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

Milestones

Received: 14 April 2020

Revised: 12 January 2021

Accepted: 27 January 2021

Published: 17 March 2021