### School of Mathematics and Statistics

### Robert Howlett

University of Sydney

### Normalizers of parabolic subgroups of Coxeter groups.

**Friday 4th September, 12-1pm, Carslaw 273.**
This talk will describe joint work of the speaker and
Brigitte Brink. Preprints can be obtained on request,
or downloaded from
http://www.maths.usyd.edu.au/u/bobh/

A Coxeter group is a group W given by a generating set S
subject to defining relations which specify that each
element of S is an involution and that st raised to
the *m(s,t)*^{th} power is the identity (for various pairs of
generators {*s,t*} and integers *m(s,t) > 1*). Thus, in
particular, there are two kinds of relations: those that
involve a single generator and those that involve a pair
of generators.

The aim of the present work is to describe the normalizers
of parabolic subgroups. These are the subgroups of *W* which
are generated by subsets of *S*. "Describe" means (at
least) to give presentations for these normalizers;
preferably, the presentations will clarify the structure of
the normalizers. It turns out to be best to describe instead
groupoids whose elements correspond to elements *w* in *W* and
pairs *A, B* of parabolic subgroups for which *wA = Bw*; there
will be one such groupoid for each class of conjugate
parabolic subgroups.

Let *P(J)* be the parabolic subgroup generated by the subset
*J* of *S*. Fix a class of conjugate *P(J)*'s. It turns out that
the corresponding groupoid is generated by certain elements
*v[s,J]* corresponding to subsets *J* (with *P(J)* in the fixed
class) and elements s of *S* not in *J*. Each generator carries
with it a bijection from *J* to another set *K* such that *P(K)*
is also in the class, and *K* is a subset of the union of
J and {*s*}. Defining relations can be be given in terms of
these generators. They are of two kinds. If
*v[s J]P(J)* = *P(K)v[s,J]* then *v[s,J]* is the inverse of *v[t,K]*,
where t is such that the union of *K* and {t} equals the union
of *J* and {*s*}. (So if *K* = *J* then *v[s,J]* is an involution.)
These relations live inside the parabolic subgroup
*P(J,s)*=*P(K,t)*, in which *P(J)* and *P(K)* have corank 1. The other
kind of defining relations are derived from parabolic
subgroups *P(J,s,t)* in which *P(J)* has corank two; each such
subgroup contributes at most one relation, and the relation is
easily computed from a little knowledge of properties of
maximal length elements of finite Coxeter groups.