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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 For each infinite series of the classical Lie groups of type \(B\), \(C\) or \(D\), we construct a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a finite Weyl group, these polynomials can be expressed in terms of the factorial analogues of Schur's \(Q\)- or \(P\)-functions defined earlier by Ivanov. double Schubert polynomials; equivariant cohomology Ikeda, T.; Mihalcea, L.; Naruse, H., \textit{double Schubert polynomials for the classical groups}, Adv. Math., 226, 840-886, (2011) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Double Schubert polynomials for the classical groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We extend the work of Fomin and Greene on noncommutative Schur functions by defining noncommutative analogs of Schubert polynomials. If the variables satisfy certain relations (essentially the same as those needed in the theory of noncommutative Schur functions), we prove a Pieri-type formula and a Cauchy identity for our noncommutative polynomials. Our results imply the conjecture of Fomin and Kirillov concerning the expansion of an arbitrary Grothendieck polynomial on the basis of Schubert polynomials; we also present a combinatorial interpretation for the coefficients of the expansion. We conclude with some open problems related to it. Schur functions; Schubert polynomials; conjecture of Fomin and Kirillov; Grothendieck polynomial Lenart, C., Noncommutative Schubert calculus and Grothendieck polynomials.Adv. Math., 143 (1999), 159--183. Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Noncommutative Schubert calculus and Grothendieck 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 study the torus equivariant Schubert classes of the Grassmannian of non-maximal isotropic subspaces in a symplectic vector space. We prove a formula that expresses each of those classes as a \textit{sum} of multi Schur-Pfaffians, whose entries are equivariantly modified special Schubert classes. Our result gives a proof to Wilson's conjectural formula, which generalizes the Giambelli formula for the ordinary cohomology proved by Buch-Kresch-Tamvakis, given in terms of Young's raising operators. Furthermore we show that the formula extends to a certain family of Schubert classes of the symplectic partial isotropic flag varieties. Schubert classes; symplectic Grassmannians; torus equivariant cohomology; Giambelli type formula; Wilson's conjecture; double Schubert polynomials Ikeda, T.; Matsumura, T., \textit{Pfaffian sum formula for the symplectic Grassmannians}, Math. Z., 280, 269-306, (2015) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Pfaffian sum formula for the symplectic 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 prove a formula for double Schubert and Grothendieck polynomials, specialized to two re-arrangements of the same set of variables. Our formula generalizes the usual formulas for Schubert and Grothendieck polynomials in terms of RC-graphs, and it gives immediate proofs of many other important properties of these polynomials. double Schubert polynomials Buch, Anders S.; Rimányi, Richárd, Specializations of Grothendieck polynomials, C. R. Math. Acad. Sci. Paris, 339, 1, 1-4, (2004) Classical problems, Schubert calculus, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Specializations of Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [\textit{S. Fomin} and the author, in: Brylinski, Jean-Luc (ed.) et al., Advances in geometry. Boston, MA: Birkhäuser. Prog. Math. 172, 147--182 (1999; Zbl 0940.05070); \textit{A. Postnikov}, in: Brylinski, Jean-Luc (ed.) et al., Advances in geometry. Boston, MA: Birkhäuser. Prog. Math. 172, 371--383 (1999; Zbl 0944.14019)]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [\textit{E. Mukhin, V. Tarasov} and \textit{A. Varchenko}, ``Bethe Subalgebras of the Group Algebra of the Symmetric Group'', Preprint, \url{arXiv:1004.4248}].
Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang-Baxter algebra in a connection with the values of the \(\beta\)-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan-Robbins polytope. Dunkl and Gaudin operators at critical level; Catalan numbers, Schroder numbers; Schubert polynomials; Grothendieck polynomials A. N. Kirillov, \textit{On Some Combinatorial and Algebraic Properties of Dunkl Elements}, RIMS preprint, 2012. Polynomial rings and ideals; rings of integer-valued polynomials, Grassmannians, Schubert varieties, flag manifolds, Quadratic and Koszul algebras, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) On some algebraic and combinatorial properties of Dunkl elements | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [\textit{S. Fomin} and \textit{A. N. Kirillov}, Prog. Math. 172, 147--182 (1999; Zbl 0940.05070); \textit{A. Postnikov}, ibid. 172, 371--383 (1999; Zbl 0944.14019); \textit{A. N. Kirillov} and \textit{T. Maeno}, ``A note on quantum \(K\)-theory of flag varieties'', Preprint]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group [\textit{E. Mukhin} et al., ``Bethe subalgebras of the group algebra of the symmetric group'', Preprint, \url{arXiv:1004.4248}].{
}Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang-Baxter algebra in a connection with the values of the {\(\beta\)}-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan-Robbins polytope. Dunkl operators at critical level; Gaudin operators at critical level; Catalan numbers; Schröder numbers; Schubert polynomials; Grothendieck polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds On some algebraic and combinatorial properties of Dunkl elements | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 a formula for degeneracy loci of a map of flagged vector bundles. Let \(h:E\to F\) be a map of vector bundles on a variety \(X\) and consider flags \(E_ 1\subset E_ 2\subset\cdots\subset E_ s=E\) (resp., \(F=F_ t\twoheadrightarrow F_{t- 1}\twoheadrightarrow\cdots\twoheadrightarrow F_ 1)\) of subbundles (resp., quotient bundles) of \(E\) (resp., \(F)\). One can consider the degeneracy loci: \(\Omega_ r(h):=\{x\in X|\text{rk}(E_ p(x)\to F_ q(x))\leq r(p,q)\), for all \(p,q\}\), where \(r\) is a collection of rank numbers satisfying certain conditions, which guarantee that, for generic \(h\), \(\Omega_ r(h)\) is irreducible, reduced, Cohen-Macaulay. The author gives a formula for the class \([\Omega_ r(h)]\) of this locus in the Chow ring of \(X\), as a polynomial in the Chern classes of the vector bundles. When expressed in terms of Chern roots, these polynomials are the ``double Schubert polynomials'' introduced and studied by Lascoux and Schützenberger.
Special cases of this formula recover the Kempf-Laksov determinantal formula, the Giambelli-Thom-Porteous formula, as well as a formula of \textit{P. Pragacz} [Ann. Sci. Éc. Norm Supér. IV. Ser. 21, No. 3, 413- 454 (1988; Zbl 0687.14043)]. When specialized to the flag manifold of flags in an \(n\)-dimensional vector space, the formula implies that of \textit{I. N. Bernstein}, \textit{M. Gel'fand} and \textit{S. T. Gel'fand} [Russ. Math. Surveys, 28, No. 3, 1-26 (1973; Zbl 0289.57024)] and \textit{M. Demazure} [Ann. sci. Ec. Norm. Super., IV. Ser. 7, 53-88 (1974; Zbl 0312.14009)]. Doing the general case makes the proof easier. The simplicity of the proof arises from the realization of the operators considered in the papers cited above as correspondences (a fact noticed by several people, and, as the author asserts, communicated to him by R. MacPherson). double Schubert polynomials; degeneracy loci; map of flagged vector bundles; Chow ring; Chern roots; determinantal formula W. Fulton, ``Flags, Schubert polynomials, degeneracy loci, and determinantal formulas'', Duke Math. J. 65 (1992), 381--420. Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Linkage, complete intersections and determinantal ideals, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Flags, Schubert polynomials, degeneracy loci, and determinantal formulas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 fundamental and beautiful article the author introduces universal Schubert polynomials that specialize to all previously known Schubert polynomials: those of Lascoux and Schützenberger, the quantum Schubert polynomials of Fomin, Gelfand, and Postnikov, and the quantum Schubert polynomials for partial flag varieties of Ciocan-Fontanine. Also double versions of these polynomials are given, that generalize the previously known double Schubert polynomials of Lascoux, MacDonald, Kirillov and Maeno, and those of Ciocan-Fontanine and Fulton.
The universal Schubert polynomials describe degeneracy loci of maps of vector bundles, in a more general setting than that of the author's beautiful earlier article [\textit{W. Fulton}, Duke Math. J. 65, 381-420 (1992; Zbl 0788.14044)].
The setting is a sequence of maps of locally free \(\mathcal O_X\)-modules
\[
F_1\to F_2\to \cdots \to F_n \to E_n \to \cdots \to E_2\to E_1
\]
on a scheme \(X\). In contrast to the mentioned article (loc. cit.) the maps \(F_i \to F_{i+1}\) do not have to be injective and the maps \(E_{i+1} \to E_i\) do not have to be surjective. For each \(w\) in the symmetric group \(S_{n+1}\), there is a degeneracy locus
\[
\Omega_w =\{x\in X\mid \text{rank}(F_q(x) \to E_p(x)) \leq r_w(p,q) \text{ for all } 1\leq p, q\leq n\},
\]
where \(r_w(p,q)\) is the number of \(i\leq p\) such that \(w(i)\leq q\). Such degeneracy loci are described by the double form \({\mathfrak S}_w(c,d)\) of universal Schubert polynomials evaluated at the Chern classes of all the bundles involved.
The classical approaches of \textit{Demazure}, or \textit{Bernstein, Gel'fand}, and \textit{Gel'fand} do not work in this case. Instead a locus in a flag bundle is found that maps to a given degeneracy locus \(\Omega_w\), such that one has injections and surjections of the bundles involved, and such that the results of the article mentioned above can be applied. Then the formula is pushed forward to get a formula for \(\Omega_w\). universal Schubert polynomials; quantum Schubert polynomials; partial flag varieties; double Schubert polynomials; degeneracy loci; Chern classes Fulton W. (1999). Universal Schubert polynomials. Duke Math. J. 96(3): 575--594 Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Determinantal varieties, Characteristic classes and numbers in differential topology Universal Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in the parameters which we call the Littlewood-Richardson polynomials. We give a combinatorial rule for their calculation by modifying an earlier result of B. Sagan and the author [\textit{A.I. Molev} and \textit{B.E. Sagan},''A Littlewood-Richardson rule for factorial Schur functions,'' Trans. Am. Math. Soc. 351, No.\,11, 4429--4443 (1999; Zbl 0972.05053)]. The new rule provides a formula for these polynomials which is positive in the sense of \textit{W. Graham} [''Positivity in equivariant Schubert calculus,'' Duke Math. J. 109, No.\,3, 599--614 (2001; Zbl 1069.14055)]. We apply this formula for the calculation of the product of equivariant Schubert classes on Grassmannians which implies a stability property of the structure coefficients. The first manifestly positive formula for such an expansion was given by \textit{A. Knutson} and \textit{T. Tao} [''Puzzles and (equivariant) cohomology of Grassmannians,'' Duke Math. J. 119, No.\,2, 221--260 (2003).] by using combinatorics of puzzles while the stability property was not apparent from that formula. We also use the Littlewood-Richardson polynomials to describe the multiplication rule in the algebra of the Casimir elements for the general linear Lie algebra in the basis of the quantum immanants constructed by \textit{A. Okounkov} and \textit{G. Olshanski} [''Shifted Schur functions,'' St. Petersbg. Math. J. 9, No.\,2, 239--300 (1998); translation from Algebra Anal. 9, No.\,2, 73--146 (1997; Zbl 0894.05053).]. Littlewood-Richardson rule; double symmetric functions; equivariant Schubert classes; Grassmannians; quantum immanants; Schur functions; Littlewood-Richardson polynomials; combinatorics of puzzles; Casimir elements; general linear Lie algebra Alexander I. Molev, ``Littlewood-Richardson polynomials'', J. Algebra321 (2009) no. 11, p. 3450-3468 Symmetric functions and generalizations, Classical problems, Schubert calculus Littlewood-Richardson 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 These notes are based upon a series of lectures that both authors had given in a summer school at Thurnau, Germany, held from June 19 to June 23, 1995. The lectures were designed to provide an introduction to the theory of Schubert varieties, at its contemporary state of knowledge, and to the related theory of degeneracy loci of vector bundle morphisms in algebraic geometry. The text under review follows closely the lectures delivered at Thurnau, the notes of which had been circulating, since then, among the community of algebraic geometers, but it has been enhanced, in its present published from, by ten additional appendices and a few up-dating remarks or footnotes. As the authors emphasize in the preface to the book, this text is neither intended to be a textbook, nor a research monograph, nor a survey on the subject. Instead, they have tried to describe what they, in their capacity of being two of the most active and competent researches in this area of algebraic geometry, regard as essential features of the whole complex of topics, each from his own point of view. The outcome is a great, huge panorama of a fascinating subject in both classical and contemporary algebraic geometry. The present text consists of nine chapters, ten appendices to them, and an utmost rich bibliography.
Chapter I starts with the classical origin of the whole subject, that is, with the description of loci of matrices of various ranks. This is followed by discussing classical and modern solutions of these old problems, including the combinatorial framework of Schur functions and Schubert polynomials. Chapter II turns to the modern generalization of the classical background, namely to morphisms of vector bundles over algebraic varieties, their degeneracy loci, and the cohomological invariants of these degeneracy loci. The fundamental case of Grassmannians and flag manifolds, together with the Schubert subvarieties associated with them, is the central topic of this chapter. Chapter III is devoted to the crucial combinatorial tools: the various kinds of symmetric functions such as Schur \(S\)-polynomials, Schur \(Q\)-polynomials, supersymmetric polynomials, and others, together with their fundamental properties and identities. Chapter IV discusses symmetric polynomials supported on degeneracy loci of vector bundle maps. The powerful general technique of Gysin maps is also explained in this chapter, and that for the important special case of Grassmannians and flag manifolds. In addition, chapters III and IV touch upon the problem of determining those polynomials that are universally supported on degeneracy loci with an explicit description of their defining ideals. Chapter V gives an application of the technique described in chapter IV to the problem of computing topological Euler characteristics of degeneracy loci and Brill-Noether loci in Jacobians of curves. The geometry of flag manifolds and determinantal formulas for Schubert varieties in the case of general homogeneous spaces associated with various classical groups are treated in chapters VI-VII. Following the correspondence method described in chapter III, degeneracy loci for generalized vector bundles (over homogeneous spaces) are investigated, too. Chapter VIII provides a particularly important application of the general theory developed in chapter VII, namely the computation of cohomology classes of some Brill-Noether loci in Prym varieties.
Although several further applications and open problems are pointed out in the course of chapters I-VIII, the concluding chapter IX is exclusively devoted to the discussion of a huge variety of other applications, related questions, and more open problems.
The following ten appendices A-J serve the purpose of making the text as self-contained as possible, on the one hand, and of indicating some closely related work that has been done since 1995, on the other hand.
