IMPORTANT NOTICE

The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at cust-serv@ams.org or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).

 

Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

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
Full-text PDF

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


References [Enhancements On Off] (What's this?)

  • [Bo] N. Bourbaki, ``Groupes et algèbres de Lie, Chapitres IV-VI'', Hermann, Paris, 1968. MR 39:1590
  • [Br] K. S. Brown, ``Buildings'', Springer-Verlag, New York, 1989. MR 90e:20001
  • [BH] M. Burger and P. de la Harpe, Irreducible representations of discrete groups, in preparation.
  • [De] V. V. Deodhar, On the root system of a Coxeter group, Comm. Algebra 10 (1982), 611-630. MR 83j:20052a
  • [Ho] R. B. Howlett, Normalizers of parabolic subgroups of reflection groups, J. London Math. Soc. (2) 21 (1980), 62-80. MR 81g:20094
  • [Hu] J. E. Humphreys, ``Reflection groups and Coxeter groups'', Cambridge studies in advanced mathematics, vol. 29, Cambridge University Press, 1990. MR 92h:20002
  • [Kr] D. Krammer, ``The conjugacy problem for Coxeter groups'', Ph. D. Thesis, Utrecht, 1994.
  • [Ma] G. W. Mackey, ``The theory of unitary group representations'', The University of Chicago Press, 1976. MR 53:686
  • [So] L. Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976), 255-264. MR 56:3104

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

DOI: https://doi.org/10.1090/S0002-9939-97-03815-X
Received by editor(s): October 17, 1995
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society