A characterization of edge reflection positive partition functions of vertex coloring models

Szegedy (B. Szegedy, Edge coloring models and reflection positivity, Journal of the American Mathematical Society 20, 2007, 969-988.) showed that the partition function of any vertex coloring model is equal to the partition function of a complex edge coloring model. Using some results in geometric invariant theory, we characterize for which vertex coloring model the edge coloring model can be taken to be real valued that is, we characterize which partition functions of vertex coloring models are edge reflection positive. This answers a question posed by Szegedy.