Dynamical algebraic combinatorics: Actions, orbits, and averages
- Tom Roby (University of Connecticut)
Dynamical algebraic combinatorics explores actions on sets of discrete combinatorial objects, many of which can be built up by small local changes, e.g., Schützenberger's promotion and evacuation, or the rowmotion map on order ideals. There are strong connections to the combinatorics of representation theory and with Coxeter groups. Birational liftings of these actions are related to the Y-systems of statistical mechanics, thereby to cluster algebras, in ways that are still relatively unexplored.
The term "homomesy" (fka "combinatorial ergodicity") describes the following widespread phenomenon: Given a group action on a set of combinatorial objects, a statistic on these objects is called "homomesic" if its average value is the same over all orbits. Along with its intrinsic interest as a kind of "hidden invariant", homomesy can be used to prove certain properties of the action, e.g., facts about the orbit sizes. Homomesy can often be found among the same dynamics that afford cyclic sieving. Proofs of homomesy often involve developing tools that further our understanding of the underlying dynamics, e.g., by finding an equivariant bijection.
This talk will be an introduction to these ideas, giving a number of examples of such actions and pointing out connections to other areas.