Fusion-Invariant Representations for Symmetric Groups

Abstract

For a prime p, we show that uniqueness of factorization into irreducible \(\Sigma _{p^2}\)-invariant representations of \({\mathbb Z}/p \wr {\mathbb Z}/p\) holds if and only if \(p=2\). We also show nonuniqueness of factorization for \(\Sigma _8\)-invariant representations of \(D_8 \wr {\mathbb Z}/2\). The representation ring of \(\Sigma _{p^2}\)-invariant representations of \({\mathbb Z}/p \wr {\mathbb Z}/p\) is determined completely when p equals two or three.