Dissimilarity Maps on Trees and the Representation Theory of GLn(ℂ)

We revisit representation theory in type A, used previously to establish that the dissimilarity vectors of phylogenetic trees are points on the tropical Grassmannian variety. We use a different version of this construction to show that the space of phylogenetic trees Kn maps to the tropical varieties of every flag variety of GLn(C). Using this map, we find a tropical function on the space of phylogenetic trees for each semistandard tableaux, and we show that the functions satisfy the tropicalized equations which cut out GLn(C) flag varieties.