
\overline W= \{w\in W/l(w)\leq l(w_1)\text{ for all }w_1\in W'\}
\]
and taking a canonical partition of the flag variety \(G/H\) into Schubert subvarieties
\[
G/H= \bigcup_{w\in\overline W} X_w(H),\quad\dim X_w(H)= 2l(w).
\]
Then the coherent sheaves \(\Omega_w(H)\in K(G/T)\) form a basis for the \(\mathbb{Z}\)-module \(K(G/H)\). Therefore, in order to give a complete description of the ring \(K(G/T)\) (resp \(K(G/H)\)), one has to specify the structure constants \(C^w_{u,v}\in\mathbb{Z}\) (resp. \(K^w_{u,v}\in\mathbb{Z}\)) where \(u,v,w\in W\) (resp. \(\in\overline W\)), which express the product of the basis elements \(a_u\cdot a_v= \sum C^w_{u,v}\) in the first basis or
\[
\Omega_u(H)\cdot\Omega_v(H)= \sum K^w_{u,v}(H) \Omega_w(H)
\]
in the second basis.
In this paper the author proves a formula that expresses the constants \(C^w_{u,v}\) (resp \(K^w_{u,v}\)) in terms of the Cartan numbers of \(G\). These formulae are computable, in the sense that an existing algorithm for multiplying Schubert classes, can be extended to implement the \(C^w_{u,v}\) (resp. the \(K^w_{u,v}\)). Duan, Haibao: Multiplicative rule in the Grothendieck cohomology of a flag variety. (2004) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Multiplicative rule in the Grothendieck cohomology of a flag variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Perhaps the nicest multivariate orthogonal polynomials are the Macdonald and Koornwinder polynomials, respectively 2-parameter deformations of Schur functions and 6-parameter deformations of orthogonal and symplectic characters, satisfying a trio of nice properties (evaluation, norm and symmetry) known as the Macdonald ``conjectures''.
In recent work, the author has constructed elliptic analogues: a family of multivariate functions on an elliptic curve satisfying analogues of Macdonald conjectures, and degenerating to Macdonald and Koornwinder polynomials under suitable limits. This article will discuss the two main constructions for these functions. While the first construction is intrinsically complex analytic in nature, the second one is much more combinatorial and algebraic. This paper will focus on the second construction, modified somewhat to make the arguments self-contained. Macdonald polynomials; Koornwinder polynomials; elliptic curves; special functions Rains, Eric M., The noncommutative geometry of elliptic difference equations, (None) Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Elliptic curves Elliptic analogues of the Macdonald and Koornwinder 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 I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum \(A_{\infty}\)-algebras, introduced in [\textit{S. Barannikov}, Int. Math. Res. Not. 2007, No. 19, Article ID rnm075, 31 p. (2007; Zbl 1135.18006)], is represented via de Rham differential acting on the supermatrix spaces related with Bernstein-Leites simple associative algebras with odd trace \(\mathfrak q(N)\), and \(\mathfrak{gl}(N|N)\). I also show that the matrix Lagrangians from [\textit{S. Barannikov}, C. R., Math., Acad. Sci. Paris 348, No. 7--8, 359--362 (2010; Zbl 1219.81171)] are represented by equivariantly closed differential forms. cyclic homology; non-commutative geometry; matrix integrals; super Lie algebras; homotopy associative algebras; Batalin-Vilkovisky formalism; mirror symmetry. Barannikov, S., Matrix de Rham complex and quantum \textit{A}-infinity algebras, Lett. Math. Phys., 104, 4, 373-395, (2014) de Rham theory in global analysis, Permutations, words, matrices, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Homological methods in Lie (super)algebras, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Groups and algebras in quantum theory Matrix de Rham complex and quantum \(A\)-infinity 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 A proper algebraic map \(\pi: Z \to X\) between irreducible complex algebraic varieties is \textit{semi-small} if for a finite stratification of \(X\) into irreducible smooth subvarieties, the dimension of the inverse image of a point in a stratum is at most half the codimension of the stratum; it is \textit{small} if in addition the equality holds only if the stratum is dense; [see \textit{W. Borho} and \textit{R. MacPherson}, Astérisque 101--102, 23--74 (1983; Zbl 0576.14046)]. The interest of this notion stems from the elegant description of the singularities of a small morphism in terms of intersection cohomology sheaves. Fundamental examples of semi-small morphisms are the Springer resolution of the nilpotent cone and its partial versions. In [\textit{H. Nakajima}, Ann. Math. (2) 160, No. 3, 1057--1097 (2005; Zbl 1140.17015)] some varieties together with their resolutions were introduced, starting from a simply-laced Dynkin diagram with a fixed orientation. These are called graded quiver varieties and the resolutions are analogues of the Springer resolution. Nakajima translated the natural question of smallness of these resolutions into representation-theoretic terms, see loc. cit. Namely, let \(\mathcal U_q(\mathcal L(\mathfrak g))\) be the quantum loop algebra (the so-called Drinfeld realization), where \(\mathfrak g\) is the simple Lie algebra corresponding to the initial Dynkin diagram. Then the smallness of the resolutions of the graded quiver varieties is expressed in terms of characters of certain finite-dimensional \(\mathcal U_q(\mathcal L(\mathfrak g))\)-modules, called standard; those that satisfy the necessary condition for smallness of the resolution are called small.
The main result of the present paper is the characterization of the standard modules corresponding to Kirillov-Reshetikhin modules that are small. This is also extended to general simply-laced quantum affinizations. quantum affine algebras; graded quiver varieties Hernandez, D., Smallness problem for quantum affine algebras and quiver varieties, Ann. Sci. Éc. Norm. Supér. (4), 41, 2, 271-306, (2008) Quantum groups (quantized enveloping algebras) and related deformations, Singularities of surfaces or higher-dimensional varieties Smallness problem for quantum affine algebras and quiver varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(R\) be a finite dimensional algebra of finite global dimension over a field \(k\). In this paper, we will characterize a \(k\)-linear abelian category \(\mathscr{C}\) such that \(\mathscr{C}\cong \operatorname{tails} A\) for some graded right coherent AS-regular algebra \(A\) over \(R\). As an application, we will prove that if \(\mathscr{C}\) is a smooth quadric surface in a quantum \(\mathbb{P}^3\) in the sense of Smith and Van den Bergh, then there exists a right noetherian AS-regular algebra \(A\) over \(kK_2\) of dimension 3 and of Gorenstein parameter 2 such that \(\mathscr{C}\cong \operatorname{tails} A\) where \(kK_2\) is the path algebra of the 2-Kronecker quiver \(K_2\). quantum projective space; AS-regular algebra; abelian category; helix; noncommutative quadric surface Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, Derived categories and associative algebras, Abelian categories, Grothendieck categories A categorical characterization of quantum 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 Arithmetic groups have been found useful in the physics literature [\textit{E. B. Bogomolny}, \textit{B. Georgeot}, \textit{M.-J Giannoni}, and \textit{C. Schmit}, Phys. Rev. Lett. 69, No. 10, 1477-1480 (1992)]\ while investigating symptoms of chaos (energy level spacing distributions) in quantum versions of classically chaotic Hamiltonian systems.
The paper presents a review of topics pertaining to Hamiltonians which are geodesic flows on an arithmetic hyperbolic manifold. Recent numerical computations of the eigenvalues and eigenfunctions of the Laplacian for such manifolds are reported. Their fine structure (level spacing distribution) is studied by relating these problems to ones about Riemann's zeta functions and the \(L\)-functions (generalized zeta functions).
The principal mathematical material with outlines of the proofs is contained in sections devoted to arithmetic manifolds, and \(L\)-functions. Classical Lindelöf, Ramanujan and Sato-Tate conjectures are discussed in this context. quantum chaos; arithmetic groups; Riemann zeta function; Hamiltonians; geodesic flows; hyperbolic manifold; eigenvalues; eigenfunctions; Laplacian [20] Sarnak (P.).-- Arithmetic quantum chaos, Israel Math. Conf. Proc. 8, p.~183-236 (1995). &MR~13 | &Zbl~0831. Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.), Geodesic flows in symplectic geometry and contact geometry, Strange attractors, chaotic dynamics of systems with hyperbolic behavior, Spectral problems; spectral geometry; scattering theory on manifolds, \(\zeta (s)\) and \(L(s, \chi)\), Langlands \(L\)-functions; one variable Dirichlet series and functional equations Arithmetic quantum chaos | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 a recent observation that entanglement classification for qubits is closely related to local \(\text{SL}(2,\mathbb C)\)-invariants including the invariance under qubit permutations, which has been termed \(\text{SL}^*\) invariance. In order to single out the \(\text{SL}^*\) invariants, we analyze the \(\text{SL}(2,\mathbb C)\)-invariants of four resp. five qubits and decompose them into irreducible modules for the symmetric group \(S_4\) resp. \(S_5\) of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of \(\text{SL}^*\)-invariants vanishing on arbitrary product states. We find that low degree homogeneous components of this ideal can be constructed in full by using the approach introduced by Osterloh and Siewert in [Phys. Rev. A 72, 012337 (2005)]. Our analysis highlights an intimate connection between this latter procedure and the standard methods to create invariants, such as the \(\Omega\)-process. As the degrees of invariants increase, the alternative method proves to be particularly efficient.
