Markov invariants and the isotropy subgroup of a quartet tree

TL;DR

This paper uses the isotropy subgroup of leaf permutations to systematically identify Markov invariants for phylogenetic trees, providing explicit constructions for quartets and extending to general trees.

Contribution

It introduces a systematic method to identify tree-informative invariants using group representation theory, specifically for quartet trees, and extends the approach to arbitrary trees.

Findings

Explicit construction of all representations for quartet trees

Identification of linear combinations of invariants that vanish for specific quartets

Method applicable to general trees and related phylogenetic invariants

Abstract

The purpose of this article is to show how the isotropy subgroup of leaf permutations on binary trees can be used to systematically identify tree-informative invariants relevant to models of phylogenetic evolution. In the quartet case, we give an explicit construction of the full set of representations and describe their properties. We apply these results directly to Markov invariants, thereby extending previous theoretical results by systematically identifying linear combinations that vanish for a given quartet. We also note that the theory is fully generalizable to arbitrary trees and is equally applicable to the related case of phylogenetic invariants. All results follow from elementary consideration of the representation theory of finite groups.