Equivariant K-theory of Grassmannians

TL;DR

This paper develops a combinatorial rule for calculating structure coefficients in the torus-equivariant K-theory of Grassmannians, unifying previous results and confirming a conjecture with positivity properties.

Contribution

It introduces a new combinatorial rule based on genomic tableaux and generalized jeu de taquin, unifying and extending prior work in equivariant K-theory of Grassmannians.

Findings

Proved a combinatorial rule for structure coefficients in equivariant K-theory.

Unified previous Schubert calculus results into a single framework.

Confirmed the conjectural rule of Thomas and Yong with positivity.

Abstract

We address a unification of the Schubert calculus problems solved by [A. Buch '02] and [A. Knutson-T. Tao '03]. That is, we prove a combinatorial rule for the structure coefficients in the torus-equivariant K-theory of Grassmannians with respect to the basis of Schubert structure sheaves. We thereby deduce the conjectural rule of [H. Thomas-A. Yong '13] for the same coefficients. Both rules are positive in the sense of [D. Anderson-S. Griffeth-E. Miller '11] (and moreover in a stronger form). Our work is based on the combinatorics of genomic tableaux and a generalization of [M.-P. Schutzenberger '77]'s jeu de taquin.