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

TL;DR

This paper explores the connection between phylogenetic trees and tropical geometry, showing that the space of trees maps to tropical varieties of flag varieties of GL_n(C), linking representation theory and biology.

Contribution

It introduces a new perspective by mapping phylogenetic trees to tropical varieties of all flag varieties of GL_n(C), extending previous work on dissimilarity vectors.

Findings

Dissimilarity vectors of trees lie on the tropical Grassmannian.

The space of phylogenetic trees maps to tropical varieties of flag varieties.

Tropicalization of tableaux bases encodes tree invariants.

Abstract

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.