  Let V be an n-dimensional vector space and let On be the orthogonal group.
Motivated by a question of B. Szegedy (B. Szegedy, Edge coloring models and
reflection positivity, Journal of the American Mathematical Society Volume 20,
Number 4, 2007), about the rank of edge connection matrices of partition
functions of vertex models, we give a combinatorial parameterization of tensors
in V \otimes k invariant under certain subgroups of the orthogonal group. This
allows us to give an answer to this question for vertex models with values in
an algebraically closed field of characteristic zero.
