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Dynamical algebraic combinatorics and homomesy: An action-packed introduction

Tom Roby

Dynamical Algebraic Combinatorics explores actions on sets of discrete combinatorial objects, many of which can be built up by small local changes, e.g., Sch├╝tzenberger'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. Some of these actions can be extended to piecewise-linear maps on polytopes, then detropicalized to the birational setting. Here the dynamics have the flavor of cluster algebras, but this connection is still relatively unexplored.

The term "homomesy" 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 help understand certain properties of the action. This notion can be lifted to the birational setting, and the resulting identities are somewhat surprising. 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 a introduction to these ideas, giving a number of examples of such actions and pointing out connections to other areas.