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Dynamical algebraic combinatorics and the homomesy phenomenon. (English) Zbl 1354.05146
Beveridge, Andrew (ed.) et al., Recent trends in combinatorics. Cham: Springer (ISBN 978-3-319-24296-5/hbk; 978-3-319-24298-9/ebook). The IMA Volumes in Mathematics and its Applications 159, 619-652 (2016).
Summary: 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 [in: Proceedings of the 25th international conference on formal power series and algebraic combinatorics, FPSAC 2013, Paris, France, June 24–28, 2013. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). 945–956 (2013; Zbl 1285.05012)] by J. 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.
For the entire collection see [Zbl 1348.05002].

##### MSC:
 05E18 Group actions on combinatorial structures 06A11 Algebraic aspects of posets
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