Polyhedral and Tropical Geometry of Flag Positroids
Jonathan Boretsky, Christopher Eur, Lauren Williams
TL;DR
The paper studies the interplay between nonnegative tropicalizations of flag varieties, flag Dressians, and flag positroid polytopes, focusing on consecutive-rank flags to unify Grassmannian and complete-flag cases. It proves the central equivalence $TrFl_{oldsymbol r;n}^{\geq 0} = FlDr_{oldsymbol r;n}^{\geq 0}$ for $\boldsymbol r=(a,\dots,b)$ and shows that the induced coherent subdivisions $P(\underline{\mu})$ decompose into flag positroid polytopes, with all 2-dimensional faces also flag positroids; it also links to three-term relations and extendability lemmas. The authors introduce positively oriented flag matroids, establish realizability for consecutive ranks, and study projections of positive Richardsons to positroids, as well as diverse fan structures and explicit examples such as $TrFl_4^{>0}$. These results connect tropical geometry, total positivity, and Bruhat polytope theory, providing realizability criteria and a subdivision framework for flag matroid polytopes that have implications for Bruhat interval polytopes and Coxeter matroids.
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 $\boldsymbolμ = (μ_a,\ldots, μ_b)$ of TrFl$_{\boldsymbol{r},n}^{\geq 0} =$ FlDr$_{\boldsymbol{r},n}^{\geq 0}$ give rise to coherent subdivisions of the flag positroid polytope $P(\underline{\boldsymbolμ})$ 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 $(χ_1,\dots,χ_k)$ 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.