Subrings of the asymptotic Hecke algebra of type $H_4$

TL;DR

This paper characterizes the subring structure of the asymptotic Hecke algebra for the Coxeter group of type H4, identifying generators and subalgebras within it.

Contribution

It provides a detailed description of subrings of the asymptotic Hecke algebra for type H4, including generators and subalgebra structures, which was previously unknown.

Findings

Identified generators for the subring $J^{ ext{Gamma} ext{cap} ext{Gamma}^{-1}}$

Determined subalgebras spanned by subsets of basis elements

Described the structure of these subrings in detail

Abstract

The structure of subring $J^{\Gamma \cap \Gamma^{-1}}$ of the asymptotic Hecke algebra is described for $\Gamma$ a left cell of the Coxeter group of type $H_4$. A small set of generators is produced. The subalgebras spanned by a subset of the basis ${t_x}_{x\in \Gamma\cap\Gamma^{-1}}$ are determined.