Schur-Weyl duality, Verma modules, and row quotients of Ariki-Koike algebras

Abstract

We prove a Schur-Weyl duality between the quantum enveloping algebra of ${\mathfrak{gl}}_m$ and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several algebraically independent parameters and is realized through the tensor product of a parabolic universal Verma module and a tensor power of the natural representation of ${\mathfrak{gl}}_m$. We also give a new presentation by generators and relations of the generalized blob algebras of Martin and Woodcock as well as an interpretation in terms of Schur-Weyl duality by showing they occur as a particular case of our algebras.

v2: some minor modifications v3: corrections of misprints and added a brief explanation about the non-semisimple case, accepted version