Schur-Weyl reciprocity for Ariki-Koike algebras (Q1818839)

The Ariki-Koike algebra \({\mathcal H}_{n,r}\), is a generalization of the Hecke algebra \({\mathcal H}_n\). The Hecke algebra is well known to be the quantum generalization of the symmetric group for the \({\mathbf A}_n\) type Lie algebras and their Weyl groups, i.e. complex reflection groups. The authors of the present paper show that the Schur-Weyl reciprocity theorem is valid for the pair of a quantum group \(U_q ({\mathbf g})\) and the Ariki-Koike algebra \({\mathcal H}_{n,r}\). An explicit representation of the Ariki-Koike algebra is constructed, the action of this algebra on an \(n\)-fold tensor product space \(V^{\otimes^n}\) is given and it is proved that Ariki-Koike algebras are semisimple. The work is to a large extent self contained and provides full proofs.