The classification of Zamolodchikov periodic quivers

Abstract

Zamolodchikov periodicity is a property of certain discrete dynamical systems associated with quivers. It has been shown by Keller to hold for quivers obtained as products of two Dynkin diagrams. We prove that the quivers exhibiting Zamolodchikov periodicity are in bijection with pairs of commuting Cartan matrices of finite type. Such pairs were classified by Stembridge in his study of W -graphs. The classification includes products of Dynkin diagrams along with four other infinite families, and eight exceptional cases. We provide a proof of Zamolodchikov periodicity for all four remaining infinite families, and verify the exceptional cases using a computer program.

Quivers with subadditive labelings: classification and integrability

“…Of special note is the work of Hernandez [19], where he studied the occurrence of T -systems in representation theory for simply-laced quivers beyond Dynkin quivers. This is the third and final paper in the series [12,13] of works that classify bipartite recurrent quivers for which the T -system satisfies a certain algebraic property. In [12], we have shown that the T -system associated to a bipartite recurrent quiver Q is periodic if and only if Q admits a strictly subadditive labeling.…”

“…This is the third and final paper in the series [12,13] of works that classify bipartite recurrent quivers for which the T -system satisfies a certain algebraic property. In [12], we have shown that the T -system associated to a bipartite recurrent quiver Q is periodic if and only if Q admits a strictly subadditive labeling. Such quivers turn out to exactly correspond to commuting pairs of Cartan matrices which have been classified earlier by Stembridge [39].…”

R-systems

“…It was shown by Grinberg and Roby [GR16,GR15] that this is the case for rectangular posets. As it was observed by Max Glick (private communication), their result can be deduced from Zamolodchikov periodicity [Kel13,Vol07,GP16] for rectangular Y -systems by relating the values of birational rowmotion to the values of the Y -system via a monomial transformation. In essence, this can be seen as an application of the Laurentification technique introduced by Hone [Hon07] (the term "Laurentification" is taken from [HHvdKQ17]).…”

Exponents Associated with Y-Systems and their Relationship with q-Series

“…Furthermore, they are (possibly decomposable) symmetric Cartan matrices. We remark that such a pair of commuting Cartan matrices appeared in the study of W -graphs by Stembridge [26], and also in the classification of periodic mutations by Galashin and Pylyavskyy [7].…”

“…The dual Coxeter number is given by h ∨ = 5, so t( + h ∨ ) = 14. Therefore, the exponents of N B3,2 (x) is given by E(N B3,2 (x)) = (1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7,8,8,8,9,9,9,10,10,10,11,11,11,12,12,12,13,13,13).…”