T-Increasing Paths on the Bruhat Graph of Affine Weyl Groups are Self-Avoiding

Let (W, S) be a Coxeter system, T = {wsw − 1s S, w W} its set of reflections, any total reflection order, and Γ the undirected Bruhat graph. We consider the natural labeling of the edges of Γ which assigns to the edge {v, w} the reflection vw − 1. A path on Γ, i.e., a sequence v1,…,vk such that viv − 1i + 1 T for i = 1,…,k − 1, is called T-increasing if v1v − 12 ••• vk − 1v − 1k. T-increasing paths play an important role in the computation of both the Kazhdan–Lusztig and the R-polynomials of W. We prove that if W is finite or is an affine Weyl group, then any T-increasing path is self-avoiding, i.e., it has no self-intersection points.