Jim Propp: Combinatorial Ergodicity

Abstract

For many cyclic actions τ on a finite set S of combinatorial objects, and for many natural statistics φ on S, one finds that the triple (S,τ,φ) exhibits "combinatorial ergodicity": the average of φ over each τ-orbit in S is the same as the average of φ over the whole set S. (Example: Let S be the set of binary sequences s = (s₁,...,s_n) containing k 1's and n-k 0's, let τ be the cyclic shift, and let φ(s) be the inversion number #{i < j: s_i > s_j}.) This phenomenon was first noticed by Panyushev in 2007 in the context of antichains in root posets; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. In this talk, describing joint work with Tom Roby, I'll describe a theoretical framework for results of this kind, and discuss in detail combinatorial ergodicity for rowmotion and promotion (in Striker and Williams' terminology) acting on order ideals of minuscule posets of type A (products of two chains).