Sum rules for the supersymmetric eight-vertex model

The eight-vertex model on the square lattice with vertex weights a, b, c, d obeying the relation (a² + ab)(b² + ab) = (c² + ab)(d² + ab) is considered. Its transfer matrix with L = 2n + 1, n ⩾ 0, vertical lines and periodic boundary conditions along the horizontal direction has the doubly-degenerate eigenvalue Θ_n = (a + b)²ⁿ⁺¹. A basis of the corresponding eigenspace is investigated. Several scalar products involving the basis vectors are computed in terms of a family of polynomials introduced by Rosengren and Zinn-Justin. These scalar products are used to find explicit expressions for particular entries of the vectors. The proofs of these results are based on the generalisation of the eigenvalue problem for Θ_n to the inhomogeneous eight-vertex model. (paper)