A an element w in a Coxeter system (W, S) may admit many different reduced expressions in the generators S, and by Matsumoto’s theorem any reduced expression for w can be made into any other by repeatedly applying the defining braid relations for (W, S). Another way to say this is that the set of reduced expressions form the vertices of an undirected graph, with edges representing braid moves; Matsumoto’s theorem is equivalent to this graph being connected.

Mousing over each group element w below will show the reduced expression graph for w.