We study the higher spin analogs of the six-vertex model on the basis of its symmetry under the quantum affine algebra [Formula: see text]. Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/ annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin 1/2, and that the n-particle space has an RSOS type structure rather than a simple tensor product of the one-particle space. This agrees with the picture proposed earlier by Reshetikhin.

We solved the Yang-Baxter equation for the R-matrices of three-state vertex models with ice condition, and obtained a complete list of solvable nineteen-vertex models and associated quantum spin Hamiltonians of spin one

Vertex operators associated with level two U q ( sl 2 ) modules are constructed explicitly using bosons and fermions. An integral formula is derived for the trace of products of vertex operators. These results are applied to give n-point spin correlation functions of an integrable S = 1 quantum spin chain, extending an earlier work of Jimbo et al for the case S = 1/2.

Any state of the box-ball system (BBS) together with its time evolution is described by the N-soliton solution (with appropriate choice of N) of the ultradiscrete KdV equation. It is shown that simultaneous elimination of all '10'-walls in a state of the BBS corresponds exactly to reducing the parameters that determine 'the size of a soliton' by one. This observation leads to an expression for the solution to the initial-value problem (IVP) for the BBS. Expressions for the solution to the IVP for the ultradiscrete Toda molecule equation and the periodic BBS are also presented.

We show that the initial value problem of a periodic box-ball system can be solved in an elementary way using simple combinatorial methods.A periodic box-ball system (PBBS) is a dynamical system of balls in an array of boxes with a periodic boundary condition [1,2]. The PBBS is obtained from the discrete KdV equation and the discrete Toda equation, both of which are known as typical integrable nonlinear discrete equations, through a limiting procedure called ultradiscretization [3,4]. Since the ultradiscretization preserves the main properties of the original discrete equations, and the solvability of the initial value problem being an important property of integrable equations, we expect that the initial value problem of the PBBS can also be solved. In fact, the initial value problem for the PBBS was first solved by inverse ultradiscretization combined with the method of inverse scattering transform of the discrete Toda equation [5] and recently by the Bethe ansatz for an integrable lattice model with quantum group symmetry at the deformation parameter q = 0 and q = 1 [6]. These two methods, however, require fairly specialized mathematical knowledge on algebraic curves or representation theory of quantum algebras.An important property which characterizes a state of the PBBS is the fundamental cycle of the state, i.e., the length of the trajectory to which it belongs. Its explicit formula as well as statistical distribution was obtained and its relation to the celebrated Riemann hypothesis was clarified [7,8,9]. To prove the formula for fundamental cycle, one of the key steps is to compare a state with its 'reduced states' constructed by the '10-elimination'. In this article, we show that the initial value problem of the PBBS is solved by simple combinatorial arguments -essentially given in Ref.[7] -with some remarkable features of the reduced states.First we quickly review the definition of the PBBS and its conserved quantities. Consider a one-dimensional array of boxes each with a capacity of one ball. A periodic boundary condition is imposed by assuming that the last box is adjacent to the first one. Let the number of boxes be N and that of balls be M . We assume M < N/2. An arrangement of M balls in N boxes is called a state of the PBBS. Denoting a vacant box by 0 and a filled box by 1, a state 1

We investigate conserved quantities of periodic box-ball systems ͑PBBS͒ with arbitrary kinds of balls and box capacity greater than or equal to 1. We introduce the notion of nonintersecting paths on the two dimensional array of boxes, and give a combinatorial formula for the conserved quantities of the generalized PBBS using these paths.

We investigate the link between a periodic box-ball system (PBBS) and a solvable lattice model. Introducing a PBBS with an integer parameter corresponding to the dimensionality of the auxiliary space for the lattice model, we prove an important relationship between the conserved quantities of states of the PBBS and eigenvectors constructed through the string hypothesis.

We present an expression for the solution to the initial value problem for the ultradiscrete periodic Toda equation. The expression provides explicit forms of all dependent variables of the equation, while the previously known solutions give only half of the dependent variables while the others have to be determined implicitly using the conserved quantities.