Editorial remark: No review copy delivered Hilbert spaces; information theory; polynomials; quantum computing; quantum entanglement Djoković, DZ; Osterloh, A, On polynomial invariants of several qubits, J. Math. Phys., 50, 033509, (2009) Quantum coherence, entanglement, quantum correlations, Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Finite-dimensional groups and algebras motivated by physics and their representations On polynomial invariants of several qubits | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials ``This, mostly expository, paper is built around the topic of the Grothendieck anabelian section conjecture. This conjecture predicts that splittings, or sections, of the exact sequence of the arithmetic fundamental group
\[
1\to \pi_1(\bar X)\to \pi_1(X)\to G_k\to 1
\]
of a proper, smooth, and hyperbolic curve \(X\), all arise from decomposition subgroups associated to rational points of \(X\)'' over a certain arithmetic base field \(k\). The author introduces a notion of \textit{(uniformly) good group-theoretical section}, that behaves nicely with local-global principle in \(k\), which can be lifted to sections into the \textit{cuspidally abelian quotient} of the absolute Galois group of the function field \(k(X)\). The main ingredients are based on a preprint whose expanded version has been published as [\textit{M. Saïdi}, Adv. Math. 230, No. 4--6, 1931--1954 (2012; Zbl 1260.14036); J. Pure Appl. Algebra 217, No. 3, 583--584 (2013; Zbl 1262.14029)]. anabelian Geometry; arithmetic fundamental groups; Section Conjecture Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Separable extensions, Galois theory Around the Grothendieck anabelian section 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 \textit{M. Haiman} [J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001)] has reduced the Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product \({\mathfrak S}_n \ltimes (\mathbb{Z} / r \mathbb{Z})^n\). He has proven the original conjecture by establishing the geometric statement about the Hilbert scheme, as a byproduct he obtained a derived equivalence between coherent sheaves on the Hilbert scheme and coherent sheaves on the orbifold quotient of \(\mathbb{A}^{2n}\) by the symmetric group \({\mathfrak S}_n\).
A short proof of a similar derived equivalence for any symplectic quotient singularity has been obtained by the first author and \textit{D. B. Kaledin} [in: Algebraic geometry. Methods, relations, and applications. Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin. Moscow: Maik Nauka/Interperiodica. 13--33 (2004; Zbl 1137.14301)] via quantization in positive characteristic. In the present note we prove various properties of these derived equivalences and then deduce generalized Macdonald positivity for wreath products. Bezrukavnikov, R., Finkelberg, M.: Wreath Macdonald polynomials and categorical McKay correspondence. arXiv:1208.3696 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], McKay correspondence, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Rings of differential operators (associative algebraic aspects), Symmetric functions and generalizations, Combinatorial aspects of representation theory, Parametrization (Chow and Hilbert schemes), Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) Wreath Macdonald polynomials and the categorical McKay correspondence. With an appendix by Vadim Vologodsky | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article formulas for the quantum product of a rational surface are given, and used to give an algebro-geometric proof of the associativity of the quantum product for strict Del Pezzo surfaces, those for which \(- K\) is very ample. An argument for the associativity in general is proposed, which also avoids resorting to the symplectic category. The enumerative predictions of \textit{M. Kontsevich} and \textit{Yu. Manin} [Commun. Math. Phys. 164, No. 3, 525--562 (1994; Zbl 0853.14020)] concerning the degree of the rational curve locus in a linear system are recovered. The associativity of the quantum product for the cubic surface is shown to be essentially equivalent to the classical enumerative facts concerning lines: there are 27 of them, each meeting 10 others. quantum product of a rational surface; Del Pezzo surfaces B. Crauder and R. Miranda, \textit{Quantum cohomology of rational surfaces}, alg-geom/9410028. Rational and ruled surfaces, Quantization in field theory; cohomological methods, (Co)homology theory in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry Quantum cohomology of rational surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article, we announce our results on the Grothendieck-Teichmüller group GT obtained in the joint work with Yuichiro Hoshi and Shinichi Mochizuki. Let \(n\ge 2\) be a positive integer and \(k\) an algebraically closed field of characteristic zero. Write \(\Pi_n\) for the étale fundamental group of the \(n\)-th configuration space of \(\mathbb{P}^1_k\setminus\{0,1,\infty\}\) and \(\mathfrak{S}_{n+3}\) for the symmetric group on \(n+3\) letters. Then our main theorem asserts that we have a direct product decomposition \(\text{Out}(\Pi_n)= \text{GT}\times\mathfrak{S}_{n+3}\). The detail of the arguments appearing in this article may be found in [\textit{Y. Hoshi} et al., ``Group-theoreticity of Numerical Invariants and Distinguished Subgroups of Configuration Space Groups'', RIMS Preprint 1970 (2017)]. Grothendieck-Teichmüller group; combinatorial anabelian geometry; fiber subgroup; generalized fiber subgroup; configuration space Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Families, moduli of curves (algebraic) On a direct product decomposition related to the Grothendieck-Teichmüller group | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(p\) be a prime number and let \(R\) be the valuation ring of a local field \(K\) containing \(\mathbb{Q}_p\). The unifying theme of this monograph is to construct classes of finite, \(p\)-elementary \(R\)-group schemes, or equivalently, commutative, cocommutative, \(p\)-power rank \(R\)-Hopf algebras of exponent \(p\). Finite cocommutative Hopf algebras over \(K\) are, in principle, well-understood: They are all \(K\)-forms of group rings of finite groups and are obtainable by Galois descent. (A \(K\)-form of \(KG\) is a \(K\)-Hopf algebra \(H\) so that \(L\otimes_KH\cong LG\) for some finite extension \(L\) of \(K.)\) In fact, the well-known correspondence between Galois extensions of \(K\) and finite \(\text{Gal} (\widehat K/K)\)-sets extends to the classification of commutative, cocommutative, finite \(K\)-Hopf algebras and their principal homogeneous spaces. However, over \(R\), finite commutative, cocommutative Hopf algebras are much less well understood. We begin with a brief survey of known results about these algebras. The purpose of this paper is to explain why studying polynomial formal groups is of interest in the study of finite, local, commutative Hopf algebras over valuation rings of local fields. polynomial formal groups; Hopf algebras over valuation rings Formal groups, \(p\)-divisible groups, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Class field theory; \(p\)-adic formal groups, Group schemes Introduction to polynomial formal groups and Hopf algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We present a brief introductin to quantum sheaf cohomology, a generalization of quantum cohomology based on the physics of the \((0,2)\) nonlinear sigma model. nonlinear sigma model; supersymmetry J. Guffin, \textit{Quantum sheaf cohomology, a precis}, \textit{Mat. Contemp.