Dissimilarity Maps on Trees and the Representation Theory of $GL_n(\mathbb{C})$

Abstract

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 $K_n$ maps to the tropical varieties of every flag variety of $GL_n(\mathbb{C}).$ 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 $GL_n(\mathbb{C})$ flag varieties.