Combinatorics behind certain induction of characters of the Coxeter group of type $B_n$

Question

Let $W_n$ be a Coxeter group of type $B_n$ with $n\geq 1$. Concretely, it is generated by a set of simple reflexions $S = \{s_1,\ldots ,s_n\}$ which satisfy the relations $s_i^2 = 1, s_is_j=s_js_i$ as long as $|i-j| \geq 2$, $(s_is_{i-1})^3 = 1$ for $2\leq i \leq n-1$ and $(s_ns_{n-1})^4 = 1$.

Given integers $a,b \geq 0$ such that $a+b = n$, consider the subgroup $H_{a,b} \simeq \mathfrak S_a \times W_b \subset W_n$ which is generated by all the simple reflexions except $s_a$.

Complex characters of the symmetric group $\mathfrak S_n$ (resp. of the group $W_n$) are classically classified by partitions of $n$ (resp. by bipartitions of $n$). Any such representation will be denoted by the letter $\rho$ with index a (bi)partition. Given a partition $\lambda \vdash a$ and a bipartition $(\alpha,\beta) \vdash b$, is there any paper or textbook reference describing the Littlewood-Richardson type formula to compute the induced character $$\mathrm{Ind}_{H_{a,b}}^{W_n} \, \rho_{\lambda}\otimes \rho_{\alpha,\beta}?$$ In the special case where $\lambda = (a)$ is the trivial partition, this induction corresponds to Pieri's rule for groups of type $B_n$, which is described in paragraph 6.1.9 of Geck and Pfeiffer's book Characters of finite Coxeter groups and Iwahori-Hecke algebras (2000). In terms of Young diagrams, the rule says that we obtain the multiplicity-free sum of all the irreducible characters $\rho_{\alpha',\beta'}$ where, for some $0 \leq d \leq a$, $\alpha'$ (resp. $\beta'$) can be obtained from $\alpha$ (resp. $\beta$) by successively adding $d$ boxes (resp. $a-d$ boxes) in the Young diagram, with no two boxes in the same column.

If anything, I am particularly interested in the case where $\lambda$ is a hook partition, ie. $\lambda = (x,1^{a-x})$ for some $1\leq x \leq a$.

Answer 1

Note that $\newcommand{\Sa}{\mathfrak{S}_a} \DeclareMathOperator{Ind}{Ind} \DeclareMathOperator{Res}{Res}$ $$\Sa \times W_b \subseteq W_a \times W_b \subseteq W_{a+b}.$$ Hence

$$ \Ind_{\Sa \times W_b}^{W_{a+b}} \rho_\lambda \boxtimes \rho_{\alpha,\beta} = \Ind^{W_{a+b}}_{W_a \times W_b} (\Ind^{W_a}_{\Sa} \rho_\lambda) \boxtimes \rho_{\alpha,\beta}.$$ The problem breaks into two parts: decomposing $\Ind^{W_a}_{\Sa} \rho_\lambda$, and then describing the functor $\Ind^{W_{a+b}}_{W_a \times W_b}$.

Induction from $\Sa$: recall that the irreducible representation $\rho_{\alpha',\beta'}$ of $W_{a}$ is given by $$ \rho_{\alpha',\beta'} = \Ind^{W_a}_{W_{|\alpha'|} \times W_{|\beta'|} } \rho_{\alpha'} \boxtimes ((-\mathbf{1})^{\otimes |\beta'|}\rho_{\beta'}) $$ where $(-\mathbf{1})$ means the nontrivial representation of $\mathbb Z/2$, and where we view $W_n = \mathfrak{S}_n \wr (\mathbb Z/2)$. Hence $$ \langle \rho_{\alpha',\beta'}, \Ind_{\Sa}^{W_a} \rho_{\lambda}\rangle = \langle \Res_{\Sa}^{W_a} \Ind^{W_a}_{W_{|\alpha'|} \times W_{|\beta'|} } \rho_{\alpha'} \boxtimes ((-\mathbf{1})^{\otimes |\beta'|}\rho_{\beta'}),\rho_{\lambda}\rangle.$$ Now we apply the Mackey theorem. Note that $\Sa \backslash W_a /( W_{|\alpha'|} \times W_{|\beta'|})$ has only one element, and $\Sa \cap (W_{|\alpha'|} \times W_{|\beta'|}) = \mathfrak{S}_{|\alpha'|} \times \mathfrak{S}_{|\beta'|}$, so we obtain $$ \Res_{\Sa}^{W_a} \Ind^{W_a}_{W_{|\alpha'|} \times W_{|\beta'|} } \rho_{\alpha'} \boxtimes ((-\mathbf{1})^{\otimes |\beta'|}\rho_{\beta'}) \cong \Ind_{\mathfrak{S}_{|\alpha'|}\times \mathfrak{S}_{|\beta'|}}^{\Sa} \rho_{\alpha'} \boxtimes \rho_{\beta'}.$$ Thus $$ \langle \rho_{\alpha',\beta'} , \Ind_{\Sa}^{W_a}\rho_{\lambda}\rangle = c_{\alpha',\beta'}^{\lambda},$$ where $c_{*,*}^*$ are the usual Littlewood-Richardson coefficients.

Induction from $W_a \times W_b$: This is the type B Littlewood-Richardson rule. We have $$ \Ind_{W_a \times W_b}^{W_{a+b}} \rho_{\alpha',\beta'} \boxtimes \rho_{\alpha,\beta} = \sum_{\gamma,\delta} c_{\alpha',\alpha}^{\gamma} c_{\beta',\beta}^{\delta} \rho_{\gamma,\delta}.$$ This analysis holds for general wreath products of finite groups with $\mathfrak{S}_n$, see e.g. Zelevinsky's book.

Conclusion: $$\Ind_{\Sa \times W_b}^{W_{a+b}} \rho_\lambda \boxtimes \rho_{\alpha,\beta} = \sum_{\alpha',\beta',\delta,\gamma} c^{\lambda}_{\alpha',\beta'} c^{\gamma}_{\alpha',\alpha} c^{\delta}_{\beta',\beta} \rho_{\gamma,\delta}.$$