Matrix group structure and Markov invariants in the strand symmetric phylogenetic substitution model

We consider the continuous time presentation of the strand symmetric phylogenetic substitution model (in which rate matrices are unchanged under Watson-Crick base conjugation). Analysis of the model's underlying structure as a matrix group leads to a presentation of the rate generator matrix in an appropriate basis in terms of a two-part block decomposition. Markov invariants are classified and enumerated, and the quadratic invariants for the two-leaf phylogenetic tree are evaluated explicitly. Two of these Markov invariants are zero, and hence are concurrently phylogenetic invariants, while the other two invariants provide independent estimates of the total of substitution rates within Watson-Crick conjugate pairs, and the total of remaining substitution rates across conjugate base pairs.