Relating homomorphism spaces between Specht modules of different degrees

TL;DR

This paper investigates the relationship between homomorphism spaces of Specht modules for symmetric groups of different degrees over fields of positive characteristic, establishing conditions under which these spaces are isomorphic.

Contribution

It provides new conditions ensuring isomorphisms between homomorphism spaces of Specht modules across different degrees, extending prior understanding in modular representation theory.

Findings

Hom spaces are isomorphic under specified conditions.

Conditions involve the characteristic p and partition parameters.

Results apply to fields with odd characteristic p.

Abstract

Let $K$ be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions of $n$, where $\lambda=(\lambda_1,...,\lambda_n)$ and $\mu=(\mu_1,..,\mu_n)$. By $S^{\lambda}$ we denote the Specht module corresponding to $\lambda$ 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 $\Hom_{\mathfrak{S}_n}(S^{\mu}, S^{\lambda})$ and $\Hom_{\mathfrak{S}_{n'}}(S^{\mu^+}, S^{\lambda^+})$, where $n'=n+kp^d$, $\lambda^+ =\lambda+(kp^{d})$, $\mu^+=\mu+(kp^{d})$, and $d, k$ are positive integers. We show that these are isomorphic if $p$ is odd, $p^d >\min\{\lambda_2, \mu_1-\lambda_1\}$ and $\mu_2 \le \lambda_1$.