Modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras

TL;DR

This paper studies the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras, establishing equivalences with tensor products of affine Hecke algebras and classifying irreducible representations over fields of characteristic p.

Contribution

It introduces an equivalence of categories linking Yokonuma-Hecke algebras to tensor products of affine Hecke algebras and classifies their irreducible modules in characteristic p.

Findings

Classified irreducible representations over algebraically closed fields of characteristic p.

Established modular branching rules for these algebras.

Identified modular branching graphs with crystal graphs of affine Lie algebra representations.

Abstract

We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type $A$ (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic $p$. Secondly, the modular branching rules for these algebras are obtained; moreover, the resulting modular branching graphs for cyclotomic Yokonuma-Hecke algebras are identified with crystal graphs of irreducible integrable representations of affine Lie algebras of type $A.$