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Inspired by the study of the minimal excludant in integer partitions by G.E. Andrews and D. Newman, we introduce a pair of new partition statistics, squrank and recrank. It is related to a polynomial bosonic form of statistical configuration sums for an integrable cellular automaton. For all nonnegative integer $n$, we prove that the partitions of $n$ on which squrank or recrank takes on a particular value, say $r$, are equinumerous with the partitions of $n$ on which the odd/even minimal exclutant takes on the corresponding value, $2r+1$ or $2r+2$., Comment: 23 pages, 6 figures

We present two types of systems of differential equations that can be derived from a set of discrete integrable systems which we call the closed geometric crystal chains. One is a kind of extended Lotka-Volterra systems, and the other seems to be generally new but reduces to a previously known system in a special case. Both equations have Lax representations associated with what are known as the loop elementary symmetric functions, which were originally introduced to describe products of affine type A geometric crystals for symmetric tensor representations. Examples of the derivations of the continuous time Lax equations from a discrete time one are described in detail, where a novel method of taking a continuum limit by assuming asymptotic behaviors of the eigenvalues of the Lax matrix in Puiseux series expansions is used., Comment: v2: 29 pages, many corrections

Inspired by G. Frieden's recent work on the geometric R-matrix for affine type A crystal associated with rectangular shaped Young tableaux, we propose a method to construct a novel family of discrete integrable systems which can be regarded as a geometric lifting of the generalized periodic box-ball systems. By converting the conventional usage of the matrices for defining the Lax representation of the discrete periodic Toda chain, together with a clever use of the Perron-Frobenious theorem, we give a definition of our systems. It is carried out on the space of real positive dependent variables, without regarding them to be written by subtraction-free rational functions of independent variables but nevertheless with the conserved quantities which can be tropicalized. We prove that, in this setup an equation of an analogue of the `carrier' of the box-ball system for assuring its periodic boundary condition always has a unique solution. As a result, any states in our systems admit a commuting family of time evolutions associated with any rectangular shaped tableaux, in contrast to the case of corresponding generalized periodic box-ball systems where some states did not admit some of such time evolutions., Comment: 39 pages, 5 figures. v2: added 2 references

A new generalized Wick theorem for interacting fields in 2D conformal field theory is described. We briefly discuss its relation to the Borcherds identity and its derivation by an analytic method. Examples of the calculations of the operator product expansions by using the generalized Wick theorems including fermionic fields are also presented., Comment: 12 pages, to appear in the Proceedings of the International Workshop "CQIS-2016" (St. Petersburg, Russia)

As a counterpart of the well-known generalized Wick theorem by Bais et. al. in 1988 for interacting fields in two dimensional conformal field theory, we present a new contour integral formula for the operator product expansion of a normally ordered operator and a single operator on its right hand. Quite similar to the original Wick theorem for the opposite order operator product, it expresses the contraction i.e.the singular part of the operator product expansion as a contour integral of only two terms, each of which is a product of a contraction and a single operator. We discuss the relation between these formulas and the Borcherds identity satisfied by the quantum fields associated with the theory of vertex algebras., Comment: 28 pages

The tropical periodic Toda lattice (trop p-Toda) is a dynamical system attracting attentions in the area of the interplay of integrable systems and tropical geometry. We show that the Young diagrams associated with trop p-Toda given by two very different definitions are identical. The first definition is given by a Lax representation of the discrete periodic Toda lattice, and the second one is associated with a generalization of the Kerov-Kirillov-Reshetikhin bijection in the combinatorics of the Bethe ansatz. By means of this identification, it is shown for the first time that the Young diagrams given by the latter definition are preserved under time evolution. This result is regarded as an important first step in clarifying the iso-level set structure of this dynamical system in general cases, i.e. not restricted to generic cases., Comment: 27 pages, 9 figures

The tropical (ultradiscrete) periodic Toda lattice is a dynamical system derived from a time-discretized version of the periodic Toda lattice through a limiting procedure called tropicalization. We propose a new formulation for this dynamical system with a representation by two-colored strips. Based on this formulation, a family of its commuting time evolutions is constructed., Comment: 16 pages

The box-ball system is an integrable cellular automaton on one dimensional lattice. It arises from either quantum or classical integrable systems by the 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 article, we demonstrate these integrable structures of the box-ball system and its generalizations based on the developments in the last two decades., Comment: 73 pages

Fermionic formulas in combinatorial Bethe ansatz consist of sums of products of q-binomial coefficients. There exist refinements without a sum that are known to yield partition functions of box-ball systems with a prescribed soliton content. In this paper, such a refined fermionic formula is extended to the periodic box-ball system and a q-analogue of the Bethe root counting formula for XXZ chain at $\Delta=\infty$., Comment: 24 pages

