Richard P. Stanley Seminar in Combinatorics: Zamolodchikov Periodic Cluster Algebras

In 2007, Fomin and Zelevinsky introduced the bipartite belt, a sequence of bipartite cluster mutations whose exchange relations form a discrete dynamical system. When the system is periodic, we call this Zamolodchikov periodicity. In this talk, we discuss the classification of all Zamolodchikov periodic cluster algebras, with connections to W-graphs, root systems, and maximal green sequences. We observe that there is a natural red-blue statistic on the associated tropical dynamics that is homomesic, and use laminations to prove it for finite type. This gives a natural bijection between colored mutations in the dynamical system and roots in the associated root system.

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