}\textbf{41} (2012) 17 [arXiv:1101.1305 ] [INSPIRE]. Kähler manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Quantum field theory on curved space or space-time backgrounds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum sheaf cohomology, a précis | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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{A. Kohnert}'s algorithm [Bayreuther Math. Schr. 38, 1--97 (1991; Zbl 0755.05095)], we associate a polynomial to any cell diagram in the positive quadrant, simultaneously generalizing Schubert polynomials and \(\mathrm{GL}_n\) Demazure characters. We survey properties of these Kohnert polynomials and their stable limits, which are quasisymmetric functions. As a first application, we introduce and study two new bases of Kohnert polynomials, one of which stabilizes to the skew-Schur functions and is conjecturally Schubert-positive, the other stabilizes to a new basis of quasisymmetric functions that contains the Schur functions. Schubert polynomials; Demazure characters; monomial slide polynomials Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Skew polynomials and extended 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 review our recent results on the noncommutative geometry of \(\mathbb Q\)-lattices modulo commensurability. We discuss the cases of one-dimensional and two-dimensional \(\mathbb Q\)-lattices. In the first case, we show that, by considering commensurability classes of one-dimensional \(\mathbb Q\)-lattices up to scaling, one recovers the Bost-Connes quantum statistical mechanical system, whose zero temperature KMS states intertwine the symmetries of the system with the Galois action of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\). In the two-dimensional case, commensurability classes of \(\mathbb Q\)-lattices up to scaling give rise to another quantum statistical mechanical system, whose symmetries are the automorphisms of the modular field, and whose (generic) zero temperature KMS states intertwine the action of these symmetries with the Galois action on an embedding in \(\mathbb C\) of the modular field. Following our joint work with \textit{N. Ramachandran} [KMS states and complex multiplication. Sel. Math., New Ser. 11, No. 3-4, 325--347 (2005; Zbl 1106.58005), Part II. in: Bratteli, Ola (ed.) et al., Operator algebras. The Abel symposium 2004. Proceedings of the first Abel symposium, Oslo, Norway, September 3--5, 2004. Berlin: Springer. Abel Symposia 1, 15--59 (2006; Zbl 1123.58004)], we then show how the noncommutative spaces associated to commensurability classes of \(\mathbb Q\)-lattices up to scale have a natural geometric interpretation as noncommutative versions of the Shimura varieties \(Sh(\text{GL}_1, \{\pm 2\})\) (in the Bost-Connes case and \(Sh(\text{GL}_2,\mathbb H^\pm)\) in the case of the \(\text{GL}_2\) system. We also show how this leads naturally to the construction of a system generalizing the Bost-Connes system that fully recovers the explicit class field theory of imaginary quadratic fields. \(\mathbb Q\)-lattices; quantum statistical mechanics; KMS states; class field theory; modular field; Shimura varieties A. Connes and M. Marcolli, ''Quantum statistical mechanics of \(\mathbb{Q}\)-lattices,'' in Frontiers in Number Theory, Physics, and Geometry, I pp. 269--350 (Springer Verlag, 2006). Noncommutative geometry (à la Connes), Applications of selfadjoint operator algebras to physics, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Phase transitions (general) in equilibrium statistical mechanics \(\mathbb Q\)-lattices: quantum statistical mechanics and Galois theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper we prove a refined version of a theorem by \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135--194 (1997; Zbl 0899.14007)] and \textit{S. Mochizuki} [J. Math. Kyoto Univ. 47, No. 3, 451--539 (2007; Zbl 1143.14305)] on isomorphisms between (tame) arithmetic fundamental groups of hyperbolic curves over finite fields, where one ``ignores'' the information provided by a ``small'' set of primes. Varieties over finite and local fields, Coverings of curves, fundamental group A refined version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Solutions of the KP hierarchy associated with polynomial \(\tau\)-functions are recovered in an elementary way, using a geometric approach to the linearization of the KP equations. It starts from the geometry of bi-Hamiltonian manifolds and allows obtaining a system of infinite-matrix Riccati equations, which are linearized by elementary methods. Here the authors explicitly determine the solutions of the Sato system where the initial condition has only a finite number of nonzero entries, and show that these solutions yield solutions of the KP hierarchy associated with polynomial \(\tau\)-functions. \(\tau\)-function; KP hierarchy; bi-Hamiltonian manifold; Riccati equation G. Falqui, F. Magri, M. Pedroni, J.P. Zubelli, An elementary approach to the polynomial {\(\tau\)}-functions of the KP hierarchy, in: Proceedings of NEEDS'98, Theor Math. Phys., in press. Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relationships between algebraic curves and integrable systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, KdV equations (Korteweg-de Vries equations) An elementary approach to the polynomial \(\tau\)-functions of the KP-hierarchy | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. We show that it represents the weighted Schubert classes of the cohomology of the weighted Grassmannian introduced by Corti-Reid, and we regard it as an analogue of the Schur polynomials. Furthermore, we prove that these polynomials are the characters of certain representations, and hence, we give an interpretation of the Schubert structure constants of the weighted Grassmannians as the (rational) multiplicities of the tensor products of the representations. We also derive two determinantal formulas for the weighted Schubert classes: One is in terms of the special weighted Schubert classes, and the other is in terms of the Chern classes of the tautological orbi-bundles. Schur polynomial; weighted Grassmannian; orbifold; Schubert calculus; representation Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Combinatorial aspects of representation theory, Topology and geometry of orbifolds, Classical problems, Schubert calculus Schur polynomials and weighted Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The existence theorem in this paper is named after Grothendiecks so called existence theorem in algebraic geometry, which gives an equivalence of the category of coherent sheaves with proper supports on a relative scheme with that of the same sheaves on a formal completion of the scheme, see EGA III [\textit{A. Grothendieck}, Éléments de géométrie algébrique. III. (1962; Zbl 0118.362), théorème 5.1.4]. Although it is not a direct analogue of the latter it gives a strong statement on the algebraic nature of formal sheaves or spaces with proper supports with applications to deformation theory. Its formulation is as follows: Let A be a formal \({\mathbb{C}}\)-algebra, i.e. an algebra \({\mathbb{C}}<x>^{\wedge a}/b\), where \({\mathbb{C}}<x>={\mathbb{C}}<x_ 1,...,x_ m>\) is a convergent power series ring, \({\mathbb{C}}<x>^{\wedge a}\) is the completion with respect to a proper ideal \(a\), and where \(b\) is an ideal in the completion. Define the category An(A) of complex spaces over A. f.e. by local models. A smooth local model over A is a domain D in some \({\mathbb{C}}^ n\) with the structure sheaf \({\mathcal O}_ D\wedge a={\mathcal O}_ D<x_ 1,...,x_ m>^{\wedge a}/b {\mathcal O}_ D<x_ 1,...,x_ m>^{\wedge a}\), and a local model over A is defined as Supp(\({\mathcal O}_ D\wedge a/{\mathcal I})\), where \({\mathcal I}\subset {\mathcal O}_ D\wedge a\) is some coherent sheaf of ideals. Note that the category An(A) is slightly more general than just the category of formal complex spaces over A. There is an obvious base change functor An(A)\(\to An(B)\) for each homomorphism \(A\to B\) of formal \({\mathbb{C}}\)-algebras, and there are finite fibre products in An(A). Let now (S,o) be a usual complex space germ and \({\mathcal O}_{S,o}\to A\) be a formal algebra over \({\mathcal O}_{S,o}\), and let \(X\to_{f}S\) be a morphism of complex spaces. If \(X_ A\in An(A)\) denotes the pullback of X under base change, the following functors on the category of formal algebras over \({\mathcal O}_{S,o}\) are considered: \(Coh(X_ A)\), \(Coh_ p(X_ A)\), \(Coh_{p,f}(X_ A)\) are the categories of all coherent sheaves on \(X_ A\), those with compact supports, and those which are in addition flat over A, respectively. If B is any convergent \({\mathcal O}_{S,o}\)-subalgebra of A, f.e. one which is generated by an element, there is the pullback functor \(Coh(X_ B)\to Coh(X_ A)\) whose image consists of pullbacks of usual coherent modules.
The existence theorem obtained essentially says that \(Coh_ p(X_ A)\) entirely consists of such pullbacks. More precisely: If a family \(A_ i\), \(i\in I\), of usual analytic \({\mathcal O}_{S,o}\)-subalgebras of A is chosen such that \(A=\lim_{\to}A_ i\), then the natural functor
\[
\lim_{\to}Coh_ p(X_{A_ i})\to Coh_ p(X_ A)
\]
is an equivalence of categories. This theorem answers a question of \textit{P. Deligne} and \textit{J. Bingener}, see \textit{J. Bingener}'s paper in Ann. Sci. Ec. Norm. Super., IV. Ser. 13, 317-347 (1980; Zbl 0454.32017) where the existence theorem for the functor \(Coh_{p,f}\) had been proved, which is related to the existence of flat versal deformations of coherent sheaves with compact supports.
Whilst the existence theorem for \(Coh_{p,f}\) has an elegant proof, the case \(Coh_ p\) needs the enormous machinery of the so called PO-spaces developed by \textit{J. Bingener} in his books ``Lokale Modulräume in der analytischen Geometrie. Band 1 und 2.'' in the series ``Aspekte der Mathematik'', Vieweg Verlag (1987), using ideas of Palamodov. To follow the highly technical proof the reader has to know that apparatus. Details cannot be given in this review.
