Phylogenetic invariants for group-based models

TL;DR

This paper studies algebraic varieties in group-based phylogenetic models, proposing a method to generate invariants, proving completeness for certain models, and exploring properties of these varieties, including non-normal cases.

Contribution

It introduces a new method for generating phylogenetic invariants and proves its completeness for the binary Jukes-Cantor model, advancing understanding of algebraic properties of these models.

Findings

Complete invariants for binary Jukes-Cantor model

Conjecture that the method yields all invariants for any tree

First example of a non-normal group-based model

Abstract

In this paper we investigate properties of algebraic varieties representing group-based phylogenetic models. We propose a method of generating many phylogenetic invariants. We prove that we obtain all invariants for any tree for the binary Jukes-Cantor model. We conjecture that our method can give all phylogenetic invariants for any tree. We show that for 3-Kimura our conjecture is equivalent to the conjecture of Sturmfels and Sullivant. This, combined with the results of Sturmfels and Sullivant, would make it possible to determine all phylogenetic invariants for any tree for 3-Kimura model, and also other phylogenetic models. Next we give the (first) example of a non-normal general group-based model for an abelian group. Following Kubjas we also determine some invariants of group-based models showing that the associated varieties do not have to be deformation equivalent.