Appendix A provides some background material from general intersection theory and the representation theory of degeneracy loci by symmetric polynomials. Appendix B gives a recent improvement of Fulton's theorem on the characterization of vexillary permutations in terms of degeneracy loci. Appendix C points to the relation between degeneracy loci, Demazure's resolution scheme for singularities, and the so-called Bott-Samelson schemes, just so for the sake of completeness. Appendix D compiles the definition and basic properties of Pfaffians, while appendix \(E\) sketches the relevant background material from the group-theoretic approach to Schubert varieties. Appendix F explains a useful Gysin formula for Grassmannian bundles, and appendix G discusses a general criterion for computing the classes of relative diagonals. A special construction for vector bundles, which is well-known and due to D. Mumford, is explained in appendix H (and used in chapter VIII). Appendix I provides a little bit of the relevant representation theory of groups and the combinatorics of Young tableaux, though this is not needed anywhere in the text. Finally, appendix J points to the very recent developments in quantum cohomology, in particular to the significance of the so-called ``quantum double Schubert polynomials'' introduced by \textit{I. Ciocan-Fontanine} and \textit{W. Fulton} (cf.: ``Quantum double Schubert polynomials'', Inst. Mittag-Leffler Report No. 6 (1996-97). Throughout the entire, highly enlightening and inspiring text, the authors have focused on careful explanations of the treated material, with lots of included examples and hints to the original papers. Proofs are mostly just indicated, but always come with precise references to the original papers. The omittance of technical details is to the benefit of the non-expert reader, because this makes the beauty of the entire panorama drawn here more transparent and enjoyable. It should be mentioned that another beautiful, recent introduction to the topic of Schubert varieties and symmetric polynomials is given by the lecture notes ``Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence'' by \textit{L. Manivel} [Cours Spécialisés, No. 3, Paris (1998; Zbl 0911.14023)]. Schubert varieties; quantum double Schubert polynomials; Schur functions; Schubert polynomials; morphisms of vector bundles; degeneracy loci; Grassmannians; flag manifolds; symmetric functions Fulton, W., Pragacz, P.: Schubert Varieties and Degeneracy Loci. Lecture Notes in Mathematics, vol. 1689. Springer, Berlin (1998) Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Determinantal varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], (Equivariant) Chow groups and rings; motives Schubert varieties and degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 is a remarkable formula for the principal specialization of a type A Schubert polynomial as a weighted sum over reduced words. Taking appropriate limits transforms this to an identity for the backstable Schubert polynomials recently introduced by \textit{T. Lam} et al. [``Back stable Schubert calculus'', Preprint, \url{arXiv:1806.11233}]. This note identifies some analogues of the latter formula for principal specializations of Schubert polynomials in classical types B, C, and D. We also describe some more general identities for Grothendieck polynomials. As a related application, we derive a simple proof of a pipe dream formula for involution Grothendieck polynomials. Schubert polynomials; Grothendieck polynomials; Coxeter systems; reduced words Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Principal specializations of Schubert polynomials in classical types | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Matrix Schubert varieties are the closures of the orbits of \(B \times B\) acting on all \(n \times n\) matrices, where \(B\) is the group of invertible lower triangular matrices. Extending work of \textit{W. Fulton} [Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)], \textit{A. Knutson} and \textit{E. Miller} [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] identified a Gröbner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes, and derived a related primary decomposition in terms of reduced pipe dreams. These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many other applications. We define skew-symmetric matrix Schubert varieties to be the nonempty intersections of matrix Schubert varieties with the subspace of skew-symmetric matrices. In analogy with Knutson and Miller's work, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gröbner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams. We show that these initial ideals are likewise the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials. Our methods differ from \textit{A. Knutson} and \textit{E. Miller}'s [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] and can be used to give new proofs of some of their results, as we explain at the end of this article. Schubert varieties; Gröbner bases; Grothendieck polynomials; simplicial complexes Combinatorial aspects of algebraic geometry, Combinatorial aspects of simplicial complexes, Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Permutations, words, matrices Gröbner geometry for skew-symmetric matrix Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The factorial flagged Grothendieck polynomials are defined by flagged set-valued tableaux of \textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)]. We show that they can be expressed by a Jacobi-Trudi type determinant formula, generalizing the work of \textit{T. Hudson} and the first author [Eur. J. Comb. 70, 190--201 (2018; Zbl 1408.14030)]. As an application, we obtain alternative proofs of the tableau and the determinant formulas of vexillary double Grothendieck polynomials, which were originally obtained by Knutson et al. [loc. cit.] and Hudson and the first author [loc. cit.] respectively. Furthermore, we show that each factorial flagged Grothendieck polynomial can be obtained by applying \(K\)-theoretic divided difference operators to a product of linear polynomials. factorial Grothendieck polynomials; flagged partitions; flagged set-valued tableaux; vexillary permutations; Jacobi-Trudi formula; double Grothendieck polynomials Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry Factorial flagged Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors go back to the original motivation of Yang for his introduction of the Yang-Baxter equation. He had been led to study certain elements of the group algebra of the symmetric group, and to expand them on the basis of permutations. They obtain some new results in the study of these objects by replacing elementary transpositions with generators of a Hecke algebra. They define a new basis of the Hecke algebra and a bilinear form on it. In some cases they give formulas for the coefficients of this basis when expanded in the usual basis. This involves Schubert and Grothendieck polynomials which were originally defined as canonical bases of the cohomology and Grothendieck rings of flag manifolds. Yang-Baxter equations; Hecke algebra; flag varieties; Schubert polynomials; Grothendieck polynomials Lascoux, A.; Leclerc, B.; Thibon, J. -Y: Flag varieties and the Yang--Baxter equation. Lett. math. Phys. 40, No. 1, 75-90 (1997) Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Flag varieties and the Yang-Baxter equation | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials are explicit representatives for Schubert classes in the cohomology ring of a flag variety. Those of type \(A_n\) were introduced by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [Polynômes de Schubert, C. R. Acad. Sci., Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)]. The \(K\)-theory of flag varieties, which is the next level of complexity after singular cohomology in encoding the topological structure of these varieties, leads to a generalization of the Schubert calculus. The analogs of Schubert polynomials, as representatives for the \(K\)-theory classes determined by the structure sheaves of Schubert varieties, are the Grothendieck polynomials which where introduced by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [Symmetry and flag manifolds, Lect. Notes Math. 996, 118-144 (1983; Zbl 0542.14031)]. In the paper under review Grothendieck polynomials indexed by Grassmannian permutations are studied. Transition matrices between these polynomials and Schur polynomials are given. Moreover, a Pieri formula for the Grothendieck polynomials indexed by Grassmannian permutations is given. Grothendieck polynomials; Schur polynomials; Pieri formula; Schubert polynomials; Schubert classes; Grassmannian permutations Lenart, C., Combinatorial aspects of the \(K\)-theory of Grassmannians, Ann. Comb., 4, 1, 67-82, (2000) Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Combinatorial aspects of the \(K\)-theory 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 give a combinatorial interpretation of a Pieri formula for double Grothendieck polynomials in terms of an interval of the Bruhat order. Another description had been given by Lenart and Postnikov in terms of chain enumerations. We use Lascoux's interpretation of a product of Grothendieck polynomials as a product of two kinds of generators of the 0-Hecke algebra, or sorting operators. In this way, we obtain a direct proof of the result of Lenart and Postnikov and then prove that the set of permutations occurring in the result is actually an interval of the Bruhat order. double Grothendieck polynomials; key polynomials; 0-Hecke algebra; sorting operators; Bruhat order; Pieri formula Pons, V.: Interval structure of the Pieri formula for Grothendieck polynomials, Internat. J. Algebra comput. 23, 123-146 (2013) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Interval structure of the Pieri formula for Grothendieck 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 Subword complexes were defined by \textit{A. Knutson} and \textit{E. Miller} [Adv. Math. 184, No. 1, 161--176 (2004; Zbl 1069.20026)] to describe Gröbner degenerations of matrix Schubert varieties. Subword complexes of a certain type are called pipe dream complexes. The facets of such a complex are indexed by pipe dreams, or, equivalently, by monomials in the corresponding Schubert polynomial. In [Adv. Math. 306, 89--122 (2017; Zbl 1356.14039)] \textit{S. Assaf} and \textit{D. Searles} defined a basis of slide polynomials, generalizing Stanley symmetric functions, and described a combinatorial rule for expanding Schubert polynomials in this basis. We describe a decomposition of subword complexes into strata called slide complexes. The slide complexes appearing in such a way are shown to be homeomorphic to balls or spheres. For pipe dream complexes, such strata correspond to slide polynomials. flag varieties; Schubert polynomials; Grothendieck polynomials; simplicial complexes Classical problems, Schubert calculus, Reflection and Coxeter groups (group-theoretic aspects), Simplicial sets and complexes in algebraic topology, Combinatorial aspects of simplicial complexes, \(K\)-theory in geometry Slide polynomials and subword complexes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author first introduces \(A\) type double Schubert polynomials, then discusses flagged Schur polynomials, and type \(A\) duality. Finally, he considers the geometry of the single and double Schubert polynomials, focusing on the symplectic case. double Schubert polynomials; flagged Schur polynomials, reverse double Schubert polynomials Classical problems, Schubert calculus, Symmetric functions and generalizations, Combinatorial aspects of representation theory Schubert polynomials and degeneracy locus formulas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 algebraic aspects of the equivariant quantum cohomology algebra of the flag manifolds. We introduce and study the quantum double Schubert polynomials \(\widetilde{\mathfrak{S}}_w(x,~y)\), which are the Lascoux-Schützenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials \(\widetilde{\mathfrak{S}}_w(x)\) as the Gram-Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Using the quantum Cauchy identity, we prove that \(\widetilde{\mathfrak{S}}_w(x)=\widetilde{\mathfrak{S}}_w(x,y)|_{y=0}\) and as a corollary we obtain a simple formula for the quantum Schubert polynomials \(\widetilde{\mathfrak{S}}_w(x)=\partial_{ww_0}^{(y)}\widetilde{\mathfrak{S}}_{w_0}(x,~y)|_{y=0}\). We also prove the higher genus analog of Vafa-Intriligator's formula for the flag manifolds and study the quantum residues generating function. We introduce the Ehresmann-Bruhat graph on the symmetric group and formulate the equivariant quantum Pieri rule. quantum double Schubert polynomials; Ehresmann-Bruhat graph; quantum Pieri's rule; cohomology; flag manifold Kirillov, A. N.; Maeno, T.: Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-intriligator formula. Discrete math. 217, No. 1-3, 191-223 (2000) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study six combinatorial Hopf algebras, which fit into a diagram generalizing a diagram describing the relations between four well-known combinatorial Hopf algebras: \(Sym\) of symmetric functions [\textit{R. P. Stanley}, Enumerative combinatorics. Vol. 2. Cambridge: Cambridge Univ. Press (1999; Zbl 0928.05001)], \(NSym\) of noncommutative symmetric functions [\textit{I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh} and \textit{J.-Y. Thibon}, Adv. Math. 112, No. 2, 218--348 (1995; Zbl 0831.05063)], \(QSym\) of quasisymmetric functions [\textit{I. M. Gessel}, Contemp. Math. 34, 289--301 (1984; Zbl 0562.05007)] and \(MR\), the Malvenuto-Reutenauer Hopf algebra of permutations [\textit{C. Malvenuto} and \textit{C. Reutenauer}, J. Algebra 177, No. 3, 967--982 (1995; Zbl 0838.05100)].
The diagram for these four Hopf algebras is
\[
\begin{matrix} &Sym&\twoheadleftarrow &NSym&\twoheadrightarrow &MR\\ &|& &|& &|\\ &Sym&\twoheadrightarrow &QSym&\twoheadleftarrow &MR\end{matrix}
\]
where \(\twoheadleftarrow\) denotes a surjection, \(\twoheadrightarrow\) an injection, and \(|\) duality of Hopf algebras. The six Hopf algebras studied are denoted \(mSym\), \(mQSym\), \(mMR\) and \(MSym\), \(MNSym\) and \(MM\)R. They fit in the diagram
\[
\begin{matrix} &MSym&\twoheadleftarrow &MNSym&\twoheadrightarrow &MMR\\ &|& &|& &|\\ &mSym&\twoheadrightarrow &mQsym&\twoheadleftarrow &mMR\end{matrix}
\]
We cannot give the definitions of these six Hopf algebras here, but will make a few comments. In the three prefixed by \(m\), the classical bases are the lowest degree components of the new basis, and products in these new bases are infinite (except in \(mSym\)), and both the product and coproduct contain classical terms plus terms of higher degree. In the three prefixed by \(M\), the classical bases are the highest degree components of the new bases, and products and coproducts are both finite and contain classical terms plus terms of lower degree.
To explain the title, \textit{A. Lascoux} and \textit{M.-P. Schützenberger} introduced Grothendieck polynomials as representatives of K-theory classes of structure sheaves of Schubert varieties [C. R. Acad. Sci., Paris, Sér. I 295, 629--633 (1982; Zbl 0542.14030)]. \textit{S. Fomin} and \textit{A. N. Kirillov} introduced stable Grothendieck polynomials, which are symmetric power-series obtained as limits of Grothendieck polynomials [Discrete Math. 153, No. 1--3, 123--143 (1996; Zbl 0852.05078)]. \textit{A. S. Buch} gave a combinatorial expression for stable Grothendieck polynomials as generating series of set-valued tableaux [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)]. He showed that the stable Grothendieck polynomials play the role of Schur functions in the \(K\)-theory of Grassmannians, and studied a bialgebra spanned by the stable Grothendieck polynomials. \(mSym\) is the completion of this bialgebra.
Besides studying the Hopf algebra structure of these six Hopf algebras, the main results include a theory of set-valued \(P\)-partitions in the context of \(mQSym\); and three new families of symmetric functions in the context of \(MSym\) and \(mSym\). These symmetric functions are weight-generating functions of weak set-valued tableaux, valued-set tableaux and reverse plane partitions. The last of these are the dual stable Grothendieck polynomials. combinatorial Hopf algebras; noncommutative symmetric functions; quasisymmetric functions; Malvenuto-Reutenauer Hopf algebras; dualities; coproducts; Grothendieck polynomials; K-theory; Schubert varieties; generating series; set-valued tableaux; Schur functions; bialgebras Lam, Thomas; Pylyavskyy, Pavlo, Combinatorial Hopf algebras and \(K\)-homology of grassmanians, Int. Math. Res. Not., 2007, 24, (2007) Hopf algebras and their applications, Connections of Hopf algebras with combinatorics, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Combinatorial Hopf algebras and \(K\)-homology of Grassmanians. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 positive integer and \(OG = OG(n-k, 2n)\) be the Grassmannian that parametrizes isotropic subspaces of dimension \(n-k\) in the vector space \(\mathbb C^{2n}\), equipped with an orthogonal form. The eta polynomials \(H_{\lambda}(c)\) of Buch, Kresch, and the author are Giambelli polynomials that represent the Schubert classes in the cohomology ring of \(OG\).
In this paper using Young raising operators the author defines double eta polynomials \(H_{\lambda}(c|t)\), which represent the equivariant Schubert classes in the equivariant cohomology ring \({H^\ast}_T (OG)\), where \(T\) is a maximal torus of the complex even orthogonal group. Eta polynomials; double eta polynomials; Giambelli polynomials; Young raising operators; Schubert calculus; equivariant cohomology Tamvakis, H., \textit{double eta polynomials and equivariant Giambelli formulas}, J. Lond. Math. Soc. (2), 94, 209-229, (2016) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Double eta polynomials and equivariant Giambelli formulas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0511.00010.]
In this article the authors study the cohomology and Grothendieck ring of flag manifolds. It starts with a quick and elegant introduction to Schubert functions using symmetry operators. The origin of this approach is works of Demazure and Gelfand-Gelfand-Bernstein. The authors give a remarkable formula for some of the symmetrizing operators in terms of Vandermonde determinants that generalizes Jacobi's expression for Schur functions, Weyl's character formula for linear groups and Bott's theorem for line bundles on flag manifolds. Then, extending work of Ehresman, Chevalley and Monk they introduce the Ehresmanoeder and treat formulas for muliplication of Schubert polynomials, like the Pieri formula. The main reason for studying the Schubert polynomials is that they give a basis for the cohomology ring of the flag manifold. The authors go on studying this ring and show how to express an element of the ring in terms of Schubert polynomials and then give an effective way of expressing polynomials in terms of elementary symmetric polynomials. The latter method gives a strengthening of straightening methods of Bott and Rota. - The authors also show how their methods can be used to express the projective degree of a Schubert cycle (Schubert polynomial) in terms of the number of paths in the Ehresmanoeder.
Finally they indicate how their methods can give interesting information about (i) The representation of the linear and symmetric groups. (ii) The enumerative geometry of flag manifolds. (iii) Root systems and Coxeter groups. (vi) Determinants. (v) Ferrers diagrams and Young tableaux. - They show that their methods give analogous results in the cohomology ring and the Grothendieck ring of a flag manifold. Grothendieck polynomials; projective degree of Schubert cycles; flag manifolds; symmetrizing operators; Ehresmanoeder; Schubert polynomials; Pieri formula; enumerative geometry; Root systems; Coxeter groups; Young tableaux; cohomology ring; Grothendieck ring Lascoux, Alain and Schützenberger, Marcel-Paul, Symmetry and flag manifolds, Invariant Theory ({M}ontecatini, 1982), Lecture Notes in Math., 996, 118-144, (1983), Springer, Berlin Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Cohomology theory for linear algebraic groups, Homogeneous spaces and generalizations, Representation theory for linear algebraic groups Symmetry and flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The results of the paper contribute to the combinatorial understanding of degeneracy loci formulas. \par For a sequence of vector bundle maps $E_1\to E_2 \to \ldots \to E_{n-1} \to H_{n-1} \to \ldots \to H_2\to H_1$, and a permutation in $S_n$ one can associate a degeneracy locus, a subvariety in the base, consisting of points over which the induced maps $E_\bullet \to H_\bullet$ are of certain ranks. The fundamental cohomology class represented by this locus, in terms of characteristic classes of the bundles, is called double Schubert polynomial, and their $K$-theory analogue (the class of the structure sheaf) is called double Grothendieck polynomial. Both of these sets of polynomials are defined recursively. \par Combinatorial formulas are often sought for these classes, sometimes for just special classes of permutations. These formulas also depend on the basic classes the formulas are expressed in terms of. \par In the present paper the authors prove determinant formulas for the $K$-theory classes of the structure sheaves of the degeneracy loci, in case the permutation is $2143$-avoiding (vexillary), in terms of Grothendieck polynomials of one-row partitions. The proof depends on the careful study of a resolution of the loci. In the last section the formula is generalized to algebraic cobordism. vexillary permutations; Lascoux-Schützenberger's double Grothendieck polynomials; degeneracy loci Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, \(K\)-theory in geometry Vexillary degeneracy loci classes in \(K\)-theory and algebraic cobordism | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 singular cohomology of the Grassmann variety of \(k\)-planes in \(\mathbb{C}^n\) has a basis \(\{s_\nu\}\) indexed by partitions. The classical Pieri formula is an explicit rule for determining the coefficients in the expansion of the cup product \(s_{1^m}\cup S_\lambda= \sum c^\mu_{1^m,\lambda^S\mu}\), where \(1^m\) is a column of length \(m\) and \(s_{1^m}\) is the \(m\)th Chern class of the tautological bundle. \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci. Paris 294, 447--450 (1982; Zbl 0495.14031)] formulated a generalization of Pieri's formula to the cohomology of the flag variety \(\text{SL}_n (\mathbb{C})/B\) and briefly indicated an algebraic proof [see e.g., \textit{L. Manivel}, ``Symmetric functions Schubert polynomials and degeneracy loci'', Cours Spécialisés 3, Paris (1998; Zbl 0911.14023) for details of this proof]. A geometric proof was given by \textit{F. Sottile} [Ann. Inst. Fourier (Grenoble) 46, 89--110 (1996; Zbl 0837.14041)].
