# Positivity for partial tropical flag varieties

Jorge Alberto Olarte  CUNEF Universidad, Departamento de métodos cuantitativos
<EMAIL_ADDRESS>
###### Abstract.

We study positivity notions for the tropicalization of type A flag varieties
and the flag Dressian. We focus on the hollow case, where we have one
constituent of rank 1 and another of corank 1. We characterize the three
different notions that exist for this case in terms of Plücker coordinates.
These notions are the tropicalization of the totally non-negative flag
variety, Bruhat subdivisions and the non-negative flag Dressian. We show the
first example where the first two concepts above do not coincide. In this case
the non-negative flag Dressian equals the tropicalization of the Plücker non-
negative flag variety and is equivalent to flag positroid subdivisions. Using
these characterizations, we provide similar conditions on the Plücker
coordinates for flag varieties of arbitrary rank that are necessary for each
of these positivity notions, and conjecture them to be sufficient. In
particular we show that regular flag positroid subdivisions always come from
the non-negative Dressian. Along the way, we show that the set of Bruhat
interval polytopes equals the set of twisted Bruhat interval polytopes and we
introduce valuated flag gammoids.

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Background on valuated flag matroids
  4. 4 Bruhat polytopes
  5. 5 The geometry of hollow flag matroids
  6. 6 Bruhat polytopes of hollow rank
  7. 7 Flag gammoids
  8. 8 Positivity for hollow flag matroids
  9. 9 Arbitrary rank

##  1\. Introduction

The flag variety $\operatorname{Fl}(\textbf{d},\mathbb{K}^{n})$ of rank
$\textbf{d}=(d_{1},\dots,d_{s})$ over a field $\mathbb{K}$ consists of all
flags of linear spaces $L_{1}\subset\dots L_{s}\subseteq\mathbb{K}^{n}$ of
dimensions $d_{1}<\dots<d_{s}$ respectively. Using the Plücker embedding we
have that $\operatorname{Fl}(\textbf{d},\mathbb{K}^{n})$ is a projective
variety satisfying

$\sum_{t_{i}\in T\backslash S}^{s}(-1)^{i}x_{St_{i}}x_{T\backslash t_{i}}=0$
(1.1)

for every $j\in[s-1]$ and pair of subsets $(S,T)$ where $S\in{[n]\choose
d_{j}-1}$ and $T$ is either in ${[n]\choose d_{j}+1}$ (Plücker relations) or
$T$ is in ${[n]\choose d_{j+1}+1}$ (incidence relations), where
$t_{1}<\dots<t_{s}$ are the elements of $T\backslash S$ in order.

Positivity in the flag variety and its tropicalization has recently gathered a
lot of attention [1, 10, 13, 7, 22, 24, 42]. There are two well studied
notions of non-negativity for the flag variety over the reals. The first,
notion simply asks that all all the Plücker coordinates are non-negative. This
is called Plücker non-negative in [10] or naive non-negative in [1]. This
notion comes as a generalization of poistroids and positroid polytopes which
have been well studied [35, 4, 25]

The second notion, was originally introduced by Lustzig [27, 28]. For the
complete flag variety, this notion is equivalent to Plücker non-negativity
[13, 10], and therefor this second notion is equivalent to asking whether the
flag can be extended to a complete flag which is Plücker non-negative. The
part of the flag variety satisfying this condition is called the totally _non-
negative_ (or tnn) _flag variety_.

The support vector of a flag in
$\operatorname{Fl}(\textbf{d},\mathbb{K}^{n})$, that is, the set of $B$ such
that $x_{B}\neq 0$ are the bases of a flag matroid $(M_{1},\dots,M_{s})$ of
rank d (see Section 3.2 for a precise definition of flag matroid). A flag
matroid M can be encoded in a polytope $P_{\textbf{M}}$ which is the Minkowski
sum $P_{M_{1}}+\dots+P_{M_{s}}$ of the polytopes of the matroid constituents.
The polytopes of flag matroids in the tnn flag variety were shown to be of as
special kind, namely Bruhat (interval) polytopes [24]. These are polytopes
associated to intervals in the poset of permutations given by the Bruhat order
which were further studied in [42]. They are in correspondence with the
Richardson varieties that were shown to subdivide the tnn flag variety [36]
and are also indexed by Bruhat intervals. Parametrizations of Richardson
varieties were first provided by Marsh and Rietsch in [31]. There have been
two different definitions of Bruhat polytopes (see Definitions 4.1 and 4.4)
which were known to give the same class of polytopes for complete rank, but
this was not previously known for general rank. This is our first result:

######  Theorem 1.1 (Theorem 4.6).

The set of twisted Bruhat polytopes is the same as the set of untwisted Bruhat
polytopes.

The tropical flag variety $\operatorname{TFl}(\textbf{d},n)$ was introduced in
[8] as the tropicalization of the flag variety via its Plücker embedding. The
corresponding prevariety $\operatorname{Fl}\operatorname{Dr}(\textbf{d},n)$ is
called flag Dressian and it is defined by

$\bigoplus_{i\in T\backslash S}{\bm{\mu}}(Si)\odot{\bm{\mu}}(T\backslash
i)\text{ achieves the minimum at least twice.}$ (1.2)

for all pairs of sets $(S,T)$ as in (1.1). For $T\in{[n]\choose d_{j}+1}$
these are the tropical Plücker relations introduced in [39] for the tropical
Grassmannian. For $T\in{[n]\choose d_{j+1}+1}$ these are the tropical
incidence relations which were first shown by Haque [20] to characterize
containment of tropical linear spaces. Much like Dressians characterize
regular subdivisions of matroid polytopes into matroid polytopes [38], the
flag Dressian characterizes coherent mixed subdivisions of flag matroid
polytopes into flag matroid polytopes [8].

The study of positivity of the tropical Grassmannian was initiated by Speyer
and Williams [40] where it was parametrized using a tropicalized version of
the paramterizations used by Postnikov for positroid cells [35]. It was
simultaneously proven in [2] and [41] that the positive tropical Grassmannian
is determined by the three-term positive relations and that it contained all
valuated matroids inducing positroid subdivisions. Positroid subdivisions are
of particular interest as they induce subvidisions of the amplithuedron
introudced by Arkani-Hamed and Trnka [3] for the study of scattering
amplitudes in the $\mathcal{N}=4$ SYM model [1, 2, 26].

The positivity notions of the flag variety can be taken to the tropical
context, by looking at the image under the valuation map of the flag variety
defined over the Puiseux series. In this paper, we consider the following
notions of non-negativity for valuated flag matroids:

  1. (a)

The tropicalization of the tnn flag variety,
$\operatorname{TFl}^{\textsl{tnn}}(\textbf{d},n)$. These are the valuated flag
matroids that can be realized by a flag of linear spaces coming from a matrix
where all the minors are positive. We call such valuated flag matroids
_totally non-negative_.

  2. (b)

Valuated flag matroids inducing subdivisions consisting exclusively of Bruhat
polytopes. We call such subdivisions _Bruhat subdivisions_.

  3. (c)

The tropicalization of the Plücker non-negative flag variety,
$\operatorname{TFl}^{\geq 0}(\textbf{d},n)$. These are the flag matroids that
can be realized by a flag of linear spaces whose $d_{i}\times d_{i}$ minors
using the first $d_{i}$ rows are positive. Such flag matroids were _called
flag positroids_ in [7].

  4. (d)

Valuated flag matroids inducing subdivisions consisting exclusively of flag
positroid polytopes. We call such subdivisions _flag postroid subdivisions_.

  5. (e)

The _non-negative flag Dressian_ $\operatorname{FlDr}^{\geq 0}(\textbf{d},n)$,
consisting of all flag valuated matroids ${\bm{\mu}}$ satisfying

$\bigoplus_{\begin{subarray}{c}t_{i}\in T\backslash S\\\ i\text{
odd}\end{subarray}}{\bm{\mu}}(Si)\odot{\bm{\mu}}(T\backslash
i)=\bigoplus_{\begin{subarray}{c}t_{i}\in T\backslash S\\\ i\text{
even}\end{subarray}}{\bm{\mu}}(Si)\odot{\bm{\mu}}(T\backslash i)$ (1.3)

for all $(S,T)$ as in (1.1). This definition can be rephrased as flag matroids
over the hyperfield of signed tropical numbers [21] which are positively
signed.

All of these notions of positivity were shown to be the same for the
Grassmannian [41, 2]. Then it was proved in [22] that these notions agree for
the flag variety with full support (i.e. when the flag matroid polytope is the
permutahedron). Both of this cases were then generalized for flags of
consecutive rank in [7] without necessarily full support. The main goal of
this paper is to look at what happens to these notions whenever there are gaps
d.

It is immediate that a is stronger than c, b is stronger than d and c is
stronger than both d and 1.3. It follows from [24] (see Theorem 4.2) that a is
stronger than b. An example is given in [7, Example 4.6] showing that c does
not imply a or b for non-consecutive flags. We provide the first example
showing that a is stronger than b for non-consecutive flags (see Example 5.2).
Moreover, this example shows that total positivity can can not be read from
the matroid subdivision. It is asked in [7, Question 1.6] whether c and 1.3
are equivalent for unvaluated matroids. This can be thought of as a flag
generalization of the fact that positivity implies realizability for oriented
matroids [5].

To study non consecutive flags, it makes sense to focus on the extremal case
of rank $(1,n-1)$. For Dressians, all the geometric information is
concentrated in the octahedron because it is the matroid polytope of
$U_{2,4}$, the smallest matroid where where a non-trivial tropical Plücker
relation appears. Similarly hexagons are the polytope of $(U_{1,3},U_{2,3})$,
the smallest flag matroid containing a non-trivial tropical incidence
relation, so most of the geomteric information is contained in them (although
the subdivision is not determined by them, see [22, Figure 3]). So the idea is
that copies of the polytope of $(U_{1,n},U_{n-1,n})$, the smallest flag
matroid where the equations above have $n$ terms, contain enough information
to determine all these positivity concepts, just like in the consecutive rank
case. We use the adjective _holllow_ for matroids of rank
$\textbf{d}=(1,n-1)$.

In the hollow case, there is only one incidence relation (and no non-trivial
Plücker relations). To simplify notations, we write

$\lambda_{i}:={\bm{\mu}}(i)\odot{\bm{\mu}}([n]\backslash i)$

for the terms that appear in the incidence relation. Formally, we have a
vector $\lambda\in\mathbb{TP}^{n-1}$ in tropical projective space. It is
invariant under translation of tropical linear spaces (scaling of coordinates)
as it is homogenous. Geometrically, we can think of the lambda values as the
relative heights of the labeled points in the mixed subdivision induced by
${\bm{\mu}}$ that land in the center of $\Delta(1,n-1;n)$. For what lies
below, we also let $\lambda_{0}=\lambda_{n+1}=\infty$.

Our next main result, is completely characterizing all positivity notions for
hollow rank in terms of the quadratic terms of the Plücker relations (see
Figure 4). In particular we answer [7, Question 1.6] positively for this case.

######  Theorem 1.2 (Theorems 6.6, 8.2 and 8.4).

Let ${\bm{\mu}}\in\operatorname{FlDr}(1,n-1;n)$ be a flag valuated matroid of
hollow rank.

  1. 1.

${\bm{\mu}}$ is totally non-negative if and only if

$\lambda_{i}\geq\lambda_{i-1}\oplus\lambda_{i+1}$ (1.4)

for every $i\in[n]$.

  2. 2.

${\bm{\mu}}$ induces a Bruhat subdivision if and only if

$\lambda_{i}=\lambda_{i-1}\oplus\lambda_{i+1}$ (1.5)

for every $i$ such that $\lambda_{i}=\lambda_{1}\oplus\dots\oplus\lambda_{j}$.

  3. 3.

The following are equivalent:

    1. i.

${\bm{\mu}}$ is a valuated flag positroid

    2. ii.

${\bm{\mu}}$ induces a flag positroid subdivison

    3. iii.

${\bm{\mu}}\in\operatorname{FlDr}^{\geq 0}(1,n-1;n)$, i.e.

