### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 038, 21 pages      arXiv:1606.00542      https://doi.org/10.3842/SIGMA.2018.038
Contribution to the Special Issue on the Representation Theory of the Symmetric Groups and Related Topics

### Homomorphisms from Specht Modules to Signed Young Permutation Modules

Kay Jin Lim a and Kai Meng Tan b
a) Division of Mathematical Sciences, Nanyang Technological University, SPMS-PAP-03-01, 21 Nanyang Link, 637371 Singapore
b) Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, 119076 Singapore

Received July 14, 2017, in final form April 18, 2018; Published online April 25, 2018

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.

Key words: symmetric group; Specht module; signed Young permutation module; homomorphism.

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