Relating homomorphism spaces between Specht modules of different degrees

Abstract

Let $K$ be an infinite field of characteristic $p>0$ and let $\hat{I}»,\hat{I}¼$ be partitions of $n$, where $\hat{I}»=(\hat{I}{»}_1,...,\hat{I}{»}_n)$ and $\hat{I}¼=(\hat{I}{¼}_1,..,\hat{I}{¼}_n)$. By $S^{\hat{I}}»$ we denote the Specht module corresponding to $\hat{I}»$ for the group algebra $K{\mathfrak{S}}_n$ of the symmetric group ${\mathfrak{S}}_n$. D. Hemmer has raised the question of relating the homomorphism spaces $\Hom_{\mathfrak{S}_n}(S^μ, S^λ)$ and $\Hom_{\mathfrak{S}_{n'}}(S^{μ^+}, S^{λ^+})$, where $n^{\prime }=n+k{p}^d$, $\hat{I}{»}^{+}=\hat{I}»+(k{p}^d)$, $\hat{I}{¼}^{+}=\hat{I}¼+(k{p}^d)$, and $d,k$ are positive integers. We show that these are isomorphic if $p$ is odd, $p^d>\min \{\hat{I}{»}_2,\hat{I}{¼}_1-\hat{I}{»}_1\}$ and $\hat{I}{¼}_2\le \hat{I}{»}_1$.

This paper is superseded by arXiv:2211.06978