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

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.

23 pages, 0 figures