Centers of Hecke Algebras of Complex Reflection Groups

TL;DR

This paper extends the understanding of the centers of Hecke algebras to complex reflection groups, providing new matrix models and explicit bases for specific cases, facilitating concrete algebraic computations.

Contribution

It offers a dual version of the Geck--Rouquier Theorem for complex cases and introduces faithful matrix models for eight rank 2 complex reflection groups.

Findings

Dual version of the Geck--Rouquier Theorem established

New faithful matrix models constructed for eight rank 2 groups

Explicit integral bases of the centers computed for these groups

Abstract

We provide a dual version of the Geck--Rouquier Theorem on the center of an Iwahori--Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank $2$, for which the symmetrising trace conjecture is known to be true, we provide a new faithful matrix model for their Hecke algebra $H$. These models enable concrete calculations inside $H$. For each of the eight groups, we compute an explicit integral basis of the center of $H$.