Structure constants for K-theory of Grassmannians, revisited

TL;DR

This paper revisits the computation of structure constants in the K-theory of Grassmannians, establishing a combinatorial framework involving Yamanouchi set-valued tableaux and providing a dual proof of Buch's K-theoretic Littlewood-Richardson rule.

Contribution

It introduces a new combinatorial interpretation of K-theoretic structure constants using skew reverse plane partitions and Yamanouchi set-valued tableaux, and offers a dual proof of Buch's rule.

Findings

Established the expansion of skew reverse plane partitions in terms of dual stable Grothendieck polynomials.

Proved that Yamanouchi set-valued tableaux determine the K-theoretic structure constants.

Provided a dual approach proof for Buch's K-theoretic Littlewood-Richardson rule.

Abstract

The problem of computing products of Schubert classes in the cohomology ring can be formulated as the problem of expanding skew Schur polynomials into the basis of ordinary Schur polynomials. In contrast, the problem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to its basis of Schubert structure sheaves is not equivalent to expanding skew stable Grothendieck polynomials into the basis of ordinary stable Grothendiecks. Instead, we show that the appropriate K-theoretic analogy is through the expansion of skew reverse plane partitions into the basis of polynomials which are Hopf-dual to stable Grothendieck polynomials. We combinatorially prove this expansion is determined by Yamanouchi set-valued tableaux. A by-product of our results is a dual approach proof for Buch's K-theoretic Littlewood-Richardson rule for the product of stable Grothendieck polynomials.