A classification of commutative parabolic Hecke algebras

TL;DR

This paper provides a comprehensive classification of all commutative parabolic Hecke algebras associated with Coxeter systems, identifying when these algebras are commutative across various types.

Contribution

It offers the first complete classification of commutative parabolic Hecke algebras for all Coxeter types, filling a significant gap in the understanding of their structure.

Findings

Classification of commutative parabolic Hecke algebras across all Coxeter types

Identification of conditions for commutativity in these algebras

Complete characterization of the algebraic structure in the Coxeter framework

Abstract

Let $(W,S)$ be a Coxeter system with $I\subseteq S$ such that the parabolic subgroup $W_I$ is finite. Associated to this data there is a \textit{Hecke algebra} $\scH$ and a \textit{parabolic Hecke algebra} $\scH^I=\mathbf{1}_I\scH\mathbf{1}_I$ (over a ring $\ZZ[q_s]_{s\in S}$). We give a complete classification of the commutative parabolic Hecke algebras across all Coxeter types.