\documentclass[finalversion]{FPSAC2023}
\articlenumber{60}
\usepackage[utf8]{inputenc}
\usepackage{mathtools}
\usepackage{amsfonts}
\usepackage{float}
\usepackage{thmtools}
\usepackage{bm}
\usepackage{tikz}
\usetikzlibrary{positioning}
\usetikzlibrary{arrows.meta}
\usepackage{indentfirst}
\usepackage{blkarray}
\usepackage{biblatex}
\addbibresource{biblio.bib}


 %packages
\usepackage{amsxtra, amsfonts}
\usepackage{mathrsfs}
\usepackage[normalem]{ulem}
\usepackage{bbm}
\renewcommand{\familydefault}{ppl}
\usepackage[T1]{fontenc}
%table stuff:
\usepackage{array,multirow}
\usepackage{arydshln}

\usepackage{tikz}
\tikzset{
  dep u/.style={insert path={-- ++(0,15) node{}}},
  dep r/.style={insert path={-- ++(15,0) node{}}},
  dep d/.style={insert path={-- ++(0,-15) node{}}},
  dep l/.style={insert path={-- ++(-15,0) node{}}},
  recurse lattice path/.code args={#1#2}{
    \ifx#1.\else\tikzset{dep #1,recurse lattice path=#2}\fi
  },
  lattice path/.style={recurse lattice path=#1.}
}
%\usetikzlibrary{external}
%\tikzexternalize
\usetikzlibrary{arrows,calc,intersections,matrix,positioning,through}

\usepackage{tkz-euclide}



%letters
\newcommand{\QQ}{\mathbb Q}
\newcommand{\RR}{\mathbb R}
\newcommand{\CC}{\mathbb C}
\newcommand{\PP}{\mathbb P}
\newcommand{\ZZ}{\mathbb Z}
\newcommand{\TT}{\mathbb T}
\renewcommand{\SS}{\mathbb S}
\newcommand{\KK}{\mathbb K}
\newcommand{\kk}{\mathbbm k}
\newcommand{\C}{\mathcal C}
\newcommand{\I}{\mathcal I}
\newcommand{\be}{\mathbf e}
\newcommand{\M}{{\boldsymbol M}}
\newcommand{\br}{\mathbf r}
\newcommand{\bw}{\boldsymbol w}
\newcommand{\bmu}{{\boldsymbol \mu}}

%term defining format
\newcommand{\newterm}{\emph}

%multiline cells in table command
\newcommand{\centered}[1]{\begin{tabular}{l} #1 \end{tabular}}


%theorem environments
\theoremstyle{definition}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{eg}[thm]{Example}
\newtheorem{rem}[thm]{Remark}
\newtheorem{ques}[thm]{Question}
\newtheorem{exerc}[thm]{Exercise}
\newtheorem{mainthm}{Theorem}
\newtheorem{maincor}[mainthm]{Corollary}
\renewcommand{\themainthm}{\Alph{mainthm}}


%Main object names
\DeclareMathOperator{\distance}{dist}
\DeclareMathOperator{\val}{val}
\DeclareMathOperator{\spn}{Span}
\DeclareMathOperator{\Dr}{Dr}
\DeclareMathOperator{\Conv}{Conv}
\DeclareMathOperator{\Trop}{Trop}
\DeclareMathOperator{\trop}{trop}
\DeclareMathOperator{\TrFl}{TrFl}
\DeclareMathOperator{\FlDr}{FlDr}
\DeclareMathOperator{\TrGr}{TrGr}
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\Gl}{GL}
\DeclareMathOperator{\PPP}{P}
\DeclareMathOperator{\Fl}{Fl}
\DeclareMathOperator{\Matroid}{\mathcal{M}}
\DeclareMathOperator{\Matroidkn}{\mathcal{M}_{k;n}}
\DeclareMathOperator{\Positroidkn}{{Pos}_{k;n}}
\DeclareMathOperator{\FlMatroid}{{FlM_n}}
\DeclareMathOperator{\FlPos}{{FlPos_n}}
\DeclareMathOperator{\FlPosRe}{{FlPos_n^{realiz}}}
\DeclareMathOperator{\FlMatroidRe}{{FlM_n^{realiz}}}


%common operators
\DeclareMathOperator{\Perm}{Perm}
\DeclareMathOperator{\rk}{rk}
\newcommand{\myrho}[1]{\operatorname{rk}_{M_{#1}}|_{\binom{[n]}{#1}}}



\title{Polyhedral and Tropical Geometry of Flag Positroids}
\author{Jonathan Boretsky\thanks{\href{mailto:jboretsky@math.harvard.edu}{jboretsky@math.harvard.edu}. J.B. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number 557353-2021]. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de référence 557353-2021].}\addressmark{1}, Christopher Eur\thanks{\href{mailto:ceur@g.harvard.edu}{ceur@g.harvard.edu}. C.E. is partially supported by the US National Science Foundation (DMS-2001854)}\addressmark{1}, and Lauren Williams\thanks{\href{mailto:williams@math.harvard.edu}{williams@math.harvard.edu}. L.W. is partially supported by 
the National Science Foundation (DMS-1854512 and DMS-2152991).}\addressmark{1}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Harvard University, Cambridge, MA, 02138}
\keywords{Flag varieties, Tropical varieties, Positroids, Polytopal subdivisons, Matroid polytopes, Oriented flag matroids}

\received{\today}


\begin{document}

 
\abstract{
   A \emph{flag positroid}  of ranks $\mathbf{r}:=(r_1<\dots <r_k)$ 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 abstract, 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 
	$\textnormal{TrFl}_{\mathbf{r},n}^{\geq 0}$ equals the 
nonnegative flag Dressian $\textnormal{FlDr}_{\mathbf{r},n}^{\geq 0}$,  and that the points 
 $\boldsymbol{\mu}  = (\mu_a,\ldots, \mu_b)$ of 
$\textnormal{TrFl}_{\mathbf{r},n}^{\geq 0} = 
 \textnormal{FlDr}_{\mathbf{r},n}^{\geq 0}$ give rise to coherent subdivisions of  flag positroid polytopes into (smaller) 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 \emph{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.}

\maketitle


\section{Introduction}



In recent years there has been a great deal of interest in the 
tropical Grassmannian \cite{SS04, HJJS08, HJS14, Cachazo, Bosstrop}, 
and subdivisions of matroid polytopes \cite{Spe08, AFF, Early}. Further, there has been research into ``positive'' 
\cite{Pos, SW05, ARW, LeFraser, LPW, SW21, AHLS}
and ``flag'' 
\cite{TW15, BEZ21,  Bossinger, JL, JLLO, Bor}
versions of the above objects.
The aim of this paper is to illustrate the beautiful relationships between the
nonnegative tropical flag variety, the nonnegative flag Dressian, 
and subdivisions of flag positroid polytopes, unifying and 
generalizing some of the existing results.  We will particularly 
focus on the case of flag varieties (respectively, flag positroids)
consisting of subspaces (respectively, positroids)
of \emph{consecutive} ranks.  This includes both Grassmannians
and complete flag varieties.



