Fusion-invariant representations for symmetric groups

Abstract

For a prime $p$, we show that uniqueness of factorization into irreducible $Σ_{p^2}$-invariant representations of $\mathbb{Z}/p \wr \mathbb{Z}/p$ holds if and only if $p=2$. We also show nonuniqueness of factorization for $Σ_8$-invariant representations of $D_8 \wr \mathbb{Z}/2$. The representation ring of $Σ_{p^2}$-invariant representations of $\mathbb{Z}/p \wr \mathbb{Z}/p$ is determined completely when $p$ equals two or three.