W-GRAPHS OVER NON-COMMUTATIVE ALGEBRAS
Alexander Diaz-Lopez, Matthew Dyer.
University of Notre Dame, Notre Dame, IN.
Given a Coxeter system (W, S), a W-graph is a graph, together with additional information that encode a representation (denoted τ-representation) of the Hecke algebra associated to W. We generalize this work by defining W-graphs over non-commutative algebras, which give rise to new representations of Hecke algebras. Various examples are discussed that give rise to several representations of Hecke algebras on quotients of path algebras (over suitable quivers). We discuss the relationship between these representations and the τ-representations. The most interesting example comes from a quotient path algebra that is isomorphic to an ideal of Lusztig's asymptotic Hecke algebra (when defined). This work suggests that the conjecture regarding the existence of the asymptotic Hecke algebra for all Coxeter groups is true.