Branching functions of A((1))(n-1) and Jantzen-Seitz problem for Ariki-Koike algebras

We study the restrictions of simple modules of Ariki-Koike algebras H-m(v) with set of parameters upsilon = (zeta; zeta(upsilon 0), ..., zeta(upsilon l-1)), where zeta is an nth root of unity, to their subalgebras Hm-j(v). Us ing a theorem of Ariki and the crystal basis theory of Kashiwara, we relate this problem to the calculation of tensor product multiplicities of highes t weight irreductible representations of the affine Lie algebra A(n-1)((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 H-m(v) restric t 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 an A(n-1)((1))-module of level l and one of level 1. We also consider natural generalisations of the Jantzen-Seitz problem corre sponding to the product of a level l module by a level l' > 1 module, and f rom the commutativity of tensor products, we deduce a remarkable symmetry b etween the generalised Jantzen-Seitz conditions and the sets of parameters of the Ariki-Koike algebras. (C) 1999 Academic Press.