Goro Hatayama

Paths, Crystals and Fermionic Formulae

We introduce a fermionic formula associated with any quantum affine algebra U q (X N (r) . Guided by the interplay between corner transfer matrix and the 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, Lepowsky—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 2 (1) -module related to the embedding G 2 (1) ↪ B 3 (1) ↪ D 4 1 .

The AM(1) automata related to crystals of symmetric tensors

A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra Uq′(AM(1)) 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 Uq′(AM−1(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.A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra Uq′(AM(1)) 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 Uq′(AM−1(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.

Character formulae of sl̂n-modules and inhomogeneous paths

Abstract 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(μ1) ⊗…⊗ 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 U q ′ ( 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.

Soliton cellular automata associated with crystal bases

Abstract We introduce a class of cellular automata associated with crystals of irreducible finite dimensional representations of quantum affine algebras U′ q ( g n ) . They have solitons labeled by crystals of the smaller algebra U′ q ( g n−1 ) . We prove stable propagation of one soliton for g n =A (2) 2n−1 ,A (2) 2n ,B (1) n ,C (1) n ,D (1) n and D (2) n +1 . For g 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.

Factorization of Combinatorial R Matrices and Associated Cellular Automata

AbstractSolvable 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′

Combinatorial R Matrices for a Family of Crystals: C n (1) and A 2n-1 (2) Cases

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Scattering rules in soliton cellular automata associated with crystal bases

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