On double centralizer properties between quantum groups and Ariki–Koike algebras

In this paper, the double centralizer properties for tensor spaces as bimodules over quantum groups and Ariki–Koike algebras as well as their relations with Shojiʹs algebra (see [J. Algebra 226 (2000) 818–856]) are studied. We proved that, under the separation condition (see [J. Reine Angew. Math. 513 (1999) 53–69]), the natural homomorphism from the Ariki–Koike algebra to the endomorphism algebra of tensor space as module over quantum group is surjective. We also show that the natural homomorphism from Shojiʹs algebra to that endomorphism algebra is always surjective, and is actually isomorphic to a direct sum of some matrix algebras over some Hecke algebras of type A.