String functions and cyclotomic Hecke algebras

Abstract

In this paper we study the Kleshchev multipartitions that label simple modules of certain cyclotomic Hecke algebras with parameters $1$ and $-1$. The $2$-residue statistic divides these multipartitions into blocks. We deduce a bivariate generating function for the blocks using representation theory. We also relate the generating functions of multipartitions in a fixed block to string functions associated by Kac and Peterson to irreducible highest weight modules of a type $A_1^{(1)}$ affine Lie algebra. By considering matrices formed by these generating functions, we reprove some results of Kac--Peterson on string functions using the Macdonald identities of Rosengren--Schlosser. As an application, we obtain new infinite product formulas for these generating functions.