On partition functions for 3-graphs
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 `-join' of cubic cyclic graphs (for all ). 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.
Keywords
Cite
@article{arxiv.1503.00337,
title = {On partition functions for 3-graphs},
author = {Guus Regts and Alexander Schrijver and Bart Sevenster},
journal= {arXiv preprint arXiv:1503.00337},
year = {2016}
}