Taichiro Takagi

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 .

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.

Inverse scattering method for a soliton cellular automaton

Abstract 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 solve initial value problems of this automaton is presented. By considering partition functions for this system a new interpretation of a fermionic character formula is obtained.

Integrable structure of box?ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry

The box–ball system is an integrable cellular automaton on a one-dimensional lattice. It arises from either quantum or classical integrable systems by procedures called crystallization and ultradiscretization, respectively. The double origin of the integrability has endowed the box–ball system with a variety of aspects related to Yang–Baxter integrable models in statistical mechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz, geometric crystals, classical theory of solitons, tau functions, inverse scattering method, action-angle variables and invariant tori in completely integrable systems, spectral curves, tropical geometry and so forth. In this review, we demonstrate these integrable structures of the box–ball system and its generalizations based on the developments in the last two decades. Dedicated to the memory of Professor Miki Wadati

Characters of Demazure modules and solvable lattice models

We study the path realization of Demazure crystals related to solvable lattice models in statistical mechanics. Various characters are represented in a unified way as the sums over one-dimensional configurations which we call unrestricted, classically restricted and restricted paths. As an application, characters of Demazure modules are obtained in terms of q-multinomial coefficients for several level-1 modules of classical affine algebras.

Bethe ansatz and inverse scattering transform in a periodic box–ball system

Abstract We formulate the inverse scattering method for a periodic box–ball system and solve the initial value problem. It is done by a synthesis of the combinatorial Bethe ansatzes at q = 1 and q = 0 , which provides the ultradiscrete analogue of quasi-periodic solutions in soliton equations, e.g., action–angle variables, Jacobi varieties, period matrices and so forth. As an application we establish explicit formulas counting the states characterized by conserved quantities and the generic and fundamental period under the commuting family of time evolutions.

Paths, Demazure crystals, and symmetric functions

The path realization of Demazure crystals is reviewed and Demazure characters in the light of symmetric functions are discussed.

Geometric crystal and tropical R for Dn(1)

We construct a geometric crystal for the affine Lie algebra D^{(1)}_n in the sense of Berenstein and Kazhdan. Based on a matrix realization including a spectral parameter, we prove uniqueness and explicit form of the tropical R, the birational map that intertwines products of the geometric crystals. The tropical R commutes with geometric Kashiwara operators and satisfies the Yang-Baxter equation. It is subtraction-free and yields a piecewise linear formula of the combinatorial R for crystals upon ultradiscretization.

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′