Polyhedral and Tropical Geometry of Flag Positroids

TL;DR

This paper investigates the polyhedral and tropical geometry of flag positroids, establishing key equalities and subdivisions, and demonstrating realizability of positively oriented flag matroids, with applications to Bruhat interval polytopes.

Contribution

It proves the equality of nonnegative tropical flag variety and flag Dressian for consecutive ranks, and shows all positively oriented flag matroids of such ranks are realizable.

Findings

Nonnegative tropical flag variety equals flag Dressian for consecutive ranks.

Flag positroid polytopes can be subdivided into flag positroid polytopes via tropical points.

Positively oriented flag matroids of consecutive ranks are always realizable.

Abstract

A flag positroid of ranks $\boldsymbol{r}:=(r_1<\dots <r_k)$ on $[n]$ is a flag matroid that can be realized by a real $r_k \times n$ matrix $A$ such that the $r_i \times r_i$ minors of $A$ involving rows $1,2,\dots,r_i$ are nonnegative for all $1\leq i \leq k$. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when $\boldsymbol{r}:=(a, a+1,\dots,b)$ is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety TrFl$_{\boldsymbol{r},n}^{\geq 0}$ equals the nonnegative flag Dressian FlDr$_{\boldsymbol{r},n}^{\geq 0}$, and that the points $\boldsymbol{\mu} = (\mu_a,\ldots, \mu_b)$ of TrFl$_{\boldsymbol{r},n}^{\geq 0} =$ FlDr$_{\boldsymbol{r},n}^{\geq 0}$ give rise to coherent subdivisions of the flag positroid polytope $P(\underline{\boldsymbol{\mu}})$ into flag positroid polytopes. Our results have applications to Bruhat interval polytopes: for example, we show that a complete flag matroid polytope is a Bruhat interval polytope if and only if its $(\leq 2)$-dimensional faces are Bruhat interval polytopes. Our results also have applications to realizability questions. We define a positively oriented flag matroid to be a sequence of positively oriented matroids $(\chi_1,\dots,\chi_k)$ which is also an oriented flag matroid. We then prove that every positively oriented flag matroid of ranks $\boldsymbol{r}=(a,a+1,\dots,b)$ is realizable.