Binomiality of colored Gaussian models

TL;DR

This paper characterizes when colored Gaussian graphical models have binomial vanishing ideals, linking algebraic properties to combinatorial structures like Jordan schemes, and refutes a previous conjecture about automorphism groups.

Contribution

It provides a necessary and sufficient condition for binomiality in colored Gaussian models using Jordan schemes, advancing understanding of their algebraic and combinatorial structure.

Findings

Characterizes binomiality using Jordan schemes

Refutes the conjecture linking binomiality to automorphism group orbits

Establishes new algebraic-combinatorial connections in Gaussian models

Abstract

Following earlier work by Coons-Maraj-Misra-Sorea and Misra-Sullivant, we study colored, undirected Gaussian graphical models, and present a necessary and sufficient condition for such a model to have binomial vanishing ideal. These conditions involve Jordan schemes, a variant of association schemes, well-known structures in algebraic combinatorics. Using association schemes without transitive group action, we refute the conjecture by Coons-Maraj-Misra-Sorea that binomiality implies that the color classes must be orbits under the automorphism group of the colored graph.