On partition functions for 3-graphs

TL;DR

This paper characterizes which real-valued functions on cubic cyclic graphs can be realized as partition functions of real vertex models, using concepts like weak reflection positivity and tools from representation theory and geometric invariant theory.

Contribution

It provides a complete characterization of partition functions for cubic cyclic graphs through positive semidefiniteness conditions and advanced algebraic tools.

Findings

Characterization of partition functions via weak reflection positivity

Use of symmetric group representation theory and invariant theory

Application of Hanlon-Wales and Procesi-Schwarz theorems

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.