# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Commensurators of parabolic subgroups of Coxeter groupsHTML articles powered by AMS MathViewer

by Luis Paris
Proc. Amer. Math. Soc. 125 (1997), 731-738 Request permission

## 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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