A class of representations of Hecke algebras II

Abstract: Let $W$ be a Coxeter group whose proper parabolic subgroups are finite. According to Theorem~1.12 of [1], if the module of a finite $W$-digraph $\Gamma$ is isomorphic to the module of a $W$-graph over $Q$, then $\Gamma$ is acyclic. We extend this result to Coxeter groups with finite dihedral parabolic subgroups and $W$-graphs over arbitrary fields $F$ of $C$. Also, an example is provided showing the converse of this theorem is false. That is, there is an example of a finite, acyclic $W$-digraph whose module does not afford a $W$-graph.