Subregular $J$-rings of Coxeter systems via quiver path algebras

TL;DR

This paper explores the structure of subregular J-rings in Coxeter systems, representing them via quiver path algebras, and analyzes their module categories to classify properties like semisimplicity and finiteness of simple modules.

Contribution

It introduces a novel quiver algebra model for subregular J-rings and applies representation theory to classify module categories of these algebras.

Findings

J_C is isomorphic to a quotient of a double quiver path algebra.

Classified Coxeter systems with semisimple or finite simple module categories.

Connected group algebras of free products of cyclic groups to Coxeter-based algebras.

Abstract

We study the subregular $J$-ring $J_C$ of a Coxeter system $(W,S)$, a subring of Lusztig's $J$-ring. We prove that $J_C$ is isomorphic to a quotient of the path algebra of the double quiver of $(W,S)$ by a suitable ideal that we associate to a family of Chebyshev polynomials. As applications, we use quiver representations to study the category mod-$A_K$ of finite dimensional right modules of the algebra $A_K=K\otimes_\Z J_C$ over an algebraically closed field $K$ of characteristic zero. Our results include classifications of Coxeter systems for which mod-$A_K$ is semisimple, has finitely many simple modules up to isomorphism, or has a bound on the dimensions of simple modules. Incidentally, we show that every group algebra of a free product of finite cyclic groups is Morita equivalent to the algebra $A_K$ for a suitable Coxeter system; this allows us to specialize the classifications to the module categories of such group algebras.