A Reduction theorem for the $W$-graph decomposition conjecture

TL;DR

This paper proves a reduction theorem for the $W$-graph decomposition conjecture, showing it holds for a product of Coxeter groups if it holds for each factor, advancing understanding of the conjecture's structure.

Contribution

The paper establishes a reduction theorem that simplifies the $W$-graph decomposition conjecture to indecomposable Coxeter groups, enabling easier verification for complex groups.

Findings

Proves the $W$-graph decomposition conjecture for product groups if it holds for factors.

Provides a framework to reduce the conjecture to indecomposable Coxeter groups.

Advances the theoretical understanding of $W$-graph algebras and their decomposition.

Abstract

Let $W$ be a finite Coxeter group and $\Omega$ be its $W$-graph algebra as defined by Gyoja. The author's previous paper \cite{hahn2016wgraphs} considered this algebra in some detail, proposed, and proved in some small cases the $W$-graph decomposition conjecture. The purpose of the current paper is to prove a reduction theorem for (a slightly stronger version of) that conjecture to indecomposable Coxeter groups in the sense that the conjecture is true for $W=W_1\times W_2$ if it holds for $W_1$ and $W_2$.