$\bigoplus_{i\text{ odd}}\lambda_{i}=\bigoplus_{i\text{ even}}\lambda_{i}$
(1.6)

In the process of proving Theorem 1.2 we introduce valuated flag gammoids (see
Theorem 7.1 and Definition 7.2). They generalize valuated gammoids introduced
in [18], which are in turn a valuated generalization of the matroid class
introduced by Mason in [30]. We use them to simplify the Marsh-Rietsch
parametrization of Richardson cells, which was first given a graphic
interpretation by Boretsky [13].

As our results are all related to the quadratic monomials that appear in
(1.2), we introduce the following notation:

######  Definition 1.3.

Given a valuated flag matroid ${\bm{\mu}}$ of rank
$\textbf{d}=(d_{1},\dots,d_{s})$, a _Plücker pair_ $(S,T)$ consists of a pair
of subsets $S\subseteq T$ such that $S\in{[n]\choose d_{i}-1}$ and either
$T\in{[n]\choose d_{i+1}+1}$ for $i\in[s-1]$, or $T\in{[n]\choose d_{i}+1}$
for $i\in[s]$. The _lambda-values_ for the Plücker pair $(S,T)$ are

$\lambda^{S,T}_{j}:={\bm{\mu}}(St_{j})\odot{\bm{\mu}}(T\backslash t_{j}).$

where $t_{1}<\dots<t_{k}$ are the elements of $T\backslash S$. Formally,
$\lambda\in\mathbb{T}\mathbb{P}^{|T\backslash S|-1}$ is a vector in tropical
projective space.

Again we let $\lambda_{0}=\lambda_{|T\backslash S|+1}=\infty$. With this
notation we have that the Plücker-incidence relations (1.2) can be written
simply as

$\bigoplus_{i}\lambda_{i}^{S,T}\text{ achieves the minimum at least twice.}$
(1.7)

With this language we can formulate our main result for arbitrary rank which
is proved using Theorem 1.2 and the fact that one can associate to each
Plücker pair a $(S,T)$ a certain hollow flag matroid encoding the positivity
conditions for that pair (see Definition 9.3).

######  Theorem 1.4.

Let ${\bm{\mu}}\in\operatorname{FlDr}(\textbf{d},n)$ be any valuated flag
matroid.

  1. 1.

If ${\bm{\mu}}$ is totally non-negative then

$\lambda_{i}^{S,T}\geq\lambda^{S,T}_{i-1}\oplus\lambda^{S,T}_{i+1}$ (1.8)

for every Plücker pair $(S,T)$ and $i\leq|T\backslash S|$.

  2. 2.

If ${\bm{\mu}}$ induces a Bruhat subdivision then

$\lambda_{i}^{S,T}=\lambda^{S,T}_{i-1}\oplus\lambda^{S,T}_{i+1}$ (1.9)

for every every Plücker pair $(S,T)$ and $i$ such that
$\lambda_{i}=\lambda_{1}\oplus\dots\oplus\lambda_{j}$.

  3. 3.

If ${\bm{\mu}}$ induces a flag positroid subdivision then
${\bm{\mu}}\in\operatorname{FlDr}^{\geq 0}(\textbf{d},n)$, i.e.

$\bigoplus_{i\text{ odd}}\lambda^{S,T}_{i}=\bigoplus_{i\text{
even}}\lambda^{S,T}_{i}$ (1.10)

From this theorem it follows immediately that b implies 1.3, but we still do
no know whether b implies c. However, we feel bold to conjecture that the
situation in hollow rank extends to arbitrary rank and the converses of
Theorem 1.4 also hold:

######  Conjecture 1.5.

Let ${\bm{\mu}}\in\operatorname{FlDr}(\textbf{d},n)$ be any valuated flag
matroid.

  1. 1.

${\bm{\mu}}$ is totally non-negative if and only if (1.8) holds for every
Plücker pair.

  2. 2.

${\bm{\mu}}$ induces a Bruhat subdivision if and only if (1.9) holds for every
Plücker pair.

  3. 3.

The following are equivalent:

    1. i.

${\bm{\mu}}$ is a valuated flag positroid

    2. ii.

${\bm{\mu}}$ induces a flag positroid subdivison

    3. iii.

${\bm{\mu}}\in\operatorname{FlDr}^{\geq 0}(\textbf{d};n)$.

We want to remark that the first part of 1.5 implies the second (see Corollary
9.7). This follows from the fact that it is enough for the second part to
prove it for unvaluated flag matroids (see Proposition 9.6) and that both
parts are equivalent for unavluated matroids. Similarly, for the equivalence
of $ii.$ and $iii.$ in the third part it is enough to prove it for unvaluated
matroids, which is precisely [7, Question 1.6]. This conjecture would
characterize all positive notions for partial flag varieties in terms of the
quadratic terms appearing in the Plücker-incidence relations. The cases where
we know 1.5 hold are the Grassmannian case [2, 41], consecutive flags [7, 22]
(equations (1.8), (1.9) and (1.10) are all equivalent to the three term
Plücker-incidence relations for these cases) and hollow rank (Theorem 1.2).

Our results only concern type A partial flag varieties, but positivity and
Bruhat polytopes have also been studied for generalized flag varieties $G/P$
[15, 24, 42]. A more general approach would involve Coxeter flag matroids and
their polytopes as described in [9]. The tropicalization of type D
Grassmannians has nice interpretation in terms of $\Delta$-matroids [37], so
we believe similar results can be obtained for this case. This is in stark
contrast with the type C Grassmannian, where the polyhedral geometry is poorly
understood [12].

The paper is organized as follows. Section 3 introduces the necessary
background on valuated flag matroids. Bruhat polytopes are studied in Section
4, which is where Theorem 1.1 is proven. The matroid subdivisions of
$\Delta(1,n-1;n)$ are explained in Section 5, where we give Example 5.2
showing that notions a and b differ for non-consecutive rank. We prove part 2.
of Theorem 1.2 in Section 6. We introduce valuated flag gammoids in Section 7
and use this language to reformulate the parametrization of the
tropicalization of Richardson cells in
$\operatorname{TFl}^{\textsl{tnn}}(1,n-1;n)$ in a way that does not make use
of Bruhat intervals. Then we finish the proof of Theorem 1.2 in Section 8.
Finally we prove Theorem 1.4 in Section 9, where also give some additional
remarks for 1.5 in general rank and some final questions.

### Acknowledgements

The author is grateful to Michael Joswig for the support during the initial
stages of this project. The author would also like to thank Lauren Williams
for useful discussions about Bruhat polytopes.

##  2\. Notation

The following notation will be used throughout:

  * •

$[n]=\\{1,2,\dots,n\\}$

  * •

${[n]\choose d}=\\{\text{subsets of }[n]\text{ of size }d\\}$

  * •

$Si=S\cup\\{i\\}$ and $T\backslash i=T\backslash\\{i\\}$.

  * •

$\mathbb{T}=\mathbb{R}\cup\\{\infty\\}$ is the tropical semifield where
$\oplus=\min$ and $\odot=+$.

  * •

$\mathbb{TP}^{n-1}=\mathbb{R}^{n}/\mathbb{R}(1,\dots,1)$ is the tropical
projective space (we keep the all $\infty$ vector because the lambda values
from Definition 1.3 can be all $\infty$).

  * •

Letters in bold correspond to flags of objects, for example
${\bm{\mu}}=(\mu_{1},\dots,\mu_{s})$ a valuated flag matroid of rank
$\textbf{d}=(d_{1},\dots,d_{s})$.

  * •

$\textbf{n}=(1,\dots,n)$ is the rank of the complete flag.

  * •

For a subset $B$ of $[n]$,

$e_{B}:=\sum_{i\in B}e_{i}$

is the indicator vector of $B$.

  * •

For a flag of sets $\textbf{B}=(B_{1},\dots,B_{s})$,
$e_{\textbf{B}}=e_{B_{1}}+\dots+e_{B_{s}}$.

  * •

$\Delta(d,n)=[0,1]^{n}\cap\left\\{\sum\limits_{i=1}^{n}x_{i}=d\right\\}$ is
the hypersimplex.

  * •

$\Delta(\textbf{d},n)=\sum\limits_{i=1}^{s}\Delta(d_{i},n)$ is the polytope of
the uniform flag matroid of rank d.

  * •

$\Pi_{n}=\Delta(\textbf{n},n)$ is the permutahedron of dimension $n-1$.

  * •

For flag varieties or hypersimplices, we use as arguments either
$(\textbf{d},n)$ or equivalently $(d_{1},\dots,d_{s};n)$.

  * •

$\operatorname{Sym}_{n}$ is the symmetric group.

  * •

$\tau_{i}$ is the transposition $(i,i+1)\in\operatorname{Sym}_{n}$.

  * •

Other permutations are written in one line notation without parenthesis, for
example $\sigma=312$ is the permutation $\sigma\in\operatorname{Sym}_{n}$
where $\sigma(1)=3$, $\sigma(2)=1$ and $\sigma(3)=2$.

##  3\. Background on valuated flag matroids

We assume basic knowledge of tropical geometry, matroid theory and polyhedral
subdivisions. For an introduction to these fields and further details we point
the reader to [23, 32] for tropical geometry, [17] for polyhedral subdivisions
and [34, 9] for matroid theory.

###  3.1. Mixed subdivisions

Recall that a function $w:V(P)\to\mathbb{R}$ on the vertices of a polytope
$P\subset\mathbb{R}^{n}$ induces a regular subdivision $\mathcal{S}_{w}(P)$.
The cells in $\mathcal{S}_{w}(P)$ are the convex hull of all $v\in V$ that
minimize $w(v)-x\cdot v$ for some $x\in(\mathbb{R}^{*})^{n}$, which we write
$\mathcal{S}_{w}(P)^{x}$ . We also get a polyhedral subdivision $\Sigma_{w}$
of $(\mathbb{R}^{*})^{n}$, where for every polytope $Q\in\mathcal{S}_{w}(P)$
there is a cell $Q^{\vee}\in\Sigma_{w}$ coonsisting of all
$x\in(\mathbb{R}^{*})^{n}$ such that $\mathcal{S}_{w}(P)^{x}=Q$.

Now consider a Minkowski sum of polytopes $P_{1}+\dots+P_{k}$. Given a tuple
of functions $\textbf{w}=(w_{1},\dots w_{k})$ where
$w_{i}:V(P_{i})\to\mathbb{R}$, the (coherent) mixed subdivision
$\mathcal{S}_{\textbf{w}}(P_{1}+\dots+P_{k})$ consists of all polytopes of the
form

$\mathcal{S}_{w_{1}}(P_{1})^{x}+\dots+\mathcal{S}_{w_{k}}(P_{k})^{x}.$

The dual complex $\Sigma_{\textbf{w}}$ is the common refinement of all
$\Sigma_{i}$. Equivalently, mixed subdivisions are the subdivisions induced by
the projection of a product of simplices to the Minkowski sum.

###  3.2. Flag matroids

Matroids have many equivalent definitions. Of most importance to us, is the
polyhedral approach intiated by Gelfand, Goresky, MacPherson and Serganova
[19]. We begin by introducing the polytopes that will play a role. First we
have the _hypersimplex_

$\Delta(d,n)=[0,1]^{n}\cap\\{x_{1}+\dots+x_{n}\\}=\operatorname{conv}\left\\{e_{B}\mid
B\in{[n]\choose d}\right\\}$

where $e_{B}$ is the indicator vector of the subset $B$. The _permutahedron_
is the convex hull of the orbit of $(1,\dots,n)$ under permuting its
coordinates, or, equivalently:

$\Pi_{n}=\Delta(1,n)+\dots+\Delta(n,n).$

A _matroid_ $M$ with underlying set $[n]$ is a _subpolytope_ (the convex hull
of a subset of vertices) of a hypersimplex $\Delta(d,n)$ such that every edge
of $M$ is an edge of the hypersimplex. The number $d$ is the _rank_ of a
matroid and any $B$ such that $e_{B}$ is a vertex of $M$ is called a _basis_ ,
from which you can recover the more traditional notions of matroids. We write
$\mathcal{B}(M)$ for the set of bases of $M$.