We study an integrable vertex model with a periodic boundary condition associated with U_q(A_n^{(1)}) at the crystallizing point q=0. It is an (n+1)-state cellular automaton describing the factorized scattering of solitons. The dynamics originates in the commuting family of fusion transfer matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the periodic box-ball system. Combining Bethe ansatz and crystal theory in quantum group, we develop an inverse scattering/spectral formalism and solve the initial value problem based on several conjectures. The action-angle variables are constructed representing the amplitudes and phases of solitons. By the direct and inverse scattering maps, separation of variables into solitons is achieved and nonlinear dynamics is transformed into a straight motion on a tropical analogue of the Jacobi variety. We decompose the level set into connected components under the commuting family of time evolutions and identify each of them with the set of integer points on a torus. The weight multiplicity formula derived from the q=0 Bethe equation acquires an elegant interpretation as the volume of the phase space expressed by the size and multiplicity of these tori. The dynamical period is determined as an explicit arithmetical function of the n-tuple of Young diagrams specifying the level set. The inverse map, i.e., tropical Jacobi inversion is expressed in terms of a tropical Riemann theta function associated with the Bethe ansatz data. As an application, time average of some local variable is calculated.

Based on a group theoretical setting a sort of discrete dynamical system is constructed and applied to a combinatorial dynamical system defined on the set of certain Bethe ansatz related objects known as the rigged configurations. This system is then used to study a one-dimensional periodic cellular automaton related to discrete Toda lattice. It is shown for the first time that the level set of this cellular automaton is decomposed into connected components and every such component is a torus.

Motivated by an attempt to develop a method for solving initial value problems in a class of one dimensional periodic cellular automata (CA) associated with crystal bases and soliton equations, we consider a generalization of a simple proposition in elementary mathematics. The original proposition says that any sequence of letters 1 and 2, having no less 1's than 2's, can be changed into a ballot sequence via cyclic shifts only. We generalize it to treat sequences of cells of common capacity s > 1, each of them containing consecutive 2's (left) and 1's (right), and show that these sequences can be changed into a ballot sequence via two manipulations, cyclic and "quasi-cyclic" shifts. The latter is a new CA rule and we find that various kink-like structures are traveling along the system like particles under the time evolution of this rule., Comment: 31 pages. Section 1 changed and section 5 added

We formulate the inverse scattering method for a periodic box-ball system and solve the initial value problem. It is done by a synthesis of the combinatorial Bethe ansa"tze at q=1 and q=0, which provides the ultradiscrete analogue of quasi-periodic solutions in soliton equations, e.g., action-angle variables, Jacobi varieties, period matrices and so forth. As an application we establish explicit formulas counting the states characterized by conserved quantities and the generic and fundamental period under the commuting family of time evolutions., Comment: 43 pages

The Kerov-Kirillov-Reshetikhin (KKR) bijection is the crux in proving fermionic formulas. It is defined by a combinatorial algorithm on rigged configurations and highest paths. We reformulate the KKR bijection as a vertex operator by purely using combinatorial R in crystal base theory. The result is viewed as a nested Bethe ansatz at q=0 as well as the direct and the inverse scattering (Gel'fand-Levitan) map in the associated soliton cellular automaton., Comment: 28 pages

We present an algorithm to reduce the coloured box-ball system, a one dimensional integrable cellular automaton described by motions of several colour (kind) of balls, into a simpler monochrome system. This algorithm extracts the colour degree of freedom of the automaton as a word which turns out to be a conserved quantity of this dynamical system. It is based on the theory of crystal basis and in particular on the tensor products of sl_n crystals of symmetric and anti-symmetric tensor representations., Comment: 19 pages

A set of action-angle variables for a soliton cellular automaton is obtained. It is identified with the rigged configuration, a well-known object in Bethe ansatz. Regarding it as the set of scattering data an inverse scattering method to solve initial value problems of this automaton is presented. By considering partition functions for this system a new interpretation of a fermionic character formula is obtained., Comment: 29 pages, LaTeX

We present a derivation of a formula that gives dynamics of an integrable cellular automaton associated with crystal bases. This automaton is related to type D affine Lie algebra and contains usual box-ball systems as a special case. The dynamics is described by means of such objects as carriers, particles, and antiparticles. We derive it from an analysis of a recently obtained formula of the combinatorial R (an intertwiner between tensor products of crystals) that was found in a study of geometric crystals., Comment: LaTeX, 21 pages, 2 figures

Tropical R is the birational map that intertwines products of geometric crystals and satisfies the Yang-Baxter equation. We show that the D^{(1)}_n tropical R introduced by the authors and its reduction to A^{(2)}_{2n-1} and C^{(1)}_n are equivalent to a system of bilinear difference equations of Hirota type. Associated tropical vertex models admit solutions in terms of tau functions of the BKP and DKP hierarchies., Comment: LaTeX2e, 26pages

We construct a geometric crystal for the affine Lie algebra D^{(1)}_n in the sense of Berenstein and Kazhdan. Based on a matrix realization including a spectral parameter, we prove uniqueness and explicit form of the tropical R, the birational map that intertwines products of the geometric crystals. The tropical R commutes with geometric Kashiwara operators and satisfies the Yang-Baxter equation. It is subtraction-free and yields a piecewise linear formula of the combinatorial R for crystals upon ultradiscretization., Comment: 41 pages

