Excited Young diagrams, equivariant K-theory, and Schubert varieties

TL;DR

This paper provides combinatorial formulas for restrictions of structure sheaves of Schubert varieties in equivariant K-theory of Grassmannians, using excited Young diagrams and set-valued tableaux, extending previous results and offering new computational tools.

Contribution

It introduces new combinatorial descriptions and formulas for K-theoretic restrictions and Hilbert series of Schubert varieties, applicable to various types and general flag varieties.

Findings

Explicit positive formulas for restrictions to T-fixed points.

Formulas for Hilbert series and polynomials at T-fixed points.

Extension of combinatorial models to general cominuscule flag varieties.

Abstract

We give combinatorial descriptions of the restrictions to T-fixed points of the classes of structure sheaves of Schubert varieties in the T-equivariant K-theory of Grassmannians and of maximal isotropic Grassmannians of orthogonal and symplectic types. We also give formulas, based on these descriptions, for the Hilbert series and Hilbert polynomials at T-fixed points of the corresponding Schubert varieties. These descriptions and formulas are given in terms of two equivalent combinatorial models: excited Young diagrams and set-valued tableaux. The restriction fomulas are positive, in that for a Schubert variety of codimension d, the formula equals (-1)^d times a sum, with nonnegative coefficients, of monomials in the expressions (e^{-\alpha}-1), as \alpha runs over the positive roots. In types A_n and C_n the restriction formulas had been proved earlier by [Kreiman 05], [Kreiman 06] by a different method. In type A_n, the formula for the Hilbert series had been proved earlier by [Li-Yong 12]. The method of this paper, which relies on a restriction formula of [Graham 02] and [Willems 06], is based on the method used by [Ikeda-Naruse 09] to obtain the analogous formulas in equivariant cohomology. The formulas we give differ from the K-theoretic restriction formulas given by [Ikeda-Naruse 11], which use different versions of excited Young diagrams and set-valued tableaux. We also give Hilbert series and Hilbert polynomial formulas which are valid for Schubert varieties in any cominuscule flag variety, in terms of the 0-Hecke algebra.