Pavel Galashin: Zamolodchikov periodicity and integrability.

Abstract

T-systems are certain discrete dynamical systems associated with quivers. Keller showed in 2013 that the T-system is periodic when the quiver is a product of two finite Dynkin diagrams. We prove that the T-system is periodic if and only if the quiver is a finite x finite quiver. Such quivers correspond to pairs of commuting Cartan matrices which have been classified by Stembridge in the context of Kazhdan-Lusztig theory. We show that if the T-system is linearizable then the quiver is necessarily an affine x finite quiver. We classify such quivers and conjecture that the T-system is linearizable for each of them. Next, we show that if the T-system has algebraic entropy zero then the quiver is an affine x affine quiver, and classify them as well. We pay special attention to the tropical version of the problem. This is joint work with Pavlo Pylyavskyy.