The complete system of algebraic invariants for the sixteen-vertex model

In a previous paper, the partition function of the 16-vertex model was shown to be invariant under a group of linear transformations in the space of the vertex weights. According to a theorem by Hilbert, every algebraic invariant such as the partition function for a finite lattice can be expressed algebraically in terms of a finite set of basic algebraic invariants, which are sums of products of the vertex weights. We construct this set by analysing the structural properties of the transformation group (the direct product of two three-dimensional orthogonal groups). The basic set is found to consist of 21 invariants, ranging from a linear invariant up to invariants of the ninth degree. In particular cases, notably the (general or the symmetric) eight-vertex model, the six-vertex model and the free-fermion model, several invariants vanish and a number of additional algebraic relations between the basic invariants are obtained.