For positive integers $n$ and $d$ with $d<n$, we 
let $[n]$ denote the set $\{1, \ldots, n\}$ and we let 
${[n] \choose d}$ denote the collection of all $d$-element subsets of $[n]$.
Given a subset $S\subseteq [n]$ we 
let $\be_S$ denote the sum of standard basis vectors $\sum_{i\in S} \be_i$.  
For a collection $\mathcal B \subset {[n]\choose d}$, we let
$
P(\mathcal B) = \text{the convex hull of $\{\be_B : B\in \mathcal B\}$ in $\RR^n$}.
$
The collection $\mathcal B$ is said to define a \newterm{matroid $M$ 
of rank $d$ on $[n]$} if every edge of the polytope
$P(\mathcal B)$
is parallel to $\be_i - \be_j$ for some $i\neq j \in [n]$.
In this case, we call $\mathcal B$ the set of 
\newterm{bases} of $M$, and define the \newterm{matroid polytope} $P(M)$ of $M$ to be the polytope $P(\mathcal B)$.  
When $\mathcal B$ indexes the nonvanishing Pl\"ucker coordinates of 
an element $A$ of the Grassmannian $\Gr_{d,n}(\CC)$, 
we say that $A$ \emph{realizes} $M$, and it is well-known that 
%we
%call $$S_{\mathcal{B}}:=\{A\in \Gr_{k,n}(\CC) \ \vert \ p_I(A)\neq 0 \text{ if and only if }I\in
%\mathcal{B} \}$$ a \emph{matroid stratum}.  It was shown in \cite{GGMS87}  that 
$P(\mathcal B)$ is the moment map image
of the closure of the torus orbit of $A$ in the Grassmannian \cite{GGMS87}.
%$S_{\mathcal{B}}$.
We assume familiarity with the fundamentals of matroid theory as in  \cite{Oxl11} and
\cite{BGW03}.


The above definition of a matroid in terms of its 
polytope is due to \cite{GGMS87}.
Flag matroids are natural generalizations of matroids that admit the following polytopal definition.
%We can also define a flag matroid in terms of its polytope.
%and can be generalized as follows.
%\LW{Is there any advantage of saying nondecreasing here, as opposed to increasing?
%If we want to identify partial flag varieties with $G/P$ (as we do later,
%to use results of Rietsch), I think we need the 
%$r_i$'s to be distinct.}\chris{Both can be made to work, but let's change to "increasing" (initially, I wanted nondecreasing because a "minor" of a flag matroid with all distinct ranks can produce a flag matroid with non-distinct ranks, so in other papers of mine on flag matroids I go with nondecreasing.  However, this is not a problem in this paper, especially if one takes the symbol $\M^{\mathscr S}$ in Proposition 4.3 not as a flag matroid).}

%\LW{I believe our old definition of flag matroid is WRONG -- 
%check out \cite[Theorem 1.13.3]{BGW03}.  We should fix the 
%definition as follows.  
\begin{defn}\cite[Corollary 1.13.5 and Theorem 1.13.6]{BGW03}
Let $ \br = (r_1, \ldots, r_k)$ be a sequence of increasing integers in $[n]$.
A \newterm{flag matroid} of ranks $\br$ on $[n]$ is a sequence 
$\M = (M_1, \ldots, M_k)$ of matroids of ranks $(r_1, \ldots, r_k)$ on $[n]$ such that
all vertices of the polytope
\[
P(\M) = P(M_1)+ \cdots + P(M_k), \text{ the Minkowski sum of  matroid polytopes},
\]
are equidistant from the origin. This polytope is called the \newterm{flag matroid polytope} of $\M$.
\end{defn}

Flag matroids are exactly the type $A$ objects in the theory of Coxeter matroids \cite{GS87, BGW03}. Just as a realization of a matroid is a point in a Grassmannian, a realization of a flag matroid 
is a point in a flag variety.  More concretely, a \newterm{realization} of a flag matroid 
of ranks $(r_1,\dots,r_k)$ over a field
is an $r_k \times n$ matrix $A$ 
over that field such that for each $1 \leq i \leq k$, the first $r_i$ rows of $A$ is a realization of $M_i$.
For an equivalent definition of flag matroids in terms of Pl\"ucker relations on partial flag varieties, see \cite[Proposition A]{JL}. There are natural ``positive'' analogues of matroids, flag matroids, and their polytopes.

\begin{defn}\label{def:flagpositroid}
Let $\br = (r_1, \cdots ,r_k)$ be a sequence of increasing integers in $[n]$.
	We say that a flag matroid $(M_1,\dots,M_k)$  of ranks $\br$ on $[n]$ is  a \newterm{flag positroid} if it has a realization by a real matrix $A$ such that 
	the $r_i \times n$ submatrix of $A$ formed by the first $r_i$ rows of $A$ has all nonnegative minors
	for each $1 \leq i\leq k$.
\end{defn}

We refer to the flag matroid polytope of a flag positroid as a \emph{flag positroid polytope}.
It follows from our definition above that flag positroids are realizable.

Setting $k = 1$ in \Cref{def:flagpositroid} gives the well-studied notion of \newterm{positroids} 
and \emph{positroid polytopes} \cite{Pos, ARW}.  
Therefore each flag positroid is a sequence of positroids.

In recent years it has been gradually understood that the tropical 
geometry of the Grassmannian and flag variety,
and in particular, the \emph{Dressian} and \emph{flag Dressian}, are
%and their positive
  intimately connected to (flag) matroid 
polytopes and their subdivisions \cite{Spe08, HJJS08, BEZ21, MS15}.  

The tropical geometry of the \emph{positive} Grassmannian and of the flag variety
are particularly nice:
the positive tropical Grassmannian equals the positive Dressian, 
whose cones in turn parameterize 
subdivisions of the hypersimplex into positroid polytopes \cite{SW05, SW21, LPW, AHLS}, while the positive tropical complete flag variety equals
the positive complete flag Dressian, whose cones
parameterize subdivisions of the permutohedron into 
\emph{Bruhat interval polytopes} \cite{Bor, JLLO}.

