Résumé : Rowmotion is a group action on partially ordered sets which has received much attention of late in the dynamical algebraic combinatorics community. In particular, various statistics on posets turn out to be homomesic with respect to row motion, that is, the statistic has the same average over any orbit. A fence is a poset obtained from a sequence of chains by identifying maximal and minimal elements in an alternating fashion. These posets are important in the theory of cluster algebras and $q$-analogues.
We investigate rowmotion on antichains and ideals of fences. In particular, we show that orbits of antichains can be visualized using tilings. This permits us to prove various homomesy results for the number of elements of an antichain or ideal in an orbit. Rowmotion on fences also exhibits a new phenomenon, which we call homometry, where the value of a statistic is constant on orbits of the same size.
Along the way, we prove a homomesy result for all self-dual posets and show that any two Coxeter elements in certain toggle groups behave similarly with respect to homomesies which are linear combinations of ideal indicator functions. We end with some conjectures and avenues for future research. This is joint work with Sergi Elizalde, Matthew Plante, and Tom Roby. No background in dynamical algebraic combinatorics will be assumed.

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Dernière modification : Thursday 21 November 2024

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