Constructing representations of Hecke algebras for complex reflection groups

TL;DR

This paper develops computational methods to construct and analyze irreducible representations of Hecke algebras associated with complex reflection groups, verifying key conjectures for low-dimensional cases.

Contribution

It introduces a generalized $W$-graph concept and provides explicit models for all irreducible representations of certain complex reflection groups.

Findings

Constructed models for all irreducible representations in dimension ≤ 3

Verified important conjectures on Hecke algebra structures

Generalized $W$-graph concept for complex reflection groups

Abstract

We investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. We first describe computational methods for the construction of irreducible representations of these algebras, including a generalization of the concept of $W$-graph to the situation of complex reflection groups. We then use these techniques to find models for all irreducible representations in the case of complex reflection groups of dimension at most three. Using these models we are able to verify some important conjectures on the structure of Hecke algebras.