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On partition functions for 3-graphs
A {\em cyclic graph} is a graph with at each vertex a cyclic order of the
edges incident with it specified. We characterize which real-valued functions
on the collection of cubic cyclic graphs are partition functions of a real
vertex model (P. de la Harpe, V.F.R. Jones, Graph invariants related to
statistical mechanical models: examples and problems, Journal of Combinatorial
Theory, Series B 57 (1993) 207--227). They are characterized by `weak
reflection positivity', which amounts to the positive semidefiniteness of
matrices based on the `-join' of cubic cyclic graphs (for all k\in\oZ_+).
Basic tools are the representation theory of the symmetric group and
geometric invariant theory, in particular the Hanlon-Wales theorem on the
decomposition of Brauer algebras and the Procesi-Schwarz theorem on
inequalities defining orbit spaces
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