Our main theorem, which follows, unifies and generalizes the above results. In particular, \Cref{thm:eqvs} applies to the \textit{nonnegative} flag variety of rank $\br$, where $\br$ consists of any consecutive integers. This includes both the Grassmannian case, when $\br=(d)$ and the complete flag case, when $\br=(1,2\ldots,n)$. Much of the terminology appearing in this theorem statement will be defined in \Cref{defs}.  


\begin{thm}\label{thm:eqvs}
Suppose $\br$ is a sequence of consecutive integers $(a, \ldots, b)$ for some $1\leq a \leq b \leq n$.
Then, for $\bmu  = (\mu_a,\ldots, \mu_b) \in \prod_{i = a}^{b} \PP\left( \TT^{\binom{[n]}{i}}\right)$, 
	the following are equivalent:
\begin{enumerate}[label = (\alph*),nosep]
\item\label{eqvs:TrFl} $\bmu\in \TrFl_{\br,n}^{\geq 0}$, the nonnegative tropicalization of the flag variety, i.e.\ the closure of the coordinate-wise valuation of points in $\Fl_{\br,n}(\C_{\geq 0})$.
\item\label{eqvs:FlDr} $\bmu \in \FlDr_{\br,n}^{\geq 0}$, the nonnegative flag Dressian, i.e. the ``solutions" to the positive-tropical Grassmann-Pl\"ucker and incidence-Pl\"ucker relations.
\item\label{eqvs:subdiv} 
%		The support  $\underline{\bmu}$ of $\bmu$ is a flag matroid, 
		Every face in the coherent subdivision $\mathcal 
		D_{\boldsymbol\mu}$ of the polytope $P(\underline{\bmu}) = P(\underline{\mu_1}) + \cdots + P(\underline{\mu_k})$ induced by $\bmu$ is a flag positroid polytope (of rank $\br$), where $\underline{\mu_i}$ is the support of $\mu_i$ for $1\leq i\leq k$.

\item\label{eqvs:2faces} 
	Every face of dimension at most 2 in the subdivision $\mathcal D_{\boldsymbol\mu}$ of $P(\underline{\boldsymbol \mu})$ is a flag positroid polytope (of rank $\br$).
\item\label{eqvs:3terms} The support $\underline \bmu$ of $\bmu$
	is a flag matroid, and 
	$\bmu$ satisfies every three-term positive-tropical incidence relation
		(respectively, Grassmann-Pl\"ucker relation)
		when $a<b$ (respectively, $a=b$).
\end{enumerate}
\end{thm}

We present here three useful corollaries of this result.  


\begin{cor}\label{cor:bigpoly}
    Suppose $\br$ is a sequence of consecutive integers and
 let $\bmu\in \FlDr_{\br,n}^{\geq 0}$. Then $P(\underline{\bmu})$ is a flag positroid polytope. 
\end{cor}



We note that in the Grassmannian case, that is, the case that $\br=(d)$ is a single integer, 
\Cref{thm:eqvs} and \Cref{cor:bigpoly} describe the
relationship between the nonnegative tropical Grassmannian,
the nonnegative Dressian, and subdivisions of positroid polytopes (e.g. the hypersimplex,
if $\bmu$ has no coordinates equal to $\infty$)
into positroid polytopes. Moreover, when $\br=(1,2,\dots,n)$, \Cref{thm:eqvs} and \Cref{cor:bigpoly}
describe the 
relationship between the nonnegative tropical complete flag variety, 
the nonnegative complete flag Dressian, and subdivisions of Bruhat interval polytopes (e.g. the permutohedron, if $\bmu$ has no coordinates equal to $\infty$) into 
Bruhat interval polytopes. We illustrate this relationship in the case where $\bmu$ has no coordinates equal to $\infty$ in \Cref{sample}.



We now present an application of \Cref{thm:eqvs} to flag matroid polytopes. 

\begin{cor}\label{cor:2faces}
	For a flag matroid $\M=(M_a,M_{a+1},\dots,M_b)$ 
	 of consecutive ranks $\br$, 
	its flag matroid polytope $P(\M)$ is a 
 flag positroid polytope
if and only if its $(\leq 2)$-dimensional faces are 
	flag positroid polytopes (of rank $\br$).
\end{cor}



% \begin{proof}
% Let $\bmu = (\mu_a,\dots,\mu_b)$, with $\mu_i\in \{0,\infty\}^{{[n]\choose i}}$, where the coordinates of each $\mu_i$ are either $0$ or $\infty$ based on whether 
% we have a basis or nonbasis of $M_i$.
% This gives rise to the trivial subdivision of the corresponding flag matroid 
% 	polytope $P(\underline{\bmu})=P(\M)$ (using the coherent subdivision defined in \Cref{sec:subdiv}).  
% The result now follows from the equivalence of \ref{eqvs:subdiv} and \ref{eqvs:2faces} in \Cref{thm:eqvs}.
% \end{proof}

In the Grassmannian case, the flag positroid polytopes of rank $\br=(d)$
are precisely the positroid polytopes and \Cref{cor:2faces} appeared as 
\cite[Theorem 3.9]{LPW}.


It has been shown that all \textit{positively oriented matroids} are positroids, which is to say, realizable by a matrix with nonnegative maximal minors \cite{ARW17}\cite{SW21}. It is natural to ask when a sequence of positroids is a flag positroid, which is to say, realizable as in \Cref{def:flagpositroid}. Note however that questions of realizability for flag matroids are rather subtle, as demonstrated in \Cref{eg:notreal}. By working with \emph{oriented} flag matroids, we give an answer to this realizability question for flag positroids, in the case of consecutive ranks.
%A sequence $(M_1, \ldots, M_k)$ of oriented matroids is an \newterm{oriented flag matroid} every covector of $M_i$ is a covector of $M_{i+1}$ for all $i = 1, \ldots, k-1$.


\begin{cor}\label{cor:real}
Suppose $(M_1, \ldots, M_k)$ is a sequence of positroids on $[n]$ of consecutive ranks.  
Then, considered as a sequence of positively oriented matroids, $(M_1, \ldots, M_k)$ is a flag positroid if and only if it is an oriented flag matroid.
\end{cor}
	
We define
a \emph{positively oriented flag matroid} to be a sequence
of positively oriented matroids $(\chi_1,\dots,\chi_k)$ which is also 
an oriented flag matroid.  \Cref{cor:real} then says that every positively oriented flag matroid 
of consecutive ranks is realizable. We do not know whether \Cref{cor:real} holds 
if $\br$ fails to satisfy the consecutive rank condition.