In this paper we state and prove a generalization of this Pieri-type formula for the \(T\)-equivariant cohomology of the flag variety. We use the algebraic description of the \(T\)-equivariant cohomology of the flag variety due to \textit{B. Kostant} and \textit{S. Kumar} [Adv. Math. 62, 187--237 (1986; Zbl 0641.17008)] and \textit{A. Arabia} [Bull. Soc. Math. France 117, 129--165 (1989; Zbl 0706.57024)], and our new formula exposes an equality of certain structure constants in this algebra. Our proof is an induction based on the original ideas of Lascoux and Schützenberger [loc. cit.]. flag manifold; equivariant cohomology; double Schubert polynomials; Pieri formula; equivariant Schubert structure constants Shawn Robinson, A Pieri-type formula for \?*_{\?}(\?\?_{\?}(\Bbb C)/\?), J. Algebra 249 (2002), no. 1, 38 -- 58. Classical problems, Schubert calculus, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds A Pieri-type formula for \(H^*_T(SL_n(\mathbb C)/B)\) | 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 Let \(A\) be a finite alphabet (independent variables), \({\mathcal R}(A)\) the ring of rational functions on \(A\), \({\mathcal G}(A)\) the symmetric group of \(A\), and \(\mathcal E\) the group algebra of \({\mathcal G}(A)\) on the field \({\mathcal R}(A)\). The elements of \(\mathcal E\) are linear operators on \({\mathcal R}(A)\), and \(\mathcal E\) is an \({\mathcal R}(A)\)-module the ``canonical'' basis of which is formed by the permutations of \({\mathcal G}(A)\). Here, several other classical bases of \(\mathcal E\) consisting of symmetrizing operators are considered, namely Newton's divided differences, the convex symmetrizers, and the concave symmetrizers. The authors deal with the problem to determine the matrices of change of such bases explicitly. Their main result is that the elements of these transformation matrices are just specializations of Schubert or Grothendieck polynomials. An important tool is the fact that all but one specialization of the maximal (twofold) Schubert polynomial vanish. (Unfortunately, there are misprints in some formulas.) Moreover, the paper contains instructive remarks on references to algebraic geometry regarding the interpretation of the matrices (cohomology and \(K\)-theory rings of flag manifolds, Schubert varieties). group algebra; permutations; symmetrizing operators; Newton's divided differences; Grothendieck polynomials; Schubert polynomial Lascoux, Alain; Schützenberger, Marcel-Paul, Décompositions dans l'algèbre des différences divisées, Discrete Math., 99, 1-3, 165-179, (1992) Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Orthogonal polynomials [See also 33C45, 33C50, 33D45], Grassmannians, Schubert varieties, flag manifolds, \(K\)-theory and homology; cyclic homology and cohomology, Computations of higher \(K\)-theory of rings Décompositions dans l'algèbre des différences divisées. (Decompositions in the algebra of divided differences) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Authors' abstract: We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold \(G/B\). The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold and discuss related combinatorial objects: flagged Schur polynomials, \(312\)-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees. flag manifold; Schubert varieties; Bruhat order; saturated chains; harmonic polynomials; Grothendieck ring; Demazure modules; Schubert polynomials; flagged Schur polynomials; 312-avoiding permutations; Kempf elements; vexillary permutations; Gelfand-Tsetlin polytope; toric degeneration; parking functions; binary trees Postnikov, A.; Stanley, R., Chains in the Bruhat order, J. Algebr. Comb., 46, 133-174, (2009) Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Chains in the Bruhat order | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. We study two families of polynomials introduced by \textit{B. J. Wyser} and \textit{A. Yong} [Sel. Math., New Ser. 20, No. 4, 1083--1110 (2014; Zbl 1303.05212); Transform. Groups 22, No. 1, 267--290 (2017; Zbl 1400.14130)] representing the \(K\)-theory classes of the closures of these orbits. Our polynomials are analogous to the Grothendieck polynomials representing \(K\)-classes of Schubert varieties, and we show that like Grothendieck polynomials, they are uniquely characterized among all polynomials representing the relevant classes by a certain stability property. We show that the same polynomials represent the equivariant \(K\)-classes of symmetric and skew-symmetric analogues of \textit{A. Knutson} and \textit{E. Miller}'s [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] matrix Schubert varieties. We derive explicit expressions for these polynomials in special cases, including a Pfaffian formula relying on a more general degeneracy locus formula of \textit{D. Anderson} [Adv. Math. 350, 440--485 (2019; Zbl 1426.14014)]. Finally, we show that taking an appropriate limit of our representatives recovers the \(K\)-theoretic Schur \(Q\)-functions of \textit{T. Ikeda} and \textit{H. Naruse} [Adv. Math. 243, 22--66 (2013; Zbl 1278.05240)]. Schubert calculus; spherical orbits; Grothendieck polynomials; \(K\)-theory; degeneracy loci \(K\)-theory in geometry, Combinatorial aspects of groups and algebras, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds \(K\)-theory formulas for orthogonal and symplectic orbit closures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0717.00010.]
The Grothendieck ring of the flag manifold for \(Gl(\mathbb{C}^ n)\) is a quotient of the ring of polynomials generated by the classes of the so- called tautological line bundles \(a_ 1,\ldots,a_ n\). \textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand} [Russ. Math. Surv. 28, No. 3, 1-26 (1973); translation from Usp. Mat. Nauk 28, No. 3 (171), 3-26 (1973; Zbl 0286.57025)] and \textit{M. Demazure} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)] have used Bott-Samelson geometric construction to reduce computations to the case of \(\mathbb{P}^ 1\)-bundles; \textit{W. Fulton} [Duke Math. J. 65, No.\ 3, 381-420 (1991)] gives a simpler method using correspondences. The \(\mathbb{P}^ 1\)-bundles correspond to isobaric divided differences
\[
\pi_ i:f(\ldots,a_ i,a_{i+1},\ldots)\to(a_ if(\ldots,a_ i,a_{i+1},\ldots)-a_{i+1} f(\ldots,a_{i+1},a_ i\ldots))/(a_ i- a_{i+1}).
\]
Products of such operators are in bijection with permutations. Applying them to the class of a point: \((1-1/a_ 1)^{n- 1}\cdots(1-1/a_ n)^ 0\), one obtains by definition the Grothendieck polynomials \(G_ \mu\), \(\mu\in{\mathfrak S}(n)\). These polynomials are representatives of the structure sheaves of Schubert varieties.
The article under review deals with combinatorial properties of Grothendieck polynomials and shows how to express with them the Demazure character formula, the Pieri formula for the intersection of Schubert varieties with a hyperplane section, the Riemann-Roch theorem, and the postulation of line bundles. A similar combinatoric for the cohomology ring of the flag variety can be found in ``Notes on Schubert polynomials'', Publ. LACIM (Montréal 1991) by \textit{I. G. Macdonald}. Grothendieck ring; structure sheaves of Schubert varieties; Grothendieck polynomials; cohomology ring of the flag variety Lascoux, A., Anneau de Grothendieck de la variété de drapeaux, (The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, (1990), Birkhäuser Boston Boston, MA), 1-34 Grassmannians, Schubert varieties, flag manifolds, Grothendieck groups (category-theoretic aspects) Anneau de Grothendieck de la variété de drapeaux. (Grothendieck rings 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 We give an algebraic proof of the determinant formulas for factorial Grothendieck polynomials obtained by \textit{T. Hudson} et al. [Adv. Math. 320, 115--156 (2017; Zbl 1401.19008)] by \textit{T. Hudson} and \textit{T. Matsumura} [``Segre classes and Kempf-Laksov formula in algebraic cobordism'', Preprint, \url{arXiv:1602.05704}]. symmetric polynomials; Grothendieck polynomials; \(K\)-theory; Grassmannians; Schubert varieties Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, \(K\)-theory of schemes, Algebraic cycles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] An algebraic proof of determinant formulas of Grothendieck 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 describe the torus-equivariant cohomology of weighted partial flag orbifolds \({\text{w}}\Sigma\) of type A. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to as ``Schubert Calculus on \({\text{w}}\Sigma \)''. For the weighed Schubert classes in \({\text{w}}\Sigma \), we give the Chevalley's formula. In addition, we define the weighted analogue of double Schubert polynomials and give the corresponding Chevalley-Monk's formula. weighted flag varieties; equivariant cohomology; Schubert classes; double Schubert polynomials Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) The equivariant cohomology of weighted flag orbifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials 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 First the authors give the description of the double Schubert polynomials of types \(B,C\) and \(D\) using Schur \(P\) and Schur \(Q\)-functions in the rings \(R_{\infty}\) and \(R'_{\infty}\), and the actions of divided difference operators on these rings. After then they are giving the basic properties of these super-symmetric polynomials. They are also giving the ways how to calculate double Schubert polynomials.
The double Schubert polynomials represent the torus equivariant Schubert classes in the equivariant cohomology of flag manifolds \(G/B\) where \(G\) is semi-simple Lie group and \(B\) is its Borel subgroup. In this paper the authors explain for the case of the type \(C_n\). But I could not see the product formulas for these classes(or double Schubert polynomials). double Schubert polynomials; excited Young diagrams; Schur P-functions; Schur Q-functions; divided difference operators; torus equivarant cohomology T. Ikeda, H. Naruse, Double Schubert polynomials of classical type and Excited Young diagrams, Kôkyûroku Bessatsu B11 (2009). Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Double Schubert polynomials of classical type and excited Young diagrams | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 have developed a patch implementing multivariate polynomial seen as a multi-base algebra. The patch is to be released into the software Sage and can already be found within the Sage-Combinat distribution. One can use our patch to define a polynomial in a set of indexed variables and expand it into a linear basis of the multivariate polynomials. So far, we have the Schubert polynomials, the Key polynomials of types \(A\), \(B\), \(C\), or \(D\), the Grothendieck polynomials and the non-symmetric Macdonald polynomials. One can also use a double set of variables and work with specific double-linear bases like the double Schubert polynomials or double Grothendieck polynomials. Our implementation is based on a definition of the basis using divided difference operators and one can also define new bases using these operators. Sage-Combinat distribution; Schubert polynomials; key polynomials; Grothendieck polynomials; non-symmetric Macdonald polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Multivariate polynomials in Sage | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Matrix Schubert varieties are the orbit closures of \(B \times B\) acting on all \(n \times n\) matrices, where \(B\) is the group of invertible lower triangular matrices. We define skew-symmetric matrix Schubert varieties to be the orbit closures of \(B\) acting on all \(n \times n\) skew-symmetric matrices. In analogy with Knutson and Miller's work on matrix Schubert varieties, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gröbner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams, analogous to the pipe dreams of Bergeron and Billey. We show that these initial ideals are the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials. Schubert varieties; Gröbner bases; Grothendieck polynomials; simplicial complexes Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of simplicial complexes, Exact enumeration problems, generating functions Gröbner geometry for skew-symmetric matrix Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials There is a remarkable formula for the principal specialization of a type A Schubert polynomial as a weighted sum over reduced words. Taking appropriate limits transforms this to an identity for the backstable Schubert polynomials recently introduced by \textit{T. Lam} et al. [Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)]. In this paper, the authors identify some apparently new analogues of Macdonald's identity for the principal specializations of Schubert polynomials in other classical types B, C, and D and also derive some more general identities for Grothendieck polynomials. The methods used are based on the algebraic techniques of \textit{S. Fomin} and \textit{R. P. Stanley} [Adv. Math. 103, No. 2, 196--207 (1994; Zbl 0809.05091)]. Schubert polynomials; Grothendieck polynomials; Coxeter systems; reduced words Symmetric functions and generalizations, Classical problems, Schubert calculus Principal specializations of Schubert polynomials in classical types | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0728.00006.]
This article gives an informative survey of the combinatorial theory of Schubert polynomials developed by A. Lascoux and M.-P. Schützenberger. Its topic may be described by the titles of the sections and subsections: Permutations (Bruhat order, diagrams and codes, vexillary permutations); divided differences; multi-Schur functions (duality); Schubert polynomials; orthogonality; double Schubert polynomials. In many cases proofs have been omitted. Bruhat order; Schur functions; permutations; divided differences; Schubert polynomials Macdonald, I. G., Notes on Schubert polynomials, (1991), Publications du Laboratoire de Combinatoire et D'informatique Mathématique, Dép. de Mathématiques et D'informatique, Universitédu Québec à Montréal, available at Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Orthogonal polynomials [See also 33C45, 33C50, 33D45], Grassmannians, Schubert varieties, flag manifolds 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 It is well known that a Schur function is the `limit' of a sequence of Schur polynomials in an increasing number of variables, and that Schubert polynomials generalize Schur polynomials. We show that the set of Schubert polynomials can be organized into sequences, whose `limits' we call Schubert functions. A graded version of these Schubert functions can be computed effectively by the application of mixed shift/multiplication operators to the sequence of variables \(x=(x_1,x_2,x_3,\dots)\). This generalizes the Baxter operator approach to graded Schur functions of G. P. Thomas, and allows the easy introduction of skew Schubert polynomials and functions. Since the computation of these operator formulas relies basically on the knowledge of the set of reduced words of permutations, it seems natural that in turn the number of reduced words of a permutation can be determined with the help of Schubert functions: we describe new algebraic formulas and a combinatorial procedure, which allow the effective determination of the number of reduced words for an arbitrary permutation in terms of Schubert polynomials. Schur function; Schubert polynomials; Schur polynomials; Schubert functions; Baxter operator; reduced words of permutations Winkel, R.: Schubert functions and the number of reduced words of permutations, Sém. lothar. Combin. 39, 1-28 (1997) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds Schubert functions and the number of reduced words of permutations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0719.00018.]
This paper, as a continuation of [\textit{M. Kashiwara}, The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 407-433 (1990; Zbl 0727.17013)], completes the proof of the Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebras. The proof consists of two parts: (1) the algebraic part --- the correspondence between \({\mathcal D}\)-modules on the flag variety and representations of the Kac-Moody Lie algebra, (2) the topological part --- the description of geometry of Schubert varieties in terms of the Kazhdan-Lusztig polynomials. The algebraic part is already established in the paper cited above and the paper under review is devoted to the topological part. There are two points in the proof except which the proof is similar to the finite dimensional case. The first one is the usage of the theory of mixed Hodge modules and the second one is the interpretation of the inverse Kazhdan-Lusztig polynomials as the coefficients of certain elements in the dual of the Hecke-Iwahori algebra.
Let \({\mathfrak h}\) be the Cartan subalgebra of a symmetrizable Kac-Moody Lie algebra and \(W\) the Weyl group. For \(w\in W\) define the action on \({\mathfrak h}^*\) by \(w\cdot\lambda=w(\lambda+\rho)-\rho\). Let \(P_{z,w}(q)\) be the Kazhdan-Lusztig polynomial and \(Q_{z,w}(q)\) the inverse Kazhdan- Lusztig polynomial. They are related by
\[
\sum_ w(-1)^{\ell(w)- \ell(y)}Q_{y,w}(q)P_{w,z}(q)=\delta_{y,z}.
\]
The main result of the paper is the following. For a dominant integral weight \(\lambda\in{\mathfrak h}^*\), one has
\[
ch L(w\cdot\lambda)=\sum_ z(-1)^{\ell(z)- \ell(w)}Q_{w,z}(1)ch M(z\cdot\lambda),
\]
or equivalently \(ch M(w\cdot\lambda)=\sum_ zP_{w,z}(1)ch L(z\cdot\lambda)\). Kazhdan-Lusztig conjecture; symmetrizable Kac-Moody Lie algebras; \({\mathcal D}\)-modules; flag variety; representations; geometry of Schubert varieties; Kazhdan-Lusztig polynomials; mixed Hodge modules O.J. Ganor, \textit{Supersymmetric interactions of a six-dimensional self-dual tensor and fixed-shape second quantized strings}, \textit{Phys. Rev.}\textbf{D 97} (2018) 041901 [arXiv:1710.06880] [INSPIRE]. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebra. II: Intersection cohomologies of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In [Eur. J. Comb. 25, No. 8, 1327--1344 (2004; Zbl 1062.14065); C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011)], \textit{W. Kraśkiewicz} and \textit{P. Pragacz} introduced representations of the upper-triangular Lie algebra \(\mathfrak{b}\) whose characters are Schubert polynomials. In [Eur. J. Comb. 58, 17--33 (2016; Zbl 1343.05168)], the author studied the properties of Kraśkiewicz-Pragacz modules using the theory of highest weight categories. From the results there, in particular we obtain a certain highest weight category whose standard modules are KP modules. In this paper we show that this highest weight category is self Ringel-dual. This leads to an interesting symmetry relation on Ext groups between KP modules. We also show that the tensor product operation on \(\mathfrak{b}\)-modules is compatible with Ringel duality functor. Schubert polynomials; Kraśkiewicz-Pragacz modules; highest weight categories; ringel duality; B-modules Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Ext and Tor, generalizations, Künneth formula (category-theoretic aspects), Classical problems, Schubert calculus Kraśkiewicz-Pragacz modules and ringel duality | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study certain bijection between plane partitions and \(\mathbb{N}\)-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating functions are similar to classical MacMahon's formulas; one of these statistics is equidistributed with the usual volume. We also show natural connections with the longest increasing subsequences of words. plane partitions; MacMahon's formulas; dual Grothendieck polynomials; volume generating functions Combinatorial aspects of representation theory, Exact enumeration problems, generating functions, Combinatorial aspects of partitions of integers, Partitions of sets, Grassmannians, Schubert varieties, flag manifolds Enumeration of plane partitions by descents | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 hook tableau is a semi-standard Young tableau shaped like an `L', in French notation. A hook-valued tableau is a tableau where each box contains a hook tableau, such that
\par i) if a box \(A\) is to the left of a box \(B\), but in the same row, then \(\max(A) \leqslant \min(B)\), and
\par ii) if a box \(A\) is below a box \(B\), but in the same column, then \(\max(A) < \min(C)\).
Here, \(\max(A)\) refers to the maximal entry of the hook tableau in box \(A\), and \(\min(A)\) is the minimal entry. Hook-valued tableaux generalise set-valued tableau and multiset-valued tableau: these result from the cases where the hooks consist of single columns and single rows, respectively. Just as with other sorts of tableaux, there is a crystal structure on hook-valued tableaux, introduced by \textit{G. Hawkes} and \textit{T. Scrimshaw} [Algebr. Comb. 3, No. 3, 727--755 (2020; Zbl 1441.05236)]. This specialises in the crystal structure of set-valued tableaux and multiset-valued tableaux.
For set-valued tableaux, there exists an uncrowding operator which maps a set-valued tableau to a pair consisting of a semi-standard Young tableau and a flagged increasing tableau. The operator is ``uncrowding'' in the sense that the output semistandard Young tableau has the same underlying multiset of numerical entries as the original set-valued tableau, except now we have one numerical entry per box, instead of a set of numerical entries per box. The flagged increasing tableau which is part of the output records data on how the tableau was uncrowded, thus allowing the original tableau to be reconstructed from the pair. A flagged increasing tableau is a tableau of skew shape which is increasing in both rows and columns such that entries in row \(i\) are at least \(i - 1\). An important property of the uncrowding operator on set-valued tableaux is that it intertwines with crystal operators.
The heart of the paper is the definition of an uncrowding operator for hook-valued tableaux. The output of such an operator is a set-valued tableaux and a column-flagged increasing tableaux. A column-flagged increasing tableau is the transpose of a flagged increasing tableau. This operator also has the desired property of intertwining with crystal operators. One can then uncrowded completely by uncrowding the output set-valued tableau.
