A Frobenius formula for the characters of Ariki-Koike algebras

Abstract

Let Hn,r be the Ariki-Koike algebra associated to the complex reflection group Wn,r=G(r,1,n). In this paper, we give a new presentation of Hn,r by making use of the Schur-Weyl reciprocity for Hn,r established by M. Sakamoto and T. Shoji (1999, J. Algebra, 221, 293-314). This allows us to construct various non-parabolic subalgebras of Hn,r. We construct all the irreducible representations of Hn,r as induced modules from such subalgebras. We show the existence of a partition of unity in Hn,r, which is specialized to a partition of unity in the group algebra CWn,r. Then we prove a Frobenius formula for the characters of Hn,r, which is an analogy of the Frobenius formula proved by A. Ram (1991, Invent. Math.106, 461-488) for the Iwahori-Hecke algebra of type A. © 2000 Academic Press.