We present an elementary algorithm for the dynamics of recently introduced soliton cellular automata associated with quantum affine algebra U_q(g_n) at q=0. For g_n = A^{(1)}_n, the rule reproduces the ball-moving algorithm in Takahashi-Satsuma's box-ball system. For non-exceptional g_n other than A^{(1)}_n, it is described as a motion of particles and anti-particles which undergo pair-annihilation and creation through a neutral bound state. The algorithm is formulated without using representation theory nor crystal basis theory., Comment: LaTex2e 9 pages, no figure. For proceedings of SIDE IV conference

For coherent families of crystals of affine Lie algebras of type B^{(1)}_n, D^{(1)}_n, A^{(2)}_{2n} and D^{(2)}_{n+1} we describe the combinatorial R matrix using column insertion algorithms for B,C,D Young tableaux., Comment: 39 pages, LaTeX. This is a continuation of the authors' work appeared in "Physical Combinatorics", ed. M.Kashiwara and T.Miwa, Birkha"user, Boston, 2000

Solvable vertex models in statistical mechanics give rise to soliton cellular automata at q=0 in a ferromagnetic regime. By means of the crystal base theory we study a class of such automata associated with non-exceptional quantum affine algebras U'_q(\hat{\geh}_n). Let B_l be the crystal of the U'_q(\hat{\geh}_n)-module corresponding to the l-fold symmetric fusion of the vector representation. For any crystal of the form B = B_{l_1} \otimes ... \otimes B_{l_N}, we prove that the combinatorial R matrix B_M \otimes B \xrightarrow{\sim} B \otimes B_M is factorized into a product of Weyl group operators in a certain domain if M is sufficiently large. It implies the factorization of certain transfer matrix at q=0, hence the time evolution in the associated cellular automata. The result generalizes the ball-moving algorithm in the box-ball systems., Comment: 19pages, LaTeX2e, no figure. For proceedings of The Baxter revolution in mathematical physics. (revised version)

A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra U'_q(A^{(1)}_M) is introduced. It 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(A^{(1)}_{M-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., Comment: 45 pages, latex2e, 2 figures

The combinatorial R matrices are obtained for a family {B_l} of crystals for U'_q(C^{(1)}_n) and U'_q(A^{(2)}_{2n-1}), where B_l is the crystal of the irreducible module corresponding to the one-row Young diagram of length l. The isomorphism B_l\otimes B_k \simeq B_k \otimes B_l and the energy function are described explicitly in terms of a C_n-analogue of the Robinson-Schensted-Knuth type insertion algorithm. As an application a C^{(1)}_n-analogue of the Kostka polynomials is calculated for several cases., Comment: 38 pages, latex2e, no figure

We introduce a class of cellular automata associated with crystals of irreducible finite dimensional representations of quantum affine algebras U'_q(\hat{\geh}_n). They have solitons labeled by crystals of the smaller algebra U'_q(\hat{\geh}_{n-1}). We prove stable propagation of one soliton for \hat{\geh}_n = A^{(2)}_{2n-1}, A^{(2)}_{2n}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}. For \gh_n = C^{(1)}_n, we also prove that the scattering matrices of two solitons coincide with the combinatorial R matrices of U'_q(C^{(1)}_{n-1})-crystals., Comment: 29 pages, 1 figure, LaTeX2e

We consider a category of finite crystals of a quantum affine algebra whose objects are not necessarily perfect, and set of paths, semi-infinite tensor product of an object of this category with a certain boundary condition. It is shown that the set of paths is isomorphic to a direct sum of infinitely many, in general, crystals of integrable highest weight modules. We present examples from C_n^{(1)} and A_{n-1}^{(1)}, in which the direct sum becomes a tensor product as suggested from the Bethe Ansatz., Comment: 15 pages, LaTeX2e, submitted to the proceedings of the RIMS98 program "Combinatorial Methods in Representation Theory"

Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general non-twisted quantum affine algebra U_q(X^{(1)}_n) and discuss several aspects related to representation theories and combinatorics. They include crystal base theory, one dimensional sums, spinon character formulae, Q-system and combinatorial completeness of the string hypothesis for arbitrary X_n., Comment: 49 pages, no figures, LaTeX2e using package amsmath

Let B_{(l)} be the perfect crystal for the l-symmetric tensor representation of the quantum affine algebra U'_q(\hat{sl(n)}). For a partition mu = (mu_1,...,mu_m), elements of the tensor product B_{(mu_1)} \otimes ... \otimes B_{(mu_m)} can be regarded as inhomogeneous paths. We establish a bijection between a certain large mu limit of this crystal and the crystal of an (generally reducible) integrable U_q(\hat{sl(n)})-module, which forms a large family depending on the inhomogeneity of mu kept in the limit. For the associated one dimensional sums, relations with the Kostka-Foulkes polynomials are clarified, and new fermionic formulae are presented. By combining their limits with the bijection, we prove or conjecture several formulae for the string functions, branching functions, coset branching functions and spinon character formula of both vertex and RSOS types., Comment: 42 pages, LaTeX2.09