  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.
