String functions and cyclotomic Hecke algebras

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.