On the equivalence of two stability conditions of FB-modules

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.