There also exists an uncrowding operator on multiset-valued tableaux. The authors prove that their uncrowding operator on hook-valued tableaux generalises this operator. They also provide an inverse to their uncrowding map, giving a ``crowding'' map, which reassembles the original hook-valued tableau from a set-valued tableau and a column-flagged increasing tableau. This crowding map can only be applied to pairs that are compatible with each other in a certain way.
The authors also introduce an alternative uncrowding map on hook-valued tableaux which outputs a multiset-valued tableau and flagged increasing tableau. This uncrowding operator uncrowds the legs of the hooks in the hook-valued tableau, rather than the arms, as it were. It likewise intertwines with crystal operators.
In the final section, the authors apply their results to canonical Grothendieck polynomials. They use the uncrowding map to show that canonical Grothendieck polynomials have a tableau Schur expansion. Canonical Grothendieck polynomials are symmetric polynomials that can be expressed as generating functions of hook-valued tableaux. A symmetric function is said to have a tableau Schur expansion if it is the weighted sum of the Schur functions of a particular set of tableaux. A corollary of this result is an expansion of canonical Grothendieck polynomials in terms of stable symmetric Grothendieck polynomials and dual stable symmetric Grothendieck polynomials. Here, stable symmetric Grothendieck polynomials are generating functions of set-valued tableau and dual stable symmetric Grothendieck polynomials are generating functions of reverse plane partitions. stable (canonical) Grothendieck polynomials; hook-valued tableaux; crystal bases; uncrowding algorithm Combinatorial aspects of representation theory, Symmetric functions and generalizations, Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus, Quantum groups (quantized function algebras) and their representations Uncrowding algorithm for hook-valued tableaux | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 considers the Schubert polynomials \(X_{\pi}\in {\mathbb{Z}}[x_1,x_2,\ldots]\) associated with the permutations \(\pi\) contained in the symmetric groups \(S_n\). There are many possible ways to introduce the Schubert polynomials via divided difference operators, recursive generation without divided differences based on the Monk rule and the Bruhat order on permutations, via nil-Coxeter relations, the formula of Billey-Jockusch-Stanley, via sums of mixed shift and multiplication operators, via balanced labeled tableaux, via configurations of labeled pseudo-lines, via flagged Schur modules associated to a Rothe diagram, etc. The theme of the paper is the combinatorial generation of Schubert polynomials via sets of box diagrams. The main reasons to expect a combinatorial rule in terms of box diagrams are: the coefficients in \(X_{\pi}\) are non-negative integers and should count some discrete objects; in the special case of Grassmannian permutations the Schubert polynomial is equal to a Schur function in a finite number of variables and the well-known combinatorial properties of Schur functions should extend to Schubert polynomials. The main result of the paper is the proof of a very elegant and easily applicable combinatorial rule for the generation of Schubert polynomials conjectured in 1990 in the thesis by Kohnert. A similar type of combinatorial rule was given by Bergeron. As an intermediate step in the proof of the Kohnert conjecture the author also obtains a simplified proof of the Bergeron rule. In particular, he shows that the Bergeron rule may be also simplified to a version which is very similar to the recent combinatorial rule of Magyar proved with algebro-geometric methods. The author obtains a direct combinatorial proof of the Magyar rule as well. This shows that there is an algebro-geometric meaning of the Bergeron rule. On the other hand it makes apparent the possibility to give fully combinatorial proofs of other results concerning, e.g., the fact that the Schubert polynomial is the character of the flagged Schur module associated to a Rothe diagram. box diagrams; diagram rules; tableaux; Schubert polynomials; symmetric functions; Schur functions R. Winkel. ''Diagram rules for the generation of Schubert polynomials''. J. Combin. Theory Ser. A 86 (1999), pp. 14--48.DOI. Symmetric functions and generalizations, Polynomials, factorization in commutative rings, Grassmannians, Schubert varieties, flag manifolds Diagram rules for the generation of Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author studies algebraic structure of the double affine Hecke algebra connected with the Askey-Wilson \(q\)-hypergeometric polynomials. This algebra \(H\) was studied, in particular, by \textit{S. Sahi} [Ann. Math. (2) 150, No. 1, 267-282 (1999; Zbl 0941.33013)] and \textit{J. V. Stokman} [Sel. Math., New Ser. 9, No. 3, 409-494 (2003; Zbl 1048.33015)]. The author finds the spectrum of the center \(Z(H)\) of \(H\); it is an affine cubic surface in \(\mathbb{C}^3\). The algebra \(Z(H)\) has a Poisson structure. Irreducible representations of \(H\) are studied for the case \(q=1\). Another subject touched in the paper is the study of the double affine Hecke algebra as a deformation of a certain simpler algebra. Its Hochschild cohomology, as well as that for \(H\) itself, are investigated. double affine Hecke algebras; Askey-Wilson polynomials; Poisson cohomology; Hochschild cohomology; universal deformations Oblomkov, A., \textit{double affine Hecke algebras of rank 1 and affine cubic surfaces}, Int. Math. Res. Not. IMRN, 2004, 877-912 Hecke algebras and their representations, Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Orthogonal polynomials [See also 33C45, 33C50, 33D45], Families, moduli, classification: algebraic theory, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Double affine Hecke algebras of rank 1 and affine cubic 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 paper, we express the class of the structure sheaves of the closures of Deligne-Lusztig varieties as explicit double Grothendieck polynomials in the first Chern classes of appropriate line bundles on the ambient flag variety. This is achieved by viewing such closures as degeneracy loci of morphisms of vector bundles. Deligne-Lusztig varieties; \(K\)-theory; Grothendieck polynomials; degeneracy loci Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over finite fields, \(K\)-theory of schemes On the \(K\)-theoretic fundamental class of Deligne-Lusztig 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 In the spirit of Alain Lascoux, the authors propose the use of Schubert polynomials for (certain) computations with polynomials in several variables. The idea comes from situations like doing computations with symmetric functions: There, computations are (usually) not done with monomials, but with a basis adapted to the specific problem that we are dealing with, such as Schur functions, for example.
Most of the paper is devoted to survey the background and basic facts about Schubert polynomials. When we regard the complete ring of polynomials in \(x_1,x_2,\dots,x_n\) as a ring over the ring of symmetric polynomials in \(x_1,x_2,\dots,x_n\), then the Schubert polynomials indexed by permutations in \(S_n\) (the symmetric group on \(n\) elements) constitute a linear basis. Similarly, the ring of polynomials in \(x_1,x_2,\dots,x_n\) with coefficients that are polynomials in \(y_1,y_2,\dots,y_n\) has as a linear basis the double Schubert polynomials. In order to use Schubert polynomials efficiently for computations in such rings, one of the first things we need is a rule for multiplying Schubert polynomials. No general formula for multiplying Schubert polynomials has been found yet (in contrast to Schur functions, where we have the Littlewood-Richardson rule). At least, at the very basic level, there is Monk's formula for the multiplication of a Schubert polynomial in \(x_1,x_2,\dots,x_n\) by one of the variables. However, this formula (possibly) involves Schubert polynomials which are indexed by permutations in \(S_{n+1}\) (and not just \(S_n\)). The authors show how to modify the formula so that one obtains, within the ring of polynomials in \(x_1,x_2,\dots,x_n\), regarded as a ring over the symmetric polynomials, expansions consisting of Schubert polynomials indexed by permutations in \(S_n\). A ``Monk's formula'' for double Schubert polynomials is proved as well. Schubert polynomials; symmetric functions; Monk's formula; divided differences Kohnert, A.; Veigneau, S.: Using Schubert basis to compute with multivariate polynomials. Adv. appl. Math. 19, 45-60 (1997) Symmetric functions and generalizations, Polynomials, factorization in commutative rings, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds Using Schubert basis to compute with multivariate polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(W=(W,S)\) be a Coxeter group with \(S\) its distinguished generator set. Denote by \(\leqslant\) the Bruhat ordering on \(W\). \textit{D. Kazhdan} and \textit{G. Lusztig} defined polynomials \(P_{x,y}\in\mathbb{Z}[q]\) for each \(x\leqslant y\) in \(W\) [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. These polynomials play an important role in representation theory. \textit{M. Dyer} conjectured [Compos. Math. 78, No. 2, 185-191 (1991; Zbl 0784.20019)] that \(P_{x,y}\) depends only on the isomorphism type of the poset \([x,y]=\{z\in W\mid x\leqslant z\leqslant y\}\) for the Bruhat ordering.
The main result of the present paper is to give an affirmative answer to the conjecture for a class of groups, which can be stated as follows. Let \(W,W'\) be two Coxeter groups. Let \(w\in W\) and \(w'\in W'\) be such that the posets \([e,w]\) and \([e',w']\) are isomorphic for the Bruhat orderings on \(W,W'\), where \(e,e'\) are the identity elements of \(W,W'\), respectively. Then any poset isomorphism \(\phi\colon[e,w]\to[e',w']\) preserves Kazhdan-Lusztig polynomials, (in the sense that \(P_{\phi(x),\phi(y)}=P_{x,y}\) for any \(x\leqslant y\) in \([e,w]\)) in the case where one of the two groups \(W,W'\) has the property that the Coxeter graph of each of its irreducible constituents is either a tree or affine of type \(\widetilde A_n\). In particular, the result holds for all finite or affine Coxeter groups.
Note that the above result was also obtained by \textit{F. Brenti} [The intersection cohomology of Schubert varieties is a combinatorial invariant, preprint (2002)] in the case where \(W,W'\) are both of type \(A_n\), as a corollary of a purely combinatorial construction of the \(P_{x,y}\). Bruhat orderings; Kazhdan-Lusztig polynomials; Schubert varieties; Coxeter groups F. du Cloux, ''Rigidity of Schubert closures and invariance of Kazhdan-Lusztig polynomials,'' Adv. in Math. 180 (2003), 146--175. Representation theory for linear algebraic groups, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations, Combinatorics of partially ordered sets Rigidity of Schubert closures and invariance of Kazhdan-Lusztig polynomials. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper of exceptional importance, the authors have been able to ``explain'' many combinatorial phenomena that are associated to a Weyl group in different contexts. The key discovery is that of a set of certain polynomials \(P_{x,y}\) in one variable with integral coefficients associated to a pair \((x,y)\) of elements of a Coxeter group \(W\). These polynomials are used extensively to (1) construct certain representations of the Hecke algebra of \(W\) thereby obtaining important information on representations of \(W\), (2) give a formula (conjecturally) for the multiplicities in the Jordan-Hölder series of Verma modules or equivalently for the formal characters of irreducible highest weight modules (extending the celebrated Weyl character formula to ``nondominant'' weights), (3) give a complete description for the inclusion-relations between various primitive ideals in the enveloping algebra of a complex semisimple Lie algebra \(\mathfrak g\); in case \(\mathfrak g\) is of type \(A_n\), this is further tied up with the dimensions of certain representations of the corresponding Weyl group (via the ``Jantzen conjecture'' -- cf. a paper by \textit{A. Joseph} [Lect. Notes Math. 728, 116--135 (1979; Zbl 0422.17004)], (4) give a measure of the failure of local Poincaré duality in the geometry of Schubert cells in flag varieties. (In a later paper [\textit{D. Kazhdan} and \textit{G. Lusztig}, Proc. Symp. Pure Math. 36, 185--203 (1980; Zbl 0461.14015)] the authors give a more precise interpretation of the coefficients of \(P_{x,y}\) in terms of a certain cohomology theory called ``middle intersection cohomology'' associated with the geometry of Schubert cells.)
Considering the various topics involved and their importance, it seems worthwhile to give a detailed review in order to give some idea of the wealth of information contained in this paper.
We first describe the combinatorial setup involved. Let \((W,S)\) be a Coxeter group. Let \(\mathbb Z[q^{1/2},q^{-1/2}]\) be the ring of Laurent polynomials in the indeterminate \(q^{1/2}\) over \(\mathbb Z\). Let \(\mathcal H\) be the free \(\mathbb Z[q^{1/2},q^{-1/2}]\)-module with \(\{T_y\mid y\in W\}\) as a basis; the multiplication in \(\mathcal H\) is given by: for \(s\in S\), \(y\in W\), \(T_s\cdot T_y=T_{sy}\) if \(l(sy)\geq l(y)\) and \(T_s\cdot T_y=(q-1)T_y+q\cdot T_{sy}\) if \(l(sy)\leq l(y)\). (Classically, one considers \(\mathbb Z[q]\)-coefficients only; the algebra \(\mathcal H\) thus obtained, called the Hecke algebra of \(W\), is isomorphic to the space of intertwining operators on the ``\(1_B^G\)''-representation of a finite Chevalley group with \(W\) as the Weyl group.)
It can be seen that \(T_y\) is invertible in \(\mathcal H\) and so \(\mathcal H\) has an involution \({}^-\) under which \(T_y\) goes to \(T_{y^{-1}}^{-1}\) and \(q^{1/2}\) goes to \(q^{-1/2}\). The main discovery of the paper can now be stated as the theorem: For any \(y\in W\), there is a unique element \(C_y\in\mathcal H\) such that (i) \(\overline C_y=C_y\) and (ii) \(C_y=\sum_{x\in W}(-1)^{l(x)+l(y)}\cdot(q^{1/2})^{l(y)}\cdot q^{-l(x)}\overline P_{x,y}\cdot T_x\), where \(P_{x,y}\in\mathbb Z[q]\) with \(P_{y,y}=1\) and \(\deg P_{x,y}\leq(l(y)-l(x)-1)/2\) if \(x\mathop{<}\limits_{\neq}y\) (\(\leq\) is the Bruhat ordering on \(W\)) and \(P_{x,y}=0\) otherwise.
The authors give an inductive formula for \(P_{x,y}\); however, no closed formula is available as yet. It is conjectured that the coefficients of \(P_{x,y}\) are nonnegative; the authors have proved it in the case of Weyl groups and affine Weyl groups by showing them to be dimensions of certain cohomology groups [cf. the authors, op. cit.].
We now describe the various applications of the polynomials \(P_{x,y}\).
(1) Representations of \(\mathcal H\): In order to obtain certain representations of \(H\), the authors introduce the notion of a \(W\)-graph as follows: It is a graph \(\Gamma\) without loops such that to each vertex \(x\in\Gamma\) is associated a subset \(I_x\) of \(S\) (the set of simple reflections in \(W\)) and to each edge \((x,y)\) is associated a nonzero integer \(\mu(x,y)\) which is required to satisfy certain compatibility conditions. (These conditions ensure that one can define a representation of \(\mathcal H\).) Define a preorder \(x\leq_\Gamma y\) on vertices of \(\Gamma\) by: \(x\leq_\Gamma y\) if there exist \(x=x_0,x_1,\dots,x_n=y\in\Gamma\) such that for all \(i\), \((x_i,x_{i+1})\) is an edge of \(\Gamma\) with \(I_{x_i}\not\subset I_{x_{i+1}}\). Let \(\sim\) be the equivalence relation associated with \(\leq_\Gamma\) (i.e. \(x\sim y\) if \(x\leq_\Gamma y\) and \(y\leq_\Gamma x\)). Then each equivalence class considered as a full subgraph of \(\Gamma\) and the assignments ``\(I_x\) and \(\mu(x,y)\)'' coming from \(\Gamma\) is a \(W\)-graph itself and thus one gets many representations of \(\mathcal H\) (e.g., the ``reflection'' representation of \(\mathcal H\) can be obtained in this way). The authors construct a \(W\)-graph from the polynomials \(P_{x,y}\) in the following way: The set \(W\) is the set of vertices and \((x,y)\) is an edge if either \(x\mathop{<}\limits_{\neq}y\) with \(\deg P_{x,y}=(l(y)-l(x)-1)/2\) or \(y<x\) with \(\deg P_{y,x}=(l(x)-l(y)-1)/2\) (one denotes such pairs by \(x\prec y\) or \(y\prec x\) as the case may be). For \(x\in W\), assign \(I_x=L_x=\{s\in S\mid l(sx)\leq l(x)\}\) and for an edge \((x,y)\), assign \(\mu(x,y)=\) leading coefficient of \(P_{x,y}\) (or \(P_{y,x}\) as the case may be). The authors show that this gives a \(W\)-graph and the corresponding representation is in fact the left-regular representation of \(\mathcal H\). The equivalence classes of this \(W\)-graph are called left cells (``left'' because the set \(L_x\) involves multiplication on the left by the elements of \(S\)). One has a similar \(W\)-graph by considering multiplication on the right and a \(W\times W^0\)-graph (\(W^0\) is the opposite group) by considering the left and right multiplications simultaneously. The equivalence classes are called right cells and two-sided cells, respectively. It turns out that the configuration of these cells forms important combinatorial data from which much information can be obtained. In case \(W=S_n\), the representations of \(W\) obtained from left cells of \(W\) by specializing \(q=1\) cover all complex representations equipped with a distinguished basis.
(2) Characters of highest weight representations of a complex semisimple Lie algebra \(\mathfrak g\): Fix a Cartan subalgebra \(\mathfrak h\) and a set of simple roots \(\Pi\) for the root system of \((\mathfrak g,\mathfrak h)\). Let \((W,S)\) be the corresponding Coxeter system. For \(x\in W\), let \(M_x\) be the Verma module with highest weight \(x\rho-\rho\) (\(\rho\) is the half-sum of positive roots) and let \(L_x\) denote the (unique) irreducible quotient of \(M_x\). For \(x,y\in W\), let \(\text{mtp}(x,y)\) be the multiplicity with which \(L_y\) occurs in a Jordan-Hölder series of \(M_x\). It is then known that \(\text{mtp}(x,y)\neq 0\) if and only if \(x\leq y\) (\(\leq\) being the Bruhat ordering). The problem of determining \(\text{mtp}(x,y)\) has been considered by several people (e.g., [\textit{J. Lepowsky} and the reviewer, J. Algebra 49, 512--524 (1977; Zbl 0381.17004); \textit{J. C. Jantzen}, Moduln mit einem höchsten Gewicht. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0426.17001)]). The authors conjecture that \(\text{mtp}(x,y)=P_{x,y}(1)\). (This conjecture has been proved recently by Brylinski and Kashiwara and, independently, by Beilinson and Bernstein.) This has an equivalent formulation in terms of the formal character of \(L_x\). (Recall: If \(x=\text{id}\) then one has the Weyl character formula for a finite-dimensional representation of \(\mathfrak g\).) In his talk at the AMS Santa Cruz Conference on Finite Groups [\textit{G. Lusztig}, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 313--317 (1980; Zbl 0453.20005)] the second author proposed a modular analogue analogue the above conjecture from which the character formula of a rational irreducible representation of a Chevalley group \(G\) over an algebraically closed field of positive characteristic can be obtained.
