Polyhedral and Tropical Geometry of Flag Positroids

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.