The structure of this abstract is as follows. In \Cref{defs}, we define the terms and concepts appearing in \Cref{thm:eqvs}. In \Cref{apps}, we present further results related to \Cref{thm:eqvs}, including an application to generalized Bruhat interval polytopes and comments on what can be said about general $\br$. This is an extended abstract of the full article \cite{BEW}.  



\begin{figure}\centering
 \begin{tikzpicture}[scale=1, every node/.style={scale=1}]
                \draw(-2.7,1) node {\footnotesize
                Let $n=4,\, \br=(2),$};
                \draw(-2.7,0.25) node {\footnotesize
                $(\mu_I)\in \RR^{{[4] \choose 2}}$ such that};
\draw(-2.7,-0.5) node {\footnotesize
        $\mu_{13}+\mu_{24} = \mu_{23}+\mu_{14} < \mu_{12}+\mu_{34}$.};
     \tkzDefPoint(0,0){e14}
     \tkzDefPoint(1,0){e13}
     \tkzDefPoint(.5,.4){e24}
     \tkzDefPoint(1.5,.4){e23}
     \tkzDefPoint(.7,1.4){e12}
     \tkzDefPoint(.7,-1){e34}
         \filldraw[fill=gray!30!white](e14)--(e24)--(e23)--(e13)--(e14);
                \filldraw[fill=blue!60!white](e14)--(e13)--(e12);
                \filldraw[fill=blue!60!white](e13)--(e23)--(e12);
                \filldraw[fill=magenta](e13)--(e14)--(e34);
                \filldraw[fill=magenta](e13)--(e23)--(e34);
\tkzDrawPolygon(e24, e14,e13,e23,e24,e12,e14,e13,e12,e23,e34,e14,e13,e34,e23)
                \tkzDrawPolygon(e24,e14,e12,e24,e23,e12,e13,e14,e13,e23);
                \node[shift=(e24), anchor=north] {\tiny $e_{24}$};
                \node[shift=(e14), anchor= east] {\tiny $e_{14}$};
                \node[shift=(e23), anchor=west] {\tiny $e_{23}$};
                \node[shift=(e13), anchor=north east] {\tiny $e_{13}$};
                \node[shift=(e12), anchor=east] {\tiny $e_{12}$};
                \node[shift=(e34), anchor=east] {\tiny $e_{34}$};
        \filldraw[color=black,fill=black] (e24) circle (1pt);
        \filldraw[color=black,fill=black] (e14) circle (1pt);
        \filldraw[color=black,fill=black] (e23) circle (1pt);
        \filldraw[color=black,fill=black] (e13) circle (1pt);
        \filldraw[color=black,fill=black] (e12) circle (1pt);
        \filldraw[color=black,fill=black] (e34) circle (1pt);
\end{tikzpicture}
     \begin{tikzpicture}[scale=1, every node/.style={scale=1}]
\draw(-3.4,0.75) node {\footnotesize
Let $n=3$,\, $\br=(1,2,3),$};
\draw(-3.4,0) node {\footnotesize
$(\mu_I)\in \RR^{2^3}$ such that};
\draw(-3.4,-0.75) node {\footnotesize
        $\mu_{2}+\mu_{13} = \mu_{1}+\mu_{23} < \mu_{3}+\mu_{12}$.
                };
     \tkzDefPoint(0,.35){Z1}
     \tkzDefPoint(0,-.35){newZ1}
     \tkzDefPoint(1.2,0.35){Z3}
     \tkzDefPoint(1.2,-0.35){newZ3}
                \tkzDefPoint(.6,1){Z2}
     \tkzDefPoint(.6,-1){Z4}
                \filldraw[fill=blue!60!white](newZ1)--(Z1)--(Z2)--(Z3);
                \filldraw[fill=magenta](newZ1)--(Z3)--(newZ3)--(Z4);
                \tkzDrawPolygon(newZ1, Z1, Z2, Z3, newZ3, Z4, newZ1)
                %\tkzDrawPolygon(e24,e14,e12,e24,e23,e12,e13,e14,e13,e23);
                \node[shift=(Z1), anchor=east] {\tiny $(2,3,1)$};
                \node[shift=(newZ1), anchor=east] {\tiny $(1,3,2)$};
                \node[shift=(Z2), anchor=south] {\tiny $(3,2,1)$};
                \node[shift=(Z3), anchor=west] {\tiny $(3,1,2)$};
                \node[shift=(newZ3), anchor=west] {\tiny $(2,1,3)$};
                \node[shift=(Z4), anchor=north] {\tiny $(1,2,3)$};
        \filldraw[color=black,fill=black] (Z1) circle (1pt);
        \filldraw[color=black,fill=black] (Z2) circle (1pt);
        \filldraw[color=black,fill=black] (Z3) circle (1pt);
        \filldraw[color=black,fill=black] (Z4) circle (1pt);
\end{tikzpicture} 

\caption{Coherent subdivision of the $n=4,\,r=2$ hypersimplex into positroid polytopes induced by a point $\bmu \in \FlDr^{\geq 0}_{(2),4}$ (left) and coherent subdivision of the $n=3$ permutohedron into flag positroid  polytopes induced by a point $\bmu\in \FlDr^{\geq 0}_{(1,2,3),3}$ (right).}
\label{sample}
\end{figure}




\section{Background and Definitions}\label{defs}


\subsection{Nonnegative Flag Varieties} \label{background}
Let $n\in \ZZ_+$ and let $\br = \{r_1 < \dots < r_k\} \subseteq [n].$ 
For a field $\kk$, let  $G=\Gl_n(\kk)$, and let 
$\PPP_{\br; n}(\kk)$ denote the parabolic subgroup of 
$G$ of block upper-triangular
matrices with diagonal blocks of sizes $r_1, r_2-r_1,\dots, r_k - r_{k-1}, n-r_k$.
We define the \emph{partial flag variety} 
$$\Fl_{\br; n}(\kk):=\Gl_n(\kk)/\PPP_{\br;n}(\kk).$$  
As usual, we identify $\Fl_{\br; n}(\kk)$ with the variety of partial flags of subspaces in $\kk^n$:
\[
\operatorname{Fl}_{\br;n}(\kk) = \{(V_1 \subset \cdots \subset V_k) : \text{$V_i$ a linear subspace of $\kk^n$ of dimension $r_i$ for $i = 1, \ldots, k$}\}
\]
%be the corresponding partial flag variety.  
We write $\operatorname{Fl}_n(\kk)$ for the complete flag variety $\operatorname{Fl}_{1,2,\ldots, n;n}(\kk)$. Note that 
$\Fl_n(\kk)$ can be identified with $\Gl_n(\kk)/B(\kk)$, where
$B(\kk)$ is the subgroup of upper-triangular matrices.  
There is a natural projection $\pi$ from $\Fl_n(\kk)$ to 
any partial flag variety by simply forgetting some of the 
subspaces.

