|
This article is cited in 1 scientific paper (total in 1 paper)
Homomorphisms from Specht Modules to Signed Young Permutation Modules
Kay Jin Lima, Kai Meng Tanb 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
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 $\phi \in \mathrm{Hom}_{{\mathbb{Z}}\mathfrak{S}_n}\big(S_{\mathbb{Z}}^\lambda, M_{\mathbb{Z}}(\alpha|\beta)\big)$ 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.
Keywords:
symmetric group; Specht module; signed Young permutation module; homomorphism.
Received: July 14, 2017; in final form April 18, 2018; Published online April 25, 2018
Citation:
Kay Jin Lim, Kai Meng Tan, “Homomorphisms from Specht Modules to Signed Young Permutation Modules”, SIGMA, 14 (2018), 038, 21 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1337 https://www.mathnet.ru/eng/sigma/v14/p38
|
Statistics & downloads: |
Abstract page: | 305 | Full-text PDF : | 31 | References: | 21 |
|