W-graphs and Gyoja's W-graph algebra

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$-graphs and Gyoja proved that every irreducible representation of the Iwahori-Hecke algebra $H(W,S)$ can be realized as a $W$-graph. Gyoja defined an auxiliary algebra for this purpose which – to the author’s best knowledge – was never explicitly mentioned again in the literature after Gyoja’s proof (although the underlying ideas were reused). The purpose of this paper is to resurrect this $W$-graph algebra and study its structure and its modules. A new explicit description of it as a quotient of a certain path algebra is given. A general conjecture is proposed that – if it turns out to be true – would imply strong restrictions on the structure of $W$-graphs. This conjecture is then proven for Coxeter groups of type $I_2(m)$, $B_3$ and $A_1$ – $A_4$.