Subregular J-rings of Coxeter systems via quiver path algebras

We use quivers and their representations to bring new perspectives on 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 suitable quotient of the path algebra of the double quiver of (W, S). Up to Morita equivalence, such quotients include the group algebras of all free products of finite cyclic groups. We then use quiver representations to study the category mod-A(K) of finite dimensional right modules of the algebra A(K) = K circle times(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. Crown Copyright (c) 2022 Published by Elsevier Inc. All rights reserved.