A Class of Rigid Coxeter Groups

A Coxeter group W is said to be rigid if, given any two Coxeter systems (W,S) and (W,S′), there is an automorphism ρ : W −→ W such that ρ(S) = S′. We consider the class of Coxeter systems (W,S) for which the Coxeter graph ΓS is complete and has only odd edge labels (such a system is said to be of “type Kn”). It is shown that if W has a type Kn system, then any other system for W is also type Kn. Moreover, the multi-set of edge labels on ΓS and ΓS′ agree. In particular, if all but one of the edge labels of ΓS are identical, then W is rigid. 1 Coxeter Group Preliminaries 1.1 Coxeter Systems A Coxeter system is a triple (W,S,m) where W is a group, S is a subset of W, and m : S × S −→ {1, 2, 3, . . . ,∞} is a function satisfying (a) m(s, t) = 1 if and only if s = t; (b) m(s, t) = m(t, s) for all (s, t) ∈ S × S; (c) W has a presentation of the form < S : (st) = 1 ; (s, t) ∈ S × S > (if m(s, t) = ∞ for some s, t ∈ S, then the elements st and ts have infinite order in W, and there is no corresponding relation included in the presentation). The group W is called a Coxeter group. When there is no confusion, the “m” will often be omitted from the notation and the pair (W,S) will be referred to as a Coxeter system. We assume throughout that S is a finite set. Let RS = {wsw−1 : w ∈ W, s ∈ S}. If r ∈ RS , then r is called a reflection (with respect to the generating set S). 1991 Mathematics Subject Classification. Primary 20F55.