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Factorized dynamics in soliton cellular automata with quantum group symmetry is identified with a motion of particles and anti-particles exhibiting pair creation and annihilation. An embedding scheme is presented showing that the…
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…
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…
A solvable vertex model in ferromagnetic regime gives rise to a soliton cellular automaton which is a discrete dynamical system in which site variables take on values in a finite set. We study the scattering of a class of soliton cellular…
We calculate the combinatorial $R$ matrix for all elements of $\mathcal{B}_l\otimes \mathcal{B}_1$ where $\mathcal{B}_l$ denotes the $G_2^{(1)}$-perfect crystal of level $l$, and then study the soliton cellular automaton constructed from…
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…
We review and generalize the recent progress in a soliton cellular automaton known as the periodic box-ball system. It has the extended affine Weyl group symmetry and admits the commuting transfer matrix method and the Bethe ansatz at q=0.…
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…
A cellular automaton that is a generalization of the box-ball system with either many kinds of balls or finite carrier capacity is proposed and studied through two discrete integrable systems: nonautonomous discrete KP lattice and…
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…
A cellular automaton is a deterministic and exactly computable dynamical system which mimics certain fundamental aspects of physical dynamics such as spatial locality and finite entropy. CA systems can be constructed which have additional…
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…
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…
Solvable vertex models in a ferromagnetic regime give rise to soliton cellular automata at q=0. By means of the crystal base theory, we study a class of such automata associated with the quantum affine algebra U_q(g_n) for non exceptional…
A soliton cellular automaton on a one dimensional semi-infinite lattice with a reflecting end is presented. It extends a box-ball system on an infinite lattice associated with the crystal base of U_q(sl_n). A commuting family of time…
An L operator is presented related to an infinite dimensional limit of the fusion R matrices for U_q(A^{(1)}_{n-1}) and U_q(D^{(1)}_n). It is factorized into the local propagation operators which quantize the deterministic dynamics of…
In terms of the crystal base of a quantum affine algebra $U_q(\mathfrak{g})$, we study a soliton cellular automaton (SCA) associated with the exceptional affine Lie algebra $\mathfrak{g}=D_4^{(3)}$. The solitons therein are labeled by the…
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…
We propose a box and ball system with a periodic boundary condition (pBBS). The time evolution rule of the pBBS is represented as a Boolean recurrence formula, an inverse ultradiscretization of which is shown to be equivalent with the…
We investigate a soliton cellular automaton (Box-Ball system) with periodic boundary conditions. Since the cellular automaton is a deterministic dynamical system that takes only a finite number of states, it will exhibit periodic motion. We…