Diameters of graphs of reduced words and rank-two root subsystems

Abstract

We study the diameter of the graph G ( w ) G(w) of reduced words of an element w w in a Coxeter group W W whose edges correspond to applications of the Coxeter relations. We resolve conjectures of Reiner–Roichman [Trans. Amer. Math. Soc. 365 (2013), pp. 2279-2802] and Dahlberg–Kim [Diameters of graphs on reduced words of 12 and 21-inflations, arXiv:2010.15758, 2020] by proving a tight lower bound on this diameter when W = S n W=S_n is the symmetric group and by characterizing the equality cases. We also give partial results in other classical types which illustrate the limits of current techniques.