Bender-Knuth Billiards in Coxeter Groups

Let (W,S) be a Coxeter system, and write S={s_i : i is in I}, where I is a finite index set. Consider a nonempty finite convex subset L of W. If W is a symmetric group, then L is the set of linear extensions of a poset, and there are important Bender--Knuth involutions BK_i (indexed by I) defined on L. For arbitrary W and for each i in I, we introduce an operator τ_i on W that we call a noninvertible Bender-Knuth toggle; this operator restricts to an involution on L that coincides with BK_i when W is a symmetric group. Given an ordering i₁,...,i_n of I and a starting element u₀ of W, we can repeatedly apply the toggles in the order τ_i1,...,τ_in,τ_i1,...,τ_in,.... This produces a sequence of elements of W that can be viewed in terms of a beam of light that bounces around in an arrangement of transparent windows and one-way mirrors. Our central questions concern whether or not the beam of light eventually ends up in the convex set L. We will discuss several situations where this occurs and several situations where it does not. This is based on joint work with Grant Barkley, Eliot Hodges, Noah Kravitz, and Mitchell Lee.