I know that we can induce irreducible representations of $B\_n$ to $B\_{n+k}$ using the Ariki-Koike branching rule.


The irreducible representations of $B\_n \times S\_k$ are parametrised by tuples 2-partitions of $n$ and partitions of $k$.
I just haven't found an analogue to the branching rule for the these types of subgroups.



Inducing irreducible $B_n \times S_k$ characters to $B_{n+k}$
Expanding on my [comment](https://mathoverflow.net/questions/392499/inducing-irreducible-b-n-times-s-k-characters-to-b-nk#comment1001600_392499), here's what the two steps look like combinatorially.


Induction from $S\_k$ to $B\_k$ sends $S^\lambda$ to $\bigoplus\_{\mu, \gamma} W(\mu,\gamma)^{\oplus c^\lambda\_{\mu,\gamma}}$ where $c^\lambda\_{\mu,\gamma} $ denotes a Littlewood–Richardson coefficient. This I think is easy to see using Frobenius reciprocity and the fact that $W(\mu,\gamma)$ restricts to $\bigoplus\_\lambda (S^\lambda)^{c^\lambda\_{\mu,\gamma}}$, which is pretty clear from the construction of the $W(\mu,\gamma)$'s.


Induction from $B\_n \times B\_k$ to $B\_{n+k}$ is also governed by Littlewood–Richardson coefficients essentially independently in the two factors — this is true for all wreath products $G \wr S\_n$. Inducing $W(\mu,\gamma) \otimes W(\alpha,\beta)$ from $B\_n \times B\_k$ to $B\_{n+k}$ decomposes as $\bigoplus W(\lambda, \tau) ^{c^\lambda\_{\mu,\alpha}c^\tau\_{\gamma,\beta}}$. This can probably be found anywhere that talks about representations of wreath products, I am particularly fond of Zelevinski's book on positive self adjoint Hopf algebras for this stuff.