If $A$ is an $r_k \times n$ matrix such that $V_{r_i}$ is the span of the first $r_i$ rows,
we say that $A$ is a \emph{realization} of $V:=(V_1 \subset \dots \subset V_k) \in \Fl_{\br;n}$.
Given any realization $A$ of $V$ and any $1\leq i \leq k$, we have the 
\emph{Pl\"ucker coordinates} or \emph{flag minors}
$p_I(A)$ where $I\in {[n] \choose r_i}$; concretely, $p_I(A)$ is the determinant of the 
submatrix of $A$ occupying the first $r_i$ rows and columns $I$.
This gives the Pl\"ucker embedding of $\Fl_{\br;n}(\kk)$ into $\PP^{{[n]\choose r_1}-1} 
\times \dots \times \PP^{{[n]\choose r_k}-1}$ taking 
$V$ to $\left( \big(p_I(A)\big)_{I\in {[n]\choose r_1}}, \dots, \big(p_I(A)\big)_{I\in {[n]\choose r_k}} \right)$.


\begin{defn}
For integers $0 < r\leq s< n$, the \newterm{(single-exchange) Pl\"ucker relations of type $(r,s;n)$} are polynomials in variables $\{x_I : I \in \binom{[n]}{r} \cup \binom{[n]}{s}\}$ defined as
\[
\mathscr {P}_{r,s;n}=\left\{\sum_{j\in J\setminus I} \operatorname{sign}(j,I,J) x_{I\cup j}x_{J\setminus j }\left| I \in {\binom{[n]}{r-1}},\: J\in {\binom{[n]}{s+1}} \right.\right\},
\]
where $\operatorname{sign}(j,I,J)=(-1)^{|\{k\in J|k<j\}|+|\{i\in I|j<i\}|}$.
When $r = s$, the elements of $\mathscr P_{r,r;n}$ are called the \newterm{Grassmann-Pl\"ucker relations} (of type $(r;n)$), and when $r< s$, the elements of $\mathscr P_{r,s;n}$ are called the \newterm{incidence-Pl\"ucker relations} (of type $(r,s;n)$).
\end{defn}

As in the introduction, let 
$\br = (r_1 < \cdots < r_k)$ be a sequence of increasing integers in $[n]$.
We let
$
\mathscr P_{\br;n} = \bigcup_{\substack{r\leq s \\ r,s\in \br}} \mathscr P_{r,s;n},
$
and let $\langle \mathscr P_{\br;n}\rangle$ be the ideal generated by $\mathscr P_{\br;n}$.
It is well-known that for any field $\kk$ the ideal 
$\langle \mathscr P_{\br;n}\rangle$ 
set-theoretically carves out the partial flag variety $\operatorname{Fl}_{\br;n}(\kk)$ embedded in $\prod_{i = 1}^k \PP\left(\kk^{\binom{[n]}{r_i}}\right)$ via the Pl\"ucker embedding \cite[\S9]{Ful97}. 

We refer to the subset of relations in $\mathscr P_{\br;n}$ which are polynomial relations containing precisely three terms as the \emph{three-term 
Pl\"ucker relations}. We will now often drop the $\kk$ from our notation, with the understanding that $\kk=\RR$.


\begin{defn}
	We say that a real matrix is \emph{totally positive} if all of its minors are positive.
	We let $\Gl_n^{>0}$ denote the subset of $\Gl_n$ of totally positive matrices.
\end{defn}

There are two natural ways to define positivity for partial flag varieties.
The first notion comes from work of Lusztig \cite{lusztig}.  The second notion
uses Pl\"ucker coordinates, and was initiated in work of Postnikov \cite{Pos}.

\begin{defn}\label{def:2positive}
%Let $n\in \ZZ_+$ and let $\br = \{r_1 < \dots < r_k\} \subset [n-1]$.
	We define the \emph{(Lusztig)  positive part} of $\Fl_{\br; n}$,
	denoted by $\Fl_{\br;n}^{>0}$, as the image of $\Gl_n^{>0}$ inside 
	$\Fl_{\br;n} = \Gl_n/ \PPP_{\br; n}$. We define the 
	\emph{(Lusztig)  nonnegative part} of $\Fl_{\br; n}$, denoted 
	 by $\Fl_{\br;n}^{\geq 0}$, as the closure of 
	  $\Fl_{\br;n}^{>0}$ in the Euclidean topology.  

	We define the \emph{Pl\"ucker positive part} (respectively,
	\emph{Pl\"ucker nonnegative part}) of 
	$\Fl_{\br; n}$ to be the subset of 
	$\Fl_{\br; n}$ where all Pl\"ucker coordinates are 
	 positive (respectively, nonnegative).\footnote{The reader who is concerned
	 about the fact that we are working with projective
	 coordinates can replace  ``all Pl\"ucker coordinates are nonnegative'' by 
	  ``all nonzero Pl\"ucker coordinates have the same sign''.}
\end{defn}

It is well-known that 
the Lusztig positive part of 
$\Fl_{\br;n}$ is a subset of the Pl\"ucker positive part of 
$\Fl_{\br;n}$, and that the two notions agree in the case of the Grassmannian
\cite[Corollary 1.2]{TW13}.  
The two notions also agree in the case of the complete flag variety \cite[Theorem 5.21]{Bor}. 
More generally, we have the following.

\begin{thm}\cite[Theorem 1.1]{BK}\label{thm:BK}
	The Lusztig positive (respectively, Lusztig nonnegative) part of 
	$\Fl_{\br;n}$ equals the Pl\"ucker positive (respectively, Pl\"ucker
	nonnegative) part of 
$\Fl_{\br;n}$  if and only if 
the set $\br$ consists of consecutive integers.
\end{thm}
See \cite[Section 1.4]{BK} for more references and 
a nice discussion of the history. Since
in this abstract we will be mainly studying the case where $\br$ consists of consecutive integers,
we will use the two notions interchangeably when there is no ambiguity.

