Phylogenetic invariants for $\mathbb{Z}_3$ scheme-theoretically

TL;DR

This paper investigates the algebraic structure of phylogenetic invariants for models with a cyclic symmetry group of order three, proving a key equivalence between certain projective schemes related to these invariants.

Contribution

It proves that the projective schemes of the ideal of phylogenetic invariants and its degree-3 generated subideal are identical for the $bZ_3$ model, supporting a conjecture by Sturmfels and Sullivant.

Findings

The ideals I and I' define the same projective scheme.

Degree-3 generators suffice to describe the invariants.

Supports the conjecture that I equals I'.

Abstract

We study phylogenetic invariants of models of evolution whose group of symmetries is the cyclic group with 3 elements. We prove that projective schemes corresponding to the ideal I of phylogenetic invariants of such a model and to its subideal I' generated by elements of degree at most 3 are the same. This is motivated by a conjecture of Sturmfels and Sullivant, which would imply that I = I'.