(3) Theory of primitive ideals in the enveloping algebra \(U(\mathfrak g)\) of a complex semisimple Lie algebra \(\mathfrak g\): A two-sided ideal \(I\) in \(U(\mathfrak g)\) is said to be ``primitive'' if it is the annihilator of an irreducible \(\mathfrak g\)-module. One is then interested in the space \(X\) of primitive ideals (equipped with the Jacobson topology). As every irreducible \(\mathfrak g\)-module has a central character, one gets a map \(X\rightarrow Z(\hat{\mathfrak g})\) (\(=\text{Hom}_{\mathbb C-\text{alg}}(Z(\mathfrak g),{\mathbb C})\) where \(Z(\mathfrak g)\) is the centre of \(U(\mathfrak g)\)). By a well-known theorem of Harish-Chandra \(Z(\hat{\mathfrak g})\simeq\mathfrak h^\ast|_W\). Let \(\Lambda\) be an equivalence class. One is interested in the fibre \(X_\Lambda\) over \(\Lambda\). Let \(\lambda\in\Lambda\) and \(J(\lambda)=\text{Ann}\,L(\lambda)\), where \(L(\lambda)\) is the irreducible \(\mathfrak g\)-module with highest weight \(\lambda\). Then \(J(\lambda)\in X_\lambda\) and in fact a deep result of Duflo asserts that every element of \(X_\Lambda\) is of this form. There are various results known about the structure of fibres, ``similarity'' of two fibres, etc. The most important problem is to determine the inclusion relation between two elements of the same fibre. Known results by Borho, Duflo, Jantzen, Joseph, Vogan, etc. give sufficient conditions for it. However, since the conjecture ``\(\text{mtp}(x,y)=P_{x,y}(1)\)'' is proved to be true, the left cells of \(W\) determine the inclusion completely. To be more precise, let \(\lambda\) be an antidominant integral element. Let \(\Lambda\) be the equivalence class to which it belongs. Then every element of \(X_\Lambda\) is of the form \(J(x\cdot\lambda)\) where \(x\cdot\lambda=x(\lambda+\rho)-\rho\). Then \(J(x\cdot\lambda)\subseteq J(y\cdot\lambda)\) if and only if \(y\leq_Lx\). (The inclusion relations in other fibres can be written down with the help of a ``suitable'' subgroup \(W\) which corresponds to a sub-root-system.) It may be recalled that the primitive spectrum \(X\) is related to nilpotent orbits in \(\mathfrak g\).
(4) Geometry of Schubert cells in flag varieties: Let \(G\) be a complex semisimple algebraic group, \(B\) a Borel subgroup and \(G/B\) be the corresponding flag variety. \(B\) acts on \(G/B\) and one has the Bruhat decomposition \(G/B=\bigcup_{x\in W}BxB\). For \(y\in W\), let \(X(y)=\overline{ByB}=\bigcup_{x\leq y}BxB\) (\(\leq\) is the Bruhat ordering). Let \(e_x\) be the point \(xB\in G/B\). Then one is interested in determining the nature of the singularity at \(e_x\) when considered as a point of \(X(y)\;(x\leq y)\). The authors show that the condition ``\(P_{x',y}\equiv 1\) for all \(x\leq x'\leq y\)'' is related to the singularity of \(e_x\) in \(X(y)\). A closer tie-up is given in a later paper [the authors, loc. cit.].
Considering the central role played by the polynomials \(P_{x,y}\) in above-mentioned contexts, it is desirable to have an explicit knowledge of their coefficients in terms of certain combinatorial data. The relation \(x\prec y\) is another key notion which should be investigated further. Weyl groups; Coxeter groups; representations of Hecke algebras; Jordan-Hölder series of Verma modules; irreducible highest weight modules; Weyl character formula; primitive ideals in enveloping algebras; complex semisimple Lie algebras; local Poincaré duality; geometry of Schubert cells; flag varieties; intersection cohomology; Laurent polynomials; intertwining operators; finite Chevalley groups; affine Weyl groups; cohomology groups; simple reflections; highest weight representations; Cartan subalgebras D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, \textit{Invent.} \textit{Math.}, 53 (1979), no. 2, 165--184.Zbl 0499.20035 MR 560412 Reflection and Coxeter groups (group-theoretic aspects), Representation theory for linear algebraic groups, Hecke algebras and their representations, Universal enveloping (super)algebras, Linear algebraic groups over arbitrary fields, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Group actions on varieties or schemes (quotients) Representations of Coxeter groups and Hecke algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We associate a polynomial to any diagram of unit cells in the first quadrant of the plane using Kohnert's algorithm for moving cells down. In this way, for every weak composition one can choose a cell diagram with corresponding row-counts, with each choice giving rise to a combinatorially-defined basis of polynomials. These \textit{Kohnert bases} provide a simultaneous generalization of Schubert polynomials and Demazure characters for the general linear group. Using the monomial and fundamental slide bases defined earlier by the authors, we show that Kohnert polynomials stabilize to quasisymmetric functions that are nonnegative on the fundamental basis for quasisymmetric functions. For initial applications, we define and study two new Kohnert bases. The elements of one basis are conjecturally Schubert-positive and stabilize to the skew-Schur functions; the elements of the other basis stabilize to a new basis of quasisymmetric functions that contains the Schur functions. Schubert polynomials; Demazure characters; key polynomials; fundamental slide polynomials Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Symmetric functions and generalizations Kohnert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This monograph is based on the author's lectures at the CBMS conference at North Carolina State University in 2001. It covers the application of symmetric functions to algebraic identities related to the Euclidean algorithm. It does not require extensive knowledge of symmetric functions, although some familiarity with the basic properties would be useful.
Many texts on symmetric functions view the functions as polynomials. In constrast, this monograph uses the method of \(\lambda\)-rings, in which symmetric functions are viewed as operators on the ring of polynomials. This approach is particularly natural for these applications, because the construction of the symmetric functions and all necessary properties follow from a few fundamental results.
An arbitrary polynomial can be viewed as a symmetric function in terms of its roots. In this framework, the successive remainders in the Euclidean algorithm can be expressed in terms of symmetric functions; examples include Sturm sequences and continued fractions. Divided difference operators also act on polynomials, and operations involving partial or full symmetrization can be expressed in terms of divided differences; one example is a symmetric function identity which arises in the cohomology of Grassmannians. Another viewpoint is that of orthogonal polynomials; if the ``moments'' are the complete symmetric functions, the resulting orthogonal polynomials are Schur functions indexed by square partitions, and other Schur functions appear in contexts such as Christoffel determinants.
A similar approach can be used to study generalizations of symmetric functions. Schubert polynomials are constructed in two ways: by generalizing Newton's interpolation to multiple variables, and from a non-symmetric Cauchy kernel. The book concludes with a brief discussion of further generalizations to non-commutative Schur functions and Schubert polynomials. Schur functions; \(\lambda\)-rings; Cauchy kernel; Euclidean algorithm; continued fractions; Padé approximation; divided differences; cohomology of Grassmannian; orthogonal polynomials; Schubert polynomials Lascoux, A.: Symmetric functions \& combinatorial operators on polynomials. CBMS reg. Conf. ser. Math. 99 (2003) Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups, Approximation by polynomials, Padé approximation, Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis, Orthogonal polynomials [See also 33C45, 33C50, 33D45] Symmetric functions and combinatorial operators on polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This fine paper delivers a combinatorial formula for the characters of the homogeneous components of the coinvariant algebra---\(\mathbb{Q}[x_1,\dots, x_n]\) factored out by those polynomials invariant under the action of the symmetric group with no constant term.
It begins with a clear and concise exposition of all the ingredients needed including Schubert polynomials, Monk's formula, and Kazhdan-Lusztig cells. From here the derivation of the action of the symmetric group on Schubert polynomials and some related inner product results lead to a swift proof of the aforementioned character formula, which is also shown to be equivalent to the decomposition of homogeneous components of the coinvariant algebra into irreducible representations.
Finally we are given a taster for two subsequent works ``Deformation of the coinvariant algebra'' and ``Major index of shuffles and restriction of representations'', the former of which yields a \(q\)-analogue of the character formula in this paper, the latter an algebraic interpretation of the set of all permutations of length \(k\) in the symmetric group. coinvariant algebra; Kazhdan-Lusztig cells; Schubert polynomials; character formula; irreducible representations; symmetric group Y. Roichman, Schubert polynomials, Kazhdan--Lusztig basis and characters, (with an, appendix: On characters of Weyl groups, co-authored with, R. M. Adin, and, A. Postnikov, ), Discrete Math, to appear. Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Symmetric groups Schubert polynomials, Kazhdan-Lusztig basis and characters | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial \(K\)-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We show that the Newton polytopes of these Grothendieck polynomials and their homogeneous components have SNP. Moreover, the Newton polytope of each homogeneous component is a permutahedron. This addresses recent conjectures of C. Monical-N. Tokcan-A. Yong and of A. Fink-K. Mészáros-A. St. Dizier in this special case. symmetric Grothendieck polynomials; Newton polytopes Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Symmetric functions and generalizations, Toric varieties, Newton polyhedra, Okounkov bodies Newton polytopes and symmetric Grothendieck 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 Schubert polynomials are the building blocks of several cohomological degeneracy locus formulas. Their role in \(K\)-theory is played by so-called Grothendieck polynomials. In particular, ``stable'' versions of Schubert and Grothendieck polynomials turn up naturally, for example in degeneracy locus problems associated with certain quiver representations.
The paper under review studies the expansion of stable Grothendieck polynomials in the basis of stable Grothendieck polynomials for partitions. This generalizes the result of Fomin-Green in the cohomological setting, where the analogues of stable Grothendick polynomials for partitions are the Schur polynomials. The existence of such a finite, integer linear combination expansion was proved by Buch, and the sign of the coefficients were determined by Lascoux. In this paper the authors give a new, non-recursive combinatorial rule for the coefficients. Namely, they prove that the coefficients, up to explicit sign, count the number of increasing tableau of a given shape, with an associated word having explicit combinatorial properties stemming from the combinatorics of the 0-Hecke monoid. The main ingredient of the proof is a generalized, so-called Hecke insertion algorithm.
The main application showed in the paper is a \(K\)-theoretic analogue of the factor sequence formula of Buch-Fulton for the cohomological quiver polynomials (of equioriented type A). Grothendieck polynomials; \(K\)-theory; 0-Hecke monoid; insertion algorithm; factor sequence formula Buch, A.; Kresch, A.; Shimozono, M.; Tamvakis, H.; Yong, A., Stable Grothendieck polynomials and \textit{K}-theoretic factor sequences, Math. Ann., 340, 359-382, (2008) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], \(K\)-theory of schemes, Symmetric functions and generalizations Stable Grothendieck polynomials and \(K\)-theoretic factor sequences | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 prove a tableau formula for the double Grothendieck polynomials associated to 321-avoiding permutations. The proof is based on the compatibility of the formula with the \(K\)-theoretic divided difference operators. Our formula specializes to the one obtained by \textit{W. Y. C. Chen} et al. [Eur. J. Comb. 25, No. 8, 1181--1196 (2004; Zbl 1055.05149)] for the (double) skew Schur polynomials. symmetric polynomials; Grothendieck polynomials; \(K\)-theory; set-valued tableaux; 321-avoiding permutations Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices A tableau formula of double Grothendieck polynomials for 321-avoiding permutations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is the second edition of a work that first appeared in 1987. The first printing was reviewed extensively by \textit{V. L. Popov}, see Zbl 0654.20039. Let me just say that Jantzen's book has been an indispensable reference for anyone involved with representations of algebraic groups, in particular of reductive groups in positive characteristic.
The second edition differs substantially from the first. Again it is a book one must have. About one third is new. The old part has been sprinkled with new comments, but has been left intact as much as is feasible. New chapters at the end, identified with capital letters, take care of several developments in the intervening years.
The following topics are now also covered in the familiar clear manner. Chapter A: Truncated categories and Schur algebras. Here the truncations are those of Donkin, associated to a finite `saturated' set of dominant weights. Chapter B: Results over the integers. This globalizes several cohomological results. Chapter C: Lusztig's Conjecture and some consequences. Chapter D: Radical filtrations and Kazhdan-Lusztig polynomials. These two chapters belong together and discuss various aspects of this central conjecture concerning characters of certain simple modules. Chapter E: Tilting modules. Chapter F: Frobenius splitting. Chapter G: Frobenius splitting and good filtrations. This chapter gives the proof by Mathieu of the theorem about tensor products of modules with good filtration. Logically chapters F and G should be read early. Chapter H: Representations of quantum groups. This chapter is in a different style. It is just a survey.
It is a pity that the author has still not adopted certain widely used terminology. For instance, the usual form of `sum formula' is `Jantzen sum formula'. representation theory; reductive algebraic groups; simple modules; highest weights; character formulas; Weyl's character formula; affine group schemes; injective modules; injective resolutions; derived functors; Hochschild cohomology groups; hyperalgebra; split reductive group schemes; Steinberg's tensor product theorem; irreducible representations; Kempf's vanishing theorem; Borel-Bott-Weil theorem; characters; linkage principle; dominant weights; filtrations; Steinberg modules; cohomology rings; rings of regular functions; Schubert schemes; line bundles; Schur algebras; quantum groups; Kazhdan-Lusztig polynomials J. C. Jantzen, \textit{Representations of Algebraic Groups. Second edition}, Amer. Math. Soc., Providence (2003). Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Group schemes, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Affine algebraic groups, hyperalgebra constructions, Linear algebraic groups over arbitrary fields Representations of 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 We study the problem of expanding the product of two Stanley symmetric functions \(F_w \cdot F_u\) into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial \(F_w = \lim_{n\to \infty} \mathfrak{S}_{1^n\times w}\), and study the behavior of the expansion of \(\mathfrak{S}_{1^n\times w} \cdot \mathfrak{S}_{1^n\times u}\) into Schubert polynomials, as \(n\) increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability. Stanley symmetric functions; Schubert polynomials; Littlewood-Richardson rule Symmetric functions and generalizations, Classical problems, Schubert calculus A canonical expansion of the product of two Stanley symmetric 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 Recall that a resolution \(p:\widetilde X \to X\) of an irreducible complex projective variety \(X\) is said to be small if, for all \(i>0\), \(\text{codim}_X \{x\in X : \dim p^{-1} (x)\geq i\}>2i\). Let \(G=SO(2n)\) or \(Sp(2n)\) and let \(P_n\) be the maximal parabolic subgroup obtained by deleting the right end root (following the Bourbaki convention). In an earlier work, ``Small resolutions of Schubert varieties in symplectic and orthogonal Grassmannians'' in Publ. Res. Inst. Math. Sci. 30, No. 3, 443-458 (1994), the authors exhibited `Bott-Samleson type' small resolutions \(p:\widetilde X(\lambda) \to X(\lambda)\) of certain Schubert varieties \(X(\lambda)\) in \(G/P_n\).
The authors, in the paper under review, give an inductive formula (following similar works of Zelevinskii in the case of Grassmannian Schubert varieties) to determine the Poincaré polynomials of the fibres of \(p\) over \(T\)-fixed points (where \(T\) is the maximal torus of \(G\) acting on \(G/P_n\) via the left multiplication). They use this result (and some results of Zelevinskii) to show that the Kazhdan-Lusztig (KL for short) polynomials \(P_{\theta, \lambda_0}\) (for the Weyl group associated to \(G)\), for certain pairs \(\theta\leq \lambda_0\), are equal to the KL-polynomials \(P_{\theta', \lambda_0'}'\) (where \(P'\) denotes the KL-polynomials for \(SL(M)\), for certain integer \(M\) and certain \(\theta'\leq \lambda_0'\) in the Weyl group of \(SL(M))\).
The authors also exhibit small resolutions for certain Schubert varieties in \(E_6/P_6\) and calculate the Poincaré polynomials of the fibres over \(T\)-fixed points for them (where \(P_6\) is again the maximal parabolic subgroup obtained by deleting the right end root). In addition, they explicitly determine the singular locus for most of the Schubert varieties in \(E_6/P_6\). Kazhdan-Lusztig polynomials; Schubert varieties; small resolutions Grassmannians, Schubert varieties, flag manifolds, Global theory and resolution of singularities (algebro-geometric aspects) Small resolutions of Schubert varieties and Kazhdan-Lusztig polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Motivated by a recent conjecture of \textit{R. P. Stanley} [``Some Schubert shenanigans'', Preprint, \url{arXiv:1704.00851}] we offer a lower bound for the sum of the coefficients of a Schubert polynomial in terms of 132-pattern containment. Schubert polynomials; permutation patterns Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Schubert polynomials, 132-patterns, and Stanley's 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 Suppose some complex veector bundles are given over a manifold \(M\), together with some (sufficiently generic) vector bundle maps among them. Over a point in \(M\) then one has a quiver: some vector spaces with linear maps among them. Quivers may degenerate -- e.g. dimensions of intersections of images, kernels may drop, and more general degenerations also happen. A quiver degeneracy locus is the set a points in \(M\) over which the quiver degenerates in a particular way.
It is known that the fundamental class of quiver degeneracy loci can be expressed by a universal polynomial (the quiver polynomial) in the charactersitic classes of the bundles involved, both in cohomology and in \(K\)-theory. The paper under review presents such universal polynomials in \(K\)-theory if the quiver is of Dynkin type.
The formula presented is a non-conventional generating function (named Iterated Residue generating function, pioneered by Bérczi-Szenes, Kazarian, Feher-Rimányi). Advantages of the presented formula for quiver loci include that stabilization properties are explicit, and the expansion in terms of Grothendieck polynomials is well motivated.
