Simple modules as submodules and quotients of symmetric powers (Q6588193)

Let \(G\) be a finite group. Given a faithful (linear) action of \(G\) on a finite-dimensional vector space \(V\) over a field \(\mathbb{F}\), a classic problem is the study of invariants of \(\mathbb{F}[V]^{G}\) or semi-invariants, i.e. the one-dimensional \(G\)-submodules in \(\mathbb{F}[V]\). In particular one can ask whether and when a given irreducible \(\mathbb{F}G\)-module \(W\) occurs, as a submodule or as a quotient of \(\mathrm{Sym}^{m}(V)\) for suitable \(m\). An affirmative answer to the first question was given in [\textit{R.M. Bryant}, J. Algebra 154, No. 2, 416--436 (1993; Zbl 0828.20002)].\N\NThe main result in the paper under review is the following:\N\NTheorem 3: Let \(\mathbb{F}\) be any field, \(G\) any finite group, and \(V=\mathbb{F}^{n}\) any finite-dimensional, faithful \(\mathbb{F}G\)-module. Let \(W\) be any irreducible \(\mathbb{F}G\)-module. Then there is some integer \(1 \leq m \leq |G|\) (depending on \(W\)) such that the following statements hold: \N\begin{enumerate}\N\item[(i)] \(W\) is isomorphic to a submodule of \(\mathrm{Sym}^{m}(V)\)\N\item[(ii)] \(W\) is isomorphic to a quotient of \(\mathrm{Sym}^{m}(V)\). \N\item[(iii)] In fact, for any \(k \in \mathbb{Z}_{\geq 0}\), \(W\) is isomorphic to a submodule of \(\mathrm{Sym}^{m+k|G|}(V)\), and to a quotient of \(\mathrm{Sym}^{m+k |G|}(V)\).\N\end{enumerate}