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arXiv · 1503.00337

On partition functions for 3-graphs

Guus Regts·Alexander Schrijver·Bart Sevenster
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Abstract

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 `$k$-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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Guus Regts, Alexander Schrijver, Bart Sevenster. 2015-03-01. On partition functions for 3-graphs. https://arxiv.org/abs/1503.00337

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