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I am reading the paper "R. Dipper and G. D. James, Representations of Hecke algebras of type $B_n$, J. Algebra (146) 1992, 454–481".

Theorem 4.18 says that the category of modules of the hyperoctahedral group $W_n$ is Morita equivalent to the category of modules of the direct sum of the tensor product of two symmetric groups $\bigoplus_{i=0}^n S_i \otimes S_{n-i}$. If $V$ is a $\bigoplus_{i=0}^n S_i \otimes S_{n-i}$-module, then $e_{i,n-i}V$ is the corresponding $W_n$-module.

Therefore, I would like to understand the following: Let $U$ be a $ W_n$ module, then what will be its image in the category of $\bigoplus_{i=0}^n S_i \otimes S_{n-i}$-modules? Its image should be the direct sum of the tensor product of two modules $\bigoplus_{i=0}^{n} A_i \otimes B_i $, where $A_i$ is an $S_i$-module and $B_i$ is an $S_{n-i}$-module. I would like to know the description of $A_i$ and $B_i$ for $i=0,\dotsc, n $.

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    $\begingroup$ You used both $d$ and $n$, as far as I could tell with the same meaning. I changed $d$ to $n$. I hope that that was all right. $\endgroup$
    – LSpice
    Commented Sep 13 at 14:02
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    $\begingroup$ It is easier to go in the reverse direction: a Specht module $𝑆^πœ†βŠ—π‘†^πœ‡$ for $𝑆_𝑖\times 𝑆_{π‘›βˆ’1}$ is sent to the Specht module $𝑆^{(πœ†,πœ‡)}$ for $W_𝑛$. More generally, $π‘ˆ\otimes 𝑉$ is sent to $\operatorname{Ind}^{W_𝑛}_{𝑆_𝑖×𝑆_{π‘›βˆ’π‘–}}(π‘ˆ\otimes 𝑉)$. See Prop. 4.11 of Dipper-Mathas "Morita equivalences of Ariki–Koike algebras", Math. Z, 240 (2002), 579-610. The inverse equivalence should be given by suitable restriction functors. $\endgroup$
    – Andrew
    Commented Sep 15 at 22:03
  • $\begingroup$ The Specht module of $W_n$ is spanned by all polytabloids [Theorem 2.18, numdam.org/item/?id=AST_1981__87-88__267_0]. In the above-mentioned paper, the Specht module is defined by $z_{\lambda} W_n$. Could you please help me to understand the connection between them. $\endgroup$
    – noone
    Commented Oct 24 at 14:30

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