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

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 $$1\leq x \leq a$$.

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