However it should be mentioned that the same method also yields the existence theorem for the category \(An_ p(A)\) of compact complex spaces over A: The functors
\[
\lim_{\to}An_{p,f}(A_ i)\to An_{p,f}(A),\quad \lim_{\to}An_ p(A_ i)\to An_ p(A)
\]
are equivalences of categories. The author also considers the local situation with respect to X, which is not as satisfactory: If \(Coh'(X_ A)\) is the category of germs of modules whose non-free locus is finite, then
\[
\lim_{\to}Coh'(X_{A_ i})\to Coh'(X_ A)
\]
is in general not an equivalence, but it is still surjective. analytic moduli problem; formal spaces; deformations; formal sheaves S. Kosarew, Grothendieck's Existence Theorem in Analytic Geometry and Related Results. Regensburger Mathematische Schriften Bd.14 (1987), (to appear in Trans. AMS). Complex-analytic moduli problems, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Abstract manifolds and fiber bundles (category-theoretic aspects), Formal methods and deformations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Grothendieck's existence theorem in analytic geometry and related results | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Grothendieck's and Knutson's existence theorem states, that certain sheaves on formal completions of algebraic spaces are just arising as formal completions of certain sheaves on these algebraic spaces. In analytic geometry one has some sorts of analoga in connection with deformation theory when deforming ``analytically'' or ``formally''. The results are formulated in terms of equivalence of functors (mostly) between certain categories coming from ``analytic objects'' (for example certain complex spaces) or ``formal objects'' (for example certain formal complex spaces: these are roughly complex spaces with --- transversally - -- some additional formal parts). The surjectivities of these functors are essentially existence theorems of the type: ``formally'' implies ``analytically'', giving especially the existence of certain convergent formally semiuniversal deformations. The paper is a big step forward in the already highly sophisticated and developed deformation theory (essentially). It heavily relies on much work recently done by Polamodov, Bingener, Kosarew, especially on the new book of Bingener-Kosarew [\textit{J. Bingener}, ``Lokale Modulräume in der analytischen Geometrie'' (1987; Zbl 0644.32001)]. Without good knowledge there this paper is very hard to read.
Two sidemarks: a) In differentiable geometry one has differentiable spaces also with some ``formal parts'': so called Whitney-spaces for example. b) In complex analysis one has another, however similar principle as in this paper, saying for example that ``finite differentiability'' implies ``analyticity''. analytic formal complex spaces; deformation; universality; convexity; moduli-problem Complex-analytic moduli problems, Formal methods and deformations in algebraic geometry, Deformations of complex structures Grothendieck's existence theorem in analytic geometry and related results | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, the authors study the relationship between the quantum cohomology of a monotone closed symplectic manifold \(M\) and the symplectic cohomology of the complement of a simple normal crossings (or SC) divisor \(D\), mainly under the assumption that the pair \((M,D)\) is logarithmic effective. They prove that the former is a deformation of the latter in a suitable sense. They also prove some rigidity results about the isotropic skeleton of \(X=M-D\).
A closed symplectic manifold \((M,\omega)\) is called monotone if the cohomology class \([\omega]\) is a multiple of \(c_1(TM)\). In particular, \([\omega]\) is a rational cohomology class, and Donaldson's construction of a (smooth) symplectic divisor Poincare dual to a large multiple of \([\omega]\) applies. In this paper, the authors consider an SC divisor \(D=\bigcup_{i=1}^N D_i\) such that \(c_1(TM)\) is a weighted sum
\[
c_1(TM)=\textbf{D}:=\sum_{i=1}^N a_i D_i
\]
with positive rational weights (Note that \(\lambda_i=2a_i\) in the paper's notation). The last identity and the monotonicity imply that \(X\) is a convex exact symplectic manifold with a vanishing first Chern class. Furthermore, if \([\omega]\) is a positive multiple of \(c_1(TM)\), the positivity of \(D\) implies that \(X\) is a Weinstein domain and retracts to an isotropic skeleton. In Hypothesis A, the authors assume that \(a_i \leq 1\), which means
\[
K_M+D=D-\textbf{D}=\sum_{i=1}^n (1-a_i)D_i
\]
is effective (in the algebraic geometric language). Here, \(K_M\) is the canonical line bundle of \((M,\omega)\) and \(K_M+D\) is the logarithmic canonical line bundle of the pair \((M,D)\). The borderline case is \(a_i\equiv 1\), where \((X,D)\) is a symplectic log Calabi-Yau pair.
The vague conjecture in the literature has been that under certain assumptions on \((M,D)\), the Novikov field-valued quantum cohomology \(QH^*(M,\Lambda)\) of \(M\) coincides the cohomology of a natural deformation \(\big(SC^*(X;\Lambda),\partial\big)\) of the symplectic cochain complex \(\big(SC^*(X;\Lambda),d\big)\). The former is defined using closed pseudoholomorphic curves, and the latter involves periodic orbits of a Hamiltonian function on \(X\) and ceratin pseudoholomorphic-type curves connecting them. In Theorem C, the authors prove this conjecture when \(M\) is monotone and \((M,D)\) is log-effective. Furthermore, they prove that the spectral sequence of the deformed filtered complex converges on the first page to the quantum cohomology. As a corollary (Corollary 1.7), they conclude that \(X\) has non-vanishing symplectic cohomology.
Over the course of the proof, they study compact subsets of \(X\) and their symplectic homology relative to \(M\). For instance, if \((M,D)\) is log-effective, they prove that the isotropic skeleton \(L\) of \(X\) is SH-full, meaning that the symplectic homology of any compact set \(K\) in \(X-L\) is zero. They conclude that \(L\) can not be displaced from any ``Floer-theoretically essential monotone'' Lagrangian in \(X\). symplectic cohomology; quantum cohomology; divisor complement; spectral sequences Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Symplectic aspects of Floer homology and cohomology Quantum cohomology as a deformation of symplectic cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that Grothendieck's Hodge standard conjecture holds for abelian varieties in arbitrary characteristic if the Hodge conjecture holds for complex abelian varieties of CM-type. For abelian varieties with no exotic algebraic classes, we prove the Hodge standard conjecture unconditionally. quotient categories; motives over finite fields; abelian varieties; Hodge standard conjecture Transcendental methods, Hodge theory (algebro-geometric aspects), Arithmetic ground fields for abelian varieties Polarizations and Grothendieck's standard conjectures. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper offers a more or less self-contained introduction into the theory of the Grothendieck-Teichmüller group and Drinfeld associators using the theory of operads and graph complexes. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Operads (general) Grothendieck-Teichmüller group, operads and graph complexes: a survey | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Near the end of the 1990's, \textit{D. Abramovich} and \textit{A. Vistoli} developed some very natural and powerful algebraic techniques for compactifying the space of stable maps into a Deligne-Mumford (DM) stack, or orbifold, by allowing the source curves of the maps to be orbifolds themselves [J. Am. Math. Soc. 15, No. 1, 27--75 (2002; Zbl 0991.14007)]. Such maps are called twisted stable maps. At roughly the same time, \textit{W. Chen} and \textit{Y. Ruan}, inspired by physicists' orbifold string theories, began to develop an analytic form of orbifold Gromov-Witten theory [in: Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math. 310, 25--85 (2002; Zbl 1091.53058)]. A key part of that theory was the recognition that the usual cohomology of the underlying space was inadequate for the theory. What is needed instead is a larger ring \(H^{*}_{\text{orb}}\), called the orbifold cohomology, which, as a vector space, is simply the direct product of the cohomology of various ``sectors'' (roughly, the fixed point loci of the isotropy groups) of the orbifold. Chen and Ruan developed a ``stringy product'' on this larger space \(H^{*}_{\text{orb}}\) which, even in degree zero, is an important invariant of the orbifold.
