When is a Coxeter System Determined by its Coxeter Group?

Abstract

A Coxeter system is a pair (W, S) where W is a group and where S is a set of involutions in W such that W has a presentation of the form W=〈S|(st)m(s,t)〉 Here m(s, t) denotes the order of st in W and in the presentation for W, (s, t) ranges over all pairs in S × S such that m(s, t) ≠ ∞. We further require the set S to be finite. W is a Coxeter group and S is a fundamental set of generators for W. Obviously, if S is a fundamental set of generators, then so is wSw−1, for any w∈W. Our main result is that, under certain circumstances, this is the only way in which two fundamental sets of generators can differ. In Section 3, we will prove the following result as Theorem 3.1.