Commensurators of parabolic subgroups of Coxeter groups

Mathematics – Group Theory

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

Plain tex version, 9 pages no figures

Scientific paper

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 $wW_Xw^{-1}\cap W_X$ has finite index in both $W_X$ and $wW_Xw^{-1}$. 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 $X^0=Y^\infty$.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Commensurators of parabolic subgroups of Coxeter groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Commensurators of parabolic subgroups of Coxeter groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Commensurators of parabolic subgroups of Coxeter groups will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-232799

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.