Two matroids $M_{1}$ and $M_{2}$ on the same ground set and of ranks
$d_{1}<d_{2}$ form a _quotient_ if every flat of $M_{1}$ is also a flat
$M_{2}$. The motivation behind this definition is that if a linear space sits
inside another linear space, then the corresponding representable matroids
form a quotient. A _flag matroid_ is $\textbf{M}=(M_{1},\dots,M_{s})$ is a
chain of matroids such that any pair of them form a quotient. The matroids
$M_{i}$ are the _constituents_ of M. The _polytope_ $P_{\textbf{M}}$ of a flag
matroid M is the Minkowski sum $M_{1}+\dots+M_{s}$. For example, if M consists
of a complete chain of uniform matroids $(U_{1,n},\dots,U_{n,n})$, then
$P_{\textbf{M}}$ is the permutahedron $\Pi_{n}$. The _rank_ of M is the tuple
$(d_{1},\dots,d_{s})$ consisting of the ranks of the constituent matroids. A
polytope is a flag matroid polytope if and only if it is a _generalized
permutahedron_ (all edge directions are of the form $e_{i}-e_{j}$) and it is
the subpolytope of a Minkowski sum of hypersimplices [9, Theorem 1.11.1]. The
_bases_ of a flag matroid M are

$\mathcal{B}(\textbf{M}):=\bigcup_{i=1}^{s}\mathcal{B}(M_{i}).$

A _flag of subsets_ of rank $\textbf{d}=(d_{1},\dots,d_{s})$ is a chain of
subsets of $[n]$, $\textbf{B}=B_{1}\subset\dots\subset B_{s}$ where
$|B_{i}|=d_{i}$. Given a flag of subsets B, we define
$e_{\textbf{B}}:=e_{B_{1}}+\dots+e_{B_{s}}$. The polytope of a flag matroid
with uniform constituents $\Delta(\textbf{d},n)$ is the convex hull of
$e_{\textbf{B}}$ for all flags of subsets B.

A _flag of linear subspaces_ of rank d over a field $\mathbb{K}$ is a chain
$L_{1}\subset\dots\subset L_{s}$ of linear subspaces of a vector space
$\mathbb{K}^{n}$ such that $\dim(L_{i})=d_{i}$. A flag matroid
$\textbf{M}=(M_{1},\dots,M_{s})$ is _representable_ (or _realizable_) over
afield $\mathbb{K}$ if there is a flag of subspaces
$(L_{1},\dots,L_{s})\in\operatorname{Fl}(\textbf{d},\mathbb{K}^{n})$ such that
the matroid of $M(L_{i})$ of $L_{i}$ is $M_{i}$. We write
$\textbf{M}(\textbf{L})$ for the flag matroid of the flag L.

When the rank of a flag, whether it is of matroids, sets or linear spaces, is
$\textbf{n}=(1,\dots,n)$, we call the flag _complete_. We call the flag
_consecutive_ if the rank is of the form $(d+1,\dots,d+s)$. On the other
extreme, we call the flag _hollow_ if $\textbf{d}=(1,n-1)$.

###  3.3. Valuated matroids

A _valuated matroid_ with underlying matroid $M$ is a function
$\mu:\mathcal{B}(M)\to\mathbb{R}$ such that the regular subdivision it induces
on $P_{M}$ consists exclusively of matroids. Equivalently [38], $\mu$
satisfies the _Plücker relations_ (1.2) for pairs of subsets $S\in{[n]\choose
d-1}$ and $T\in{[n]\choose d+1}$. An _initial matroid_ of $\mu$ is a matroid
appearing in the subdivision, and we write
$\mu^{x}:=\mathcal{S}_{\mu}(P_{M})^{x}$. A valuated matroid can be interpreted
as a function from ${[n]\choose d}\to\mathbb{T}:=\mathbb{R}\cup\\{\infty\\}$
by setting the value of all non-bases to infinity. A matroid can be considered
as a valuated matroid with trivial valuation, that is, by setting $0$ to all
of its bases and $\infty$ to the non-bases. _The Dressian_
$\operatorname{Dr}(d,n)$ is the space of all valuated matroids of rank $d$
over $[n]$. It can be given a fan structure as a subfan of the secondary fan
of $\Delta(d,n)$, or, equivalently, by the terms in which the three-term
Plücker relations achieve the minimum [33].

A pair of valuated matroids $(\mu_{1},\mu_{2})$ of ranks $d_{1}<d_{2}$ form a
_quotient_ if they satisfy the _tropical incidence relations_ (1.2), for every
pair of subsets $S\in{[n]\choose d_{1}-1}$ and $T\in{[n]\choose d_{2}+1}$. If
we restrict to trivially valuated matroids we recover the concept of matroid
quotient described above.

A _valuated flag matroid_ ${\bm{\mu}}=(\mu_{1},\dots,\mu_{s})$ is a chain of
valuated matroids such that every pair of them form a quotient. _The flag
Dressian_ $\operatorname{Fl}\operatorname{Dr}(\textbf{d};n)$ consists of all
valuated flag matroids of rank d. A sequence of valuated matroids
${\bm{\mu}}=(\mu_{1},\dots,\mu_{s})$ is a flag valuated matroid if and only if
the supports $(M_{1},\dots,M_{s})$ form a flag matroid polytope and the mixed
subdivision $\mathcal{S}_{\bm{\mu}}(M_{1}+\dots+M_{s})$ consists exclusively
of flag matroid polytopes [8, Theorem 4.4.2]. Therefore a flag matroid can
also be thought of as a function from the vertices of a flag matroid polytope
to $\mathbb{R}$. Again we write
${\bm{\mu}}^{x}:=\mathcal{S}_{\bm{\mu}}(P_{\textbf{M}})^{x}$ for _initial flag
matroids_.

The _tropical linear space_ $L_{\mu}$ given by a valuated matroid $\mu$ is the
set of points $x\in\mathbb{T}^{n}$ that satisfy that

$\bigoplus_{i\in[n]}x_{i}\odot\mu_{T\backslash i}\enspace\text{ achieves the
minimum at least twice.}$

It is the subcomplex of $\Sigma_{\mu}$ (as defined in Section 3.1), consisting
of all cells $P_{M}^{\vee}$ where $M$ is a loop free matroid in the
subdivision of induced by $\mu$ [38].

Since all tropical linear spaces contain $\mathbb{R}(1,\dots,1)$ in the
lineality space, we think of tropical linear spaces as sitting inside the
_tropical projective plane_

$\mathbb{TP}^{n-1}:=\mathbb{T}^{n}/\mathbb{R}(1,\dots,1).$

Similar to the non-tropical case, a tropical linear space is contained in
another tropical linear space if and only if the corresponding valuated
matroids form a quotient [20]. Hence valuated flag matroids
$(\mu_{1},\dots,\mu_{s})$ are cryptomorphic to flags of tropical linear spaces
$L_{\mu_{1}}\subseteq\dots L_{\mu_{s}}$.

A valuated matroid $\mu$ is said to be _representable_ (or _realizable_) by a
linear subspace $L$, if $L_{\mu}=\operatorname{trop}(L)$. A beautiful fact
about the tropicalization of linear spaces is that tropicalizing commutes with
taking Plücker coordinates, i.e. the tropicalization of a linear space is
given by the valuated matroid obtained by taking the valuation of the Plücker
coordinates the linear space [39].

###  3.4. Positivity

The following two notions of positivity exist for the flag variety. The
_Plücker non-negative flag variety_ $\operatorname{Fl}^{\geq
0}(\textbf{d},\mathbb{R}^{n})$ consists of all flag of linear spaces whose
Plücker cooridnates are non-negative. In other words, $\operatorname{Fl}^{\geq
0}(\textbf{d},\mathbb{R}^{n})$ consists of flags of spaces that come from the
row span of a matrix in $\mathbb{R}^{n\times n}$ such that all top
$d_{i}\times d_{i}$ minors are non-negative for every $i\in[s]$. A stronger
notion, is the _totally non-negative_ (or just tnn) _flag variety_ ,
$\operatorname{Fl}^{\textsl{tnn}}(\textbf{d},\mathbb{R}^{n})$, which consists
of all flags of linear spaces which can be extended to a complete flag in
$\operatorname{Fl}^{\geq 0}(\textbf{n},\mathbb{R}^{n})$, i.e. flags that can
be realized as the row span of a totally non-negative matrix. These two
notions agree if and only if the rank d consists of consecutive integers [10].

A flag matroid M such that $\textbf{M}=\textbf{M}(\textbf{L})$ for a flag
$\textbf{L}\in\operatorname{Fl}^{\geq 0}(\textbf{d},\mathbb{R}^{n})$ is called
a _flag positroid_. For the Grassmannian case $\textbf{d}=(d)$, this recovers
the notion of _positroid_ introduced by Postnikov [35]. If
$\textbf{L}\in\operatorname{Fl}^{\textsl{tnn}}(\textbf{d},\mathbb{R}^{n})$,
then M belongs to a smaller class of matroids (when d is not consecutive)
which corresponds to flag matroids whose polytope is a Bruhat polytope (see
Theorem 4.2).

The study of positivity for valuated matroids started with [40], where the
following framework for the positive part of any tropical variety. Let
$\mathcal{C}=\mathbb{C}\\{\\!\\!\\{t\\}\\!\\!\\}$ be the field of generalized
Puiseux series with complex coefficients111The standard field with non-trivial
valuation which is mostly used is in tropical geometry is the field of Puiseux
series. However, this field has the disadvantage that the valuation is not
surjective; the value group is only $\mathbb{Q}$ and not all of $\mathbb{R}$.
Therefore, only valuated matroids with rational are representable over this
field. We can avoid dealing with this inconvenience by working with a larger
field whose valuation is surjective. (see [29] for details on this field) and
let $\mathcal{R}_{\geq 0}$ subset of $\mathcal{C}$ consisting of all series
whose leading coefficient is a positive real number (and $0$). Given a
subvarierty $X=V(I)$ of $\mathcal{C}^{n}$, the non-negative part
$\operatorname{trop}^{+}(X)$ of the tropicalization of
$\operatorname{trop}(X)$ has the following two equivalent interpretations
[40]:

  * •

The image under the valuation map of $X\cap\mathcal{R}_{\geq 0}^{n}$.

  * •

The intersection

$\bigcap_{f\in I}\operatorname{trop}^{+}(f)$ (3.1)

where $\operatorname{trop}^{+}(f)$ is the set of $w\in\mathbb{R}^{n}$ such
that $\operatorname{in}_{w}(f)$ does not contain any non-zero polynomial in
$\mathcal{R}_{\geq 0}$.

This idea can be extended to projective varieties and their tropicalizations
landing in tropical projective space. Just like with usual tropicalization,
the intersection in (3.1) may require infinite elements $f\in I$. A subset of
$I$ such that the intersection of $\operatorname{trop}^{+}(f)$ for every
element of that subset produces $\operatorname{trop}^{+}(X)$ is called a
_positive-tropical basis_ (see [11] for a comparison with usual tropical
bases).

When applied to the flag variety, we obtain the _non-negative tropical flag
variety_ $\operatorname{TFl}^{\geq
0}(\textbf{d},n):=\operatorname{trop}^{+}(\operatorname{Fl}(\textbf{d},\mathcal{C}_{\geq^{0}}^{n}))$.
We call a valuated flag matroid ${\bm{\mu}}\in\operatorname{TFl}^{\geq
0}(\textbf{d},n)$ a _valuated flag positroid_. By the observations above,
$\operatorname{TFl}^{\geq 0}(\textbf{d},n)$ is the image under the valuation
map of $\operatorname{Fl}^{\geq 0}(\textbf{d},\mathbb{R}^{n})$. The _totally
non-negative tropical flag variety_
$\operatorname{TFl}^{\textsl{tnn}}(\textbf{d},n)$ is the image under the
valuation map of
$\operatorname{Fl}^{\textsl{tnn}}(\textbf{d},\mathbb{R}^{n})$. A valuated flag
matroid ${\bm{\mu}}$ is called tnn if it is in
$\operatorname{TFl}^{\textsl{tnn}}(\textbf{d},n)$.

The _non-negative flag Dressian_ is

$\operatorname{FlDr}^{\geq
0}(\textbf{d},n):=\cap_{f}\operatorname{trop}^{+}(f)$

where the intersection ranges through all Plücker and incidence relations $f$.
This corresponds to flag matroids over the hyperfield of signed tropical
numbers with a positive orientation [21].