\subsection{Tropical Geometry}
Before discussing tropical varieties, we introduce the tropical hyperfield and what it means to find a solution of a tropical polynomial. For more details, see \cite{MS15} and \cite{SW05}.




\begin{defn}\label{def:tropicalsemifield}
Let $\TT = \RR \cup \{\infty\}$ be the set underlying the tropical hyperfield, endowed with
%	(also called the tropical hyperfield \chris{technically these two notions are different in that in the semifield the addition is not set-valued.  Maybe we can say "Let $\TT = \RR\cup \{\infty\}$ be the set underlying the tropical semifield (and also the tropical hyperfield)?" Or just not mention hyperfield here.})
	the topology such that $-\log: \RR_{\geq 0} \to \TT$ is a homeomorphism.  
Given a point $w \in \TT^{\binom{[n]}{r}}$, 
we  define the \newterm{support} of $w$ 
	to be $\underline w = \{S\in \binom{[n]}{r} : w_S\neq \infty\}$.
	When $\underline w$ is the set of bases of a matroid,
	we identify $\underline w$ with that matroid.
	Let $\PP\left(\TT^{\binom{[n]}{r}}\right)$ be the tropical projective space of $\TT^{\binom{[n]}{r}}$, which is defined as $\big(\TT^{\binom{[n]}{r}}\setminus \{(\infty, \ldots, \infty)\}\big) / \sim$, where $w \sim w'$ if $w = w' + (c, \ldots, c)$ for some $c\in \RR$.
\end{defn}

For a point $w = (w_1, \ldots, w_m) \in \TT^m\setminus \{(\infty,\ldots,\infty)\}$, we write $\overline w$ for its image in $\PP(\TT^m)$.  For $a = (a_1, \ldots, a_n) \in \ZZ^m$, write $a \bullet w = a_1 w_1 + \cdots + a_m w_m$.

\begin{defn}\label{defn:trophyper}
For a real homogeneous polynomial
\[
f = \sum_{a \in \mathcal A} c_{a} x^{a} \in \RR[x_1, \ldots, x_m], \quad\text{where $\mathcal A\subset \ZZ_{\geq 0}^m$ is finite and $c_{a}\in \RR\setminus \{0\}$ for all $a\in\mathcal{A}$,}
\]
the \newterm{extended tropical hypersurface} $V_{\trop}(f)$ and the \newterm{nonnegative tropical hypersurface} $V_{\trop}^{\geq 0}(f)$ are subsets of the tropical projective space $\PP(\TT^m)$ defined by
\begin{align*}
V_{\trop}(f)  &= \left\{ \overline w \in \PP(\TT^m) \  \middle| \ \text{the minimum in 
	$\min_{a\in \mathcal A}(a\bullet w)$, if finite, occurs at least twice}\right\},\\
\text{and}\\
V_{\trop}^{\geq 0}(f)  &= \left\{ \overline w \in \PP(\TT^m) \  \middle| \ \begin{matrix} \text{the minimum in $\displaystyle\min_{a\in \mathcal A}(a\bullet w)$, if finite, occurs at least twice,}\\ \text{  at some $a, a'\in \mathcal A$ such that $c_a$ and $c_{a'}$ have opposite signs} \end{matrix}\right\}.
\end{align*} 
We say that a point \newterm{satisfies the tropical relation} of $f$ if it is in $V_{\trop}(f)$, and that it \newterm{satisfies the positive-tropical relation} of $f$ if it is in $V_{\trop}^{\geq 0}(f)$.
\end{defn}

When $f$ is a multihomogeneous real polynomial, we define $V_{\trop}(f)$ and $V_{\trop}^{\geq 0}(f)$ similarly as subsets of a product of tropical projective spaces.
We will consider tropical hypersurfaces of Pl\"ucker relations, yielding tropical analogues of partial flag varieties.

\begin{defn}\label{defn:tropFl}
The \newterm{tropicalization $\TrFl_{\br;n}$} of $\operatorname{Fl}_{\br;n}$, the \newterm{nonnegative tropicalization $\TrFl_{\br;n}^{\geq 0}$} of $\operatorname{Fl}_{\br;n}$, the \newterm{flag Dressian $\FlDr_{\br;n}$}, and the \newterm{nonnegative flag Dressian} $\FlDr_{\br;n}^{\geq 0}$ are subsets of $\prod_{i = 1}^k \PP\left(\TT^{\binom{[n]}{r_i}}\right)$ defined as
\begin{align*}
\TrFl_{\br;n} & = \bigcap_{f\in \langle \mathscr P_{\br;n}\rangle} V_{\trop}(f)
\quad	\text{ and } \quad
	\TrFl_{\br;n}^{\geq 0}  = \bigcap_{f\in \langle \mathscr P_{\br;n} \rangle} V_{\trop}^{\geq 0}(f), \\
\FlDr_{\br;n} &=\bigcap_{f\in \mathscr P_{\br;n}} V_{\trop}(f) \quad \text{ and }\quad
\FlDr_{\br;n}^{\geq 0} = \bigcap_{f\in \mathscr P_{\br;n}} V_{\trop}^{\geq 0}(f).
\end{align*}
\end{defn}

When $\br=(d)$ consists of a single integer, one obtains the (nonnegative) tropicalization of the Grassmannian and the (nonnegative) Dressian studied in \cite{SS04, SW05, SW21, AHLS}.

\begin{rem}
% In \cite[\S6]{JLLO}, the authors define the ``positive flag Dressian'' to consist of the elements $\bmu = (\mu_1, \ldots, \mu_k) \in \FlDr_{\br;n}$ whose constituents $\mu_i$ are each in the strictly positive Dressian.  
% 	In our language, this is equal to considering the points of
% \[
% \bigcap_{f\in \bigcup_{i = 1}^k \mathscr P_{r_i,r_i;n}} V_{\trop}^{\geq 0}(f) \cap \bigcap_{f\in \bigcup_{r_i < r_j} \mathscr P_{r_i,r_j;n}} V_{\trop}(f)
% \]
% that have no $\infty$ coordinates.
% In a similar vein, we could consider defining the ``nonnegative flag Dressian'' 
% to be the elements of the flag Dressian whose constituents are in the 
% nonnegative Dressian.  This gives a strictly 
% larger set than our definition of the nonnegative flag Dressian,
% and has the shortcoming that %, unlike with our definition, 
% the equivalence of \ref{eqvs:TrFl} and \ref{eqvs:FlDr} in \Cref{thm:eqvs} would no longer hold.  See \Cref{eg:notreal}.
% \end{rem}

