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.
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