It was simultaneously proven in [2] and [41] that the 3-term Plücker relations
form a positive-tropical basis for the Grassmannian. Similarly, it was
simultaneously proven in [22] and [13] that the 3-term incidence relations
form a positive-tropical basis for the complete flag variety. This was
generalized to flag varieties of consecutive rank [7]. The positive
tropicalization of the three term incidence relations,

$\mu(Sj)\odot\mu(Sik)=\mu(Si)\odot\mu(Sjk)\oplus\mu(Sl)\odot\mu(Sij)$ (3.2)

for $i<j<l$, are equivalent to Equations (1.8), (1.9) and (1.10). Whether the
incidence relations together with the Plücker relations form a tropical basis
of the flag variety for any rank d is essentially the content of part 3. of
1.5.

We say that a subdivision of flag matroid polytope M is a _flag positroid
subdivision_ if all the polytopes are flag positroids and it is a _Bruhat
subdivision_ if all the polytopes are Bruhat polytopes. For consecutive rank,
the non-negative flag Dressian is equal to the non-negative tropical flag
variety and it corresponds exactly to valuated flag matroids inducing a flag
positroid subdivision, or equivalent a Bruhat subdivision [2, 7, 22, 41]. As
Bruhat polytopes differ from flag positroid polytopes for non-consecutive
rank, we have that valuated flag matroids inducing Bruhat subdivisions form a
strictly smaller class than subdivisions inducing flag positroid subdivisions.

######  Remark 3.1.

In [6] the term flag positroid is used for positroids which form a matroid
quotient. However one can easily construct positroids which form a quotient
but do not consitute a flag positroid in our terms, for example the flag
matroid with bases ${1,2,3,12,13}$ (see [22, Example 3.3]. Therefore we
restrict the term flag positroid to this stronger realization condition
following [7].

###  3.5. Matroid constructions

Given a valuated matroid $\mu$ of rank $d$, its dual $\mu^{*}$ is a valuated
matroid of rank $n-d$ such that

$\mu^{*}(B):=\mu([n]\backslash B).$

Given an element $i\in[n]$, which is not a coloop, the _deletion_ of $i$ from
$\mu$ is the valuated matroid $\mu\backslash i$ over $n\backslash i$ such that

$\mu\backslash i(B):=\mu(B)$

where $i\notin B$. If $i$ is not loop, the _contraction_ of $i$ from $\mu$ is
the valuated matroid $\mu/i$ over $n\backslash i$ such that

$\mu/i(B):=\mu(B\backslash i)$

where $i\in B$. If $i$ is either a loop or a coloop, $\mu\backslash
i=\mu/i(B)$. Given a subset $I\subseteq[n]$, the valuated matroid
$\mu\backslash I$ is the successive deletion of all elements in $I$, and
similarly $\mu/I$ is the succesive contraction by all elements in $I$. Any
valuated matroid of the form $\mu/I\backslash I$ is called a _minor_.
Sometimes is convenient for notation purposes to write
$\mu|_{I}:={\bm{\mu}}\backslash([n]\backslash I)$ the _restriction_ of
${\bm{\mu}}$ to $I$.

The _translation_ of tropical linear space $L_{\mu}+x$ by a vector
$x\in\mathbb{R}^{n}$ is also tropical linear space. It corresponds to the
valuated matroid $\mu+x$ where

$(\mu+x)(B):=\mu(B)+e_{B}\cdot x.$

All of the above constructions can be extended to valuated flag matroids by
applying it to each constituent. That is

  * •

The _dual_ is ${\bm{\mu}}^{*}:=(\mu_{s}^{*},\dots,\mu_{1}^{*})$ with rank
$\textbf{d}^{*}=(n-d_{s},\dots,n-d_{1})$.

  * •

The _minors_ ${\bm{\mu}}/I\backslash J=(\mu_{1}/I\backslash
J,\dots,\mu_{s}/I\backslash J)$, after possibly removing repeated elements.

  * •

The _translation_ ${\bm{\mu}}+x=(\mu_{1}+x,\dots,\mu_{s}+x)$.

The details that these are indeed flag valuated matroids can be found in [8]
or [16]. We will make use of the following theorem:

######  Theorem 3.2.

[8, Theorem 5.1.2] Two matroids $\mu_{1}$ and $\mu_{2}$ of rank $d$ and $d+1$
form a valuated flag matroid $(\mu_{1},\mu_{2})$ if an only if there exists a
valuated matroid $\nu$ over $[n+1]$ such that $\mu_{1}=\nu/(n+1)$ and
$\mu_{2}=\nu\backslash(n+1)$

For realizable matroids, these construction have all an interpretation at the
level of linear spaces. If $\mu=\mu(L)$, then

  * •

$\mu^{*}$ is the the valuated matroid of the orthogonal complement of $L$.

  * •

$\mu\backslash i$ is the valuated matroid of the projection of $L$ into
$\mathbb{R}^{[n]\backslash i}$,

  * •

$\mu/i$ is the valuated matroid of $L\cap\\{x_{i}=0\\}$

  * •

$\mu+x$ is the valuated matroid of the result of scaling $L$ in the coordinate
$i$ by $t^{x_{i}}$ for every $i$.

A crucial fact is that all of these notions preserve positivity [4,
Proposition 3.5]. So this implies that if ${\bm{\mu}}$ is tnn, minors, duals
or translations of ${\bm{\mu}}$ are also tnn and if ${\bm{\mu}}$ is a flag
positroid, minors, duals and translations of ${\bm{\mu}}$ are also flag
positroids.

Finally, we want to remark that the homogenous nature of the lambda-values of
${\bm{\mu}}$ as defined in Definition 1.3 implies that they are invariant
under translation. This is a fact that we will often make use of ahead.

##  4\. Bruhat polytopes

The following is an ad hoc introduction to Bruhat (interval) polytopes for
type A, in a purely combinatorial fashion. For simplicity, we drop the
“interval” and call them just Bruhat polytopes, following [22]. For details on
generalized Bruhat polytopes for a general flag variety $G/P$ we suggest [42,
Section 6].

A _word_ for a permutation $v$ is sequence of $w=(w_{1},\dots,w_{m})$ where
$v=w_{1}\cdots w_{m}$ using the precomposition product and each $w_{i}$ is a
transposition of the form $\tau_{j}:=(j,j+1)$. We say the word is _reduced_ if
it is length $m$ is minimal among all words for $v$. A subword of $w$ is a
(not necessarily consecutive) subsequence $w^{\prime}=(w_{i_{1}},\dots
w_{i_{m^{\prime}}})$ where $1\leq i_{1}<\dots<i_{m^{\prime}}\leq m$. The
_Bruhat order_ on the symmetric group $\operatorname{Sym}(n)$ is given by
$u\leq v$ if and only if for any (equivalently every) reduced word $w$ for $v$
there is a subword $w^{\prime}$ for $u$. The _length_ of a permutation $v$ is
the minimal length of a word for $v$.

Two different definitions have been used simultaneously in the literature to
assign a polytope to a Bruhat interval. These are based on two different ways
of identifying $\operatorname{Sym}(n)$ with the vertices of the permutahedron
$\Pi_{n}$.

$\displaystyle\phi:\operatorname{Sym}_{n}$
$\displaystyle\to\operatorname{Vert}(\Pi_{n})$ $\displaystyle x$
$\displaystyle\mapsto(x(1),\dots,x(n))$

and

$\displaystyle\psi:\operatorname{Sym}_{n}$
$\displaystyle\to\operatorname{Vert}(\Pi_{n})$ $\displaystyle x$
$\displaystyle\mapsto x\cdot e_{\textbf{n}}$

where $e_{[\textbf{n}]}:=e_{[1]}+\dots+e_{[n]}=(n,\dots,1)$ and $x\cdot
e_{\textbf{n}}$ is given by the action of $\operatorname{Sym}_{n}$ on
$\mathbb{R}^{n}$ of permuting coordinates. Following terminology introduced in
[7], given two permutations $u\leq v$, we call

$Q_{u,v}:=\operatorname{conv}(\phi([u,v]))=\operatorname{conv}\\{(x(1),\dots,x(n))\mid
x\in[u,v]\\}$

the _Bruhat polytope_ and

$\widetilde{Q}_{u,v}:=\operatorname{conv}(\psi([u,v]))=\operatorname{conv}\\{x\cdot
e_{\textbf{n}}\mid x\in[u,v]\\}$

the _twisted Bruhat polytope_.

For general flag varieties $G/P$, the twisted Bruhat polytope has been
generalized, sometimes called _generalized Bruhat interval polytopes_. This
works as follows for type A. Let $\textbf{d}=(d_{1},\dots,d_{s})$ be a rank
vector and let $e_{\textbf{d}}:=e_{[d_{1}]}+\dots+e_{[d_{s}]}$. Consider the
function

$\displaystyle\pi^{\textbf{d}}:\operatorname{Vert}(\Pi_{n})$
$\displaystyle\to\operatorname{Vert}(\Delta(\textbf{d},n))$ $\displaystyle p$
$\displaystyle\mapsto(f^{\textbf{d}}(p(1)),\dots,f^{\textbf{d}}(p(1))).$

where $f:[n]\to[s]$ is the function where $\hat{\phi}^{\textbf{d}}(i)$ is the
$(n+1-i)$-coordinate of $e_{\textbf{d}}$, i.e. the number of times $n+1-i$
appears in the flag $[d_{1}]\subset\dots\subset[d_{s}]$. At the level of flag
matroid polytopes, $\pi^{\textbf{d}}$ is essentially a forgetful morphism,
taking $P_{\textbf{M}}$ to $P_{\textbf{M}^{\prime}}$ where
$\textbf{M}^{\prime}$ is the result of dropping from M all constituents whose
rank is not in d. The stabilizer of $\operatorname{Sym}_{n}$ with respect to
$e_{\textbf{d}}$ subudivides $\operatorname{Sym}_{n}$ into cosets which are
the fibers of the map $\pi^{\textbf{d}}\circ\psi$. We call
$v\in\operatorname{Sym}_{n}$ a _minimal coset representative_ of rank d if it
is the smallest element under the Bruhat order of its coset, i.e. its fiber
under $\pi^{\textbf{d}}\circ\psi$. Let
$\widetilde{\operatorname{BI}}(\textbf{d},n)$ be the set of Bruhat intervals
$[u,v]$ where $v$ is a minimal coset representative of rank d.

######  Definition 4.1.

Given an interval $[u,v]\in\widetilde{\operatorname{BI}}(\textbf{d},n)$, the
(generalized) twisted Bruhat polytope $\widetilde{Q}_{u,v}^{\textbf{d}}$ is

$\widetilde{Q}_{u,v}^{\textbf{d}}:=\operatorname{conv}(\pi^{\textbf{d}}\circ\psi([u,v]))=\operatorname{conv}\\{x\cdot
e_{\textbf{d}}\mid x\in[u,v]\\}.$

Because $\phi$ is the most natural way of identifying permutations with the
vertices of the permutahedron, the untwisted variant of Bruhat polytopes is
usually used as definition in the complete flag case, [24, Definition A.5] and
[24, Definition 2.2]. However, for the general case, the twisted variant is
used because of the connection to Richardson varieties, [24, Definition 6.9]
and [24, Definition 7.8]. These are cells in a partition of
$\operatorname{Fl}^{\textsl{tnn}}(\mathbb{R}^{n},\textbf{d})$ which are
indexed by a pair $u\leq v$ with $v$ a minimal coset representative.
Concretely, we have:

######  Theorem 4.2.

[24, Corollary 6.11] Let L be a flag of linear spaces in the Richardson
variety
$\mathcal{R}_{u,v}\subseteq\operatorname{Fl}^{\textsl{tnn}}(\mathbb{R}^{n},\textbf{d})$.
If M is the flag matroid of L, then $P_{\textbf{M}}=\widetilde{Q}_{u,v}$.

We have in particular that twisted Bruhat polyopes are flag matroid polytopes
and when d is consecutive, twisted Bruhat polytopes are exactly the same as
flag positroid polytopes. Another important result about twisted Bruhat
polytopes which we will use later is the following:

######  Theorem 4.3.

[42, Theorem 7.13] The face of twisted Bruhat polytope is a twisted Bruhat
polyope.

