#### Vol. 311, No. 1, 2021

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

### Abel Lacabanne and Pedro Vaz

Vol. 311 (2021), No. 1, 113–133
DOI: 10.2140/pjm.2021.311.113
##### Abstract

We prove a Schur–Weyl duality between the quantum enveloping algebra of ${\mathfrak{𝔤}\mathfrak{𝔩}}_{m}$ and certain quotient algebras of Ariki–Koike algebras, which we describe explicitly. This duality involves several algebraically independent parameters and the module underlying it is a tensor product of a parabolic universal Verma module and a tensor power of the standard representation of ${\mathfrak{𝔤}\mathfrak{𝔩}}_{m}$. 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 special case of our algebras.

##### Keywords
Schur–Weyl duality, Verma modules, Ariki–Koike algebras
##### Mathematical Subject Classification
Primary: 20C08, 20G42