Monomial stability of Frobenius images

Abstract

We study representation stability in the sense of Church, Ellenberg, and Farb \cite{FI-module} through the lens of symmetric function theory and the different symmetric function bases. We show that a sequence, $(F_n)_n$, where $F_n$ is a homogeneous symmetric function of degree $n$, has stabilizing Schur coefficients if and only if it has stabilizing monomial coefficients. More generally, we develop a framework for checking when stabilizing coefficients transfer from one symmetric function basis to another. We also see how one may compute representation stable ranges from the monomial expansions of the $F_n$.\parspace As applications, we reprove and refine the representation stability of diagonal coinvariant algebras, $DR_n$. We also observe new representation stability phenomena of the Garsia-Haiman modules. This establishes certain stability properties of the modified Macdonald polynomials, $\tilde{H}_{\mu^{(n)}}[X;q,t]$ and the modified $q,t$-Kostka numbers, $\tilde{K}_{\mu^{(n)},\nu[n]}(q,t)$, for arbitrary sequences of partitions with $\mu^{(n)}\vdash n$ and $\mu^{(n)}\subseteq \mu^{(n+1)}$.