Even though $Q_{u,v}$ may differ from $\widetilde{Q}_{u,v}$, having these two
different definitions is not a big issue for the complete flag case because
the set of Bruhat polytopes is the same as the set of twisted Bruhat
polytopes. To see this,notice that $\phi^{-1}\circ\psi$ is the function
$\iota:\operatorname{Sym}_{n}\to\operatorname{Sym}_{N}$ given by

$\iota:x\mapsto(i\mapsto n+1-x^{-1}(i)).$ (4.1)

This function consists of the composition of the order preserving involution
$x\mapsto x^{-1}$ and the order reversing involution $x\mapsto(i\mapsto
n+1-x(i))$. Therefor $\iota$ reverses the Bruhat order. As cosets are of size
1 in the complete flag case, we do not impose any conditions on $v$ to form a
twisted or untwisted Bruhat polytope. So we have
$\widetilde{Q}_{u,v}=Q_{\iota(u),\iota(v)}$ and hence the set of Bruhat
polytopes is the same as the set of untwisted Bruhat polytopes for this case.

For non-complete flags, we can also define untwisted Bruhat polytopes using
$\phi$. Namely, let $\operatorname{BI}(\textbf{d},n)$ be the set of Bruhat
intervals $[u,v]$ where $v$ is the minimal element in its fiber of
$\pi^{\textbf{d}}\circ\phi$.

######  Definition 4.4.

Given an interval $[u,v]\in\operatorname{BI}(\textbf{d},n)$, the (generalized)
_Bruhat polytope_ $Q_{u,v}^{\textbf{d}}$ is

$Q^{\textbf{d}}_{u,v}:=\operatorname{conv}(\pi^{\textbf{d}}\circ\phi([u,v]))$

######  Example 4.5.

Consider the Bruhat interval $[2134,3241]$, using one line notation for
permutations. This interval consists of eight permutations

$[2134,3241]=\\{2134,2314,2143,3124,2341,3142,3214,3241\\}.$

For rank $\textbf{d}=(1,3)$, we have that the corresponding untwisted Bruhat
polytope is

$Q^{(1,3)}_{2134,3241}:=\operatorname{conv}((1,0,1,2),(1,0,2,1),(1,1,0,2),(1,1,2,0))$

and the twisted variant is

$\widetilde{Q}^{(1,3)}_{2134,3241}:=\operatorname{conv}((1,2,1,0),(1,2,0,1),(1,1,2,0),(0,2,1,1),(1,0,2,1),(0,1,2,1))$

Figure 1. The Bruhart polytopes $Q^{(1,3)}_{2134,3241}$ (left) and
$\widetilde{Q}^{(1,3)}_{2134,3241}$ (right).

The same argument used in the complete flag case does not work to prove that
twisted and untwisted Bruhat polytopes give the same set for general d,
because given $u\leq v$ where $v$ is a minimal coset representative, then
$\iota(v)\leq\iota(u)$ does not necessarily satisfy that $\iota(u)$ is the
minimal element in its fiber under $\pi^{\textbf{d}}\circ\phi$. However, we do
have the following:

######  Theorem 4.6.

The set of Bruhat polytopes and the set of twisted Bruhat polytopes are the
same for arbitrary rank d.

###### Proof.

Let $[u,v]\in\widetilde{\operatorname{BI}}_{\textbf{d},n}$ and consider the
twisted Bruhat polytope $\widetilde{Q}^{\textbf{d}}_{u,v}$. We will construct
another Bruhat interval
$[u^{\prime},v^{\prime}]\in\operatorname{BI}_{\textbf{d},n}$ such that
$\widetilde{Q}^{\textbf{d}}_{u,v}=Q^{\textbf{d}}_{u^{\prime},v^{\prime}}$.

Let M be the flag matroid such that
$P_{\textbf{M}}=\widetilde{Q}^{\textbf{d}}_{u,v}$. By Theorem 4.2, M is tnn.
Therefor the dual $\textbf{M}^{*}$ is tnn, and hence there is a Bruhat
interval $[a,b]\in\widetilde{\operatorname{BI}}_{\textbf{d}^{*},n}$ such that
$P_{\textbf{M}^{*}}=\widetilde{Q}^{\textbf{d}^{*}}_{a,b}$.

The key observation is that from (4.1) we obtain the following commutative
diagram:

${\operatorname{Sym}_{n}}$${\operatorname{Vert}(\Pi_{n})}$${\operatorname{Vert}(\Delta(\textbf{d}^{*},n))}$${\operatorname{Sym}_{n}}$${\operatorname{Vert}(\Pi_{n})}$${\operatorname{Vert}(\Delta(\textbf{d},n))}$$\scriptstyle{\psi}$$\scriptstyle{\\_^{-1}}$$\scriptstyle{\pi^{\textbf{d}^{*}}}$$\scriptstyle{\\_^{*}}$$\scriptstyle{\\_^{*}}$$\scriptstyle{\phi}$$\scriptstyle{\pi^{\textbf{d}}}$
(4.2)

Because $b$ is the minimal element in its fiber of
$\pi^{\textbf{d}^{*}}\circ\psi$ and $\\_^{*}$ is bijective, we have that
$b^{-1}$ is minimal in the fiber of $\pi^{\textbf{d}}\circ\phi$. Hence
$[a^{-1},b^{-1}]\in\operatorname{BI}(\textbf{d},n)$. Therefor we have that

$\displaystyle P_{\textbf{M}}$
$\displaystyle=(\widetilde{Q}^{\textbf{d}^{*}}_{a,b})^{*}$
$\displaystyle=\operatorname{conv}(\pi^{\textbf{d}^{*}}\circ\psi([a,b]))^{*}$
$\displaystyle=\operatorname{conv}(\pi^{\textbf{d}}\circ\phi([a^{-1},b^{-1}]))$
$\displaystyle=Q^{\textbf{d}}_{a^{-1},b^{-1}}$

is an untwisted Bruhat polytope, as we wanted.

We have now proven that every twisted Bruhat polytope is also an untwisted
Bruhat polytope, but the other direction follows using the same construction
since the vertical arrows in (4.2) are bijections. ∎

######  Example 4.7.

Recall the Bruhat polytopes computed in Example 4.5 for the interval
$[2134,3241]$. The matroid M whose polyope
$P_{\textbf{M}}=Q^{(1,3)}_{2134,3241}$ has bases

$\mathcal{B}(\textbf{M})=\\{3,4,123,124,134\\}.$

Its dual $\textbf{M}^{*}$ has bases

$\mathcal{B}(\textbf{M}^{*})=\\{2,3,4,123,124\\}.$

A simple verification yields that for the Bruhat polytope for the interval
$[2314,2431]$ satisfies $Q^{(1,3)}_{[2314,2431}=P_{\textbf{M}^{*}}$. The
inverses of $[2314,2431]$ are

$[3214,4132]=\\{3214,3142,4123,4132\\}.$

and indeed we have that

$Q^{(1,3)}_{2134,3241}=\widetilde{Q}^{(1,3)}_{3214,4132}.$

Similarly the flag matroid $\textbf{M}^{\prime}$ that satisfies
$P_{\textbf{M}^{\prime}}=\widetilde{Q}^{(1,3)}_{2134,3241}$ has bases

$\mathcal{B}(\textbf{M}^{\prime})=\\{2,3,123,124,134,234\\}.$

The polytope of its dual is the twisted Bruhat polytope of the interval
$[1243,4132]$ and by taking inverses we obtain

$\widetilde{Q}^{(1,3)}_{2134,3241}=Q^{(1,3)}_{1243,2431}.$

##  5\. The geometry of hollow flag matroids

In this section we study hollow valuated flag matroids
$(\mu,\nu)\in\operatorname{Dr}(1,n-1;n)$. These correspond to a point inside
the tropical hyperplane in $\mathbb{TP}^{n-1}$. The vertices of
$\Delta(1,n-1;n)$ are all points with one coordinate equal to $0$, one
coordinate equal to $2$ and the other $(n-2)$ coordinates equal to $1$.
However, to make notation simpler and more intuitive, we are going to work
with a translation of $\Delta(1,n-1;n)$ (and all matroid polytopes of rank
$(1,n-1)$) by $(-1,\dots,-1)$ so that points have only two non-zero
coordinates, which are equal to $-1$ and $1$. We make a slight abuse of
notation by also calling this polytope $\Delta(1,n-1;n)$. This way
$\Delta(1,n-1;n)$ is a centrally symmetric polytope and its matroid
subdivisions are induced by the coordinate hyperplanes as we explain below. In
Figure 2 we see a finest subdivision of the cuboctahedron $\Delta(1,3;4)$
where two of the coordinate hyperplanes each simultaneously slice up
$\Delta(1,3;4)$ in half across a hexagon. Two of the polytopes are the
Minkowski sum of a tetrahedron with a segment while the other two are the
Minkowski sum of two triangles.

Figure 2. A finest matroid subdivision of the cuboctahedron $\Delta(1,3;4)$.

Every subpolytope of the simplices $\Delta(1,n)$ and $\Delta(n-1,n)$ is a
matroid. By equation (1.2), flag matroids $\textbf{M}=(M,N)$ of rank $(1,n-1)$
are indexed by pairs of subsets $(I,J)$ of $[n]$ such that $|I\cap J|\neq 1$,
where $I$ is the set of non-loops of $M$ and $J$ is the set of non-coloops of
$N$. The polytope $P_{\textbf{M}}$ is full dimensional if and only if $I\cup
J=[n]$ and $I\cap J\neq\emptyset$.

The most simple type of non-trivial polytopal subdivision is splitting a
polytope in two by a hyperplane. The matroid subdivisions that arise in
$\Delta(1,n-1;n)$ are relatively simple, in that they all consists of the a
matroid polytope being simultaneously subdivided by $k$ hyperplanes into
$2^{k}$ pieces for some $k\leq n-2$.

######  Proposition 5.1.

Let $\textbf{M}=(M,N)$ be a hollow flag matroid. Then all of its matroid
subdivisions are induced by a hyperplane arrangement with hyperplanes of the
form $\\{x_{i}=0\\}$.

###### Proof.

Consider a valuated flag matroid
${\bm{\mu}}=(\mu,\nu)\in\operatorname{Dr}(1,n-1;n)$ over M. Lets begin with
the full support case, i.e. when M is the uniform matroid $U_{1,n-1;n}$.

In this case the tropical linear space $L_{\nu}$ is a standard tropical
hyperplane in $\mathbb{TP}^{n-1}$ centered at $-(\nu([n]\backslash
1),\dots,\nu([n-1]))$ and $L_{\mu}$ is just the point $(\mu(1),\dots,\mu(n))$.
The complex $\Sigma_{\mu}$ is the inner normal fan of $\Delta(1,n)$ centered
at the point $L_{\mu}$ and $\Sigma_{\nu}$ is the inner normal fan of
$\Delta(n-1,n)$ centered at the apex of $L_{\nu}$ (after taking the quotient
by the lineality space $\mathbb{R}(1,\dots,1)$). In particular, the rays of
$\Sigma_{\nu}$ are in direction $e_{i}$ while the rays of $\Sigma_{\mu}$ are
in direction $-e_{i}$. The facets of the mixed subdivision $S_{\bm{\mu}}$
correspond to vertices of $\Sigma_{\bm{\mu}}$, that is, intersections of cones
$C_{1}\in\Sigma_{\mu}$ and $C_{2}\in\Sigma_{\nu}$ which consist of a single
point. So the subdivision $S_{\bm{\mu}}$ only depends on the cell of $L_{\nu}$
where $L_{\mu}$ is located.

