Commensurators of parabolic subgroups of Coxeter groups

Author:
Luis Paris

Journal:
Proc. Amer. Math. Soc. **125** (1997), 731-738

MSC (1991):
Primary 20F55

DOI:
https://doi.org/10.1090/S0002-9939-97-03815-X

MathSciNet review:
1377001

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Abstract | References | Similar Articles | Additional Information

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 $X^{0}= Y^{\infty }$.

- N. Bourbaki,
*Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines*, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR**0240238** - Kenneth S. Brown,
*Buildings*, Springer-Verlag, New York, 1989. MR**969123** - M. Burger and P. de la Harpe,
*Irreducible representations of discrete groups*, in preparation. - Vinay V. Deodhar,
*On the root system of a Coxeter group*, Comm. Algebra**10**(1982), no. 6, 611–630. MR**647210**, DOI https://doi.org/10.1080/00927878208822738 - Robert B. Howlett,
*Normalizers of parabolic subgroups of reflection groups*, J. London Math. Soc. (2)**21**(1980), no. 1, 62–80. MR**576184**, DOI https://doi.org/10.1112/jlms/s2-21.1.62 - James E. Humphreys,
*Reflection groups and Coxeter groups*, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR**1066460** - D. Krammer,
*“The conjugacy problem for Coxeter groups”*, Ph. D. Thesis, Utrecht, 1994. - George W. Mackey,
*The theory of unitary group representations*, University of Chicago Press, Chicago, Ill.-London, 1976. Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955; Chicago Lectures in Mathematics. MR**0396826** - Louis Solomon,
*A Mackey formula in the group ring of a Coxeter group*, J. Algebra**41**(1976), no. 2, 255–264. MR**444756**, DOI https://doi.org/10.1016/0021-8693%2876%2990182-4

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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

Email:
lparis@satie.u-bourgogne.fr

Received by editor(s):
October 17, 1995

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 1997
American Mathematical Society