A modular branching rule for the generalized symmetric groups

We give a modular branching rule for certain wreath products as a generalization of Kleshchev's modular branching rule for the symmetric groups. Our result contains a modular branching rule for the complex reflection groups $G(m,1,n)$ (which are often called the generalized symmetric groups) in splitting fields for $\mathbb{Z}/m\mathbb{Z}$. Especially for $m=2$ (which is the case of the Weyl groups of type $B$), we can give a modular branching rule in any field. Our proof is elementary in that it is essentially a combination of Frobenius reciprocity, Mackey theorem, Clifford's theory and Kleshchev's modular branching rule.Comment: 10 page

A Fibonacci variant of the Rogers-Ramanujan identities via crystal energy

We define a length function for a perfect crystal. As an application, we derive a variant of the Rogers-Ramanujan identities which involves (a $q$-analog of) the Fibonacci numbers.Comment: This is an early draft for a talk for the conference "Recent developments in Combinatorial Representation Theory" at RIM

A proof of the second Rogers-Ramanujan identity via Kleshchev multipartitions

We give another proof of the second Rogers-Ramanujan identity by Kashiwara crystals.Comment: 6 pages, a submitted versio