A New Proof of a Theorem of Solomon

Abstract

The main purpose of this note is to give another proof and a different view of a theorem of Solomon [3] on Coxeter groups (finite groups of symmetries of R generated by reflections). A secondary purpose is to give more publicity to the fascinating special case of symmetric groups. This case was handled by Etienne whose recent paper [2] kindled my interest in this subject; I owe to Michelle Wachs my awareness of Solomon's earlier and more general theorem. The new proof is shorter and apparently more elementary than the proof of [3] (and the proof in the appendix to [3] due to Tits); nor does it reduce to Etienne's proof in the special case of the symmetric group. We begin by recalling some of the terminology of Coxeter groups. By definition they are finite groups of symmetries of n-dimensional real Euclidean space generated by those elements which are reflections in hyperplanes. Each reflecting hyperplane has two unit vectors r and — r orthogonal to it called roots and the set of roots has a reasonably canonical partition into positive roots and negative roots. The set of positive roots has a distinguished subset called the set of fundamental roots and the associated set of fundamental reflections (the Coxeter generators) also generates the group; with respect to this generating set the group has a very simple set of defining relations. Suppose that {T15 T2, ..., rm} is a set of Coxeter generators for a Coxeter group G associated with the set of fundamental roots {rx, r2,.. . , rm}. For any geG let X(g) denote the length of a minimal word in the generators which represents g. It is known [1, 6.1.1] that X(Zig) = X(g)± 1, with the positive sign if rtg is a positive root and the negative sign if rtg is a negative root. If S is any subset of {1, 2, ..., m) the signature class associated with S is the set