The paper ends with interesting comments comparing \(K\)-theory and cohomology Iterated Residue formulas, as well as remarks on Buch's positivity conjecture on the coefficients of the Grothendieck polynomial expansion. quiver polynomials; Grothendieck polynomials; iterated residues; equivariant K theory Allman, Justin, Grothendieck classes of quiver cycles as iterated residues, Michigan Math. J., 63, 4, 865-888, (2014) Grassmannians, Schubert varieties, flag manifolds, Equivariant \(K\)-theory, Symmetric functions and generalizations, Representations of quivers and partially ordered sets Grothendieck classes of quiver cycles as iterated residues | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Castelnuovo-Mumford regularity of a graded module is an important invariant that can be computed, for instance, from its minimal free resolution and can be viewed as a measure of its complexity. In this paper the authors focus on graded modules consisting of the quotient of the polynomial ring by the ideal of different classes of determinantal varieties, and they give explicit combinatorial formulas for the Castelnuovo-Mumford regularity. Specifically, they consider one-sided mixed ladder determinantal ideals and certain Kazhdan-Lusztig ideals, including those coming from open patches of Schubert varieties in Grassmannians. They achieve their results using Schubert calculus techniques. Throughout this paper the algebras they consider are all Cohen-Macaulay. Their formula for regularity of patches of Grassmannian Schubert varieties provides a correction to a conjecture of Kummini, Lakshmibai, Sastry and Seshadri, that the current authors conjectured in an earlier paper. Castelnuovo-Mumford regularity; ladder determinantal ideal; matrix Schubert variety; Grassmannian; Grothendieck polynomial Determinantal varieties, Linkage, complete intersections and determinantal ideals Castelnuovo-Mumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author shows how to associate to any polynomial \(P\), of degree \(d\), with non-negative integer coefficients and constant term equal to 1, a pair of elements \(y_P\) and \(w_P\) in the symmetric group \(S_n\) where \(n=d+P(1)+1\). Then he proves that \(P\) is indeed the Kazhdan-Lusztig polynomial of those two elements, by reducing the problem to the case when \(P-1\) is a monomial and by using intersection cohomology of Schubert varieties. Kazhdan-Lusztig polynomials; Schubert varieties; intersection cohomology P. Polo, Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups, \textit{Repre-} \textit{sent. Theory}, 3 (1999), 90--104.Zbl 0968.14029 MR 1698201 Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Representations of finite symmetric groups, Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We use the modules introduced by \textit{W. Kraskiewicz} and \textit{P. Pragacz} [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011); Eur. J. Comb. 25, No. 8, 1327--1344 (2004; Zbl 1062.14065)] to show some positivity properties of Schubert polynomials. We give a new proof to the classical fact that the product of two Schubert polynomials is Schubert-positive, and also show a new result that the plethystic composition of a Schur function with a Schubert polynomial is Schubert-positive. The present submission is an extended abstract on these results and the full version of this work will be published elsewhere. Schubert polynomials; Schubert functors; Kraśkiewicz-Pragacz modules; Schubert calculus Classical problems, Schubert calculus, Algebraic combinatorics Kraśkiewicz-Pragacz modules and some positivity properties of Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that if \(X\) is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring \(\mathrm{QH}(X)\) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring \(H^*(X)\) that multiplies with nonnegative structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants of \(X\) are uniquely determined by Witten's presentation of \(\mathrm{QH}(X)\) and the fact that they are nonnegative. We conjecture that the same is true for any flag variety \(X=G/P\) of simply laced Lie type. For the variety of complete flags in \(\mathbb{C}^n\), this conjecture is equivalent to Fomin, Gelfand, and Postnikov's conjecture that the quantum Schubert polynomials of type A are uniquely determined by positivity properties. Our proof for Grassmannians answers a question of Fulton. quantum cohomology; Grassmannians; positivity; Gromov-Witten invariant; Schubert basis; quantum Schubert polynomials; flag varieties; symmetric functions; Seidel representation Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Positivity determines 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 To each polynomial \(P\) with integral non-negative coefficients and constant term equal to 1, of degree \(d\), we associate a pair of elements \((y,w)\) in the symmetric group \(S_n\), where \(n=1+d+P(1)\), for which we prove that the Kazhdan-Lusztig polynomial \(P_{y,w}\) equals \(P\). This pair satisfies \(\ell(w)-\ell(y)=2d+P(1)-1\), where \(\ell(w)\) denotes the number of inversions of \(w\). -- For details see the author's paper: \textit{P. Polo}, Represent. Theory 3, No. 4, 90-104 (1999; Zbl 0968.14029). Kazhdan-Lusztig polynomials; Schubert varieties; intersection cohomology Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Representations of finite symmetric groups, Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Construction of arbitrary Kazhdan-Lusztig polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials By a diagram the authors mean a finite collection of cells in \({\mathbb Z}\times {\mathbb Z}\). They consider balanced labellings of diagrams. Special cases of these objects are the standard Young tableaux and the balanced tableaux introduced by \textit{P. Edelman} and \textit{C. Greene} [Adv. Math. 63, 42-99 (1987; Zbl 0616.05005)]. It turns out that for certain diagrams associated with permutations of the symmetric group \(\Sigma_n\), the set of balanced labellings has a remarkable rich structure. Balanced labellings of permutation diagrams yield the symmetric functions introduced by \textit{R. P. Stanley} [Eur. J. Comb. 5, 359-372 (1984; Zbl 0587.20002)] in the same way as the Schur functions can be constructed from column-strict tableaux. Balanced labelled diagrams can be also viewed as encodings of reduced decompositions of permutations. Imposing flag conditions, the authors obtain the Schubert polynomials of Lascoux and Schützenberger. Finally the authors construct an explicit basis for the Schubert module introduced in 1986 by W. Kraskiewicz and P. Pragacz (i.e. the representation of the upper triangular group with formal character equal to the corresponding Schubert polynomial). diagram; tableaux; Subert polynomials; symmetric functions; Schubert module S. Fomin, C. Greene, V. Reiner, and M. Shimozono, ''Balanced labellings and Schubert polynomials,'' European J. Combin. 18 (1997), no. 4, 373--389. the electronic journal of combinatorics 25(3) (2018), #P3.4622 Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Balanced labellings and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a new operator formula for Grothendieck polynomials that generalizes Magyar's Demazure operator formula [\textit{P. Magyar}, Comment. Math. Helv. 73, No. 4, 603--636 (1998; Zbl 0951.14036)] for Schubert polynomials. Our proofs are purely combinatorial, contrasting with the geometric and representation theoretic tools used by Magyar. We apply our formula to prove a necessary divisibility condition for a monomial to appear in a given Grothendieck polynomial. Schubert polynomials; Demazure operator formula Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds An orthodontia formula for Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors prove a theorem analogous to Smith's theorem but for matrices with Laurent polynomials as entries. Then they show that this result is equivalent to Grothendieck's theorem about vector bundles on the projective line. The main theorem is in Section 8.2 and Grothendieck's theorem is in Section 8.3. Smith's theorem; Laurent polynomials; Grothendieck's theorem; vector bundles Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] An elementary proof of Grothendieck's theorem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The goal of this paper is at least two-fold. First we attempt go give a survey of some recent applications of the theory of symmetric polynomials and divided differences to intersection theory. Secondly, taking this opportunity, we complement the story by either presenting some new proofs of older results or providing some new results which arose as by-products of the author's work in this domain during last years. Being in the past a good part of the classical algebraic knowledge (related for instance to the theory of algebraic equations and elimination theory), the theory of symmetric functions is rediscovered and developed nowadays. Here, we discuss only some geometric applications of symmetric polynomials which are related to the present interest of the author. In particular, the theory of polynomials universally supported on degeneracy loci is surveyed in Section 1.
Divided differences appeared already in the interpolation formulas of I. Newton. Their appearance in intersection theory is about twenty years old starting with the papers [Russ. Math. Surv. 28, No. 3, 1-26 (1973; Zbl 0286.57025)] of \textit{I. N. Bernstein}, \textit{I. M. Gelfand} and \textit{S. I. Gelfand} and [Invent. Math. 21, 287-301 (1973; Zbl 0269.22010) and Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)] of \textit{M. Demazure}. A recent work [Duke Math. J. 65, No. 3, 381-420 (1992; Zbl 0788.14044)] of \textit{W. Fulton} has illuminated the importance of divided differences to flag degeneracy loci. This was possible thanks to the algebraic theory of Schubert polynomials developed recently by A. Lascoux and M.-P. Schützenberger.
The geometrical objects we study are: (ample) vector bundles, degeneracy loci of vector bundle homomorphisms, flag varieties, Grassmannians including isotropic Grassmannians, i.e. the parameter spaces for isotropic subspaces of a given vector space endowed with an antisymmetric or symmetric form, Schubert varieties and the parameter spaces of complete quadrics.
The algebro-combinatorial tools we use are: Schur polynomials including supersymmetric and \(Q\)-polynomials, binomial determinants and Pfaffians, divided differences, Schubert polynomials of Lascoux and Schützenberger, reduced decompositions in the Weyl groups and Young-Ferrers' diagrams.
The content of the article is as follows: (1) Polynomials universally supported on degeneracy loci, (2) Some explicit formulas for Chern and Segre classes of tensor bundles with applications to enumerative geometry, (3) Flag degeneracy loci and divided differences, (4) Gysin maps and divided differences, (5) Fundamental classes, diagonals and Gysin maps, (6) Intersection rings of spaces \(G/P\), divided differences and formulas for isotropic degeneracy loci, (7) Numerically positive polynomials for ample vector bundles with applications to Schur polynomials of Schur bundles and a vanishing theorem.
Apart of surveyed results, the paper contains also some new ones. Perhaps the most valuable contribution, contained in Section 5, is provided by a method of computing the fundamental class of a subscheme using the class of the diagonal of the ambient scheme. The class of the diagonal can be determined with the help of Gysin maps (see Section 5). This method has been applied successfully in [the author and \textit{J. Ratajski}, Formulas for Lagrangian and orthogonal degeneracy loci, to appear in Compos. Math.] and seems to be useful also in other settings. Other results that appear to be new are contained in Proposition 1.3(ii), Proposition 2.1 and Corollary 7.2. Moreover, the paper is accompanied by a series of appendices which contain an original material but of more technical nature than the main text of the paper. Some proofs in the Appendices use an operator approach and the operators involved are mostly divided differences. This point of view leads to more natural proofs of many results than the ones known before, and we hope to develop it in our forthcoming book (in collaboration with Laksov, Lascoux, and Thorup). symmetric polynomials; divided differences; intersection theory; symmetric functions; polynomials universally supported on degeneracy loci; flag degeneracy loci; flag varieties; Grassmannians; Schubert varieties; Schur polynomials; \(Q\)-polynomials; determinants; Pfaffians; Weyl groups; Young-Ferrers' diagrams; Segre classes; tensor bundles; Gysin maps; vector bundles; Schur bundles; vanishing theorem P. Pragacz, ''Symmetric polynomials and divided differences in formulas of intersection theory,'' in Parameter Spaces, Warsaw: Polish Acad. Sci., 1996, vol. 36, pp. 125-177. Symmetric functions and generalizations, (Equivariant) Chow groups and rings; motives, Algebraic cycles Symmetric polynomials and divided differences in formulas of intersection 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 author determines a factorization of a double specialization of Schubert polynomials from which he derives a factorization of a specialization of \(q\)-factorial Schur functions. Schubert polynomials; factorial and \(q\)-factorial Schur functions; factorization Prosper, V.: Factorization properties of the q-specialization of Schubert polynomials, Ann. comb. 4, 91-107 (2000) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Factorization properties of the \(q\)-specialization of Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This monograph includes much of the author's previous work on Schur and Schubert polynomials and generalizations, and considerable background material. \textit{G. P. Thomas} [Frames, Young tableaux, and Baxter sequences; Adv. Math. 26, 275-289 (1977; Zbl 0375.05005) and Further results on Baxter sequences and generalized Schur functions; Lect. Notes Math. 579, 155-167 (1977; Zbl 0364.05007)] constructed the Schur functions \(S_\lambda\) combinatorially from the set of standard Young tableaux of shape \(\lambda\), using algebraic ``mixed Baxter-multiplication operators'' on the algebra of polynomials. The author [Sequences of symmetric polynomials and combinatorial properties of tableaux; Adv. Math. 134, No. 1, 46-89 (1998; Zbl 0902.05078)] generalizes this construction, providing an effective construction of Q-Schur, Hall-Littlewood, Jack, and Macdonald polynomials, all of which are generalizations of the Schur functions. The Baxter construction can also be applied to the Schubert polynomials; see \textit{R. Winkel} [A combinatorial bijection between standard Young tableaux and reduced words of Grassmannian permutations; Sémin. Lothar. Comb. 36, B36h (1996; Zbl 0886.05115) and Schubert functions and the number of reduced words of permutations; Sémin. Lothar. Comb. 39, B39a (1997; Zbl 0886.05119)]. Recursive methods for construction of the Schubert polynomials are given, and used to prove their basic properties; see \textit{R. Winkel} [Recursive and combinatorial properties of Schubert polynomials; Sémin. Lothar. Comb. 38, B38c (1996; Zbl 0886.05111)]. The original construction of the Schubert polynomials of type \(A_n\) by \textit{A. Lascoux} and \textit{M. P. Schützenberger} [Polynômes de Schubert, C. R. Acad. Sci. Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)] by divided differences, and also the construction by recursive structures, are generalized to give constructions of the cases \(B_n\), \(C_n\), and \(D_n\); see \textit{R. Winkel} [Schubert polynomials of types A--D; Manuscr. Math. 100, No. 1, 55-79 (1999; Zbl 0936.05088)]. The weak Bruhat order on Coxeter groups gives a partial order which can be partitioned into poset-isomorphic parts. This is used to give a combintorial computation of the Poincaré polynomials of the finite and some affine Coxeter groups, and a non-recursive computation of standard reduced words for signed and unsigned permutations; see \textit{R. Winkel} [A combinatorial derivation of the Poincaré polynomials of the finite irreducible Coxeter groups; Discrete Math., to appear]. Expansions of Schubert polynomials into standard elementary monomials are constructed combinatorially; see \textit{R. Winkel} [On the expansion of Schur and Schubert polynomials into standard elementary monomials; Adv. Math. 136, No. 2, 224-250 (1998; Zbl 0920.05069)]. Baxter operator; Lehmer code; Schubert polynomials; Schur functions; Young tableaux; Macdonald polynomials; weak Brunat order on Coxeter groups; Poincaré polynomials; reduced words Research exposition (monographs, survey articles) pertaining to combinatorics, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] On algebraic and combinatorial properties of Schur and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For part I see \textit{P. Pragacz} and \textit{A. Weber}, Fundam. Math. 195, No.\,1, 85--95 (2007; Zbl 1146.05049).]
We study Thom polynomials associated with Lagrange singularities. We expand them in the basis of \(\widetilde{Q}\)-functions. This basis plays a key role in the Schubert calculus of isotropic Grassmannians. We prove that the \(\widetilde{Q}\)-function expansions of the Thom polynomials of Lagrange singularities always have nonnegative coefficients. This is an analog of a result on the Thom polynomials of mapping singularities and Schur \(S\)-functions, established formerly by the last two authors. Lagrange singularities; Thom polynomials; \(\widetilde{Q}\)-functions; jets; numerical positivity; Schubert calculus; isotropic Grassmanians M. Mikosz, P. Pragacz and A. Weber, Positivity of Thom polynomials II: the Lagrange singularities, Fund. Math. 202 (2009), 65-79. Symmetric functions and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical problems, Schubert calculus, Homology of classifying spaces and characteristic classes in algebraic topology, Singularities of differentiable mappings in differential topology Positivity of Thom polynomials~II: the Lagrange singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Kohnert polynomials are polynomials indexed by unit cell diagrams in the first quadrant defined earlier by the author and Searles that give a common generalization of Schubert polynomials and Demazure characters for the general linear group. Demazure crystals are certain truncations of normal crystals whose characters are Demazure characters. For each diagram satisfying a southwest condition, we construct a Demazure crystal whose character is the Kohnert polynomial for the given diagram, resolving an earlier conjecture of the author and Searles that these polynomials expand nonnegatively into Demazure characters. We give explicit formulas for the expansions with applications including a characterization of those diagrams for which the corresponding Kohnert polynomial is a single Demazure character. Schubert polynomials; Demazure characters Combinatorial aspects of representation theory, Symmetric functions and generalizations, Classical problems, Schubert calculus Demazure crystals for Kohnert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The classical Thom-Porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of Schur polynomials. A proof of this formula was given by Pragacz by expressing this alternating sum as the Euler characteristic of a Schur complex, which gives an explanation for the signs. Fulton later generalized this formula to the situation of flags of vector bundles by using alternating sums of Schubert polynomials. Building on the Schubert functors of Kraśkiewicz and Pragacz, we introduce Schubert complexes and show that Fulton's alternating sum can be realized as the Euler characteristic of this complex, thereby providing a conceptual proof for why an alternating sum appears.
