 
Summary: PHYLOGENETIC INVARIANTS FOR THE GENERAL MARKOV
MODEL OF SEQUENCE MUTATION
ELIZABETH S. ALLMAN 1 AND JOHN A. RHODES 2
Abstract. A phylogenetic invariant for a model of biological sequence evolu
tion along a phylogenetic tree is a polynomial that vanishes on the expected
frequencies of base patterns at the terminal taxa. While the use of these invari
ants for phylogenetic inference has long been of interest, explicitly constructing
such invariants has been problematic.
We construct invariants for the general Markov model of base sequence
evolution on an ntaxon tree, for any and n. The method depends primarily
on the observation that certain matrices defined in terms of expected pattern
frequencies must commute, and yields many invariants of degree +1, regard
less of the value of n. We define strong and parameterstrong sets of invariants,
and prove several theorems indicating that the set of invariants produced here
has these properties on certain sets of possible pattern frequencies. Thus our
invariants may be sufficient for phylogenetic applications.
Note: Text in smaller type does not appear in Mathematical Biosciences version.
1. Introduction
In [1], Cavender and Felsenstein and, in an independent work [16], Lake intro
duced an approach to phylogenetic tree construction from biological sequence data
