A derivation of all linear invariants for a nonbalanced transversion model

Summary

The method of linear invariants discovered by Lake is a way of inferring phylogenies by testing statistical hypotheses. The main advantage of the method is that substitution rates for positions along the DNA sequence do not have to be identical. The assumptions and the algebraic background necessary for the applications of the method were clearly laid out in two papers by Cavender, who also described a way to derive a basis for the space of all linear invariants for rooted trees linking four species when the substitution process satisfies the assumption of balanced transversions. Cavender noted a generalization of linear invariants to certain more general substitution models. In this paper we give a simple explicit description of a basis for all linear invariants for a slight variant of Cavender's more general model, which applies to rooted trees linking any number of species. Bases for rooted trees linking five species are enumerated and the method applied to a problem concerning RNA polymerase sequence data.