Markov invariants, plethysms, and phylogenetics (the long version)

TL;DR

This paper introduces Markov invariants as group invariant polynomials for phylogenetic tree inference, providing a new algebraic framework that enhances understanding and application of these invariants in phylogenetics.

Contribution

It establishes a general algebraic framework using group representation theory for analyzing Markov processes on trees and constructs Markov invariants for any number of states and taxa.

Findings

Markov invariants form the basis of the Log-Det distance measure.

Explicit construction techniques for invariants applicable to any number of states and taxa.

Demonstrated usefulness of invariants in phylogenetic studies with three and four leaves.

Abstract

We explore model based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that the simplest Markov invariant forms the foundation of the Log-Det distance measure. We take as our primary tool group representation theory, and show that it provides a general framework for analysing Markov processes on trees. From this algebraic perspective, the inherent symmetries of these processes become apparent, and focusing on plethysms, we are able to define Markov invariants and give existence proofs. We give an explicit technique for constructing the invariants, valid for any number of character states and taxa. For phylogenetic trees with three and four leaves, we demonstrate that the corresponding Markov invariants can be fruitfully exploited in applied phylogenetic studies.