Fusion-Invariant Representations for Symmetric Groups

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