Polyhedral and Tropical Geometry of Flag Positroids

A flag positroid of ranks $\mathit{r}:=(r_1<\cdots <r_k)$ 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 $1\le i\le k$. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when $\mathit{r}:=(a,a+1,\dots ,b)$ is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety TrFl${}_{\mathit{r},n}^{\ge 0}$ equals the nonnegative flag Dressian FlDr${}_{\mathit{r},n}^{\ge 0}$, and that the points $\mathit{\mu }=({\mu }_a,\dots ,{\mu }_b)$ of TrFl${}_{\mathit{r},n}^{\ge 0}=$ FlDr${}_{\mathit{r},n}^{\ge 0}$ give rise to coherent subdivisions of the flag positroid polytope $P(\underset{‾}{\mathit{\mu }})$ 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 $(\le 2)$-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 $\mathit{r}=(a,a+1,\dots ,b)$ is realizable.