This paper gives an overview of the authors' work defining an algebraic version of Chen and Ruan's orbifold Gromov-Witten theory, with special focus on the stringy, or orbifold, quantum product. It is based primarily on the algebraic theory of twisted stable maps of the first paper cited above. The results are valid for DM stacks over any field of characteristic 0. A surprising consequence of this work is the construction of a stringy product in degree zero with integer coefficients, rather than rational coefficients. Although the paper is just an overview, it is currently the only source for much of this material. Fortunately, the authors have been very careful in their constructions and arguments, and most of the important steps are clearly sketched so the reader may fill in the details. stack; quantum product; moduli; stringy; stable map D. Abramovich, T. Graber, A. Vistoli, Algebraic orbifold quantum products. In: \textit{Orbifolds in mathematics and physics} (\textit{Madison, WI}, 2001), volume 310 of \textit{Contemp. Math}., 1-24, Amer. Math. Soc. 2002. MR1950940 Zbl 1067.14055 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Generalizations (algebraic spaces, stacks), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Algebraic orbifold quantum products. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a survey article describing the relationship between quantum curves and topological recursion. A quantum curve is a Schrödinger operator-like noncommutative analogue of a plane curve which encodes (quantum) enumerative invariants in a new and interesting way. The Schrödinger operator annihilates a wave function which can be constructed using the WKB method, and conjecturally constructed in a rather different way via topological recursion. spectral curve; quantum curve; WKB method Norbury, P., Quantum curves and topological recursion, Proc. Symp. Pure Math., 93, 41, (2015) Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory, Selfadjoint operator theory in quantum theory, including spectral analysis, Relationships between algebraic curves and physics, Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Exact enumeration problems, generating functions, Geometry and quantization, symplectic methods Quantum curves and topological recursion | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We suggest an explanation for the part of the Satake Correspondence which relates the quantum cohomology of complex Grassmannians and the quantum cohomology of complex projective space, as well as their respective Stokes data, based on the original physics approach using the tt* equations. We also use the Stokes data of the tt* equations to provide a Lie-theoretic link between particles in affine Toda models and solitons in certain sigma-models. Along the way, we review some old ideas from supersymmetric field theory, whose mathematical manifestations are becoming increasingly widespread. tt* equations; quantum cohomology; Satake correspondence Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Grassmannians, Schubert varieties, flag manifolds Topological-antitopological fusion and the quantum cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the interplay between noncommutative tori and noncommutative elliptic curves through a category of equivariant differential modules on \(\mathbb C^*\). We functorially relate this category to the category of holomorphic vector bundles on noncommutative tori as introduced by Polishchuk and Schwarz and study the induced map between the corresponding K-theories. In addition, there is a forgetful functor to the category of noncommutative elliptic curves of Soibelman and Vologodsky, as well as the forgetful functor to the category of vector bundles on \(\mathbb C^*\) with regular singular connections.
The category that we consider has the nice property of being a Tannakian category, hence it is equivalent to the category of representations of an affine group scheme. Via an equivariant version of the Riemann-Hilbert correspondence we determine this group scheme to be (the algebraic hull of) \(\mathbb Z^2\). We also obtain a full subcategory of the holomorphic vector bundles on the noncommutative torus which is equivalent to the category of representations of \(\mathbb Z\). This group is the proposed topological fundamental group of the noncommutative torus (understood as a degenerate elliptic curve) and we study Nori's notion of étale fundamental group in this context. noncommutative tori; fundamental group; semistable bundles Mahanta, S.; Suijlekom, W.D., Noncommutative tori and the Riemann-Hilbert correspondence, J. Noncommut. Geom., 3, 261-287, (2009) Noncommutative geometry (à la Connes), Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain, Noncommutative algebraic geometry, Holomorphic bundles and generalizations Noncommutative tori and the Riemann-Hilbert correspondence | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be a field, char \(k=p>0.\) Let \(n,r>0,\) and let
\[
R\left( n,r\right) =k\left[ x_{1},\dots,x_{n}\right] /\left( x_{1}^{p^{r}},\dots,x_{n}^{p^{r} }\right).
\]
Then \(R\left( n,r\right) \) represents the \(r^{\text{th}}\) Frobenius kernel on \(n\)-dimensional affine space. Let \(G\left( n,r\right) \) be the automorphism scheme of \(R\left( n,r\right) .\) Associated to each \(G\left( n,r\right) \)-representation is a canonical vector bundle obtained by twisting with the \(r^{\text{th}}\) Frobenius morphism. In \(K\)-theory, these bundles are studied using the Grothendieck ring of \(G\left( n,r\right) .\)
In the work under review, the author provides a description of the irreducible \(G\left( n,r\right) \)-representations which form a \(\mathbb{Z}\)-basis of \(K_{0}\left( G\left( n,r\right) -\text{rep}\right) \) as an abelian group. This is accomplished by taking a triangular decomposition \(G\left( n,r\right) =G^{-}\text{GL}_{n}G^{+}.\) A surjective map \(K_{0}\left( \text{GL}_{n}-\text{rep}\right) ^{\oplus r+1}\rightarrow K_{0}\left( G\left( n,r\right) -\text{rep}\right) \) is constructed recursively, and its kernel is computed. Much of the work uses the more general language of triangulated group schemes, which facilitates the recursive definition. Grothendieck rings; automorphism schemes; triangulated group schemes Group schemes, Representation theory for linear algebraic groups, \(K\)-theory of schemes, Modular Lie (super)algebras, Lie algebras of linear algebraic groups The Grothendieck ring of the structure group of the geometric Frobenius morphism | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper we give a simplified proof of the flat Grothendieck-Riemann-Roch theorem. The proof makes use of the local family index theorem and basic computations of the Chern-Simons form. In particular, it does not involve any adiabatic limit computation of the reduced eta-invariant. flat \(K\)-theory; Grothendieck-Riemann-Roch theorem; eta form Riemann-Roch theorems, Chern characters, Index theory, Riemann-Roch theorems The flat Grothendieck-Riemann-Roch theorem without adiabatic techniques | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 1988 Witten used current algebras to develop the fundaments of quantum field theory for free fermions living on an algebraic curve. The global symmetries are then given by the rational maps of the curve \(X\) to a finite-dimensional semi-simple Lie algebra over an algebraically closed field \(k\). In a previous paper, titled ``Quantum field theories on an algebraic curve'' [Lett. Math. Phys. 52, No. 1, 79--91 (2000; Zbl 1024.81016)], the author provided a solution to the problem of determining the expectation value functional for scalar fields with an abelian Lie algebra. Although abelian integrals have been studied before, so far the integral calculus on curves has not been fully developed.
In the present paper Takhtajan tries to fill this gap when the field \(k\) has characteristic zero. He gives an explicit construction of quantum field theories of (additive) bosons on an algebraic curve. The paper is organized as follows. Section 1 provides a long survey of the history of the subject. In Section 2 the author recalls the necessary basic facts from the theory of algebraic curves. This material is standard. In Section 3 he recalls the details of the differential calculus on an algebraic curve and develops a corresponding integral calculus. It is now assumed that the field \(k\) has characteristic 0 and the algebraic curve has genus \(g\geq 1\). In Section 4 local quantum field theories of additive charged bosons are formulated. Finally, in Section 5 global versions of the local QFTs -- as studied in Section 4 -- are constructed. The paper provides a stimulating account of the mathematical principles and techniques using algebraic curves and some concepts from quantum field theory though the aim is certainly not to suggest possible applications in particle physics. algebraic curves; algebraic functions; free bosons on curves; differential calculus on curves; integrals on curves; current algebra on curves; Fock space; expectation value functional; quantum field theory on curves Model quantum field theories, 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, Quantum field theory on curved space or space-time backgrounds, Operator algebra methods applied to problems in quantum theory Quantum field theories on algebraic curves. I. Additive bosons | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a rank \(r\) vector bundle over \({\mathbb P}^n\) with corresponding projective bundle \(\pi:{\mathbb P}(V)\rightarrow{\mathbb P}^n\). Denote by \(h\) and \(\xi\) the cohomology classes of a hyperplane in \({\mathbb P}^n\) and the tautological line bundle in \({\mathbb P}(V)\), respectively. For brevity, one often writes \(h\) for \(\pi^*h\). Assume \(V\) is ample and that both \(\xi\) and \(-K_{{\mathbb P}(V)}\) (\(K_{{\mathbb P}(V)}\) the canonical bundle on \({\mathbb P}(V)\)) are ample divisors. Then \({\mathbb P}(V)\) is a Fano variety with well-defined quantum cohomology ring \(H^*_{\omega}({\mathbb P}(V),{\mathbb Z})\). The simplest case arises for \(V\) of the form \({V=\bigoplus_{i=1}^r{\mathcal O}_{{\mathbb P}^n}(m_i)}\) with \(m_i>0\) for all \(i\). By twisting with \(\mathcal O_{{\mathbb P}^n}(-1)\) one can always obtain that \(\text{ min}\{m_1,\ldots,m_r\}=1\). Then \({\mathbb P}(V)\) becomes a toric variety. For this situation Batyrev conjectured:
The cohomology ring \(H^*_{\omega}({\mathbb P}(V),{\mathbb Z})\) is generated by \(h\) and \(\xi\) with two relations
\[
h^{n+1}=\prod_{i=1}^r(\xi-m_ih)^{m_i-1}\cdot e^{-t(n+1+r-\sum_{i=1}^rm_i)}\quad \text{and}\quad \prod_{i=1}^r(\xi-m_ih)=e^{-\text{tr}}.