See also the review of the journal version [\textit{S. V. Sam}, J. Algebra 337, No. 1, 103--125 (2011; Zbl 1242.13017)]. Schubert polynomials; Schubert complexes; degeneracy loci; balanced labelings; Thom-Porteous formula Syzygies, resolutions, complexes and commutative rings, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Schubert complexes and degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 a new combinatorial construction for the cohomology ring of the flag manifold. The degree 2 commutation relations satisfied by the divided difference operators corresponding to positive roots define a quadratic associative algebra. In this algebra, the formal analogues of Dunkl operators generate a commutative subring, which is shown to be canonically isomorphic to the cohomology of the flag manifold. This leads to yet another combinatorial version of the corresponding Schubert calculus. The paper contains numerous conjectures and open problems. We also discuss a generalization of the main construction to quantum cohomology. representation of the symmetric group; Pieri rule; Gromov-Witten invariants; Schubert polynomials; cohomology ring of the flag manifold; divided difference operators; quadratic associative algebra; Dunkl operators; Schubert calculus; quantum cohomology Fomin, Sergey; Kirillov, Anatol N., Quadratic algebras, Dunkl elements, and Schubert calculus. Advances in geometry, Progr. Math. 172, 147-182, (1999), Birkhäuser Boston, Boston, MA Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Quadratic algebras, Dunkl elements, and 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 Let \(w_0\) be the element of maximal length in the symmetric group \(S_n\), and let \(\text{Red}(w_0)\) be the set of all reduced words for \(w_0\). We prove the identity
\[
\sum_{(a_1,a_2,\dots)\in\text{Red}(w_0)}(x+ a_1)(x+ a_2)\cdots= {n\choose 2}! \prod_{1\leq i<j\leq n} {2x+ i+ j-1\over i+j-1},\tag{\(*\)}
\]
which generalizes Stanley's formula for the cardinality of \(\text{Red}(w_0)\), and Macdonald's formula \(\sum a_1a_2\cdots= \left(\begin{smallmatrix} n\\ 2\end{smallmatrix}\right)!\). Our approach uses an observation, based on a result by \textit{M. L. Wachs} [J. Comb. Theory, Ser. A 40, 276-289 (1985; Zbl 0579.05001)], that evaluation of certain specializations of Schubert polynomials is essentially equivalent to enumeration of plane partitions whose parts are bounded from above. Thus, enumerative results for reduced words can be obtained from the corresponding statements about plane partitions, and vice versa. In particular, identity \((*)\) follows from Proctor's formula for the number of plane partitions of a staircase shape, with bounded largest part. Similar results are obtained for other permutations and shapes; \(q\)-analogues are also given. Young tableaux; Ferrers shape; reduced words; identity; Stanley's formula; Macdonald's formula; Schubert polynomials; enumeration of plane partitions; permutations; shapes; \(q\)-analogues S. Fomin and A. N. Kirillov, \textit{Reduced words and plane partitions}, J. Algebraic Combin., 6 (1997), pp. 311--319. Combinatorial aspects of partitions of integers, Exact enumeration problems, generating functions, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups, Combinatorial aspects of representation theory Reduced words and plane partitions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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{W. Kraśkiewicz} and \textit{P. Pragacz} [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011)] defined certain modules \(\mathcal{S}_w (w\in S_\infty)\), which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of KP modules always has a KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely \(\mathcal{S}_w \otimes S^d(K^i)\) and \(\mathcal{S}_w \otimes \bigwedge^d(K^i)\), corresponding to Pieri and dual Pieri rules for Schubert polynomials. Schubert polynomials; Kraśkiewicz-Pragacz modules Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Solvable, nilpotent (super)algebras, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Determinantal varieties Kraśkiewicz-Pragacz modules and Pieri and dual Pieri rules for 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 This paper continues part I [Transform. Groups 17, No. 4, 953-987 (2012; Zbl 1318.20005)], where the authors introduced nil-DAHA as a certain limit of type \(A_1\) double affine Hecke algebra (DAHA). In the present paper, the authors define a so-called core subalgebra of nil-DAHA and study its properties and representation theory. This subalgebra can be defined either explicitly by generators and relations, or as an intersection of two other subalgebras in nil-DAHA. It is bigraded (in contrast to nil-DAHA and these subalgebras), and invariant under the natural anti-involution. The main results of the paper compare the induced representations of the core subalgebra with certain limits of the corresponding DAHA representations, studied by the first author and \textit{X. Ma} [in Sel. Math., New Ser. 19, No. 3, 737-817 (2013; Zbl 1293.20005) and ibid. 19, No. 3, 819-864 (2013; Zbl 1293.20006)]. double affine Hecke algebras; nil-DAHAs; nonsymmetric Macdonald polynomials; Whittaker functions; core subalgebras; induced representations I. Cherednik and D. Orr. ''One-dimensional nil-DAHA and Whittaker functions II''. Trans form. Groups 18 (2013), pp. 23--59.DOI. Hecke algebras and their representations, Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Filtered associative rings; filtrational and graded techniques One-dimensional nil-DAHA and Whittaker functions. II. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We present a general theory of Schubert polynomials, which are explicit representatives for Schubert classes in the cohomology ring of a flag variety with certain combinatorial properties. The starting point for this theory is a construction of Schubert classes in the cohomology ring of the flag variety of any semi-simple complex Lie group by Bernstein- Gelfand-Gelfand and Demazure. For the groups \(\text{SL}(n, \mathbb{C})\), Lascoux and Schützenberger made the crucial observation that one particular choice of representative of the top cohomology class yields Schubert polynomials simultaneously for all \(n\). In the present work we replicate the theory of \(\text{SL}(n, \mathbb{C})\) Schubert polynomials for the other infinite families of classical Lie groups and their flag varieties---the orthogonal groups \(\text{SO}(2n, \mathbb{C})\) and \(\text{SO}(2n+ 1,\mathbb{C})\) and the symplectic groups \(\text{Sp}(2n, \mathbb{C})\). We define Schubert polynomials to be elements in an inverse limit, which can be calculated as the unique solution of an infinite system of divided difference equations. The solution is derived using two equivalent formulas; one is an analog of the Billey-Jockusch-Stanley formula, while the other expresses our polynomials in terms of \(\text{SL}(n)\) Schubert polynomials and Schur \(Q\)- or \(P\)-functions. Our second formula involves the `shifted Edelman-Greene correspondences' and analogs of the Stanley symmetric functions. The Schubert polynomials form a \(\mathbb{Z}\)-basis for the ring in which they are defined. The non-negative integer coefficients that appear when they are multiplied give intersection multiplicities for Schubert varieties directly, without the need to reduce the product modulo an ideal. Schur functions; Schubert polynomials; Schubert classes; cohomology ring; flag variety; Lie group; orthogonal groups; symplectic groups; divided difference equations; Billey-Jockusch-Stanley formula; Stanley symmetric functions; Schubert varieties Billey, S.; Haiman, M., \textit{Schubert polynomials for the classical groups}, J. Amer. Math. Soc., 8, 443-482, (1995) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Schubert polynomials for the classical groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and \(k\)-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and \(k-K\)-Schur functions -- Schubert representatives for the \(K\)-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules. tableaux; Grothendieck polynomials; \(k\)-Schur functions; affine Grassmannian J. Morse. ''Combinatorics of the K-theory of affine Grassmannians''. Adv. Math. 229 (2012), pp. 2950--2984.DOI. Symmetric functions and generalizations, Combinatorial aspects of representation theory, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Infinite-dimensional Lie (super)algebras Combinatorics of the \(K\)-theory of affine 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 For a given skew shape, we build a crystal graph on the set of all reverse plane partitions that have this shape. As a consequence, we get a simple extension of the Littlewood-Richardson rule for the expansion of the corresponding dual stable Grothendieck polynomial in terms of Schur polynomials. dual stable Grothendieck polynomials; reverse plane partitions; crystal operators; Littlewood-Richardson rule Galashin, P., A Littlewood-Richardson rule for dual stable Grothendieck polynomials Symmetric functions and generalizations, Partitions of sets, Classical problems, Schubert calculus A Littlewood-Richardson rule for dual stable Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we investigate properties of modules introduced by \textit{W. Kraśkiewicz} and \textit{P. Pragacz} [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011); Eur. J. Comb. 25, No. 8, 1327--1344 (2004; Zbl 1062.14065)] which realize Schubert polynomials as their characters. In particular, we give some characterizations of modules having filtrations by Kraśkiewicz-Pragacz modules. In finding criteria for such filtrations, we calculate generating sets for the annihilator ideals of the lowest vectors in Kraśkiewicz-Pragacz modules and derive a projectivity result concerning Kraśkiewicz-Pragacz modules. Schubert polynomials; Kraśkiewicz-Pragacz modules Watanabe, M.: An approach toward Schubert positivities of polynomials using kraśkiewicz-pragacz modules. European J. Combin. 58, 17-33 (2016) Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds An approach towards Schubert positivities of polynomials using Kraśkiewicz-Pragacz modules | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using a formula of \textit{S. C. Billey}, \textit{W. Jockusch} and \textit{R. P. Stanley} [Some combinatorial properties of Schubert polynomials, J. Algebr. Comb. 2, No. 4, 345-374 (1993; Zbl 0790.05093)], \textit{S. Fomin} and \textit{A. N. Kirillov} [Yang-Baxter equation, symmetric functions, and Schubert polynomials, Proceedings of the conference on power series and algebraic combinatorics, Firenze (1993)] have introduced a new set of diagrams that encode the Schubert polynomials. In this paper, these objects are called rc-graphs. Here, two variants of an algorithm for constructing the set of all rc-graphs for a given permutation are defined and proved. This construction makes many of the identities known for Schubert polynomials more apparent, and yields new ones. In particular, we find a new proof of Monk's rule using an insertion algorithm on rc- graphs. This insertion rule is a generalization of the Schensted insertion for tableaux. We find two conjectures of analogs of Pieri's rule for multiplying Schubert polynomials. The authors also extend the algorithm to generate the double Schubert polynomials. Schubert polynomials; rc-graphs; Monk's rule; Pieri's rule N. Bergeron and S. Billey. ''RC-graphs and Schubert polynomials''. Experiment. Math. 2 (1993), pp. 257--269.DOI. Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Enumerative combinatorics, Combinatorial identities, bijective combinatorics, Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds rc-graphs and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Kohnert's algorithm for the generation of Schubert polynomials is derived from Monk's rule for the multiplication of Schubert polynomials. Schubert polynomials R. Winkel. ''A derivation of Kohnert's algorithm from Monk's rule''. Sém. Lothar. Combin. 48 (2002), Art. B48f.URL. Symmetric functions and generalizations, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds A derivation of Kohnert's algorithm from Monk's rule | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article contains some algebraic tools that can be used to make computations in the cohomology ring of Lagrangian flag manifolds, and Lagrangian degeneracy loci. The main tool is the study of several operators on a certain basis, which is orthonormal under a scalar product. This basis is useful in studying Schubert classes in Lagrangian manifolds.
The article also contains some simple proofs of previously known results, for example of the Giambelli-type formula for maximal Lagrangian Schubert classes. Lagrangian flag manifolds; Lagrangian degeneracy loci; Schubert polynomials; Giambelli-type formula Lascoux, A; Pragacz, P, Operator calculus for \({\widetilde{Q}}\)-polynomials and Schubert polynomials, Adv. Math., 140, 1-43, (1998) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Determinantal varieties, Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Operator calculus for \(\widetilde{Q}\)-polynomials and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In their work on the infinite flag variety, \textit{T. Lam} et al. [Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)] introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished manuscript of \textit{A. Lascoux} [``Chern and Yang through ice'', Preprint]. Lascoux showed how to write double Grothendieck polynomials as a sum over alternating sign matrices. We explain how to view the Lam-Lee-Shimozono formula as a disguised special case of Lascoux's alternating sign matrix formula.
\textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] gave a tableau formula for vexillary Grothendieck polynomials. We recover this formula by showing vexillary marked bumpless pipe dreams and flagged set-valued tableaux are in weight preserving bijection. Finally, we give a bijection between Hecke bumpless pipe dreams and decreasing tableaux. The restriction of this bijection to Edelman-Greene bumpless pipe dreams solves a problem of Lam, Lee, and Shimozono [loc. cit.]. Grothendieck polynomials; bumpless pipe dreams; alternating sign matrices Combinatorial aspects of algebraic geometry, Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Graph polynomials, Combinatorial aspects of representation theory, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Bumpless pipe dreams and alternating sign matrices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Finding a combinatorial rule for the multiplication of Schubert polynomials is a long-standing problem. We give a combinatorial proof of the extended Pieri rule, which says how to multiply a Schubert polynomial by a complete or elementary symmetric polynomial, and describe some observations in the direction of a general rule. Schubert polynomials; Pieri rule; symmetric polynomial Assaf, S., Bergeron, N., Sottile, F.: On the multiplication of Schubert polynomials. In preparation Symmetric functions and generalizations, Representations of finite symmetric groups, Grassmannians, Schubert varieties, flag manifolds On the multiplication of Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that the product of a monomial and a Demazure atom is a positive sum of Demazure atoms combinatorially. This result proves one particular case in a conjecture which provides an approach to a combinatorial proof of Schubert positivity property. generalized Demazure atoms; key polynomials; Schubert positivity; nonsymmetric Macdonald polynomials; skyline filings Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus Decomposition of the product of a monomial and a Demazure atom | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0665.00004.]
The authors consider a noncommutative version of Schubert polynomials. This is done by simultaneous lifting of the classical Schubert polynomials into two noncommutative algebras related by the ``Dualité de Cauchy''. As a consequence they obtain the functoriality of Schubert polynomials. noncommutative version of Schubert polynomials; functoriality A. Lascoux and M.-P. Schützenberger, Fonctorialité des polynômes de Schubert, Invariant theory (Denton, TX, 1986) Contemp. Math., vol. 88, Amer. Math. Soc., Providence, RI, 1989, pp. 585 -- 598 (French, with English summary). Representations of finite symmetric groups, Grassmannians, Schubert varieties, flag manifolds, Combinatorial identities, bijective combinatorics, Representation theory for linear algebraic groups Fonctorialité des polynômes de Schubert. (Functoriality of Schubert polynomials) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper presents an extended overview of the results obtained in the preceding years by the author and his collaborators in higher-dimensional Arakelov theory. The basic notion here is a hermitian vector bundle on a scheme (regular, projective, flat) over \({\mathbb Z}\). The author shows that many standard results of algebraic geometry can be extended to this setting, like Chern classes, Schubert calculus, Grothendieck groups, Riemann-Roch theorems and also ampleness, vanishing theorems and semi-stability. This is done in order to `compactify' arithmetic varieties at the infinite place. One expects that results on the cohomology, especially the module of sections \(H^0(X,E)\) have important arithmetic applications. The hard work in this field lies on the complex analytic side, but the main emphasis in this paper is on the analogy with the geometric case. higher-dimensional Arakelov theory; hermitian vector bundle; Chern classes; Schubert calculus; Grothendieck groups; Riemann-Roch theorems; ampleness; vanishing theorems; semi-stability Soulé, C., Hermitian vector bundles on arithmetic varieties, (Algebraic Geometry--Santa Cruz 1995. Algebraic Geometry--Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62(1), (1997), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 383-419 Arithmetic varieties and schemes; Arakelov theory; heights, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Heights, Lattices and convex bodies (number-theoretic aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Riemann-Roch theorems, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Riemann-Roch theorems, Chern characters, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Vanishing theorems, Currents in global analysis Hermitian vector bundles on arithmetic 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 goal of the present paper is to extend the mitosis algorithm, originally developed by \textit{A. Knutson} and \textit{E. Miller} [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] for the case of Schubert polynomials, to the case of Grothendieck polynomials. In addition we will also use this algorithm to construct a short combinatorial proof of Fomin-Kirillov's formula for the coefficients of Grothendieck polynomials. Grothendieck polynomials; pipe dreams; mitosis algorithm Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Linkage, complete intersections and determinantal ideals Mitosis algorithm for Grothendieck 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 relate a certain category of sheaves of \(k\)-vector spaces on a complex affine Schubert variety to modules over the \(k\)-Lie algebra (for \(\text{char\,}k>0\)) or to modules over the small quantum group (for \(k=0\)) associated to the Langlands dual root datum. As an application we give a new proof of Lusztig's conjecture on quantum characters and on modular characters for almost all characteristics. Moreover, we relate the geometric and representation-theoretic sides to sheaves on the underlying moment graph, which allows us to extend the known instances of Lusztig's modular conjecture in two directions: We give an upper bound on the exceptional characteristics and verify its multiplicity-one case for all relevant primes.
One of the fundamental problems in representation theory is the calculation of the simple characters of a given group. This problem often turns out to be difficult and there is an abundance of situations in which a solution is out of reach. In the case of algebraic groups over fields of positive characteristic we have a partial, but not yet a full answer.
In 1979, George Lusztig conjectured a formula for the simple characters of a reductive algebraic group defined over a field of characteristic greater than the associated Coxeter number; [cf. \textit{G. Lusztig}, Proc. Symp. Pure Math. 37, 313-317 (1980; Zbl 0453.20005)]. Lusztig outlined in 1990 a program that led, in a combined effort of several authors, to a proof of the conjecture for almost all characteristics. This means that for a given root system \(R\) there exists a number \(N=N(R)\) such that the conjecture holds for all algebraic groups associated to the root system \(R\) if the underlying field is of characteristic greater than \(N\). This number, however, is unknown in all but low rank cases.
One of the essential steps in Lusztig's program was the construction of a functor between the category of intersection cohomology sheaves with complex coefficients on an affine flag manifold and the category of representations of a quantum group (this combines results of \textit{M. Kashiwara} and \textit{T. Tanisaki} [Duke Math. J. 77, No. 1, 21-62 (1995; Zbl 0829.17020)], and \textit{D. Kazhdan} and \textit{G. Lusztig} [J. Am. Math. Soc. 6, No. 4, 905-947, 949-1011 (1993; Zbl 0786.17017); ibid. 7, No. 2, 335-381, 383-453 (1994; Zbl 0802.17007, Zbl 0802.17008)]). This led to a proof of the quantum (i.e. characteristic 0) analog of the conjecture. \textit{H. H. Andersen, J. C. Jantzen} and \textit{W. Soergel} then showed that the characteristic zero case implies the characteristic \(p\) case for almost all \(p\) [cf. Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): independence of \(p\). Astérisque 220 (1994; Zbl 0802.17009)].
One of the principal functors utilized in Lusztig's program was the affine version of the Beilinson-Bernstein localization functor. It amounts to realizing an affine Kac-Moody algebra inside the space of global differential operators on an affine flag manifold. A characteristic \(p\) version of this functor is a fundamental ingredient in Bezrukavnikov's program for modular representation theory [cf. \textit{R. Bezrukavnikov, I. Mirković} and \textit{D. Rumynin}, Ann. Math. (2) 167, No. 3, 945-991 (2008; Zbl 1220.17009)], and recently Frenkel and Gaitsgory used the Beilinson-Bernstein localization idea in order to study the critical level representations of an affine Kac-Moody algebra [cf. \textit{P. Fiebig}, Duke Math. J. 153, No. 3, 551-571 (2010; Zbl 1207.20040)].
