| A Coxeter system is a pair (W, S) so that W is a group which can be given by the presentation ⟨ S|R⟩ where the set R of relators consists of terms of the form (st)mst for s, t ∈ S and mst ∈ {lcub}1, 2,…, ∞{rcub}, satisfying mst = mts and m st = 1 if and only if s = t.; One may associate a labeled graph (called a Coxeter diagram) to the Coxeter system (W, S), so that captures all of the information necessary to reconstruct the presentation given above.; We define a new representation (called the pattern) of a Coxeter group, which describes much of the underlying combinatorial structure of a given Coxeter system. We then discuss how this construct can be used in order to prove “Even Rigidity”: to every Coxeter group W there corresponds at most one even diagram, up to labeled-graph isomorphism. Some consequences of this result, due to Mihalik and to the author, will be discussed in the conclusion. |