\]
One of the main results of the paper under review confirms Batyrev's conjecture if
\[
\sum_{i=1}^rm_i<\min\{2r, (n+1+2r)/2, (2n+2+r)/ 2\}.
\]
In the case of ample bundles on \({\mathbb P}^n\) which are not direct sums of line bundles the following result is obtained:
\((i)\) Let \(V\) be an ample rank \(r\) bundle on \({\mathbb P}^n\). Assume \(c_1\leq n+r\) and \(V\otimes\mathcal O_{{\mathbb P}^n}(-1)\) is nef, thus \({\mathbb P}(V)\) is a Fano variety. Then \(H^*_{\omega}({\mathbb P}(V),{\mathbb Z})\) is generated by \(h\) and \(\xi\) subject to the relations
\[
h^{n+1}=\sum_{i+j\leq c_1-r}a_{i,j}\cdot h^i\cdot\xi^j\cdot e^{-t(n+1-i-j)},\qquad \text{and}
\]
\[
\sum_{i=0}^r(-1)^ic_i\cdot h^i\cdot\xi^{r-i}=e^{-\text{tr}}+\sum_{i+j\leq c_1-n-1}b_{i,j}\cdot h^i\cdot\xi^j\cdot e^{-t(r-i-j)},
\]
where the coefficients \(a_{i,j}\) and \(b_{i,j}\) are integers depending on \(V\).
\((ii)\) If moreover \(c_1<2r\), the leading coefficient \(a_{0,c_1-r}=1\).
A third result is proved for \(V=T_{{\mathbb P}^n}\), where \(T_{{\mathbb P}^n}\) is the tangent bundle of \({\mathbb P}^n\):
The quantum cohomology ring \(H^*_{\omega}({\mathbb P}(T_{{\mathbb P}^n}),{\mathbb Z})\) with \(n\geq 2\) is generated by \(h\) and \(\xi\) subject to the relations
\[
h^{n+1}=\xi\cdot e^{-tn}\quad\text{and}\quad\sum_{i=0}^n(-1)^i c_i\cdot h^i\cdot\xi^{n-i}=(1+(-1)^n)\cdot e^{-tn}.
\]
To come to the proofs of the mentioned results, long explicit preparations are presented. Firstly there is a section on extremal rational curves, followed by the calculation of Gromov-Witten invariants of \({\mathbb P}(V)\). This sets the stage for the proof of the result for general ample bundles \(V\). Then the case of a direct sum of line bundles and the partial proof of Batyrev's conjecture is verified.
The last section deals with examples such as the tangent bundle. Many tedious calculations are at the basis of the proofs. quantum cohomology; toric variety; Batyrev's conjecture; Fano variety; Gromov-Witten invariants Z. Qin and Y. Ruan, Quantum cohomology of projective bundles over \(\mathbf{P}^{n}\) , Trans. Amer. Math. Soc. 350 , no. 9 (1998), 3615-3638. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Fano varieties, Projective techniques in algebraic geometry Quantum cohomology of projective bundles over \(\mathbb{P}^{n}\) | 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 A notion of Bernstein-Sato polynomial \(b_I(s)\) for a non-principle ideal \(I\) was introduced in [\textit{N. Budur} et al., Compos. Math. 142, No. 3, 779--797 (2006; Zbl 1112.32014)]. In the paper under review, the authors compute the Bernstein-Sato polynomials for two types of determinantal ideals. The first one is generated by maximal minors of a generic \(m\times n\) matrix, the second is generated by sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. In these cases they also verify the so-called Strong Monodromy Conjecture due to \textit{J. Denef} and \textit{F. Loeser} [J. Am. Math. Soc. 5, No. 4, 705--720 (1992; Zbl 0777.32017)], which asserts that the poles of the topological zeta function of \(I\) are roots of the Bernstein-Sato polynomial \(b_I(s)\). Bernstein-Sato polynomials; \(b\)-functions; determinantal ideals; Pfaffians; local cohomology; strong monodromy conjecture Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Determinantal varieties, Local cohomology and commutative rings, Sheaves of differential operators and their modules, \(D\)-modules, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) Bernstein-Sato polynomials for maximal minors and sub-maximal Pfaffians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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, by using the de Rham model of Chen-Ruan cohomology, we define the relative Chen-Ruan cohomology ring for a pair of almost complex orbifold \((\mathsf{G},\mathsf{H})\) with \(\mathsf{H}\) being an almost sub-orbifold of \(\mathsf{G}\). Then we use the Gromov-Witten invariants of \(\hat{\mathsf{G}}\), the blow-up of \(\mathsf{G}\) along \(\mathsf{H}\), to give a quantum modification of the relative Chen-Ruan cohomology ring \(H^\ast_{CR}(\mathsf{G},\mathsf{H})\) when \(\mathsf{H}\) is a compact symplectic sub-orbifold of the compact symplectic orbifold \(\mathsf{G}\). de Rham model; relative Chen-Ruan cohomology; relative orbifold quantum cohomology Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) A quantum modification of relative Chen-Ruan cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a vector bundle of rank \(n\) on a variety \(X\), and let \(\text{Fl}(V)\) be the bundle of complete flags in \(V\). For the universal flag of bundles \(E_1\subset \cdots \subset E_n\) on \(\text{Fl}(V)\) and given a complete flag \(F_1 \subset\cdots \subset F_n =V\) of subbundles of \(V\), one can define line bundles \(L_1= E_{n+1-i}/ E_{n-i}\) on \(\text{Fl}(V)\) and \(M_i= F_i/ F_{i-1}\) on \(X\), \(1\leq i\leq n\). For each partition \(\lambda= (\lambda_1\geq \cdots \geq \lambda_n\geq 0)\) let \(L^\lambda= L_1^{\otimes \lambda_1} \otimes \cdots \otimes L_n^{\otimes \lambda_n}\), and for each permutation \(w\) in the symmetric group \(S_n\) let \(\Omega_w\) be the corresponding Schubert variety in \(\text{Fl}(V)\).
The main result of this paper is the following formula for the class of the restriction of \(L^\lambda\) to \(\Omega_w\) in the Grothendieck ring of vector bundles on \(\text{Fl}(V)\):
\[
[L^\lambda |_{\Omega_w} ]= \sum[M^T ]\cdot [{\mathcal O}_{\Omega_{v(T, w)}} ],
\]
where the sum is over a certain set of tableaux \(T\) of shape \(\lambda\) with entries in \(\{1, \dots, n\}\). Here \([M^T ]= \bigotimes^n_{i=1} M_i^{\otimes m(i)}\), where \(m(i)\) is the number of times \(i\) occurs in \(T\), and \(v(T, w)\) is a certain permutation associated with \(T\) and \(w\). If \(L= L_1\otimes \cdots \otimes L_p\), the formula can be stated without using the languages of tableaux. If \(X\) is complete, this gives a formula for the Euler characteristic of the restriction of \(L^\lambda\) to \(\Omega_w\) as a sum of Euler characteristics of line bundles on \(X\):
\[
\chi(\Omega_w, L^\lambda |_{\Omega w})= \sum \chi(X,M^T).
\]
The general formula on the bundle of complete flags also implies corresponding formulas for all Schubert varieties on all partial flag bundles. bundle of complete flags; Schubert variety; Euler characteristic Fulton, W. \&Lascoux, A., A Pieri formula in the Grothendieck ring of a flag bundle.Duke Math. J., 76 (1994), 711--729. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves A Pieri formula in the Grothendieck ring of a flag bundle | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We discuss a surprising relationship between the partially ordered set of Newton points associated with an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the unique maximum element in this poset in terms of paths in the quantum Bruhat graph, whose vertices are indexed by elements in the finite Weyl group. Key to establishing this connection is the fact that paths in the quantum Bruhat graph encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. One important geometric application of the present work is an inequality which provides a necessary condition for nonemptiness of certain affine Deligne-Lusztig varieties in the affine flag variety. Grassmannians, Schubert varieties, flag manifolds, \(p\)-adic cohomology, crystalline cohomology, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Linear algebraic groups over local fields and their integers, Symmetric functions and generalizations Maximal Newton points and the quantum Bruhat graph | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected, simply connected, simple, complex Lie group and \(B\subset G\) a Borel subgroup. The author recovered in [Math. Res. Lett. 11, 35--48 (2004; Zbl 1062.14069)], in a purely combinatorial fashion, \textit{B.~Kim}'s presentation [Ann. Math. (2) 149, 129--148 (1999; Zbl 1054.14533)] of the quantum cohomology ring of the flag variety \(G/B\).
