  The purpose of this article is to show how the isotropy subgroup of leaf
permutations on binary trees can be used to systematically identify
tree-informative invariants relevant to models of phylogenetic evolution. In
the quartet case, we give an explicit construction of the full set of
representations and describe their properties. We apply these results directly
to Markov invariants, thereby extending previous theoretical results by
systematically identifying linear combinations that vanish for a given quartet.
We also note that the theory is fully generalizable to arbitrary trees and is
equally applicable to the related case of phylogenetic invariants. All results
follow from elementary consideration of the representation theory of finite
groups.