We record a useful equivalent description of the (nonnegative) tropicalization of a partial flag variety 
using Puiseux series. 
\begin{defn}\label{def:Puiseux}
Let $\C = \CC\{\{t\}\}$ be the field of Puiseux series with coefficients in $\CC$, with the usual valuation map $\operatorname{val}: \C \to \TT$.
	Concretely, for $f\neq 0$, $\val(f)$ is the exponent of the initial term of $f$,
	and $\val(0) = \infty$.
Let 
\begin{equation*}
	\C_{> 0} = \{f \in \C : \text{the initial coefficient of $f$ is real and positive}\} \ 
	\text{ and } \ \C_{\geq 0} = \C_{>0} \cup \{0\}.
\end{equation*}
The valuation $\val$ can also be applied coordinate wise, yielding a map $\val:\mathcal{C}^n\rightarrow \TT^n$, 
\end{defn}


\begin{prop}\cite[Theorems 3.2.3 \& 6.2.15]{MS15}\cite[Proposition 2.2]{SW05}\label{prop:closure}
We have
\begin{align*}
\TrFl_{\br;n} &= \text{the closure of } \{\operatorname{val}(p) : \{(0,\ldots,0)\}\neq  p \in \operatorname{Fl}_{\br;n}(\C)\} \text{ in }   \textstyle\prod_{i=1}^k \PP\left(\TT^{\binom{[n]}{r_i}}\right)\quad\text{and}\\
\TrFl_{\br;n}^{\geq 0} &= \text{the closure of } \{\operatorname{val}(p) : \{(0,\ldots,0)\}\neq  p \in \operatorname{Fl}_{\br;n}(\C_{\geq 0})\} \text{ in }   \textstyle\prod_{i=1}^k \PP\left(\TT^{\binom{[n]}{r_i}}\right).
\end{align*}
\end{prop}

\end{rem}

\subsection{Coherent Subdivisions}\label{sec:subdiv}

Consider a point $\bmu=(\mu_1,\dots,\mu_k) \in \prod_{i=1}^k \PP\left(\TT^{\binom{[n]}{r_i}}\right)$ 
such that its support $\underline\bmu$ is a flag matroid.  By construction, the vertices of the flag matroid polytope $P(\underline\bmu)$ have the form $\be_{B_1} + \cdots + \be_{B_k}$ where $B_i$ is a basis of the matroid $\underline\mu_i$ for each $i = 1, \ldots, k$.

\begin{defn}\label{defn:subdiv}
We define $\mathcal D_\bmu$ to be the coherent subdivision of $P(\underline\bmu)$ induced by assigning each vertex $\be_{B_1} + \cdots + \be_{B_k}$ of $P(\underline\bmu)$ the weight $\mu_1(B_1) + \cdots + \mu_k(B_k)$.  That is, the faces of $\mathcal D_\bmu$ correspond to the faces of the lower convex hull of the set of points
\[
\{(\be_{B_1} + \cdots + \be_{B_k}, \mu_1(B_1) + \cdots + \mu_k(B_k)) \in \RR^n \times \RR : \be_{B_1} + \cdots + \be_{B_k} \text{ a vertex of $P(\bmu)$}\}.
\]
\end{defn}

In \cite[Theorem A]{BEZ21}, it is shown that a point $\bmu\in \prod_{i = 1}^k \PP\left(\TT^{\binom{[n]}{r_i}}\right)$ is in the flag Dressian $\FlDr_{\br;n}$ if and only if the all faces of the subdivision $\mathcal D_\bmu$ are flag matroid polytopes.

\subsection{Oriented Flag Matroids}

Let $\SS = \{-1,0,1\}$ be the 
\emph{hyperfield of signs}, introduced in \cite{BB19}.  For a polynomial $f = \sum_{a\in \mathcal A} c_a x^a \in \RR[x_1, \ldots, x_m]$, we say that an element $\chi\in \SS^m$ is \newterm{in the null set} of $f$ if the set $\{\operatorname{sign}(c_a) \chi^a\}_{a\in \mathcal A}$ is either $\{0\}$ or contains $\{-1,1\}$. % CITE BAKER?

\begin{defn}\label{def:oriented}
	An \newterm{oriented matroid} of rank $r$ on $[n]$ is a point $\chi \in \SS^{\binom{[n]}{r}}$, called a \emph{chirotope},
	which is in the null set of each $f\in \mathscr P_{r,r;n}$.  Similarly, an \newterm{oriented flag matroid} of ranks $\br$ is a 
	point $\boldsymbol\chi = (\chi_1, \ldots, \chi_k) \in \prod_{i = 1}^k \SS^{\binom{[n]}{r_i}}$ 
	such that $\boldsymbol\chi$ is in the null set of each $f\in \mathscr P_{\br;n}$.
\end{defn}

While these definitions may seem different from those in the standard reference \cite{BLVSWZ99} on oriented matroids, one can show that \Cref{def:oriented} is equivalent to \cite[Definition 3.5.3]{BLVSWZ99}.

%\begin{rem}\chris{this can be removed}
%Like the description of elements of (flag) Dressians in terms of three-term relations in \Cref{prop:3terms}, oriented (flag) matroids admit a description in terms of three-terms relations as follows.  The proof is the same as that of \Cref{prop:3terms} when one notes that $\SS$ is a perfect hyperfield.  A point $\chi \in \SS^{\binom{[n]}{r}}$ is an oriented matroid if and only if
%\begin{itemize}
%\item the set $\{B\in \binom{[n]}{r} : \chi(B) \neq 0\}$ is a set of bases of a matroid, and 
%\item for every $S \subseteq [n]$ of cardinality $r - 2$ and $\{i<j<k<\ell\} \subseteq [n]$ disjoint from $S$, the set $\{\chi(Sij)\chi(k\ell), -\chi(Sik)\chi(Sj\ell), \chi(Si\ell)\chi(Sjk)\}$ is either $\{0\}$ or contains $\{-1,1\}$.
%\end{itemize}
%If $\br$ consists of consecutive integers, then $\boldsymbol\chi = (\chi_1, \ldots, \chi_k) \in \prod_{i = 1}^k \SS^{\binom{[n]}{r_i}}$ is an oriented flag matroid if and only if
%\begin{itemize}
%\item each $\chi_i$ is an oriented matroid for $i = 1, \ldots, k$, and
%\item for all $i =1, \ldots, k-1$, and for every $S\subseteq[n]$ of cardinality $r_i-1$ and $\{a<b<c\}\subseteq [n]$ disjoint from $S$, the set
%$\{\chi_i(Sa) \chi_{i+1}(Sbc), -  \chi_i(Sj)\chi_{i+1}(Sik), \chi_i(Sk)\chi_{i+1}(Sij)\}$ is either $\{0\}$ or contains $\{-1,1\}$.
%\end{itemize}
%\end{rem}

\begin{defn}
A \newterm{positively oriented matroid} is an oriented matroid $\chi$ such that $\chi$ only takes values 0 or 1.  
Similarly, we define a \newterm{positively oriented flag matroid} to be an oriented flag matroid $\boldsymbol\chi$ such that $\boldsymbol\chi$ only takes values 0 or 1.
\end{defn}

