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We consider novel phylogenetic models with rate matrices that arise via theembedding of a progenitor model on a small number of character states, into atarget model on a larger number of character states. Adaptingrepresentation-theoretic results from recent investigations of Markovinvariants for the general rate matrix model, we give a prescription foridentifying and counting Markov invariants for such `symmetric embedded'models, and we provide enumerations of these for low-dimensional cases. Thesimplest example is a target model on 3 states, constructed from a general 2state model; the `2->3' embedding. We show that for 2 taxa, there exist twoinvariants of quadratic degree, that can be used to directly infer pairwisedistances from observed sequences under this model. A simple simulation studyverifies their theoretical expected values, and suggests that, given theappropriateness of the model class, they have greater statistical power thanthe standard (log) Det invariant (which is of cubic degree for this case).

Markov invariants for phylogenetic rate matrices derived from embedded submodels