Representation theory of the Yokonuma-Hecke algebra

TL;DR

This paper develops an inductive framework for understanding the representation theory of Yokonuma-Hecke algebras, providing explicit formulas, semisimplicity criteria, and a canonical symmetrising form.

Contribution

It introduces an inductive approach based on Jucys-Murphy elements, explicitly describes irreducible representations, and establishes a symmetrising form for the algebra.

Findings

Explicit formulas for irreducible representations in terms of standard d-tableaux

Semisimplicity criterion for Yokonuma-Hecke algebras

Existence and calculation of a canonical symmetrising form

Abstract

We develop an inductive approach to the representation theory of the Yokonuma-Hecke algebra ${\rm Y}_{d,n}(q)$, based on the study of the spectrum of its Jucys-Murphy elements which are defined here. We give explicit formulas for the irreducible representations of ${\rm Y}_{d,n}(q)$ in terms of standard $d$-tableaux; we then use them to obtain a semisimplicity criterion. Finally, we prove the existence of a canonical symmetrising form on ${\rm Y}_{d,n}(q)$ and calculate the Schur elements with respect to that form.