A positroid $M$ defines a positively oriented matroid $\chi = \chi_M$ 
where $\chi$ takes value 1 on its bases and 0 otherwise. Every positively oriented matroid $\chi$ is realizable, i.e. 
has the form $\chi_M$ for some positroid $M$ \cite{ARW17}. Thus, each positively oriented flag matroid is a sequence of positroids which is also an oriented flag matroid. However, we remark that imposing 
the oriented flag matroid condition, as in \cref{cor:real}, is stronger than imposing that we have a realizable flag matroid whose consistent matroids are positroids.

\begin{eg}\label{eg:notreal}
We give an example of a realizable flag matroid that has positroids as its constituent matroids but is not a flag positroid.  
This example also appeared in \cite[Example 5]{JLLO} and 
	\cite[Example 6]{BK}.
	Let $(M,M')$ be matroids of ranks 1 and 2 on $[3]$ whose sets of bases are $\{1,3\}$ and $\{12,13,23\}$, respectively.  Both are positroids.  
	We can realize $(M,M')$ as a flag matroid using the matrix
\[
\begin{bmatrix}
a & 0 & b\\
c & d & e
\end{bmatrix},
\]
where the minors $a,b,ad, -bd, ae-bc$ are nonzero.
	In order to realize $(M,M')$ as a flag positroid,
	we need	all these minors to be strictly positive, which is impossible. Since $(M,M')$ does \textit{not} form an oriented flag matroid, this is consistent with \cref{cor:real}.
\end{eg}





\section{Further Results}\label{apps}


We begin by presenting two results that are crucial to the proof \cref{thm:eqvs}, and which are also interesting in their own rights. 

\begin{thm}\label{thm:almost3term}
Suppose $\boldsymbol\mu = (\mu_1, \mu_2) \in \PP\left(\TT^{\binom{[n]}{r}}\right) \times \PP\left(\TT^{\binom{[n]}{r+1}}\right)$ satisfies every three-term positive-tropical incidence relation.  If the support $\underline{\boldsymbol\mu}$ is a flag matroid, then $\boldsymbol\mu \in \FlDr^{\geq 0}_{r,r+1;n}$.
\end{thm}


We also expect that our proof of \Cref{thm:almost3term} in \cite{BEW} adapts well to arbitrary perfect hyperfields, in the language of \cite{BB19}. The following result gives a characterization of positively oriented flag matroids (and equivalently, by \cref{cor:real}, of flag positroids). 


\begin{thm}\label{lem:signembed}
The set of positively oriented flag matroids of ranks $\br$ can be identified with the set of points of the nonnegative flag Dressian $\FlDr_{\br;n}^{\geq 0}$ whose coordinates are all either 0 or $\infty$.
\end{thm}



% There are numerous combinatorial objects which are in bijection with positroids. One such object is the \textit{Grassmann necklace}, defined in \cite{Pos}. We can also give a description of when sequences of positroids form a flag positroid in terms of the Grassmann necklaces of those positroids. The construction, while too lengthy to state here, is entirely algorithmic and is spelled out in detail in \cite[Theorem 7.14]{BEW}.



We continue with a further corollary of \Cref{thm:eqvs}. A \emph{generalized Bruhat interval polytope} \cite[Definition 7.8 and Lemma 7.9]{TW15}
can be defined as the moment map image of the closure of the torus orbit of a point
$A$ in the nonnegative part $(G/P)^{\geq 0}$ (in the sense of Lusztig) of a flag variety $G/P$.
When $\br$ is a sequence of consecutive integers, it then follows from 
\cite{BK} that generalized Bruhat interval polytopes for $\Fl_{\br;n}^{\geq 0}$
are precisely the flag positroid polytopes of ranks $\br$.
In the complete flag case, a generalized Bruhat interval polytope is 
just a \emph{Bruhat interval polytope} \cite{KW15}, that is, the 
convex hull of the permutation vectors $(z(1),\dots,z(n))$ for all permutations $z$ lying
in some Bruhat interval $[u,v]$. Thus, we can now restate \Cref{cor:2faces} as follows.




\begin{cor}
For a flag matroid on $[n]$ of consecutive ranks $\br$, 
its flag matroid polytope is a 
generalized Bruhat interval polytope
if and only if its $(\leq 2)$-dimensional faces are generalized Bruhat interval polytopes.
	In particular, for a complete flag matroid on $[n]$,
	its flag matroid polytope is a Bruhat interval polytope if and 
 only if its $(\leq\!2)$-dimensional faces are  Bruhat interval polytopes.
\end{cor}

As a final point, we comment on what happens when $\br$ does not consist of consecutive integers. A key fact we use about flag positroids of consecutive ranks is that they can be extended to complete flag positroids. This is no longer necessarily true if $\br$ does not consist of consecutive integers. 

\begin{eg}
    Let the sets of bases of $M$ and $M'$ be $\{1,2,3,4\}$ and $\{123,234\}$, respectively. These form a flag positroid $(M,M')$ on $[4]$ of ranks $(1,3)$. After row reduction, a realization of this flag matroid can be written as 
    \[
\begin{bmatrix}
1 & a & b & c \\
0 & x & y & 0\\
0 & z & w & 0
\end{bmatrix}
\]
where $a,b,c>0$ and $xw-yz>0$.  The minors of the matrix formed by the first two rows include $x, y, -cx, -cy$, which cannot be all nonnegative since $c>0$ and not both of $x$ and $y$ are zero. Thus, there is no flag positroid $(M,M_2,M')$ with rank of $M_2$ equal to 2.   
\end{eg}

Passing through complete flag positroids is a key step in our proof of \Cref{thm:eqvs} and, in fact, most of \Cref{thm:eqvs} no longer holds if $\br$ does not consist of consecutive integers. However, we still have the following:

\begin{lem}
For any $\br$, the modification of \ref{eqvs:subdiv} of \Cref{thm:eqvs} to:
\begin{itemize}
\item[(c')] Every face in the coherent subdivision $\mathcal D_\bmu$ of $P(\underline\bmu)$ is the flag matroid polytope of a positively oriented flag matroid.
\end{itemize}
is equivalent to \ref{eqvs:FlDr} of \Cref{thm:eqvs}.
\end{lem}





\acknowledgements{We  are grateful to Tony Bloch, Michael Joswig, Steven Karp, Georg Loho, Dante Luber, and 
Jorge Alberto Olarte for sharing their work with us, which partially inspired this project. We also thank Yue Ren and Vasu Tewari for useful comments. 
Finally, we are grateful to Lara Bossinger for several invaluable
discussions.}

\printbibliography

\end{document}


