Matching theorems for systems of a finitely generated Coxeter group

Mihalik, Michael; Ratcliffe, John G; Tschantz, Steven T

doi:10.2140/agt.2007.7.919

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Matching theorems for systems of a finitely generated Coxeter group

Michael L Mihalik, John G Ratcliffe and Steven T Tschantz

Algebraic & Geometric Topology 7 (2007) 919–956

DOI: 10.2140/agt.2007.7.919

arXiv: math.GR/0501075

Abstract

We study the relationship between two sets $S$ and $S^{'}$ of Coxeter generators of a finitely generated Coxeter group $W$ by proving a series of theorems that identify common features of $S$ and $S^{'}$ . 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 $S^{'}$ have maximum rank, then there is a bijection between the complete subsets of $S$ and the complete subsets of $S^{'}$ so that corresponding subsets generate isomorphic Coxeter systems. In particular, the Coxeter matrices of $(W, S)$ and $(W, S^{'})$ have the same multiset of entries.

Keywords

Coxeter groups

References

Publication

Received: 31 March 2006

Published: 20 June 2007

Authors

Michael L Mihalik
Mathematics Department Vanderbilt University Nashville TN 37240 USA

John G Ratcliffe
Mathematics Department Vanderbilt University Nashville TN 37240 USA

Steven T Tschantz
Mathematics Department Vanderbilt University Nashville TN 37240 USA

Volume 7, issue 2 (2007)