Clifford theory for Yokonuma--Hecke algebras and deformation of complex reflection groups

TL;DR

This paper introduces new algebraic structures related to Yokonuma--Hecke algebras, explores their connections to braid groups and complex reflection groups, and analyzes their representation theories using Clifford theory.

Contribution

It defines actions of the symmetric group on Yokonuma--Hecke algebras, leading to new algebra classes and deformations of complex reflection groups, with detailed presentations and representation analysis.

Findings

Established connections with braid group algebras and ties algebra.

Provided presentations for the new algebra classes.

Analyzed their representation theories using Clifford theory.

Abstract

We define and study an action of the symmetric group on the Yokonuma--Hecke algebra. This leads to the definition of two classes of algebras. The first one is connected with the image of the algebra of the braid group inside the Yokonuma--Hecke algebras, and in turn with an algebra defined by Aicardi and Juyumaya known as the algebra of braids and ties. The second one can be seen as new deformations of complex reflection groups of type G(d,p,n). We provide several presentations for both algebras and a complete study of their representation theories using Clifford theory.