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 `$k$-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.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2015
- DOI:
- 10.48550/arXiv.1503.00337
- arXiv:
- arXiv:1503.00337
- Bibcode:
- 2015arXiv150300337R
- Keywords:
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- Mathematics - Quantum Algebra;
- Mathematics - Combinatorics;
- Mathematics - Representation Theory;
- 15A72;
- 17Bxx;
- 20C30;
- 57M27;
- 05C25