Relating homomorphism spaces between Specht modules of different degrees

Abstract

Let $K$ be an infinite field of characteristic $p>0$ and let $λ, μ$ be partitions of $n$, where $λ=(λ_1,...,λ_n)$ and $μ=(μ_1,..,μ_n)$. By $S^λ$ we denote the Specht module corresponding to $λ$ 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 $n'=n+kp^d$, $λ^+ =λ+(kp^{d})$, $μ^+=μ+(kp^{d})$, and $d, k$ are positive integers. We show that these are isomorphic if $p$ is odd, $p^d >\min\{λ_2, μ_1-λ_1\}$ and $μ_2 \le λ_1$.