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 𝔤𝔩m 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 𝔤𝔩m. 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
Authors
Abel Lacabanne
Institut de Recherche en Mathématique et Physique
Université Catholique de Louvain
Louvain-la-Neuve
Belgium
Pedro Vaz
Institut de Recherche en Mathématique et Physique
Université catholique de Louvain
Louvain-la-Neuve
Belgium