Positivity for partial tropical flag varieties

TL;DR

This paper explores positivity notions in tropical flag varieties, characterizing their relationships and providing conditions for positivity in type A flag varieties, with implications for flag positroid subdivisions.

Contribution

It characterizes three positivity notions for tropical flag varieties in the hollow case and establishes their relationships, introducing valuated flag gammoids and conjecturing conditions for general ranks.

Findings

First example where two positivity notions do not coincide

Non-negative flag Dressian equals tropicalization of Pl"ucker non-negative flag variety

Regular flag positroid subdivisions derive from the non-negative Dressian

Abstract

We study positivity notions for the tropicalization of type A flag varieties and the flag Dressian. We focus on the hollow case, where we have one constituent of rank 1 and another of corank 1. We characterize the three different notions that exist for this case in terms of Pl\"ucker coordinates. These notions are the tropicalization of the totally non-negative flag variety, Bruhat subdivisions and the non-negative flag Dressian. We show the first example where the first two concepts above do not coincide. In this case the non-negative flag Dressian equals the tropicalization of the Pl\"ucker non-negative flag variety and is equivalent to flag positroid subdivisions. Using these characterizations, we provide similar conditions on the Pl\"ucker coordinates for flag varieties of arbitrary rank that are necessary for each of these positivity notions, and conjecture them to be sufficient. In particular we show that regular flag positroid subdivisions always come from the non-negative Dressian. Along the way, we show that the set of Bruhat interval polytopes equals the set of twisted Bruhat interval polytopes and we introduce valuated flag gammoids.