Branching Functions ofA(1)n−1and Jantzen–Seitz Problem for Ariki–Koike Algebras

Abstract

We study the restrictions of simple modules of Ariki–Koike algebras Hm(v) with set of parametersv=(ζ; ζv0, …, ζvl−1), whereζis annth root of unity, to their subalgebras Hm−j(v). Using a theorem of Ariki and the crystal basis theory of Kashiwara, we relate this problem to the calculation of tensor product multiplicities of highest weight irreductible representations of the affine Lie algebraA(1)n−1. These multiplicities have a combinatorial description in terms of higher level paths or highest-lift multipartitions. This enables us to solve the Jantzen–Seitz problem for Ariki–Koike algebras, that is, to determine which irreducible representations of Hm(v) restrict to irreducible representations of Hm−1(v). From a combinatorial point of view, this problem is identical to that of computing the tensor product of anA(1)n−1-module of levelland one of level 1. We also consider natural generalisations of the Jantzen–Seitz problem corresponding to the product of a levellmodule by a levell′>1 module, and from the commutativity of tensor products, we deduce a remarkable symmetry between the generalised Jantzen–Seitz conditions and the sets of parameters of the Ariki–Koike algebras.