# Tabloid Construction of Permutation Representation of Hyperoctahedral Group

For a partition $\lambda \vdash n$, the permutation representation $M^{\lambda}$ of the symmetric group can be constructed in two ways. First, it may be written as the induced representation $M^{\lambda} = 1\uparrow_{S_{\lambda}}^{S_n}$. Second, it may be written as the $S_n$-module spanned by tabloids of shape $\lambda$, $M^{\lambda} = \mathbb{C}\{\{t_1\},\ldots, \{t_n\} \}$, with action given by $\pi\{t_i\} = \{ \pi t_i\}$

I have been working a bit with the permutation representations of the Hyperoctahedral group indexed by bi-partions $M^{\lambda,\mu}$, where $(\lambda, \mu) \vdash n$. In several places (for instance in Geissinger and Kinch), I have seen these representations defined as $M^{\lambda,\mu} = 1\uparrow_{S_{\lambda} \times B_{\mu}}^{B_n}$, where $B_{\mu}$ is the semidirect product of $S_{\mu}$ and $E(m)$, with $\mu$ a partition of $m$ and $E(m)$ the subgroup of $m \times m$ diagonal matrices with diagonal entries $\{\pm 1\}$.

This definition is clearly analogous to the first definition for $M^{\lambda}$. Is there an equivalent definition of $M^{\lambda, \mu}$ as a module of B(n) spanned by some sort of bi-tabloid or other tableaux like object? My end goal here is to understand the specific permutation modules $M^{h_k, \emptyset}$ and $M^{\emptyset, h_k}$ (what does it look like evaluated at some element?). The second definition for $M^{h_k}$ led to a particularly intuitive view of the module, so I was hoping something similar existed for $M^{\lambda, \mu}$

• I think you meant $M^{\lambda,\mu} = 1\uparrow_{S_\lambda\times B_\mu}^{B_n}$, and $|\lambda|+|\mu|=n$, and $E_m=\{\pm1\}^m$, right? Jan 25, 2018 at 8:21
• Updated to reflect typo and missing definitions. Thanks! Jan 25, 2018 at 16:14
• George Melvin's answer to this question -- mathoverflow.net/questions/25625/… -- seems to directly answer your question. He cites a paper of Morris that apparently "provides a construction of all the irreducible representations of the hyperoctahedral group (in characteristic zero) via an extension of the usual 'polytabloid' combinatorial machinery". I can't access the paper but it sounds promising.... Jan 25, 2018 at 16:45
• I looked up Morris' paper on ZBMath -- zbmath.org/?q=au%3Amorris++ai%3Amorris.alun-o+se%3A00003762 -- and it reinforces my impression that this paper is the one you want. It is published in Asterisque, 1981, and is called Representations of Weyl Groups over an arbitrary field. Jan 25, 2018 at 16:51
• Hmm, it looks like this may potentially be a continued and newer version of the same ideas? Morris has a 1993 paper on "Specht Modules for Weyl Groups" arxiv.org/abs/math/0312102. I'm struggling to find a copy of the other one at the moment, but perhaps this will have similar information Jan 25, 2018 at 17:01