Diameter of a commutation class on reduced words

Any permutation w of the symmetric group can be generated by a product of adjacent transpositions, and a reduced word for w is a sequence of generators of minimal length whose product is w. The main result in this paper gives a formula to compute the diameter of a commutation class of the graph G(w), whose vertices are reduced words for w and whose edges are braid relations. To do so, we use a set-valued metric on the set of all reduced words of a given permutation which turn out to induce the usual distance in any commutation class. If a permutation is fully commutative, i.e. it has only one commutation class, then the formula gives the diameter of G(w). The diameter for a Grassmanian permutation is also given in terms of its Lehmer code.