The Stembridge Equality for Skew Stable Grothendieck Polynomials and Skew Dual Stable Grothendieck Polynomials

TL;DR

This paper proves a Stembridge equality analogue for skew stable and dual stable Grothendieck polynomials, revealing symmetry properties in K-theoretic invariants of Grassmannians using Hopf algebra and Littlewood-Richardson rules.

Contribution

It establishes the first known symmetry relations for skew stable and dual stable Grothendieck polynomials analogous to the Stembridge equality for Schur polynomials.

Findings

Proved $G_{ ho/ u} = G_{ ho/ u^T}$ for skew stable Grothendieck polynomials.

Proved $g_{ ho/ u} = g_{ ho/ u^T}$ for skew dual stable Grothendieck polynomials.

Extended symmetry properties to K-theoretic invariants of Grassmannians.

Abstract

The Schur polynomials $s_{\lambda}$ are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For $\rho = (n, n-1, \dots, 1)$ a staircase shape and $\mu \subseteq \rho$ a subpartition, the Stembridge equality states that $s_{\rho/\mu} = s_{\rho/\mu^T}$. This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials $G_{\lambda}$, and the dual stable Grothendieck polynomials $g_{\lambda}$, developed by Buch, Lam, and Pylyavskyy, are variants of the Schur polynomials and describe the $K$-theory of the Grassmannians. Using the Hopf algebra structure of the ring of symmetric functions and a generalized Littlewood-Richardson rule, we prove that $G_{\rho/\mu} = G_{\rho/\mu^T}$ and $g_{\rho/\mu} = g_{\rho/\mu^T}$, the analogues of the Stembridge equality for the skew stable and skew dual stable Grothendieck polynomials.