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}$ . In several places (for instance in Geissinger and Kinch), I have seen these representations defined as $ M^{\lambda} = 1\uparrow_{S_{\lambda} \times B_{\mu}}^{S_n}$ , where $ B_{\mu}$ is the semidirect product of $ S_{\mu}$ and $ E(m)$ with $ \mu$ a partition of $ m$ .

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}$