Dissimilarity maps on trees and the representation theory of $GL_n(\C)$

We revisit the 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 $K_n$ maps to the tropical varieties of every flag variety of $GL_n(\C).$ Using this map, we interpret the tropicalization of the semistandard tableaux basis of an irreducible representation of $GL_n(\C)$ as combinatorial invariants of phylogenetic trees.