Centralizers in the Hecke algebras of complex reflection groups

TL;DR

This paper explores the structure of centralizers in Hecke algebras associated with complex reflection groups, extending known results from real groups and introducing new tools to handle complex-specific challenges.

Contribution

It provides explicit relations for centralizers in complex reflection group Hecke algebras and introduces a double coset graph to address instability issues.

Findings

Derived explicit relations for centralizers of generators.

Constructed integral bases for centralizers and centers of specific Hecke algebras.

Developed a double coset graph to manage instability in complex cases.

Abstract

How far can the elementary description of centralizers of parabolic subalgebras of Hecke algebras of finite real reflection groups be generalized to the complex reflection group case? In this paper we begin to answer this question by establishing results in two directions. First, under conditions closely analogous to those existing for the real case, we give explicit relations between coefficients in an element centralizing a generator. Second, we introduce a tool for dealing with a major challenge of the complex case -- the ``instability'' of certain double cosets -- through the definition and use of a double coset graph. We use these results to find integral bases for the centralizers of generators as well as the centres of the Hecke algebras of types $G_4$ and $G(4,1,2)$. Keywords: complex reflection group; Hecke algebra; centre; centralizer; modular; double coset.