Homomorphisms from Specht Modules to Signed Young Permutation Modules

Abstract

We construct a class ${\Theta }_{\mathcal{R}}$ 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 $\varphi \in {\mathrm{Hom}}_{\mathbb{Z}{\mathfrak{S}}_n}(S_{\mathbb{Z}}^{\lambda },M_{\mathbb{Z}}(\alpha |\beta ))$ lies in the $\mathbb{Q}$-span of ${\Theta }_{\text{sstd}}$, a subset of ${\Theta }_{\mathcal{R}}$ corresponding to semistandard $\lambda$-tableaux of type $(\alpha |\beta )$. We also study the conditions for which ${\Theta }_{\mathrm{sstd}}^{\mathbb{F}}$ - a subset of ${\mathrm{Hom}}_{\mathbb{F}{\mathfrak{S}}_n}(S_{\mathbb{F}}^{\lambda },M_{\mathbb{F}}(\alpha |\beta ))$ induced by ${\Theta }_{\mathrm{sstd}}$ - is linearly independent, and show that it is a basis for ${\mathrm{Hom}}_{\mathbb{F}{\mathfrak{S}}_n}(S_{\mathbb{F}}^{\lambda },M_{\mathbb{F}}(\alpha |\beta ))$ when $\mathbb{F}{\mathfrak{S}}_n$ is semisimple.

Cited by in corpus (1)

• On signed $p$-Kostka matrices