A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebraIt is a crystal theoretic formulation of the generalized box-ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous. Scattering matrices of two solitons coincide with the combinatorial R matrices of U ′ q (AM −1 ). A piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete KP equation. A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter.(1) M ) corresponding to the k-fold symmetric tensor representation. As a set it consists of the single row semistandard tableaux of length k on letters {1, 2, . . . , M + 1}:where we have omitted the k − 1 vertical lines separating the entries. We also represent the elements by the multiplicities of their contents. Namely, b = m 1 · · · m k ∈ B k is also denoted by b = (x 1 , x 2 , · · · , x M +1 ) with x i = #{l | m l = i}.

The box-ball system is an integrable cellular automaton on a one-dimensional lattice. It arises from either quantum or classical integrable systems by procedures called crystallization and ultradiscretization, respectively. The double origin of the integrability has endowed the box-ball system with a variety of aspects related to Yang-Baxter integrable models in statistical mechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz, geometric crystals, classical theory of solitons, tau functions, inverse scattering method, action-angle variables and invariant tori in completely integrable systems, spectral curves, tropical geometry and so forth. In this review, we demonstrate these integrable structures of the box-ball system and its generalizations based on the developments in the last two decades.

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We prove the periodicities of the restricted T and Y-systems associated with the quantum affine algebra of type Br at any level. We also prove the dilogarithm identities for the Y-systems of type Br at any level. Our proof is based on the tropical Y-systems and the categorification of the cluster algebra associated with any skew-symmetric matrix by Plamondon. Using this new method, we also give an alternative and simplified proof of the periodicities of the T and Y-systems associated with pairs of simply laced Dynkin diagrams.

Abstract. The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).