Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions

TL;DR

This paper introduces a generalized family of dual stable Grothendieck polynomials with additional parameters, proves their symmetry, and constructs involutions extending Bender-Knuth involutions, advancing the understanding of these functions in algebraic combinatorics.

Contribution

It generalizes dual stable Grothendieck polynomials by adding parameters and provides two proofs of their symmetry, including new involutions on reverse plane partitions.

Findings

Generalized dual stable Grothendieck polynomials are symmetric functions.

Constructed involutions extend Bender-Knuth involutions.

Provided classification of reverse plane partitions with entries 1 and 2.

Abstract

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2.