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.

We obtain a determinant expression for the tree-level structure constant of three non-extremal single-trace operators in the SU (2) sector of planar N = 4 supersymmetric Yang-Mills theory.

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Let H be the corner-transfer-matrix Hamiltonian for the six-vertex model in the antiferroelectric regime. It acts on the infinite tensor product W = V ⊗ V ⊗ V ⊗ · · ·, where V is the 2-dimensional irreducible representation of the quantum affine Lie algebra U q sl (2) . We observe that H is the derivation of U q sl (2) , and conjecture that the eigenvectors of H form the level-1 vacuum representation of U q sl (2) . We report on checks in support of our conjecture.

Dedicated to Professor Richard Askey on the occasion of his 65th birthday.Abstract. We study combinatorial aspects of q-weighted, length-L Forrester-Baxter paths,, and p and p ′ are co-prime.We obtain a bijection between P p,p ′ a,b,c (L) and partitions with certain prescribed hook differences. Thereby, we obtain a new description of the q-weights of P p,p ′ a,b,c (L). Using the new weights, and defining s 0 and r 0 to be the smallest non-negative integers for which |ps 0 − p ′ r 0 | = 1, we restrict the discussion to P p,p ′ s 0