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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The even isomorphism theorem for Coxeter groups

Author: M. Mihalik
Journal: Trans. Amer. Math. Soc. 359 (2007), 4297-4324
MSC (2000): Primary 20F55; Secondary 20E34
Published electronically: April 16, 2007
MathSciNet review: 2309186
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Abstract: Coxeter groups have presentations $ \langle S :(st)^{m_{st}}\forall s,t\in S \rangle$ where for all $ s,t\in S$, $ m_{st}\in \{1,2,\ldots ,\infty \}$, $ m_{st}=m_{ts}$ and $ m_{st}=1$ if and only if $ s=t$. A fundamental question in the theory of Coxeter groups is: Given two such ``Coxeter" presentations, do they present the same group? There are two known ways to change a Coxeter presentation, generally referred to as twisting and simplex exchange. We solve the isomorphism question for Coxeter groups with an even Coxeter presentation (one in which $ m_{st}$ is even or $ \infty$ when $ s\ne t$). More specifically, we give an algorithm that describes a sequence of twists and triangle-edge exchanges that either converts an arbitrary finitely generated Coxeter presentation into a unique even presentation or identifies the group as a non-even Coxeter group. Our technique can be used to produce all Coxeter presentations for a given even Coxeter group.

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M. Mihalik
Affiliation: Department of Mathematics, Vanderbilt University, 1516 Stevenson Center, Nashville, Tennessee 37240

Received by editor(s): February 18, 2004
Received by editor(s) in revised form: August 6, 2005
Published electronically: April 16, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.