The A⁽¹⁾_M automata related to crystals of symmetric tensors

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 Kadomtsev-Petviashivili equation. A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter.