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 $M^{\lambda, \mu}$

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    $\begingroup$ 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? $\endgroup$ Jan 25 '18 at 8:21
  • $\begingroup$ Updated to reflect typo and missing definitions. Thanks! $\endgroup$ Jan 25 '18 at 16:14
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    $\begingroup$ 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.... $\endgroup$
    – Nick Gill
    Jan 25 '18 at 16:45
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    $\begingroup$ 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. $\endgroup$
    – Nick Gill
    Jan 25 '18 at 16:51
  • $\begingroup$ 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 $\endgroup$ Jan 25 '18 at 17:01

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