Algebra Seminar

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.