CRYSTAL STRUCTURES FOR SYMMETRIC GROTHENDIECK POLYNOMIALS
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Springer Science+Business Media New York (2020)
Transformation Groups
CRYSTAL STRUCTURES FOR SYMMETRIC GROTHENDIECK POLYNOMIALS CARA MONICAL
OLIVER PECHENIK
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, USA
Department of Combinatorics & Optimization University of Waterloo Waterloo, ON N2L 3G1, Canada
[email protected]
[email protected]
TRAVIS SCRIMSHAW School of Mathematics and Physics The University of Queensland St. Lucia, QLD 4072, Australia [email protected]
Abstract. The symmetric Grothendieck polynomials representing Schubert classes in the Ktheory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type An crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate our crystal structure to combinatorial models using excited Young diagrams, Gelfand–Tsetlin patterns via the 5-vertex model, and biwords via Hecke insertion to compute symmetric Grothendieck polynomials.
1. Introduction The set of k-dimensional linear subspaces in Cn is known as the Grassmannian X = Grk (Cn ). Grassmannians are naturally smooth projective varieties and have been well studied from numerous viewpoints; for background, see, e.g., [20], [23], [39], [67], [78] and references therein. One approach to studying the Grassmannian is through its cohomology ring, where a basis of cohomology classes appears as the Poincar´e duals to the Schubert varieties Xλ that decompose X into a cell complex. By corresponding the Schubert classes with those Schur polynomials sλ whose defining partition λ fits inside a k×(n−k) rectangle, the cohomology H ∗ (X) DOI: 10.1007/S00031-020-09623-y Received February 7, 2019. Accepted August 13, 2020. Corresponding Author: Oliver Pechenik, e-mail: [email protected]
CARA MONICAL, OLIVER PECHENIK, TRAVIS SCRIMSHAW
is isomorphic to a projection of the algebra of symmetric functions. That is to say, the Schubert structure coefficients of H ∗ (X) are either Littlewood–Richardson coefficients or 0. This identification permits the application of many combinatorial tools to the study of Grassmannian Schubert calculus. Modern Schubert calculus strives for a richer understanding of the Grassmannian through use of generalized cohomology theories, such as K-theory. In the K-theory ring K(X) of algebraic vector bundles over X, there is a canonical basis of Schubert classes given by the structure sheaves of the Schubert varieties. As in the usual cohomology, these Schubert classes may be represented by certain polynomials; in this case the symmetric Grothendieck polynomials Gλ . The combinatorics of K-theoretic Grassmannian Schubert calculus have also been well stu
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