Proceedings of the 29 th Conference on Formal Power Séminaire Lotharingien de Combinatoire XX (2017) Article YY Series and Algebraic Combinatorics (London) Zamolodchikov periodicity and integrability (extended abstract) Pavel Galashin 1 and Pavlo Pylyavskyy ∗ 2 1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2 Department of Mathematics, University of Minnesota, Minneapolis, MN 55414, USA Received April 4, 2017. Abstract. The T -system is a certain discrete dynamical system associated with a quiver. 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 ⊠ finite quiver. Such quivers 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 ⊠ finite quiver. We classify such quivers and conjecture that the T -system is linearizable for each of them. For affine ⊠ finite quivers of type ˆ A ⊗ A , that is, for the octahedron recurrence on a cylinder, we give an explicit formula for the linear recurrence coefficients in terms of the partition functions of domino tilings of a cylinder. Next, we show that if the T -system grows slower than a double exponential function then the quiver is an affine ⊠ affine quiver, and classify them as well. Additionally, we prove that the cube recurrence introduced by Propp is periodic inside a triangle and linearizable on a cylinder. Résumé. The T -system is a certain discrete dynamical system associated with a quiver. 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 ⊠ finite quiver. Such quivers 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 ⊠ finite quiver. We classify such quivers and conjecture that the T -system is linearizable for each of them. For affine ⊠ finite quivers of type ˆ A ⊗ A , that is, for the octahedron recurrence on a cylinder, we give an explicit formula for the linear recurrence coefficients in terms of the partition functions of domino tilings of a cylinder. Next, we show that if the T -system grows slower than a double exponential function then the quiver is an affine ⊠ affine quiver, and classify them as well. Additionally, we prove that the cube recurrence introduced by Propp is periodic inside a triangle and linearizable on a cylinder. Keywords: Cluster algebras, Zamolodchikov periodicity, domino tilings, linear recur- rence, cube recurrence, commuting Cartan matrices ∗ P. P. was partially supported by NSF grants DMS-1148634, DMS-1351590, and Sloan Fellowship.
Pavel Galashin and Pavlo Pylyavskyy 2 1 Introduction Cluster algebras have been introduced by Fomin and Zelevinsky in [3] and since then have been a popular subject of research. An important class of cluster algebras are those associated with quivers which are directed graphs without loops and directed 2-cycles. One can define an operation on quivers called mutation : given a quiver Q with vertex set Vert ( Q ) and its vertex v ∈ Vert ( Q ) , µ v ( Q ) is another quiver on the same set of vertices as Q but with edges modified according to a certain combinatorial rule. We say that a quiver Q is bipartite if the underlying graph is bipartite. In other words, Q is bipartite if there is a map ǫ : Vert ( Q ) → { 0, 1 } , v �→ ǫ v such that for any arrow u → v in Q we have ǫ u � = ǫ v . It is clear from the definition of a mutation that if there are no arrows between u and v then the operations µ u and µ v commute. Thus if Q is bipartite, one can define two operations µ 0 and µ 1 on Q as products µ 0 = ∏ ǫ u = 0 µ u and µ 1 = ∏ ǫ v = 1 µ v . Let us say that Q op is the same quiver as Q but with all edges reversed. Then we call a bipartite quiver Q recurrent if µ 0 ( Q ) = µ 1 ( Q ) = Q op . In other words, a bipartite quiver Q is recurrent if mutating all vertices of the same color reverses the arrows of Q but does not introduce any new arrows. We restate this definition in an elementary way in Section 3. The notion of recurrent quivers is necessary to define the T-system which we do now. For a quiver Q let x = { x v } v ∈ Vert ( Q ) be a family of indeterminates and let Q ( x ) be the field of rational functions in these variables. Then given a bipartite recurrent quiver Q , the T-system associated with Q is a family of rational functions T v ( t ) ∈ Q ( x ) for each v ∈ Vert ( Q ) and t ∈ Z satisfying the following recurrence relation for all v ∈ Vert ( Q ) and all t ∈ Z : T v ( t + 1 ) T v ( t − 1 ) = ∏ T u ( t ) + ∏ T w ( t ) . (1.1) u → v v → w One immediately observes that the parity of t + ǫ v is the same in each term of (1.1) so the T -system splits into two independent parts. Thus we restrict the elements T v ( t ) of the T -system to only the values of t for which t ≡ ǫ v ( mod 2 ) . The initial conditions for the T -system are given by T v ( ǫ v ) = x v , v ∈ Vert ( Q ) . It is clear that these initial conditions together with (1.1) determine T v ( t ) for all v ∈ Vert ( Q ) and t ≡ ǫ v ( mod 2 ) . During the past two decades, various special cases of T -systems have been studied extensively, the most popular one being the octahedron recurrence . More generally, given two ADE Dynkin diagrams Λ and Λ ′ , one can define their tensor product Λ ⊗ Λ ′ which is a bipartite recurrent quiver, see Figure 1 (a) for an example. For these quivers, the associated T system turns out to be periodic , that is, for every ADE Dynkin diagrams Λ and Λ ′ there is an integer N such that the T -system associated with Λ ⊗ Λ ′ satisfies T v ( t ) = T v ( t + 2 N ) for all v ∈ Vert ( Q ) and t ≡ ǫ v ( mod 2 ) . This result has been recently There is also a nice formula for the period N of the T -system shown by Keller [8].
Zamolodchikov periodicity and integrability 3 (a) (b) (c) (d) Figure 1: (a) A tensor product D 5 ⊗ A 3 . (b) A finite ⊠ finite quiver. (c) An affine ⊠ finite quiver. (d) An affine ⊠ affine quiver. Arrows are colored according to Definition 2.1. associated with Λ ⊗ Λ ′ , namely, N divides h ( Λ ) + h ( Λ ′ ) where h denotes the Coxeter number of the corresponding Dynkin diagram, see Section 3. Remark 1.1. The standard formulation of Zamolodchikov periodicity includes Y-systems rather than T -systems. However, the machinery of cluster algebras with principal coef- ficients [5] allows one to show that given a bipartite recurrent quiver, the T -system is periodic if and only if the Y -system is periodic. One other interesting phenomenon related to T -systems has been studied to some extent. Given a bipartite recurrent quiver Q , let us say that the T -system associated with Q is linearizable if for every vertex v ∈ Vert ( Q ) , there exists an integer N and rational functions H 0 , H 1 , . . . , H N ∈ Q ( x ) such that H 0 , H N � = 0 and ∑ N i = 0 H i T v ( t + i ) = 0 for every t ∈ Z satisfying t ≡ ǫ v ( mod 2 ) . It was shown in [1] that if every vertex of Q is either a source or a sink and the T -system associated with Q is linearizable then the underlying graph of Q is necessarily an affine ADE Dynkin diagram. Conversely, for every such quiver the T -system was shown to be linearizable in [1, 9]. 2 Main results Before we state our results, we need to define various classes of quivers. Definition 2.1. Given a bipartite quiver Q with vertex set Vert ( Q ) , we define two undi- rected graphs Γ = Γ ( Q ) and ∆ = ∆ ( Q ) on Vert ( Q ) as follows. For every arrow u → v with ǫ u = 0, ǫ v = 1, Γ contains an undirected edge ( u , v ) , and for every arrow u → v with ǫ u = 1, ǫ v = 0, ∆ contains an undirected edge ( u , v ) .
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