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Proceedings of the American Mathematical Society

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Commensurators of parabolic subgroups of Coxeter groups

Author: Luis Paris
Journal: Proc. Amer. Math. Soc. 125 (1997), 731-738
MSC (1991): Primary 20F55
MathSciNet review: 1377001
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Abstract: Let $(W,S)$ be a Coxeter system, and let $X$ be a subset of $S$. The subgroup of $W$ generated by $X$ is denoted by $W_{X}$ and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of $W_{X}$ in $W$ is the subgroup of $w$ in $W$ such that $wW_{X}w^{-1}\cap W_{X}$ has finite index in both $W_{X}$ and $wW_{X}w^{-1}$. The subgroup $W_{X}$ can be decomposed in the form $W_{X} = W_{X^{0}} \cdot W_{X^{\infty }} \simeq W_{X^{0}} \times W_{X^{\infty }}$ where $W_{X^{0}}$ is finite and all the irreducible components of $W_{X^{\infty }}$ are infinite. Let $Y^{\infty }$ be the set of $t$ in $S$ such that $m_{s,t}=2$ for all $s\in X^{\infty }$. We prove that the commensurator of $W_{X}$ is $W_{Y^{\infty }} \cdot W_{X^{\infty }} \simeq W_{Y^{\infty }} \times W_{X^{\infty }}$. In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and $W_{X}$ is its own commensurator if and only if $X^{0}= Y^{\infty }$.

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Additional Information

Luis Paris
Affiliation: Laboratoire de Topologie, Département de Mathématiques, Université de Bourgogne, U.M.R. 5584, B.P. 138, 21004 Dijon Cedex, France

Received by editor(s): October 17, 1995
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1997 American Mathematical Society