The set $I\subset[n]$ of numbers $i$ for which $\lambda_{i}$ does not attain
the minimum in (1.7) is the set of rays $e_{i}$ spawning the cone of $L_{\nu}$
whose interior contains the point $L_{\mu}$. For every subset of $J\subset I$
there is one vertex in $\Sigma_{\bm{\mu}}$, namely, $p^{J}$ where
$p^{J}_{i}=\mu(i)$ for $i\in J$ and $p_{i}=-\nu([n]\backslash i)$ for $i\notin
J$. In particular we have $2^{|I|}$ facets. The point $p^{J}$ corresponds to
the flag matroid $\textbf{M}_{J}$ with bases

$\mathcal{B}(\textbf{M}_{J})=\\{\\{i\\}\mid i\in(J\cup([n]\backslash
I))\\}\cup\\{[n]\backslash i\mid i\notin J\\}.$

So the corresponding matroid polytope satisfies the inequalities $x_{i}\leq 0$
for $i\in I\backslash J$ and $x_{i}\geq 0$ for $i\in J$. Taking all the
$2^{|I|}$ facets together we see that the subdivision is the result of
simultaneously cutting $\Delta(1,n-1;n)$ by the hyperplanes given by
$\\{x_{i}=0\\}$ for $i\in I$.

If ${\bm{\mu}}$ does not have full support, the subdivision is still given by
the simultaneously splitting with the hyperplanes $\\{x_{i}=0\\}$ for all $i$
where $\lambda_{i}$ does not attain the minimum but is finite. For infinite
$\lambda_{i}$ we have that all of $P_{\textbf{M}}$ is on the halfplane
$x_{i}\geq 0$ if $\nu_{i}=\infty$ and on the halfplane $x_{i}\leq 0$ if
$\mu_{i}=\infty$ (in particular it is inside the hyperplane if both Plücker
coordinates are infinite). ∎

The following example shows total non-negativity, is not determined by the
subdivision of the matroid polytope.‘

######  Example 5.2.

Consider $\nu=(0,0,0,0)\in\operatorname{Dr}(3,4)$,
$\mu=(2,1,0,0)\in\operatorname{Dr}(1,4)$ and
$\mu^{\prime}=(1,2,0,0)\in\operatorname{Dr}(1,4)$. We can check that both
$(\mu,\nu)$ and $(\mu^{\prime},\nu)$ satisfy (1.2), so they are both in the
flag Dressian $\operatorname{FlDr}(1,3;4)$. Moreover, as points, both $\mu$
and $\nu$ lie in the same cone of $L_{\nu}$, so the corresponding subdivision
of $\Delta_{1,3;4}$ is the same. However, by Theorem 1.2, $(\mu,\nu)$ can be
completed to a complete flag $(\mu,\theta,\nu)\in\operatorname{TFl}^{\geq
0}(1,3;4)$. For example, we can take

$\theta(ij)=\begin{cases}1,&\text{if }ij=34\\\
0,&\text{otherwise.}\end{cases}$

But $(\mu^{\prime},\nu)$ does not satisfy (1.4), as

$\lambda_{1}=1<\lambda_{0}\oplus\lambda_{2}=\infty\oplus 2=2.$

So $(\mu^{\prime},\nu)$ is not in $\operatorname{TFl}^{\geq 0}((1,3),4)$. The
key difference here is that there are different possible subdivisions of
$\Pi_{n}$ that can arise from completing $(\mu,\nu)$ and $(\mu^{\prime},\nu)$,
despite them having the same subdivision on $\Delta(1,3;4)$. This shows that
the polyhedral geometry does not determine whether a valuated flag matroid is
a valuated positroid for non-consecutive rank.

Let us see in detail what is going on with this example. The tropical linear
space $L_{\nu}$ consists of the 6 cones of dimension 2 spanned by 2 of the 4
coordinate axes in $\mathbb{T}\mathbb{P}^{3}$. Let us examine what are the
possible rank-2 tropical linear spaces $L_{\theta}$ that are inside $L_{\nu}$
and contain a point $(x,y,0,0)$. Figure 3 shows tropical linear spaces of rank
2 which complete $(\mu,\nu)$ and $(\mu^{\prime},\nu)$.

The tropical linear space $L_{\theta}$ is determined by its bounded unique
bounded cell. Lets call $S$ that bounded cell, which is either a segment in
direction $e_{1}+e_{2}$ or a point. If $S$ is a point it must be either
$(0,y)$ or $(x,0)$. If $S$ is a segment, it can be a secant of the rectangle
$R$ inside $C$ which has opposite corners $(0,0)$ and $(x,y)$ (ignoring the
last two coordinates which are constant $0$ in $C$). Otherwise, it contains
$(x,y)$ or $(0,0)$ in its interior (in the latter case the bounded cell is
also inside the cone spanned by $e_{3}$ and $e_{4}$).

Figure 3. Left: two points (red crosses) corresponding to non-Bruhat polytopes
when completing $(\mu^{\prime},\nu)$. Right: A flag completing $(\mu,\nu)$
producing a Bruhat subdivision.

If one of the extremes $p$ of $S$ lies in the interior of the side of $R$
between $(0,0)$ and $(0,y)$, then the cell in the subdivision of $\Pi_{4}$
induced by $(\mu,\theta,\nu)$ given by $p$ is the flag matroid with bases
$\\{3,4,13,14,23,24,34,123,124,234\\}$. This is not a Bruhat polytope, because
it contains the points $(1,2,3,4)$ and $(3,2,1,4)$ but not $(2,1,3,4)$ and
$(3,1,2,4)$.

If the other extreme $q$ of $S$ lies in the interior of the side of $R$
between $(x,0)$ and $(x,y)$, then the cell in the subdivision of $\Pi_{4}$
induced by $(\mu,\theta,\nu)$ given by $(x,y)$ is the flag matroid with bases
$\\{1,2,3,4,12,23,24,123,124\\}$. This is also not a Bruhat polytope, because
it contains the points $(2,3,1,4)$ and $(4,3,1,2)$ but not $(3,2,1,4)$ and
$(4,2,1,3)$. Figure 3 (left) shows with red crosses these two points that do
not correspond to Bruhat polytopes.

If $y>x$, then the subdivision necessarily has one extreme in the interior of
a vertical edge Figure 3. Hence the subdivision necessarily has one of these
two non-Bruhat polytopes. But if $x\leq y$, it is possible to have a
subdivision without these polytopes, by choosing $L_{\theta}$ to have its
0-dimensional cells in the horizontal sides of $R$.

The flag Dressian $\operatorname{Fl}\operatorname{Dr}((1,3),4)$ has a
lineality space of dimension 3. If we mod out that lineality space, we get
that $\operatorname{Fl}\operatorname{Dr}((1,3),4)$ is a cone over the complete
graph $K_{4}$. Each ray corresponds to a split over the hyperplane
$\\{x_{i}=0\\}$. In Figure 4 we see the the three different subsets of
$\operatorname{Fl}\operatorname{Dr}((1,3),4)$ listed in Theorem 1.2. The
points at the end of the blue path in the left correspond to the rays
$x_{1}=x_{2}\geq x_{3},x_{4}$ and $x_{3}=x_{4}\geq x_{1},x_{2}$.

Figure 4. Left: The tnn tropical flag variety
$\operatorname{TFl}^{\textsl{tnn}}(1,3;4)$. Center: the subset of
$\operatorname{TFl}(1,3;4)$ producing Bruhat subdivisions. Right: The non-
negative flag Dressian $\operatorname{FlDr}^{\geq
0}(1,3;4)=\operatorname{TFl}^{\geq 0}(1,3;4)$.

######  Remark 5.3.

Notice that even though $\mu^{\prime}$ and $\nu$ in the example above are
valuated positroids and in fact $(\mu^{\prime},\nu)$ is a valuated flag
positroid, it can not be completed to a positive full flag. There are several
positroids $\theta\in\operatorname{Dr}(2,4)$ such that $(\mu,\theta,\nu)$ is a
complete flag. However, none of them satisfy that $(\mu,\theta,\nu)$ is a flag
positroid. This example shows in particular that total positivity is not
detected by the flag matroid subdivision.

##  6\. Bruhat polytopes of hollow rank

In this section we look at Bruhat polytopes of rank $(1,n-1)$. We omit the
superscript $(1,n-1)$ from $Q_{u,v}^{(1,n-1)}$ and write simply $Q_{u,v}$ as
we only deal with hollow Bruhat polytopes for the next three sections.

######  Remark 6.1.

Throughout this section, we use the untwisted convention for Bruhat polytopes,
calling $v$ a minimal coset representative if it is the minimal elemnt in the
fiber of $\pi^{\textbf{d}}\circ\phi$. Similar proof probably exist when using
the usual convention of twisted Bruhat polytopes, however we can choose the
convention thanks to Theorem 4.6. In fact, the desire to avoid translating all
of this section to the twisted convention (the one used in the literature for
non-complete flag varieties) was the main driving force to prove Theorem 4.6.

Let us begin by taking care of polytopes in the boundary of $\Delta(1,n-1;n)$.

######  Proposition 6.2.

Let $\textbf{M}=(M,N)$ be a hollow flag matroid over $[n]$. The flag matroid
polytope $P_{\textbf{M}}$ is contained in the boundary of $\Delta(1,n-1;n)$ if
and only if there is no element $i\in[n]$ which is neither a loop of $M$ or a
coloop of $N$.

###### Proof.

Let $I\subset[n]$ be the set of non-loops of $M$ and $J$ the set of non-
coloops of $N$. If $I\cap J=\emptyset$, then $P_{M}$ is inside the hyperplane
$\left\\{\sum\limits_{i\in I}x_{i}=-1\right\\}$. Now if $I\cap J>0$, it must
be have at least two elements, $i$ and $j$, because of (1.2). But then the
origin is in the segment between the vertices $x$ and $y$ of $P_{M}$, where
$x_{i}=y_{j}=1$ and $x_{j}=y_{i}=-1$ and every other coordinate equal to $0$.
So in particular $P_{M}$ intersects the interior of $\Delta(1,n-1;n)$ ∎

As a consequence of the above proposition we have that every face of
$\Delta(1,n-1;n)$ is completely contained inside one of the orthants of
$\mathbb{R}^{n}$. Then by Proposition 5.1 non of the faces of
$\Delta(1,n-1;n)$ are subdivided in matroid subdivisions. Since
$\Delta(1,n-1;n)$ is the Bruhat polytope of $[\operatorname{id},(n,1)]$, we
have that all flag matroid polytopes in the boundary of $\Delta(1,n-1;n)$ are
Bruhat by Theorem 4.3.

Now we focus on hollow flag matroid polytopes intersecting the interior of
$\Delta(1,n-1;n)$, or equivalently, containing the origin

######  Definition 6.3.

Let M be a hollow flag matroid whose polytope intersects the interior of
$\Delta(1,n-1;n)$. The _symbol sequence_
$\alpha(\textbf{M})=(\alpha_{1},\dots,\alpha_{n})\in\\{0,+,-,*\\}^{n}$ of M is
given by:

  * •

$\alpha_{i}=*$ if both $i$ and $[n]\backslash i$ are bases of M

  * •

$\alpha_{i}=-$ if $i$ is a base but $[n]\backslash i$ is not

  * •

$\alpha_{i}=+$ if $[n]\backslash i$ is a base but $i$ is not

  * •

$\alpha_{i}=0$ if neither $[n]\backslash i$ nor $i$ are bases.

Given a sequence $\alpha\in\\{0,+,-,*\\}^{n}$ with at least two $*$, we write
$P_{\alpha}$ for the polytope of that unique flag matroid with symbol
$\alpha$.

Equivalently, $\alpha_{i}=0$ if $P_{\textbf{M}}$ is inside the coordinate
hyperplane $\\{x_{i}=0\\}$, $\alpha_{i}=+$ if $P_{\textbf{M}}$ is in the
positive half space $x_{i}\geq 0$, $\alpha_{i}=-$ if $P_{\textbf{M}}$ is in
the negative half space $x_{i}\leq 0$, and $\alpha_{i}=*$ if $P_{\textbf{M}}$
is in both sides of the hyperplane. Such symbol sequences satisfy that at
least two symbols are $*$ because of (1.2), but you can get any sequence in
$\\{+,-,*\\}^{n}$ with at least two $*$.

The polytope $P_{\alpha}$ is the convex hull all vertices of $\Delta(1,n-1;n)$
where the $-1$ is in coordinate $i$ where $\alpha_{i}\in\\{-,*\\}$ and the
$+1$ is in a coordinate $j$ where $\alpha_{j}\in\\{+,*\\}$. The dimension of
$P_{\alpha}$ is $n-1-m$ where $m$ is the number of $0$’s in $\alpha$.

