On the equivalence of two stability conditions of FB-modules

TL;DR

This paper proves the equivalence of two stability conditions for FB-modules, providing optimal estimates and linking polynomial character degree with weight at infinity, with applications to tensor products.

Contribution

It establishes the equivalence of representation stability and polynomial character for FB-modules, with optimal bounds and new insights into their properties.

Findings

RS and PC are equivalent for FB-modules

Optimal estimates for the stability gap

Degree of polynomial character equals weight at infinity

Abstract

We give a proof of the fact that for an FB-module the properties of being "representation stable" (RS) and "having a polynomial character" (PC) are equivalent. We obtain optimal estimates for the gap between the ranks of the polynomiality and of representation stability. As a by-product, we show that the degree of the polynomial character and the weight at infinity of an FB-module coincide, which we apply to determine the weight at infinity of a tensor product of FB-modules.