Characterizing partition functions of the vertex model

TL;DR

This paper characterizes the graph parameters that can be represented as partition functions of vertex models over algebraically closed fields of characteristic zero, including conditions for finite rank of the moment matrix.

Contribution

It provides a complete characterization of partition functions of vertex models and conditions for finite rank moment matrices, advancing understanding of vertex model representations.

Findings

Identifies which graph parameters are partition functions of vertex models.

Characterizes when the vertex model's moment matrix has finite rank.

Provides algebraic conditions over fields of characteristic zero.

Abstract

We characterize which graph parameters are partition functions of a vertex model over an algebraically closed field of characteristic 0 (in the sense of de la Harpe and Jones). We moreover characterize when the vertex model can be taken so that its moment matrix has finite rank.