
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 be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of -graphs and Gyoja proved that every irreducible representation of the Iwahori-Hecke algebra can be realized as a -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 -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 -graphs. This conjecture is then proven for Coxeter groups of type , and -- .
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