The main goal of this paper is to construct a combinatorial quantum product on \(H^*(G/B)\otimes\mathbb R[\{q_i\}_{i=1,\dots,b_2(G/B)}]\) which satisfies the usual properties of the quantum product (e.g. commutativity, associativity, Frobenius property). Next, Mare applies his result in [loc. cit.] to describe this ring in terms of relations and generators, and finds explicit quantum representatives for the Schubert classes. Finally, he proves that the combinatorial and the usual quantum product on \(H^*(G/B)\otimes\mathbb R[\{q_i\}]\) agree. flag manifolds; quantum Chevalley formula; quantum Giambelli formula Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds The combinatorial quantum cohomology ring of \(G/B\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors prove that \(n\)-point, genus zero descendent Gromov-Witten invariants and quantum \(K\)-invariants of a smooth projective variety \(X\) can be reconstructed from the corresponding 1-point invariants.
Descendent Gromov-Witten invariants are intersections of cotangent line classes \(\psi_i\) with pullbacks of cohomology classes \(\gamma_i\) of X, calculated on the Kontsevich moduli stack \(\overline{\mathcal{M}}_{0,n}(X, \beta).\) Quantum \(K\)-invariants, as described by \textit{A. Givental} [Mich. Math. J. 48, Spec. Vol., 295--304 (2000; Zbl 1081.14523)] and the first author [Duke Math. J. 121, No. 3, 389--424 (2004; Zbl 1051.14064)], are the \(K\)-invariants of products of cotangent line bundles and pullbacks of \(K\)-classes on \(X\), also calculated on \(\overline{\mathcal{M}}_{0,n}(X, \beta).\) The result is that \(n\)-point invariants can be reconstructed from 1-point invariants provided that the cohomology (resp. \(K\)-theory) classes involved belong to a subring \(R\) which is generated by elements of \(\text{Pic}(X)\), that \(R\) is nondegenerate for the cohomological (resp. \(K\)-theoretic) Poincaré pairing, and that the \(n\)-point invariants involving \((n-1)\) classes in \(R\) and one class in \(R^{\perp}\) vanish. The result in Gromov-Witten theory was independently proved by \textit{A. Bertram} and \textit{H. P. Kley} [Topology 44, No. 1, 1--24 (2005; Zbl 1083.14064)] and extends the original reconstruction result of \textit{M. Kontsevich} and \textit{Yu. I. Manin} for invariants without cotangent classes [Commun. Math. Phys. 164, No. 3, 525--562 (1994; Zbl 0853.14020)].
The main ingredients in the proof are a new pair of linear equivalences in the Picard group of \(\overline{\mathcal{M}}_{0,n}(\mathbb{P}^r, \beta)\) between the pullbacks \(\text{ev}^{*}_i(L)\) and \(\text{ev}^{*}_j(L)\) of a line bundle \(L \rightarrow \mathbb{P}^r\) via two different markings and the cotangent line bundles \(\psi_i\) and \(\psi_j\). The authors show that these elements of \(\overline{\mathcal{M}}_{0,n}(\mathbb{P}^r, \beta)\) are related via boundary divisors parametrizing maps with reducible domains and specified splitting data. These relations are of interest in their own right.
As applications, the authors reprove the famous recursion for the number of degree \(d\) rational curves in \(\mathbb{P}^2\) passing through \((3d-1)\) points and compute some quantum \(K\)-invariants of \(\mathbb{P}^1\). The results of this paper should provide a useful tool for the further study of \(n\)-point, genus zero quantum invariants and the geometry of moduli spaces. Lee, Y.-P.; Pandharipande, R., A reconstruction theorem in quantum cohomology and quantum \(K\)-theory, Amer. J. Math., 126, 6, 1367-1379, (2004) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) A reconstruction theorem in quantum cohomology and 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 The authors aim to introduce the tool of asymptotic expansion as a way of relating generating functions of Gromov- Witten invariants of birational spaces. They provide a genus zero correspondence between the equivariant Gromov-Witten theory of Deligne-Mumford stack \([\mathbb{C}^{N}/G]\) and its blowup at the origin. Gromov-Witten invariant; Deligne-Mumford stack; quantum cohomology Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Rational and birational maps Quantum cohomology of toric blowups and Landau-Ginzburg correspondences | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce the notion of integrality of Grothendieck categories as a simultaneous generalization of the primeness of noncommutative noetherian rings and the integrality of locally noetherian schemes. Two different spaces associated to a Grothendieck category yield respective definitions of integrality, and we prove the equivalence of these definitions using a Grothendieck-categorical version of Gabriel's correspondence, which originally related indecomposable injective modules and prime two-sided ideals for noetherian rings. The generalization of prime two-sided ideals is also used to classify locally closed localizing subcategories. As an application of the main results, we develop a theory of singular objects in a Grothendieck category and deduce Goldie's theorem on the existence of the quotient ring as its consequence. Grothendieck category; atom spectrum; molecule spectrum; Gabriel spectrum; weakly closed subcategory Abelian categories, Grothendieck categories, Noncommutative algebraic geometry, Module categories in associative algebras, Prime and semiprime associative rings Integrality of Noetherian Grothendieck categories | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The purpose of this paper is to construct the quantum curve which is a Schrödinger-like equation
\[
P(x,\hbar) \Psi(x,\hbar)=0
\]
for the Gromov-Witten invariants of the complex projective line \(\mathbb{P}^1\) . In article gives the first rigorous example of a direct connection between Gromov-Witten theory and quantum curves. This construction requires the fermionic Fock space representation of the Gromov-Witten invariants, and a subtle combinatorial analysis based on representation theory of symmetric groups. Gromov-Witten invariants; Schrödinger-like equation; quantum curve P. Dunin-Barkowski, M. Mulase, P. Norbury, A. Popolitov and S. Shadrin, \textit{Quantum spectral curve for the Gromov-Witten theory of the complex projective line}, arXiv:1312.5336 [INSPIRE]. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Special algebraic curves and curves of low genus Quantum spectral curve for the Gromov-Witten theory of the complex projective line | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 power structure over a ring is a method to give sense to expressions of the form \((1+a_1t+a_2t^2+\cdots )^m\), where \(a_i\), \(i=1, 2,\ldots \), and \(m\) are elements of the ring. The (natural) power structure over the Grothendieck ring of complex quasi-projective varieties appeared to be useful for a number of applications. We discuss new examples of \(\lambda \)-and power structures over some Grothendieck rings. The main example is for the Grothendieck ring of maps of complex quasi-projective varieties. We describe two natural \(\lambda \)-structures on it which lead to the same power structure. We show that this power structure is effective. In the terms of this power structure we write some equations containing classes of Hilbert-Chow morphisms. We describe some generalizations of this construction for maps of varieties with some additional structures. lambda-structure; power structure; complex quasi-projective varieties; maps; Grothendieck ring Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grothendieck groups (category-theoretic aspects), Varieties and morphisms, Finite groups of transformations in algebraic topology (including Smith theory) Power structure over the Grothendieck ring of maps | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Continuing their studies of polynomials in non-commuting variables, the authors determine in this paper the structure of a symmetric non-commutative polynomial whose Hessian has negative signature one and a non-commutative second fundamental form. Motivations from both geometry and engineering are presented. noncommutative polynomials Dym, H.; Greene, J. M.; Helton, J. W.; Mccullough, S. A.: Classification of all noncommutative polynomials whose Hessian has negative signature one and a noncommutative second fundamental form, J. anal. Math. 108, 19-59 (2009) Several-variable operator theory (spectral, Fredholm, etc.), Real polynomials: analytic properties, etc., Multilinear and polynomial operators, Eigenvalues, singular values, and eigenvectors, Real algebraic and real-analytic geometry, Rings arising from noncommutative algebraic geometry Classification of all noncommutative polynomials whose Hessian has negative signature one and a noncommutative second fundamental form | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author calculates the Alexander polynomial of the dual of the Klein quartic, of the dual of the Fermat quartic and of the dual of one further smooth quartic curve, called \(G_4\). The curve \(G_4\) has the same number of hyperflexes as the Fermat quartic. Therefore the dual curve \(G_4\) and the dual of the Fermat quartic have the same configuration of singularities. However, it turns out that the dual of the Fermat quartic and the dual of \(G_4\) have different Alexander polynomials. Hence they form a Zariski pair. The calculation of the Alexander polynomials is a straightforward application of Libgober's method to calculate Alexander polynomials. Zariski pair; Alexander polynomial; torus curve Coverings of curves, fundamental group, Singularities of curves, local rings, Special algebraic curves and curves of low genus, Global theory of complex singularities; cohomological properties Alexander polynomials of certain dual of smooth quartics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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{E. A. Rødland} [Compos. Math. 122, 135-149 (2000; Zbl 0974.14026)] constructed a Calabi-Yau threefold \(M^3\) as the rank \(4\) locus of a general skew-symmetric \(7\times 7\) matrix with coefficients in \(\mathbb{P}^6\). \(M^3\) is a non-complete intersection Calabi-Yau threefold with \(h^{1,1}(M^3)=1\). A mirror family \(W_q\) is constructed as the orbifold \(M_q^3/{\mathbb{Z}}_7\), where \(M_q^3\) is a one-parameter family of invariants of a natural \(\mathbb{Z}_7\)-action on the space of all skew-symmetric \(7\times 7\)-matrices. In this paper, it is shown that \((M^3, W_q)\) is a (topological) mirror pair, that is, \(h^{2,1}(W_q)=1\). Further, at a point of maximal unipotent monodromy, the mirror symmetry prediction of Rødland posed in the above article is proved in the affirmative. The main result is formulated as follows.
