Richard P. Stanley Seminar in Combinatorics: Colin Defant (Harvard)

Harvard Science Center, Room 232

Speaker: Colin Defant (Harvard University)Where: Harvard Science Center Room 232Title: Bender-Knuth Billiards in Coxeter GroupsAbstract: Let (W, S) be a Coxeter system, and write 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 (indexed by I) defined on L. For arbitrary W and for each i in I, we introduce an operator on W that we call a noninvertible Bender--Knuth toggle; this operator restricts to an involution on L that coincides with when W is a symmetric group. Given an ordering of I and a starting element of W, we can repeatedly apply the toggles in the order 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.