A Frobenius Formula for the Characters of Ariki–Koike Algebras,

Let n, 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 n, r by making use of the Schur–Weyl reciprocity for n, r established by M. Sakamoto and T. Shoji (1999, J. Algebra, 221, 293–314). This allows us to construct various non-parabolic subalgebras of n, r. We construct all the irreducible representations of n, r as induced modules from such subalgebras. We show the existence of a partition of unity in n, r, which is specialized to a partition of unity in the group algebra Wn, r. Then we prove a Frobenius formula for the characters of n, 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.