Consider a matroid subdivision $S_{\bm{\mu}}$ of a hollow flag matroid
polytope $P_{\textbf{M}}$ intersecting the interior of $\Delta(1,n-1;n)$. The
faces of a matroid subdivision $S_{\bm{\mu}}$ of $\Delta(1,n-1;n)$ in the
interior correspond to the symbol sequences satisfying that $\alpha_{i}=*$ for
all $i$ where $\lambda_{i}$ attains the minimum and no other $i$. So there are
$3^{k}$ of these polytopes where $k$ is the number of finite $\lambda_{i}$
that do not attain the minimum. Moreover, the $f$-vector of the subdivision
restricted to these faces is the mirror of the $f$-vector of a $k$-cube.

Given a permutation $v\in\operatorname{Sym}_{n}$, the corresponding point in
$\Delta(1,n-1;n)$ is the one where $-1$ is at position $v^{-1}(1)$, $1$ is at
position $v^{-1}(n)$. So the minimal elements of the fiber $\pi^{d}\circ\phi$
are those permutations where $\\{2,\dots,n-1\\}$ are in ascending order. If
$v$ is a minimal element of its fiber $\pi^{d}\circ\phi$ and satisfies that
$v^{-1}(n)<v^{-1}(1)$, a minimal word $w$ for $v$ can be constructed by an
ascending subword starting with $\tau_{1}$ followed by a descending subword
starting on $\tau_{n-1}$. Explicitly,

$w=(\tau_{1},\dots,\tau_{k-2},\tau_{n-1},\tau_{n-2},\dots,\tau_{m},)$ (6.1)

where $k=v^{-1}(1)$ and $m=v^{-1}(n)$. If $k=1$ then the ascending subword is
empty and if $m=n$ then the descending subword is empty. Given a subword
$w^{\prime}$ of $w$, the corresponding vertex in $\Delta(1,n-1;n)$ has the
$-1$ in position $k^{\prime}$ where $\tau_{1}\dots\tau_{k^{\prime}-1}$ is the
largest ascending subword of $w^{\prime}$ that starts with $\tau_{1}$ and the
$1$ is in position $m^{\prime}$ where $\tau_{n-1}\dots\tau_{m^{\prime}}$ is
the longest descending subword that starts with $\tau_{n-1}$.

Given a symbol sequence $\alpha$, we say it has an _isolated_ $*$ if there
exists $i$ such that $\alpha_{i}=*$ but neither $\alpha_{i-1}$ or
$\alpha_{i+1}$ are $*$ (in the extremal cases we do not consider $\alpha_{0}$
or $\alpha_{n+1}$ to be $*$). We are now ready to prove the main result of
this section. Example 6.5 showcases the constructions used in the proof.

######  Theorem 6.4.

Let M be a hollow flag matroid such that $P_{\textbf{M}}$ contains the origin.
Then $P_{\textbf{M}}$ is a Bruhat polytope if and only if $\alpha(\textbf{M})$
has no isolated $*$.

###### Proof.

First, lets suppose that if a sequence $\alpha$ has an isolated $*$ and then
$P_{\alpha}$ is not a Bruhat polytope. Consider the face $F$ of $P_{\alpha}$
given by $x_{i}=0$ for every $i$ such that $\alpha_{i}\neq*$. We show that
this $F$ is not a Bruhat polytope, which implies that $P_{\alpha}$ is not
Bruhat by Theorem 4.2.

We proceed by contradiction and suppose there are permutations $u\leq v$ such
that $F=Q_{u,v}$. Let $w$ be the reduced word of $v$ as in (6.1) and consider
$w^{\prime}$ a subword of $w$ for $u$. Let that isolated $*$ be $\alpha_{i}$.
Notice that we have that

$v^{-1}(n)=m\leq i\leq v^{-1}(1)=k-1.$

Lets suppose for now that $i$ is neither $1$ or $n$. Since none of the
vertices of $F$ have a $-1$ at the $i-1$ coordinate, any subword of $w$ for a
permutation in $[u,v]$ with an ascending subword $\tau_{1}\dots\tau_{i-2}$
must have a $\tau_{i-1}$ afterwards. This shows that there must be a
$\tau_{i-1}$ in $w^{\prime}$. Similarly, since none of the verticees of
$Q_{u,v}$ have a $1$ at the $i+1$ coordinate, there must be a $\tau_{i}$ in
$w^{\prime}$.

Suppose that $\tau_{i-1}$ appears in $w^{\prime}$ before $\tau_{i}$, so that
$\tau_{i-1}\tau_{i}\leq u$. In order to be able to get a vertex with $-1$ at
coordinate $i$, we need to be able to get a subword $w^{\prime\prime}$ of $w$
with an ascending subword that stops at $\tau_{i-1}$. Then $w^{\prime\prime}$
needs to have a $\tau_{i-1}$ before a $\tau_{i}$ and another $\tau_{i-1}$ one
after it. Moreover, $w^{\prime\prime}$ must have
$\tau_{i-1}\tau_{i}\tau_{i-2}\tau_{i}$ as a subword. However this is not a
subword of $w$ which is a contradiction (if $i=2$ already the existence of
$\tau_{i-1}\tau_{i}$ as a subword of $w^{\prime}$ make it impossible to have a
$-1$ in the second coordinate).

Now suppose that $\tau_{i}\tau_{i-1}\leq u$. The argument is similar: to have
a vertex with $1$ at coordinate $i$, we need a subword $w^{\prime\prime}$ of
$w$ with a descending subword stopping at $\tau_{i}$. But then this implies
that $\tau_{i}\tau_{i-1}\tau_{i+1}\tau_{i}$ is subword of $w^{\prime\prime}$
but again this is not a subword of $w$.

For the case $i=1$, to be able to have vertices of $P_{M}$ with $1$ at the
first coordinate we need $w$ to have $\tau_{n}\dots\tau_{1}$ as a subword. But
there can not be vertices with $1$ at the second coordinate, so we can not
allow descending subwords stopping at $\tau_{2}$. Then we must have that
$\tau_{1}$ is in $w^{\prime}$. But then this rules out the possibility of
having a $-1$ in the first coordinate, which contradicts that $\alpha_{1}=*$.
A similar argument takes care of the case $i=n$.

Now to prove the other direction, we assume that $\alpha=\alpha(\textbf{M})$
has no isolated $*$ and construct $u,v$ such that $Q_{u,v}=P_{\textbf{M}}$. By
Theorem 4.2, it is enough to consider full dimensional polytopes, that is,
$\alpha\in\\{+,-,*\\}^{n}$. The choice of $v$ is straightforward, it is given
by the word

$w=(\tau_{1},\dots,\tau_{k-2},\tau_{n-1},\tau_{n-2},\dots,\tau_{m})$

where $k$ is the largest number such that $\alpha_{k}\in\\{-,*\\}$ and $m$ is
the smallest number such that $\alpha_{m}\in\\{+,*\\}$. We call
$\tau_{1},\dots,\tau_{k-2}$ the ascending part of $w$ and
$\tau_{n-1},\tau_{n-2},\dots,\tau_{m}$ the descending part of $w$. However,
the choice of $u$ is more elaborate.

  1. (1)

First write in ascending order all $\tau_{i}$ for each $i$ such that

     * •

$\alpha_{i+1}=+$ except when $\alpha_{j}=+$ for all $j\geq i+1$ or when
$\alpha_{j}=-$ for all $j\leq i$.

     * •

$(\alpha_{i},\alpha_{i+1})=(+,*)$.

We call this the ascending part of $w^{\prime}$.

  2. (2)

Then write in descending order all $\tau_{i}$ for each $i$ such that

     * •

$\alpha_{i+1}=-$ except when $\alpha_{j}=-$ for all $j<i$.

     * •

$(\alpha_{i},\alpha_{i+1})=(-,*)$ except when $\alpha_{j}=-$ for all $j\leq
i$.

We call this the descending part of $w^{\prime}$.

It is straightforward to verify that the resulting word $w^{\prime}$ is a
subword of $w$; the fact that we exclude the cases where there are only $+$
after $\alpha_{i}$ in the ascending part and when there are only $-$ before
$\alpha_{i}$ in the descending part ensures we never get out of the bounds
given by $w$. We choose $u$ to be the permutation given by $w^{\prime}$.

Let us show that $Q_{u,v}$ indeed gives us $P_{\textbf{M}}$. First we show
that all vertices in $P_{\textbf{M}}$ are in $Q_{u,v}$. Take a vertex $x$ of
$P$ where $x_{i}=-1$ and $x_{j}=1$. We will construct a permutation in $[u,v]$
that gives us $x$ with the following procedure:

  1. (1)

Start with $w^{\prime}$.

  2. (2)

Fill all gaps in the ascending part until $\tau_{i-2}$ (so that the first
$i-2$ letter are $\tau_{1}\dots\tau_{i-2}$).

  3. (3)

Fill all gaps in the descending part until $\tau_{j}$.

  4. (4)

If $j<i$ and $(\alpha_{i-1},\alpha_{i})=(+,*)$, remove $\tau_{i-1}$ from the
ascending part.

  5. (5)

If $(\alpha_{j-1},\alpha_{j})=(-,*)$, remove $\tau_{j-1}$ from the descending
part, except when $\alpha_{l}=-$ for all $l<j$.

  6. (6)

If $i<j$ $(\alpha_{j-1},\alpha_{j})=(-,*)$, add $\tau_{j-1}$ to the ascending
part, except when $\alpha_{l}=-$ for all $l<j$.

  7. (7)

If $i<j$ and $(\alpha_{i-1},\alpha_{i})=(*,*)$, add $\tau_{i-1}$ to the
descending part.

  8. (8)

If $i<j$ and $\alpha_{l}=-$, for all $l<i$, add $\tau_{i-1}$ to the ascending
part.

We call $w^{\prime\prime}$ the resulting word and $\sigma$ the corresponding
permutation.

Let us verify that $\sigma\in[u,v]$. First we check $\sigma\leq v$. Filling
gaps in steps 2 and 3 does not add letters to $w^{\prime\prime}$ that are not
in $w$. When adding the letter $\tau_{j-1}$ to the ascending part in step 6 we
have that $\alpha_{j+1}=*$ so $k-2\geq j$. When adding the letter $\tau_{i-1}$
to the descending part in step 7 we have $\alpha_{i-1}=*$ so $m\leq i-1$. When
adding the letter $\tau_{i-1}$ to the ascending part in step 8 we know that in
$\alpha$ there are $*$ eventually so $k-2\geq i$. So all letters added are
within the bounds given by $w$ and therefor $w^{\prime\prime}$ is a subword of
$w$ and $\sigma\leq v$.

Now lets check that $u\leq\sigma$. If $\tau_{i-1}$ was removed in step 4, then
$(\alpha_{i},\alpha_{i+1})=(*,*)$ so $\tau_{i}$ is not a subword of
$w^{\prime}$. As $\tau_{i-1}$ commutes with every $\tau_{l}$ for $l>i$, we can
take $w^{\prime}$ and move the $\tau_{i-1}$ in the ascending part after all
$\tau_{l}$ for $l>i$ so that it is sitting in the corresponding place in the
descending part without changing that $u$ is the resulting permutation. Notice
that $w^{\prime\prime}$ has a $\tau_{i-1}$ in the descending part because
$j<i$ for this case. Similarly, if $\tau_{j-1}$ was removed in step 5, the
$(\alpha_{j},\alpha_{j+1})=(*,*)$ and there is no $\tau_{j}$ in $w^{\prime}$
so we can move $\tau_{j-1}$ before all $\tau_{l}$ with $l>j$. We have that
$\tau_{j-1}$ is in the ascending part of $w^{\prime\prime}$ because either
$j>i$ and we added it since step 1, or $i<j$ and we added it in step 6. This
way we obtain a word for $u$ which is a subword of $w$ and $u\leq\sigma$.
Notice that here we are already using the fact that there are no isolated $*$.

