Homomorphisms from Specht Modules to Signed Young Permutation Modules

Symmetry Integrability and Geometry-Methods and Applications

article

Kay Jin Lim¹  Kai Meng Tan² 

[1] Division of Mathematical Sciences, Nanyang Technological University;Department of Mathematics, National University of Singapore

关键词: symmetric group;    Specht module;    signed Young permutation module;    homomorphism;   

DOI  :  10.3842/SIGMA.2018.038

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown