Polyhedral and Tropical Geometry of Flag Positroids

Abstract

A flag positroid of ranks $\mathbf{r}:=(r_1<\cdots <r_k)$ 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 $\mathbf{r}:=(a,a+1,\dots ,b)$ is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety TrFl${}_{\mathbf{r},n}^{\ge 0}$ equals the nonnegative flag Dressian FlDr${}_{\mathbf{r},n}^{\ge 0}$, and that the points $\mathbf{\mu }=({\mu }_a,\dots ,{\mu }_b)$ of TrFl${}_{\mathbf{r},n}^{\ge 0}=$ FlDr${}_{\mathbf{r},n}^{\ge 0}$ give rise to coherent subdivisions of the flag positroid polytope $P(\underline{\mathbf{\mu }})$ 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 $\mathbf{r}=(a,a+1,\dots ,b)$ is realizable.

43 pages New version containing an updated version of the main theorem, which originally contained an error. An erratum, available at https://www.math.cmu.edu/~ceur/pdf/BEW_NonnegTropFlagVar_Erratum.pdf, has been submitted to supplement the published version of this article

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