Coloured Graphical Models and their Symmetries

TL;DR

This paper explores the relationship between symmetries in coloured graphical models and the linear forms that vanish on their reciprocal varieties, providing a complete description for four specific families.

Contribution

It establishes a connection between graph symmetries and linear forms in coloured Gaussian models, offering new insights into their algebraic structure.

Findings

Symmetries determine the linear forms vanishing on the reciprocal variety.

Four families of models are fully characterized by their symmetries.

The work links graph symmetries to algebraic properties of Gaussian models.

Abstract

Coloured graphical models are Gaussian statistical models determined by an undirected coloured graph. These models can be described by linear spaces of symmetric matrices. We outline a relationship between the symmetries of the graph and the linear forms that vanish on the reciprocal variety of the model. In particular, we give four families for which such linear forms are completely described by symmetries.