Abstract. Reported are two applications of the functional relations (T -system) among a commuting family of row-to-row transfer matrices proposed in the previous paper Part I.For a general simple Lie algebra X r , we determine the correlation lengths of the associated massive vertex models in the anti-ferroelectric regime and central charges of the RSOS models in two critical regimes. The results reproduce known values or even generalize them, demonstrating the efficiency of the T -system.

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

T- and Y-systems are ubiquitous structures in classical and quantum integrable systems. They are difference equations having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov–Reshetikhin modules of quantum affine algebras, cluster algebras with coefficients, periodicity conjectures of Zamolodchikov and others, dilogarithm identities in conformal field theory, difference analog of L-operators in KP hierarchy, Stokes phenomena in 1D Schrödinger problem, AdS/CFT correspondence, Toda field equations on discrete spacetime, Laplace sequence in discrete geometry, Fermionic character formulas and combinatorial completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics, analytic and thermodynamic Bethe ansätze, quantum transfer matrix method and so forth. This review is a collection of short reviews on these topics which can be read more or less independently.

We study the analytic Bethe ansatz in solvable vertex models associated with the Yangian Y (X r ) or its quantum affine analogue U q (X (1) r ) for X r = B r , C r and D r . Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations of Y (X r ). Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying the T -system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semistandard Young tableaux.

Analytic Bethe ansatz is executed for a wide class of finite dimensional U q (B (1) r ) modules. They are labeled by skew-Young diagrams which, in general, contain a fragment corresponding to the spin representation. For the transfer matrix spectra of the relevant vertex models, we establish a number of formulae, which are U q (B (1) r ) analogues of the classical ones due to Jacobi-Trudi and Giambelli on Schur functions. They yield a full solution to the previously proposed functional relation (T -system), which is a Toda equation on discrete space-time. †

We formulate the inverse scattering method for a periodic boxball system and solve the initial value problem. It is done by a synthesis of the combinatorial Bethe ansätze 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. from the following diagram:

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