The even isomorphism theorem for Coxeter groups
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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.References
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Additional Information
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 4297-4324
- MSC (2000): Primary 20F55; Secondary 20E34
- DOI: https://doi.org/10.1090/S0002-9947-07-04133-5
- MathSciNet review: 2309186