There is, however, an alternative approach that links the geometry of an algebraic variety to representation theory. It was originally developed in the case of finite-dimensional complex simple Lie algebras by \textit{W. Soergel} [J. Am. Math. Soc. 3, No. 2, 421-445 (1990; Zbl 0747.17008)]. The idea was to give a ``combinatorial description'' of both the topological and the representation-theoretic categories in terms of the underlying root system using Jantzen's translation functors. This approach gives a new proof of the Kazhdan-Lusztig conjecture, but it is also important in its own right: when taken together with the Beilinson-Bernstein localization it establishes the celebrated Koszul duality for simple finite-dimensional complex Lie algebras [cf. \textit{W. Soergel}, loc. cit., and \textit{A. Beilinson, V. Ginzburg, W. Soergel}, J. Am. Math. Soc. 9, No. 2, 473-527 (1996; Zbl 0864.17006)].
In this paper we develop the combinatorial approach for quantum and modular representations. We relate a certain category of sheaves of \(k\)-vector spaces on an affine flag manifold to representations of the \(k\)-Lie algebra or the quantum group associated to Langlands' dual root datum (the occurrence of Langlands' duality is typical for this type of approach). As a corollary we obtain Lusztig's conjecture for quantum groups and for modular representations for large enough characteristics.
The main tool that we use is the theory of sheaves on moment graphs, which originally appeared in the work on the localization theorem for equivariant sheaves on topological spaces by \textit{M. Goresky, R. Kottwitz} and \textit{R. MacPherson} [Invent. Math. 131, No. 1, 25-83 (1998; Zbl 0897.22009)] and \textit{T. Braden} and \textit{R. MacPherson} [Math. Ann. 321, No. 3, 533-551 (2001; Zbl 1077.14522)]. In particular, we state a conjecture in terms of moment graphs that implies Lusztig's quantum and modular conjectures for all relevant characteristics.
Although there is no general proof of this moment graph conjecture yet, some important instances are known: The smooth locus of a moment graph is determined by \textit{P. Fiebig} [loc. cit.], which yields the multiplicity-one case of Lusztig's conjecture in full generality. Moreover, by developing a Lefschetz theory on a moment graph we obtain in [\textit{P. Fiebig}, J. Reine Angew. Math. 673, 1-31 (2012; Zbl 1266.20059)] an upper bound on the exceptional primes, i.e. an upper bound for the number \(N\) referred to above. Although this bound is huge (in particular, much greater than the Coxeter number), it can be calculated by an explicit formula in terms of the underlying root system. Kazhdan-Lusztig polynomials; irreducible characters; highest weight modules; simple Lie algebras; quantized enveloping algebras; reductive algebraic groups; positive characteristic; root systems; intersection cohomology sheaves; Schubert varieties; character formulae; Coxeter numbers; Lusztig conjecture; affine flag manifolds; affine Kac-Moody algebras; moment graphs Fiebig, Peter, Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture, J. Amer. Math. Soc., 0894-0347, 24, 1, 133\textendash 181 pp., (2011) Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations, Modular representations and characters, Sheaf cohomology in algebraic topology Sheaves on affine Schubert varieties, modular representations, and Lusztig's 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 Stable Grothendieck polynomials can be viewed as a \(K\)-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi-Trudi-type identities, and associated Fomin-Greene operators. symmetric functions; Grothendieck polynomials; Schur polynomials Yeliussizov, D., Duality and deformations of stable Grothendieck polynomials, J. Algebraic Combin., 45, 295-344, (2017) Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Duality and deformations of stable Grothendieck 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 Schur polynomials are a special case of Schubert polynomials. In this paper, we give an algorithm to compute the product of a Schubert polynomial with a Schur polynomial on the basis of Schubert polynomials. This is a special case of the general problem of the multiplication of two Schubert polynomials, where the corresponding algorithm is still missing. The main tools for the given algorithm are a factorization property of a special class of Schubert polynomials and the transition formula for Schubert polynomials. Schur polynomials; Schubert polynomials; transition formula Kohnert, A., Multiplication of a Schubert polynomial by a Schur polynomial, Ann. Comb., 1, 367-375, (1997) Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Multiplication of a Schubert polynomial by a Schur polynomial | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Continuing their work from a previous paper [Adv. Math. 140, 1-43 (1998; Zbl 0951.14035)], the authors develop a calculus of divided differences for types \(B\) and \(D\) (i.e., for the orthogonal groups). (In the quoted paper, a corresponding calculus had been developed for type \(C\).) The main result of this paper describes the action of a certain divided difference operator on the product of a (variant of a) Schur \(P\)-polynomial times a (type \(A\)) Schubert polynomial, the result being either 0 or an explicitly described (variant of a) \(P\)-polynomial (up to a multiplicative constant). Applications of this result to type \(D\) Schubert polynomials are briefly discussed, and it is indicated that these results have also implications for the cohomological study of Schubert varieties for the orthogonal groups and the related degeneracy loci. divided differences; vertex operators; jeu de taquin; symmetric group; orthogonal groups; Schur \(P\)-polynomials; orthogonal Schubert polynomials Alain Lascoux and Piotr Pragacz, Orthogonal divided differences and Schubert polynomials, \?-functions, and vertex operators, Michigan Math. J. 48 (2000), 417 -- 441. Dedicated to William Fulton on the occasion of his 60th birthday. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Classical problems, Schubert calculus Orthogonal divided differences and Schubert polynomials, \(\widetilde P\)-functions, and vertex operators | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The concatenation product of permutations enjoys many nice properties with respect to the Schubert calculus; that is, from a combinatorial point of view, with respect to the Lascoux-Schützenberger calculus of Schubert polynomials. We give explicit formulas for the product of the Schubert cycles (resp. polynomials) which are associated to the corresponding permutations with general Schubert cycles (resp. polynomials). Those formulas complete the partial known results about the combinatorics of intersection products on flag manifolds (Monk's formula, generalized Pieri formula of Lascoux and Schützenberger, some properties of vexillary permutations). Monk formula; Lascoux-Schützenberger calculus of Schubert polynomials; Schubert cycles; permutations; flag manifolds; Pieri formula F. Patras, Le calcul de Schubert des permutations décomposables , Sém. Lothar. Combin. 35 (1995), Art. B35f, approx. 10 pp. (electronic), http://cartan.u-strasbg.fr:80/~slc/wpapers/s35patras.html. Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices The Schubert calculus of decomposable permutations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 theory of DAHA-Jones polynomials is extended from torus knots to iterated torus knot, for any reduced root systems and weights. This is inspired by \textit{P. Samuelson}'s construction for the \(\mathfrak{sl}_2\) case [``Iterated torus knots and double affine Hecke algebras'', Preprint, \url{arXiv:1408.0483}]. The paper proves polynomiality, duality and other properties, and computes several examples. They conjecture that these polynomials specialize to Khovanov-Rozansky polynomials, which was since proven by \textit{H. Morton} and \textit{P. Samuelson} [Duke Math. J. 166, No. 5, 801--854 (2017; Zbl 1369.16034)].
The same authors have since extended the DAHA-Jones polynomials to iterated torus links [\textit{A. Beliakova} (ed.) and \textit{A. D. Lauda} (ed.), Categorification in geometry, topology, and physics. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1362.81007)]. double affine Hecke algebra; Jones polynomials; HOMFLY-PT polynomial; Khovanov-Rozansky homology; iterated torus knot; cabling; MacDonald polynomial; plane curve singularity; generalized Jacobian; Betti numbers; Puiseux expansion Cherednik, I.; Danilenko, I., DAHA and iterated torus knots, Algebr. Geom. Topol., 16, 843-898, (2016) Singularities of curves, local rings, Knots and links in the 3-sphere, Hecke algebras and their representations, Braid groups; Artin groups, Compact Riemann surfaces and uniformization, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Lie algebras of linear algebraic groups, Singular homology and cohomology theory DAHA and iterated torus knots | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type \(A\) by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual \(k\)-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's \(r\)-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual \(k\)-Schur functions given by studying the affine grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual \(k\)-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially the positivity of the multiplication of a dual \(k\)-Schur function by a Schur function. Bruhat order; Schubert polynomials; \(k\)-Schur functions; Hopf algebras Symmetric functions and generalizations, Hopf algebras and their applications, Classical problems, Schubert calculus Schubert polynomials and \(k\)-Schur functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials \(\mathfrak{S}_w\) were first introduced by \textit{I. N. Bernstein} et al. [Russ. Math. Surv. 28, No. 3, 1--26 (1973; Zbl 0289.57024)] as certain polynomial representatives of cohomology classes of Schubert cycles \(X_w\) in flag varieties. They were extensively studied by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031)] using an explicit definition in terms of difference operators \({\partial}_w\). Subsequently, a combinatorial expression for Schubert polynomials as the generating polynomial for compatible sequences for reduced expressions of a permutation \(w\) was discovered by \textit{S. C. Billey} et al. [J. Algebr. Comb. 2, No. 4, 345--374 (1993; Zbl 0790.05093)]. In the special case of the Grassmannian subvariety, Schubert polynomials are Schur polynomials, which also arise as the irreducible characters for the general linear group.
The Stanley symmetric functions \(F_w\) were introduced by \textit{R. P. Stanley} [Eur. J. Comb. 5, 359--372 (1984; Zbl 0587.20002)] in the pursuit of enumerations of the reduced expressions of permutations, in particular of the long permutation \(w_0\). They are defined combinatorially as the generating functions of reduced factorizations of permutations. Stanley symmetric functions are the stable limit of Schubert polynomials
\[
F_w (x_1, x_2, \ldots ) = \lim_{m\rightarrow \infty} \mathfrak{S}_{1^m \times w}(x_1, x_2, \ldots, x_{n+m}).
\]
\textit{P. Edelman} and \textit{C. Greene} [Adv. Math. 63, 42--99 (1987; Zbl 0616.05005)] showed that the coefficients of the Schur expansion of Stanley symmetric functions are nonnegative integer coefficients. Demazure modules for the general linear group are closely related to Schubert classes for the cohomology of the flag manifold. In certain cases these modules are irreducible polynomial representations, and so the Demazure characters also contain the Schur polynomials as a special case. Lascoux and Schützenberger stated that Schubert polynomials are nonnegative sums of Demazure characters.
In this paper the authors prove the converse to limit identity above by showing that Schubert polynomials are Demazure truncations of Stanley symmetric functions. Specifically, they show that the combinatorial objects underlying the Schubert polynomials, namely the compatible sequences, exhibit a Demazure crystal truncation of the full Stanley crystal of Morse and Schilling. They prove this, in which they give an explicit Demazure crystal structure on semi-standard key tableaux, which coincide with semi-skyline augmented fillings. Also they show that the crystal operators on reduced factorizations intertwine with (weak) Edelman-Greene insertion, proves their main result. Schubert polynomials; Demazure characters; Stanley symmetric functions; crystal bases Classical problems, Schubert calculus, Combinatorial aspects of representation theory, Permutations, words, matrices, Symmetric functions and generalizations, Group actions on combinatorial structures, Quantum groups (quantized function algebras) and their representations A Demazure crystal construction for 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 A $\delta$-nodal curve is a reduced (not necessarily irreducible) projective curve having $\delta\geq 0$ nodes and no other singularity. Let $\mathcal{C}$ be a relative effective divisor on the total space of a threefold $q: \mathcal{S}\rightarrow B$ fibered in projective smooth surfaces, and consider the restriction of $q$ to $\mathcal{C}$. Let $[\overline{B(\delta)}]$ be the class of the locus in $B$ corresponding to $\delta$-nodal fibers. That there exists a universal polynomial in certain Chow classes of $B$, giving the number of $\delta$-nodal fibres for all $\delta\geq 0$, was a conjecture by \textit{S. L. Kleiman} and \textit{R. Piene} [Math. Nachr. 271, 69--90 (2004; Zbl 1066.14063)], who proved the statement only for $\delta$ ranging from $0$ to $8$, leaving open the case $\delta >8$. Good news. That conjecture is now a theorem, due to the author of the beautiful, very interesting and exceptionally well written paper under review. The result is proven by a masterly use of a large spectrum of mathematical tools, from the most classical (Schubert Calculus) to the more modern (BPS Calculus by Pandharipande and Thomas). To be more precise, let \[ \epsilon(a,b,c):=q_*(c_1(O_S(C))^ac_1(T_{S/B})^bc_2(T_{S/B})^c). \] Then Kleiman and Piene proved, under the assumption of some mild and reasonable hypotheses, that there exists a natural non negative cycle $U(\delta)$ supported on $\overline{B(\delta)}$, whose rational equivalence class $[U(\delta)]$ is given by a universal polynomial in $\epsilon(a,b,c)$ for all $0\leq \delta\leq 8$. The challenge they proposed was to prove, or disprove, the claim according which the universal polynomial should work for all $\delta \geq 0$, and not only in the limited range studied by them. The main result of the paper under review, Theorem A, shows that the KP conjecture holds. The proof is offered in Section 5 through a sequence of preliminary propositions, many of which are interesting in their own, such as Theorem 5.3 which is concerned with universal polynomials in Chern classes as well, in a situation that involves a finite sequence of vector bundles on the total space $\mathcal{S}$. \par The strategy of the author is to propose a bivariant class $\gamma(\mathcal{C})\in A^*(B)$, inspired by the BPS (as acronym of Bogomol'nyi-Prasad-Sommerfield) calculus introduced by \textit{R. Pandharipande} and \textit{R. P. Thomas} [J. Am. Math. Soc. 23, No. 1, 267--297 (2010; Zbl 1250.14035)], which shows that $\gamma(\mathcal{C})\cap [B]$ is a natural effective cycle supported on $\overline{B(\delta)}$. This can be universally expressed as a polynomial of degree $\delta$ in the classes $\epsilon(a,b,c)$. The author applies Theorem A to the enumerative geometry of plane curves in $\mathbb{P}^3$. A second main result of the paper is Theorem B (an application of Theorem A). It states that for all $\delta\geq 0$ the number of plane curves of degree $d$ in $\mathbb{P}^3$, intersecting $n:=d(d+3)/2+3-\delta$ general lines, is given by a polynomial function $N_\delta(d)$, of degree $\leq 9+2\delta$. These polynomials are explicitly computed in Appendix A. The author observes that on the basis of his computations, careful described in the paper, the degree of the polynomial function $N_\delta(d)$ actually seems to be exactly $9+2\delta$. In Section 8 some of the formula are checked in low degree. The paper ends with a few low degree checks using the classical methods provided by Schubert Calculus in the Grassmannian of lines $G(1, \mathbb{P}^3)$ (Section 8) and with two computational appendices, followed by a comprehensive reference list. classes of $\delta$-nodal curves in a $1$-parameter family; Kleiman-Piene conjecture; BPS calculus by Pandharipande and Thomas; plane curves; universal node polynomials; Schubert calculus in the Grassmannian of lines Enumerative problems (combinatorial problems) in algebraic geometry, Divisors, linear systems, invertible sheaves, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus The Kleiman-Piene conjecture and node polynomials for plane curves in \(\mathbb{P}^3\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this survey article the authors collect many of the most important results on singular loci of Schubert varieties. The article can be used as a handbook by geometers and combinatorists. It covers the topics:
1. Generalities of \(G/B\) and \(G/Q\).
2. Schubert varieties in \(SL(n)/B\).
3. Tangent spaces and smoothness.
4. Rational smoothness.
5. Determination of the singular locus of \(X(w)\).
6. Descriptions of \(T(w, \tau)\).
7. Computationally efficient criteria for smoothness and rational smoothness.
8. Irreducible components of the singular locus of a Schubert variety.
9. Groups of rank 2.
10. Factoring the Poincaré polynomial of a Schubert variety.
11. Counting smooth Schubert varieties.
The treatment is short and technical. For a more complete and leisurely presentation the interested reader should consult the book ``Singular loci of Schubert varieties'' [Prog. Math. 182 (2000; Zbl 0959.14032)] by \textit{S. Billey} and \textit{V. Lakshmibai}. singular locus; Schubert varieties; rational smoothness; Chevalley-Bruhat order; Plücker coordinates; Kazhdan-Lusztig polynomials; quotient Sara Billey and V. Lakshmibai, On the singular locus of a Schubert variety, J. Ramanujan Math. Soc. 15 (2000), no. 3, 155 -- 223. Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry On the singular locus of a Schubert 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 Any polynomial in \(\mathbb{Z}[x]\) can be expressed in terms of elementary symmetric polynomials which can in turn be expressed as linear combinations of standard elementary monomials or SEM. The paper under review examines the SEM expansions of Schur and Schubert polynomials. The SEM expansion for Schubert polynomials is of particular interest because quantum Schubert polynomials can be computed by quantizing the SEM expansion of ordinary Schubert polynomials [\textit{S. Fomin}, \textit{S. Gelfand} and \textit{A. Postnikov}, Quantum Schubert polynomials, J. Am. Math. Soc. 10, No. 3, 565-596 (1997; Zbl 0912.14018)].
In the case of Schur functions, the SEM expansion can be obtained via a variant of the Jacobi-Trudi identity, and the author demonstrates how a combinatorial rule based on posets of staircase box diagrams can be used to calculate SEM expansion coefficients.
In the case of Schubert polynomials, the SEM expansion can be obtained via the determinantal expression of \textit{A. N. Kirillov} and \textit{T. Maeno} [Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula, q-alg/9610022, 1996, preprint, 52 pp.] and the author conjectures a combinatorial rule similar to that proved for Schur functions. elementary symmetric polynomials; standard elementary monomials; Schubert polynomials; SEM expansion; Schur functions Rudolf Winkel, On the expansion of Schur and Schubert polynomials into standard elementary monomials, Adv. Math. 136 (1998), no. 2, 224-250. Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds On the expansion of Schur and Schubert polynomials into standard elementary monomials | 0 |
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