The diameter problem for reduced words and galleries

An arrangement of linear hyperplanes in real space dissects space into connected components called chambers. There is a natural connected graph structure on the set of all walks from one particular chamber to its antipodal chamber that cross each hyperplane exactly once. For the braid arrangement of hyperplanes, namely all those of the form x_i=x_j, this is the graph of all reduced words for the longest permutation in S_n, with braid relations as edges between them.

In this talk, we will define this graph generally, and discuss how little is known currently about its diameter. There is an obvious lower bound for the diameter, which we show is tight for certain classes of hyperplane arrangements (3-dimensional arrangements and supersolvable arrangements) which includes the braid arrangements. We do not whether this lower bound is tight in general-- this is a question ripe for further exploration.