Symmetric Powers

Abstract

Given a faithful finite-dimensional representation $V$ of a finite group $G$ over any field $\mathbb{F}$, we show that any irreducible $\mathbb{F}G$-module $W$ appears, as a submodule or a quotient, in ${\mathrm{Sym}}^m(V)$ for some integer $1\le m\le |G|$ (that may depend on $W$).