W-graphs and Gyoja's W-graph algebra

TL;DR

This paper revisits Gyoja's $W$-graph algebra for finite Coxeter groups, providing a new description, exploring its structure, and proving a conjecture for specific types, thereby deepening understanding of $W$-graphs and their representations.

Contribution

It explicitly describes the $W$-graph algebra as a quotient of a path algebra and proves a conjecture for certain Coxeter groups, advancing the algebraic understanding of $W$-graphs.

Findings

New explicit description of the $W$-graph algebra as a quotient of a path algebra

Proven the conjecture for Coxeter groups of types $I_2(m)$, $B_3$, and $A_1$-$A_4$

Enhanced understanding of the structure and modules of the $W$-graph algebra

Abstract

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 $A_1$ -- $A_4$.