Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Commensurators of parabolic subgroups of Coxeter groups
HTML articles powered by AMS MathViewer

by Luis Paris PDF
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 $X^{0}= Y^{\infty }$.
References
  • 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, DOI 10.1007/978-1-4612-1019-1
  • 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 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 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, DOI 10.1017/CBO9780511623646
  • D. Krammer, “The conjugacy problem for Coxeter groups”, Ph. D. Thesis, Utrecht, 1994.
  • George W. Mackey, The theory of unitary group representations, Chicago Lectures in Mathematics, 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. 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 10.1016/0021-8693(76)90182-4
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 20F55
  • Retrieve articles in all journals with MSC (1991): 20F55
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
  • © Copyright 1997 American Mathematical Society
  • 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