Combinatorics of the K-theory of affine Grassmannians

TL;DR

This paper introduces a new family of tableaux that generalize existing combinatorial models, establishing affine Grothendieck polynomials and k-K-Schur functions as Schubert representatives for the K-theory of affine Grassmannians.

Contribution

It develops a unified combinatorial framework for affine Grothendieck polynomials and k-K-Schur functions, including Pieri rules and other properties.

Findings

Defined a new family of tableaux generalizing previous models

Proved these tableaux represent affine Grothendieck polynomials and k-K-Schur functions

Established combinatorial properties such as Pieri rules

Abstract

We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k-K-Schur functions -- Schubert representatives for the K-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.