Abstract. 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 Uq(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 Xn.

We introduce a fermionic formula associated with any quantum affine algebra Uq(X (r) N ). Guided by the interplay between corner transfer matrix and Bethe ansatz in solvable lattice models, we study several aspects related to representation theory, most crucially, the crystal basis theory. They include one dimensional sums over both finite and semi-infinite paths, spinon character formulae, Lepowski-Primc type conjectural formula for vacuum string functions, dilogarithm identities, Q-systems and their solution by characters of various classical subalgebras and so forth. The results expand [HKOTY1] including the twisted cases and more details on inhomogeneous paths consisting of non-perfect crystals. As a most intriguing example, certain inhomogeneous one dimensional sums conjecturally give rise to branching functions of an integrable G

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}.

Let B (l) be the perfect crystal for the l-symmetric tensor representation of the quantum affine algebra U ′ q ( sl n). For a partition µ = (µ1, . . . , µm), elements of the tensor product B (µ 1 ) ⊗· · ·⊗B (µm) can be regarded as inhomogeneous paths. We establish a bijection between a certain large µ limit of this crystal and the crystal of an (generally reducible) integrable Uq( sl n)-module, which forms a large family depending on the inhomogeneity of µ 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.

We introduce a class of cellular automata associated with crystals of irreducible finite dimensional representations of quantum affine algebras U

Abstract. 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 Uq(gn) for non exceptional series gn = An+1 . They possess a commuting family of time evolutions and solitons labeled by crystals of the smaller algebra Uq(g n−1 ). Two-soliton scattering rule is identified with the combinatorial R of Uq(g n−1 )-crystals, and the multi-soliton scattering is shown to factorize into the two-body ones.

We present an elementary algorithm for the dynamics of recently introduced soliton cellular automata associated with quantum affine algebra Uq(gn) at q = 0. Forn , the rule reproduces the ball-moving algorithm in Takahashi-Satsuma's box-ball system. For non-exceptional gn 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.

Abstract. The combinatorial R matrices are obtained for a family {B/} of crystals for U;(ql) and U;(A~_I)' where B/ is the crystal of the irreducible module corresponding to the one-row Young diagram oflength I. The isomorphism B/ ® Bk ~ Bk ® B/ and the energy function are described explicitly in terms of a Cn-analogue of the RobinsonSchensted-Knuth-type insertion algorithm. As an application, a C!I)-analogue of the Kostka polynomials is calculated for several cases.