The rank of edge connection matrices and the dimension of algebras of invariant tensors

We characterize the rank of edge connection matrices of partition functions of real vertex models, as the dimension of the homogeneous components of the algebra of GG-invariant tensors. Here GG is the subgroup of the real orthogonal group that stabilizes the vertex model. This answers a question of Balázs Szegedy from 2007.

► We characterize the rank of edge connection matrices of partition functions.
► It is equal to the dimension of tensors invariant under a subgroup of O(n)O(n).
► The proof is based upon a theorem of A. Schrijver characterizing invariant algebras.