Homomorphisms from Specht Modules to Signed Young Permutation Modules

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