Combinatorial expansions in K-theoretic bases

TL;DR

This paper explores the combinatorial structure of symmetric functions within a specific class, providing new descriptions of their coefficients in K-theoretic bases, which connect to geometric and representation-theoretic contexts.

Contribution

It offers a combinatorial description of coefficients for functions in class $\\mathcal C$ when expanded in K-theoretic bases, linking tableau combinatorics to K-theory.

Findings

Provides combinatorial formulas for K-theoretic expansions

Connects tableau-based functions to geometric K-theory

Enhances understanding of symmetric functions in algebraic geometry

Abstract

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, $k$-atoms, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space. In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.