  We explore model based techniques of phylogenetic tree inference exercising
Markov invariants. Markov invariants are group invariant polynomials and are
distinct from what is known in the literature as phylogenetic invariants,
although we establish a commonality in some special cases. We show that the
simplest Markov invariant forms the foundation of the Log-Det distance measure.
We take as our primary tool group representation theory, and show that it
provides a general framework for analysing Markov processes on trees. From this
algebraic perspective, the inherent symmetries of these processes become
apparent, and focusing on plethysms, we are able to define Markov invariants
and give existence proofs. We give an explicit technique for constructing the
invariants, valid for any number of character states and taxa. For phylogenetic
trees with three and four leaves, we demonstrate that the corresponding Markov
invariants can be fruitfully exploited in applied phylogenetic studies.
