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.

27 pages. Comments are welcome