Theorem: At a point of maximal unipotent monodromy, the Picard-Fuchs operator for the periods (with \(D=qd/dq\)):
(a) \((1-289q-57q^2+q^3)(1-3q)^2D^4+4q(3q-1)(143+57q-87q^2+3q^3)D^3+2q(-212-473q+725q^2-435q^3+27q^4)D^2 +2q(-69-481q+159q^2-171q^3+18q^4)D+q(-17-202q-8q^2-54q^3+9q^4)\)
is equivalent to the operator
(b) \(D^2{1\over K}D^2,\quad\text{where}\quad K(q)=14+\sum_{d\geq 1} n_d d^3{q^d\over{1-q^d}}\). Here \(n_d\), the instanton number of degree \(d\) rational curves on \(M^3\), is defined using Gromov-Witten invariants by
\[
\langle p,p,p\rangle^{M^3}_d=\sum_{k\mid d} k^3 n_k.
\]
More precisely, if \(I_0,I_1,I_2,I_3\) is a basis of solutions to (a) with holomorphic solution \(I_0=1+\sum_{d\geq 1} a_d q^d\) and logarithmic solution \(I_1=\ln(q) I_0+\sum_{d\geq 1} b_d q^d\). Then \({I_0\over{I_0}},{I_1\over{I_0}},{I_2\over{I_0}},{I_3\over{I_0}}\) is a basis of solutions for (b) after change of coordinates \(q= \exp(I_1/I_0)\).
Proof is along the line of \textit{A. Givental} on complete intersections on toric manifolds [in: ``Topological field theory, primitive forms and related topics'' (Kyoto 1996), 141-175 (1998; Zbl 0936.14031) and Int. Math. Res. Not. 13, 613-663 (1996; Zbl 0881.55006)], and \textit{B. Kim} on quantum hyperplane principle [J. Korean Math. Soc. 37, 455-461 (2000; Zbl 0986.14035)]. Specifically it is built on the following observations:
(i) The degeneracy locus \(M^3\) is identified with the vanishing locus of a section of a vector bundle on a Grassmannian manifold. This vector bundle decomposes into a direct sum of vector bundles \(E\oplus H\), where \(H\) is again a direct sum of line bundles.
(ii) The quantum hyperplane principle of Kim extends to relate the \(E\)-restricted quantum cohomology with the \(E\oplus H\)-restricted one.
(iii) The \(E\)-restricted quantum cohomology can be effectively computed using localization techniques and WDVV-relations. Pfaffian; Calabi-Yau varieties; mirror symmetry; quantum cohomology; Calabi-Yau threefold; non-complete intersection; mirror family; maximal unipotent monodromy; Picard-Fuchs operator; instanton number; complete intersections; toric manifolds E.N. Tjøtta, \textit{Quantum cohomology of a Pfaffian Calabi-Yau variety: verifying mirror symmetry predictions}, math/9906119. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects) Quantum cohomology of a Pfaffian Calabi-Yau variety: Verifying mirror symmetry predictions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 second volume contains chapters 3 and 4 [cf. Astérisque 314. (Paris): Société Mathématique de France. (2007; Zbl 1146.14001)]. In chapter 3 the construction of nearby motive functors \({\Psi}_{f}\) is given. These functors are analogous to the nearby cycle functors in étale cohomology. In section 3.1 the author introduces a notion of a system of specialization and derives some formal properties of it. Section 3.2 is devoted to the construction of new systems of specializations out of the given diagram of schemes and a system of specialization. The construction of the nearby functors is a particular case of this procedure. In section 3.3 the most important theorems concerning systems of specializations and nearby functors are established. In particular the effect of the functor \({\Psi}_{f}\) for the case when \(f\) has semistable reduction is computed. In section 3.4 and 3.5 two systems of specialization \(\mathcal\Gamma\) and \({\Psi}\) are studied. It is shown that \({\Psi}_{f}\) preserve constructible motives and commute with external products and duality. In section 3.6 the monodromy operator is defined and it is proven that this operator is nilpotent.
Chapter 4 is devoted to the construction in full details of the homotopy category of \(S\)-schemes. It relies on the works of \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)], [\textit{J. F. Jardine}, J. Pure Appl. Algebra 47, 35--87 (1987; Zbl 0624.18007), Doc. Math., J. DMV 5, 445--553 (2000; Zbl 0969.19004)], \textit{M. Hoovey} [J. Pure Appl. Algebra 165, No. 1, 63--127 (2001; Zbl 1008.55006)] and others. stable homotopy; nearby cycles functors; monodromy operator; constructibilty Ayoub, Joseph, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, II, Astérisque, 315, (2007), vi+364 pp. (2008) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Homotopy theory and fundamental groups in algebraic geometry, Motivic cohomology; motivic homotopy theory The Grothendieck six operations and the vanishing cycles formalism in the motivic world. 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 It is well known that a graph can be represented by a square matrix by considering its adjacency matrix. One of the goals of this paper is to give an algebraic description of such a correspondence for directed graphs.
A directed graph can be viewed as an ordered pair \((\alpha,\beta)\) of mappings from the set of directed edges to the set of vertices in such a way that a directed edge \(e\) is the one with the initial vertex \(\alpha(e)\) and the terminal vertex \(\beta(e)\). Thus, with an appropriate morphism, we can consider the category of directed graphs whose objects are ordered pairs of mappings of finite sets. More precisely, we consider the category of directed graphs with a fixed set \(X\) of vertices whose objects are viewed as the set of ordered pairs \((\alpha,\beta)\) of mappings \(\alpha,\beta: Y\to X\) from various finite sets \(Y\) to the given set \(X\). We extend the set of isomorphism classes of the objects in this category to a set which has a ring structure and prove that the resulting ring is isomorphic to the ring of \(m\times m\) integral matrices, where \(m\) is the number of elements in \(X\) (Theorem 1). We also consider a subring of this ring corresponding to regular digraphs and show that it is isomorphic to the ring of generalized magic squares (Theorem 2).
Given a finite set \(X\), the category of single mappings \(\phi: Z\to X\) can be regarded as a Grothendieck topology on the category of finite sets. In Section 7, we discuss the action of the ring associated to directed graphs above on the objects of this category. adjacency matrix; category of directed graphs; ring structure Directed graphs (digraphs), tournaments, Orthogonal arrays, Latin squares, Room squares, Étale and other Grothendieck topologies and (co)homologies Directed graphs, magic squares, and Grothendieck topologies | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work [\textit{I. Ciocan-Fontanine}, Int. Math. Res. Not. 1995, No. 6, 263-277 (1995; Zbl 0847.14011)]. We also give a geometric proof of the quantum Monk formula. quantum cohomology; flag varieties; hyperquot schemes; degeneracy loci; quantum Monk formula Ionuţ Ciocan-Fontanine, The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2695 -- 2729. Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry, (Co)homology theory in algebraic geometry, Model quantum field theories The quantum cohomology ring of flag varieties | 0 |
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