Subregular J-rings of Coxeter systems via quiver path algebras

Abstract

We use quivers and their representations to bring new perspectives on the subregular J-ring JC of a Coxeter system (W,S), a subring of Lusztig's J-ring. We prove that JC 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-AK of finite dimensional right modules of the algebra AK=K⊗ZJC over an algebraically closed field K of characteristic zero. Our results include classifications of Coxeter systems for which mod-AK is semisimple, has finitely many simple modules up to isomorphism, or has a bound on the dimensions of simple modules.