Tom Roby

Dynamical algebraic combinatorics and the homomesy phenomenon

We survey recent work within the area of algebraic combinatorics that has the flavor of discrete dynamical systems, with a particular focus on the homomesy phenomenon codified in 2013 by James Propp and the author. In these situations, a group action on a set of combinatorial objects partitions them into orbits, and we search for statistics that are homomesic, i.e., have the same average value over each orbit. We give a number of examples, many very explicit, to illustrate the wide range of the phenomenon and its connections to other parts of combinatorics. In particular, we look at several actions that can be defined as a product of toggles, involutions on the set that make only local changes. This allows us to lift the well-known poset maps of rowmotion and promotion to the piecewise-linear and birational settings, where periodicity becomes much harder to prove, and homomesy continues to hold. Some of the examples have strong connections with the representation theory of semisimple Lie algebras, and others to cluster algebras via Y -systems.

Complementary Algorithms for Tableaux

We study four operations defined on pairs of tableaux. Algorithms for the first three involve the familiar procedures of jeu de taquin, row insertion, and column insertion. The fourth operation, hopscotch, is new, although specialised versions have appeared previously. Like the other three operations, this new operation may be computed with a set of local rules in a growth diagram, and it preserves the Knuth equivalence class. Each of these four operations gives rise to an a priori distinct theory of dual equivalence. We show that these four theories coincide. The four operations are linked via the involutive tableau operations of complementation and conjugation.

Jeux de tableaux

We study four operations defined on pairs of tableaux. Algorithms for the first three involve the familiar procedures of jeu de taquin, row insertion, and column insertion, respectively. The fourth operation of hopscotch is new, although specialised versions have appeared previously. Like the other three operations, hopscotch may be computed with a set of local rules in a growth diagram, and it preserves Knuth equivalence class. Each of these four operations gives rise to an a priori distinct theory of dual equivalence. We show that these four theories coincide. The four operations are linked via the involutive tableau operations of complementation and conjugation.