# On the subregular $J$-rings of Coxeter systems

## TL;DR

This paper explores the structure of subregular J-rings in Coxeter systems, providing a combinatorial approach to compute them and revealing their connections to Coxeter diagrams and quantum groups.

## Contribution

It introduces a new combinatorial method to compute subregular J-rings without relying on the Kazhdan--Lusztig basis and links these rings to quantum groups.

## Key findings

- Developed a combinatorial technique for computing J_C
- Connected J_C subalgebras to Coxeter diagrams
- Showed J_C contains subalgebras related to quantum groups

## Abstract

We recall Lusztig's construction of the asymptotic Hecke algebra $J$ of a Coxeter system $(W,S)$ via the Kazhdan--Lusztig basis of the corresponding Hecke algebra. The algebra $J$ has a direct summand $J_E$ for each two-sided Kazhdan--Lusztig cell of $W$, and we study the summand $J_C$ corresponding to a particular cell $C$ called the subregular cell. We develop a combinatorial method to compute $J_C$ without using the Kazhdan--Lusztig basis. As applications, we deduce some connections between $J_C$ and the Coxeter diagram of $W$, and we show that for certain Coxeter systems $J_C$ contains subalgebras that are free fusion rings in the sense of [Banica], thereby connecting the subalgebras to compact quantum groups arising from operator algebra theory.