We diagonalize the anti-ferroelectric XXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of U q sl(2) . Our method is based on the representation theory of quantum affine algebras, the related vertex operators and KZ equation, and thereby bypasses the usual process of starting from a finite lattice, taking the thermodynamic limit and filling the Dirac sea. From recent results on the algebraic structure of the corner transfer matrix of the model, we obtain the vacuum vector of the Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex operators, which act as particle creation operators in the space of eigenvectors.We check the agreement of our results with those obtained using the Bethe Ansatz in a number of cases, and with others obtained in the scaling limitthe su(2)-invariant Thirring model.

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Abstract.A new approach to the correlation functions is presented for the XXZ model in the anti-ferroelectric regime. The method is based on the recent realization of the quantum affine symmetry using vertex operators. With the aid of a boson representation for the latter, an integral formula is found for correlation functions of arbitrary local operators. As a special case it reproduces the spontaneous staggered polarization obtained earlier by Baxter.

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The relation between the charge of Lascoux-Schützenberger and the energy function in solvable lattice models is clarified. As an application, A. N. Kirillov's conjecture on the expression of the branching coefficient of sln/sln as a limit of Kostka polynomials is proved.Mathematics Subject Classification (1991). 81R50, 82B23, 17B37, 05A30.

An expression of the multivariate sigma function associated with a (n,s)-curve is given in terms of algebraic integrals. As a corollary the first term of the series expansion around the origin of the sigma function is directly proved to be the Schur function determined from the gap sequence at infinity. *

We propose that the correlation functions of the inhomogeneous eightvertex model in the anti-ferroelectric regime satisfy a system of difference equations with respect to the spectral parameters. Solving the simplest difference equation we obtain the expression for the spontaneous staggered polarization conjectured by Baxter and Kelland. We also discuss a related construction of vertex operators on the lattice. IntroductionIn [1, 2] it was recognized that the correlation functions of the inhomogeneous six-vertex model in the anti-ferroelectric regime can be expressed as a trace of products of the q-deformed vertex operators. An explicit integral formula is given in [2]. These correlators satisfy a system of q-difference equations [3], that were introduced by Smirnov in the study of form factors in massive integrable QFT [4] and correspond to the level-0 case of the q-KZ equation of Frenkel and Reshetikhin [5]. (To be precise the equations for the correlators are 'dual' to Smirnov's ones, but we do not go into such details here). As remarked in [5], if one replaces the trigonometric R matrices appearing here by the elliptic ones, the resulting equations are still 'completely integrable'. In this paper we propose that the latter are precisely those satisfied by the correlators of the eight-vertex model in the anti-ferroelectric regime.Our method is based simply on the Yang-Baxter equation and the crossing symmetry and, as such, is applicable to more general models. This construction also allows one to interpret the q-vertex operator employed in [1,2] as an operator that inserts a dislocation (an extra half infinite line) on the lattice. In this *

We formulate the basic properties of q-vertex operators in the context of the Andrews-Baxter-Forrester (ABF) series, as an example of face-interaction models, derive the q-difference equations satisfied by their correlation functions, and establish their connection with representation theory. We also discuss the q-difference equations of the Kashiwara-Miwa (KM) series, as an example of edge-interaction models.Next, the Ising model-the simplest special case of both ABF and KM series-is studied in more detail using the Jordan-Wigner fermions. In particular, all matrix elements of vertex operators are calculated.