Homomorphisms from Specht Modules to Signed Young Permutation Modules

TL;DR

This paper constructs and analyzes a class of homomorphisms from Specht modules to signed permutation modules, generalizing previous work and establishing bases for homomorphism spaces over fields, especially in semisimple cases.

Contribution

It introduces a new class of homomorphisms generalizing James's construction and characterizes bases for homomorphism spaces in semisimple cases.

Findings

Any homomorphism from a Specht module to a signed permutation module lies in the span of constructed homomorphisms.

The set of semistandard tableaux induces a basis for homomorphisms over fields when the group algebra is semisimple.

Conditions for linear independence of homomorphisms are established.

Abstract

We construct a class $\Theta_{\mathscr{R}}$ of homomorphisms from a Specht module $S_{\mathbb{Z}}^{\lambda}$ to a signed permutation module $M_{\mathbb{Z}}(\alpha|\beta)$ which generalises James's construction of homomorphisms whose codomain is a Young permutation module. We show that any $\phi \in \operatorname{Hom}_{{\mathbb{Z}}\mathfrak{S}_{n}}\big(S_{\mathbb{Z}}^\lambda, M_{\mathbb{Z}}(\alpha|\beta)\big)$ lies in the $\mathbb{Q}$-span of $\Theta_{\text{sstd}}$, a subset of $\Theta_{\mathscr{R}}$ corresponding to semistandard $\lambda$-tableaux of type $(\alpha|\beta)$. We also study the conditions for which $\Theta^{\mathbb{F}}_{\mathrm{sstd}}$ - a subset of $\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^\lambda,M_{\mathbb{F}}(\alpha|\beta)\big)$ induced by $\Theta_{\mathrm{sstd}}$ - is linearly independent, and show that it is a basis for $\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^\lambda,M_{\mathbb{F}}(\alpha|\beta)\big)$ when $\mathbb{F}\mathfrak{S}_{n}$ is semisimple.