Tropical flag varieties

TL;DR

This paper introduces the flag Dressian as a tropical analogue of the partial flag variety, establishing a correspondence with valuated flag matroids and tropical linear spaces, and explores realizability of these structures.

Contribution

It defines the flag Dressian, characterizes projective tropical linear spaces, and proves realizability results for small ground sets, advancing tropical geometry and combinatorics.

Findings

Established a correspondence between points on the flag Dressian and valuated flag matroids.

Proved all valuated flag matroids on ground sets up to size 5 are realizable.

Provided a counterexample showing non-realizability for a flag matroid on 6 elements.

Abstract

Flag matroids are combinatorial abstractions of flags of linear subspaces, just as matroids are of linear subspaces. We introduce the flag Dressian as a tropical analogue of the partial flag variety, and prove a correspondence between: (a) points on the flag Dressian, (b) valuated flag matroids, (c) flags of projective tropical linear spaces, and (d) coherent flag matroidal subdivisions. We introduce and characterize projective tropical linear spaces, which serve as a fundamental tool in our proof. We apply the correspondence to prove that all valuated flag matroids on ground set up to size 5 are realizable, and give an example where this fails for a flag matroid on 6 elements.