Combinatorics Seminar

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 using Kac diagrams. For the octahedron and the cube recurrence, we prove the converse direction of each of the three statements using explicit formulas due to Speyer and Carroll combined with our general linearizability result for cylindrical networks. This is joint work with Pavlo Pylyavskyy. Speaker(s): Pavel Galashin (MIT)