$W$-GRAPHS AND GYOJA'S $W$-GRAPH ALGEBRA. (2nd May 2017)

• Record Type: Journal Article
• Authors: HAHN, JOHANNES
• Is Part Of: Nagoya mathematical journal. Volume 226(2017)
• Page Start: 1
• Page End: 43
• Publication Date: 2017-05-02
• Subjects: Primary 20C08, -- 20F55, -- 16G30; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathematics; Wiskunde; Electronic journals; Periodicals; 510.5
• DOI: 10.1017/nmj.2016.69
• Languages: English
• ISSNs: 0027-7630

Abstract

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 best of the author's 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 to 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 which 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 $A_{1}$ – $A_{4}$ .