Matching theorems for systems of a finitely generated Coxeter group

Abstract

We study the relationship between two sets $S$ and $S^{\prime }$ of Coxeter generators of a finitely generated Coxeter group $W$ by proving a series of theorems that identify common features of $S$ and $S^{\prime }$. We describe an algorithm for constructing from any set of Coxeter generators $S$ of $W$ a set of Coxeter generators $R$ of maximum rank for $W$.

A subset $C$ of $S$ is called complete if any two elements of $C$ generate a finite group. We prove that if $S$ and $S^{\prime }$ have maximum rank, then there is a bijection between the complete subsets of $S$ and the complete subsets of $S^{\prime }$ so that corresponding subsets generate isomorphic Coxeter systems. In particular, the Coxeter matrices of $\left(W,S\right)$ and $\left(W,S^{\prime }\right)$ have the same multiset of entries.