Phylogenetic Invariants: From the General Markov to Equivariant Models

Abstract

In the last decade, some algebraic tools have been successfully applied to phylogenetic reconstruction. These tools are mainly based on the knowledge of equations describing algebraic varieties associated to Markov processes of molecular substitution on phylogenetic trees, the so called phylogenetic invariants. Although the theory involved allows for the explicit determination of these equations for all equivariant models (which include some of the most popular nucleotide substitution models), practical uses of these algebraic tools have been restricted to the case of the general Markov model. Arguably, one of the reasons for this restriction is that knowledge of linear representation theory is required before making these equations explicit. With the aim of enlarging the practical uses of algebraic phylogenetics, in this paper we prove that phylogenetic invariants for equivariant models can be derived from phylogenetic invariants for the general Markov model without the need of representation theory. Our main result states that the algebraic variety corresponding to an equivariant model on a phylogenetic tree T is an irreducible component of the variety corresponding to the general Markov model on T intersected with the linear space defined by the model. We also prove that for any equivariant model, those phylogenetic invariants that are relevant for practical uses (e.g., tree reconstruction) can be simply deduced from a single rank constraint on the matrices obtained by flattening the joint distribution at the leaves of the tree. This condition can be easily tested from singular values of the matrices, and it allows us to extend our results from trees to phylogenetic networks.