Now we need to verify that $\sigma$ satisfies that $\sigma(i)=1$ and
$\sigma(j)=n$. To see that $\sigma(i)=1$, we show that the largest ascending
subword of $w^{\prime\prime}$ stops at $\tau_{i-1}$ By step 2, there is an
initial ascending subword in $w^{\prime\prime}$ at least until $\tau_{i-2}$.
We have the following cases:

  * •

If $j<i$, then there is a $\tau_{i-1}$ in the descending part of
$w^{\prime\prime}$ and this can not be followed by a $\tau_{i}$. The only
problem that we could have is that there could already be a $\tau_{i-1}$ in
the ascending part of $w^{\prime}$. This can only happen if $\tau_{i-1}$ is
already in the ascending part of $w^{\prime}$. But since $x_{i}=-1$, this is
only possible if $(\alpha_{i-1},\alpha_{i})=(+,*)$. However we removed this
$\tau_{i-1}$ from $w^{\prime\prime}$ in step 4.

  * •

If $i<j$ and $(\alpha_{i-1},\alpha_{i})=(+,*)$, then there is a $\tau_{i-1}$
in the ascending part of $w^{\prime\prime}$. As there are no isolated $*$, we
have that $(\alpha_{i},\alpha_{i+1})=(*,*)$, so there is no $\tau_{i}$ in
$w^{\prime\prime}$.

  * •

If $i<j$ and $(\alpha_{i-1},\alpha_{i})=(*,*)$, then there are no $\tau_{i-1}$
in $w^{\prime}$ but in step 7 there was a $\tau_{i-1}$ added to the descending
part of $w^{\prime\prime}$. Since it is in the descending part it can not be
followed by a $\tau_{i}$.

  * •

If $i<j$ and $\alpha_{l}=-$ for all $l<i$, then $\tau_{i-1}$ was added to the
ascending part in step 8. In this case there is no $\tau_{i}$ in the ascending
part of $w^{\prime}$, because the only possibility would be for
$(\alpha_{i},\alpha_{i+1})=(-,+)$ but this case is excluded when
$\alpha_{l}=-$ for all $l<i$. Also because $\alpha_{l}=-$ for all $l<i$,
$\tau_{i}$ was not added in step 6.

  * •

The remaining cases are $i<j$, not all $\alpha_{l}=-$ for $l<i$ and either
$\alpha_{i}=-$ or $(\alpha_{i-1},\alpha_{i})=(-,*)$. Here we always have a
$\tau_{i-1}$ in the descending part of $w^{\prime}$ but not in the ascending
part. This remains true in $w^{\prime\prime}$ so the ascending subword stops
at $\tau_{i-1}$.

Fortunately, it is way easier to verify that $\sigma(j)=n$. By construction we
have a descending chain until $\tau_{j}$. Since $\alpha_{j}\in\\{+,*\\}$, the
only case $\tau_{j-1}$ is in the descending part of $w^{\prime}$ is when
$(\alpha_{j-1},\alpha_{j})=(-,*)$ and not all $\alpha_{l}=-$ for $l<j$, which
is why it is removed from $w^{\prime\prime}$ in step 5.

Finally, we need to show that no other points appear in $Q(u,v)$. So we must
show that for $\sigma\in[u,v]$ it can not happen that $\sigma(i)=1$ if
$\alpha_{i}=+$ or $\sigma(i)=n$ if $\alpha_{i}=-$. Let $w^{\prime\prime}$ be a
word for $\sigma$. Suppose $\alpha_{i}=+$ and the largest ascending subword
stops at $\tau_{i-1}$, so that $\sigma(i)=1$. Notice that $\alpha_{i}=+$
implies that $w^{\prime}$ has a $\tau_{i}$ as subword at the ascending or at
the descending part. So $w^{\prime\prime}$ also has $\tau_{i}$ as subword. If
$\alpha_{l}=-$ for all $l<i$, then there are no $\tau_{l}$ in the descending
part of $w$ for $l<i$. Therefor the ascending subword of $w^{\prime\prime}$
must all be in its ascending part and hence before the $\tau_{i}$. But then we
would have that $\sigma^{-1}(1)>i$, which is a contradiction. So we suppose
that not all $\alpha_{l}=-$ for $l<i$. Then there is a $\tau_{i-1}$ in the
ascending part of $w^{\prime}$ and hence $\tau_{i-1}\tau_{i}$ is a subword of
$w^{\prime\prime}$. But then the final $\tau_{i-1}$ in the ascending subword
of $w^{\prime\prime}$ must be after $\tau_{i}$ and therefor we have that
$\tau_{i-1}\tau_{i}\tau_{i-2}\tau_{i}\leq\sigma$. However
$\tau_{i-1}\tau_{i}\tau_{i-2}\tau_{i}\not\leq v$, which is a contradiction.

Similarly, if $\alpha_{i}=-$ then for $\tau_{i}$ to exist in the descending
part we have that not all $\alpha_{l}=-$ for $l<i$. So we have that
$\tau_{i}\tau_{i-1}$ is a subword of $w^{\prime\prime}$ and if the largest
descending subword ends at $\tau_{i}$ then
$\tau_{i}\tau_{i-1}\tau_{i+1}\tau_{i}\leq\sigma$ but
$\tau_{i}\tau_{i-1}\tau_{i+1}\tau_{i}\not\leq\sigma$.

∎

######  Example 6.5.

Consider the flag matroid $\textbf{M}=(M,N)$ over $[5]$ where
$\mathcal{B}(M)=\\{1,3,4,5\\}$ and $\mathcal{B}(N)=\\{1235,1245,1345\\}$. The
symbol sequence of M is $\alpha=(-,+,*,*,-)$. We get
$w=\tau_{1}\tau_{2}\tau_{3}\tau_{4}\tau_{3}\tau_{2}$ and
$w^{\prime}=\tau_{2}\tau_{4}$, so $P_{\textbf{M}}=Q_{u,v}$ for $u=(13254)$ and
$v=(25341)$. To verify that the point $x=(0,1,-1,0,0)\in Q_{u,v}$ we follow
the 8 steps to obtain $w^{\prime\prime}=\tau_{1}\tau_{4}\tau_{3}\tau_{2}$. The
resulting permutation is $\sigma=(25134)$ which indeed satisfies that
$\sigma\in[u,v]$ and $x_{\sigma(1)}=-1$ and $x_{\sigma(5)}=1$.

We get the second part of Theorem 1.2 as corollary of Theorem 6.4:

######  Theorem 6.6.

A valuated flag matroid of hollow rank induces a Bruhat subdivision if and
only if its lambda-values satisfy (1.5).

###### Proof.

Let ${\bm{\mu}}$ be a valuated flag matroid over M. We assume $P_{\textbf{M}}$
contains the origin as otherwise $S_{\bm{\mu}}$ is trivially a Bruhat
subdivision and equation (1.5) is satisfied trivially as all $\lambda_{i}$ are
infinite. Since the polytopes we get in $S_{\bm{\mu}}$ have $\alpha_{i}=*$ if
and only if $\lambda_{i}$ achieve the minimum, then if $\lambda_{i}$ achieves
the minimum by Theorem 6.4 we have that either $\lambda_{i-1}$ or
$\lambda_{i+1}$ achieve the minimum as well. This is exactly what (1.5) says.
∎

##  7\. Flag gammoids

We now introduce a language which is useful to describe the paramtrization of
tnn flag varieties given by Boretsky [13]. This is based on the class of
gammoids introduced by Mason in [30] which was generalized in [18] to valuated
matroids. We adapt this construction to (valuated) flag matroids, an call them
(valuated) flag gammoids. For details on the classical theory of gammoids and
the related construction of transversal matroids we refer to [14].

Consider a directed graph $\Gamma=(V,E)$. We identify $n$ distinct vertices
with $[n]$, so that $[n]\subseteq V$. Let $\textbf{d}=(d_{1},\dots,d_{s})$ be
a rank vector and consider a flag $S_{1}\subset\dots\subset S_{s}$ of subsets
of $V$ such that $|A_{i}|=d_{i}$ for every $i$. Let us call
$\textbf{S}=(S_{1},\dots,S_{s})$ a flag of sinks, even though we do not
require the vertices in $S_{i}$ to be sinks. Given sets $I$ and $J$ of the
same cardinality, a _linking_ from $I$ onto $J$ is a collection of $|I|$
vertex-disjoint paths where each path starts at a vertex in $I$ and ends in a
vertex in $J$. The collection of subsets $I\subseteq[n]$ for which there is a
linking onto $S_{i}$ form the bases of rank $d_{i}$ matroid (supposing this
collection is not empty). A matroid that can be constructed this way is called
a _gammoid_ [30].

Now suppose each edge $e$ has a real weight $w_{e}$, that is, we have a
function $w:E\to\mathbb{R}$. For technical reasons, we assume there are no
cycles with negative weight and there is always a subset of $[n]$ with a
linking onto $S_{i}$ for every $i$. The weight of a linking $\Phi$ is the sum
of the weights of all edges in $\Phi$. Given a subset $I\subset[n]$ of size
$d_{i}$, let $\mu_{i}(I)$ be the minimum weight over all linkings from $I$
onto $S_{i}$. Then $\mu_{i}$ is valuated matroid of rank $d_{i}$ known as
_valuated gammoid_ [18].

######  Theorem 7.1.

The valuated matroids $(\mu_{1},\dots,\mu_{s})$ defined as above form a
valuated flag matroid $\mu(\Gamma,w,\textbf{S})$ of rank d.

###### Proof.

First suppose $\textbf{d}=(1,\dots,s)$. We only need to prove that
$\mu_{i},\mu_{i+1}$ form a valuated matroid quotient. To see this, consider
giving the label $n+1$ to the unique vertex in $S_{i+1}\backslash S_{i}$. In
the case that this vertex already had a label $k$, just create a new vertex
labeled $n+1$ with a single edge to $k$ of weight $0$. Let us call $\nu$ for
the valuated matroid of rank $d_{i+1}$ over $[n+1]$ given by the linkings to
$S_{i+1}$.

Now, subsets of $I\subset[n+1]$ of size $d_{i+1}$ containing $n+1$ have a
liking to $S_{i+1}$ if and only if $I\backslash(n+1)$ has a linking to $S_{i}$
with the same weight. So we have that $\nu\backslash(n+1)=\mu_{i+1}$ and
$\nu/(n+1)=\mu_{i}$. By Theorem 3.2 we have that $(\mu_{i},\mu_{i+1})$ is a
flag valuated matroid as desired.

Now for arbitrary rank d, extend the flag of sinks S to a flag of sinks
$(S_{1}^{\prime},\dots,S_{m}^{\prime})$ of rank $(1,\dots,m)$ where
$m=|S_{s}|$. This can be done by choosing a linear ordering to $S_{s}$ where
if $x\in S_{i}$ and $y\in S_{j}\backslash S_{i}$ then $x<y$ and having
$S_{i}^{\prime}$ be the $i$ smallest elements of $S_{s}$. Now we are in the
previous case so we have a valuated flag matroid
${\bm{\mu}}^{\prime}=(\mu_{1}^{\prime},\dots,\mu_{m}^{\prime})$. We can then
obtain ${\bm{\mu}}=(\mu_{1},\dots,\mu_{s})$ from ${\bm{\mu}}^{\prime}$ by
dropping all constituents whose rank is not in d, so ${\bm{\mu}}$ is a
valuated flag matroid. ∎

######  Definition 7.2.

A (valuated) flag matroid $\mu=\mu(\Gamma,w,\textbf{S})$ which arises from a
directed graph $\Gamma$ with edge weights $w$ and a flag of sinks S is called
a _(valuated) flag gammoid_.

######  Proposition 7.3.

Valuated flag gammoids are closed under translation.

###### Proof.

Let ${\bm{\mu}}={\bm{\mu}}(\Gamma,w,\textbf{S})$ be valuated flag gammoid.
Suppose you want to translate the flag of tropical linear spaces by an amount
$r\in\mathbb{R}$ in the $i$-th coordinate. First suppose that $i$ is not a in
the flag of sinks. This corresponds to increasing $\mu_{I}$ by $r$ for every
$I$ containing $i$. To do that, consider $w^{\prime}$ given by

$w^{\prime}(e):=\begin{cases}w(e)+r&\text{if }e\text{ is an outgoing edge from
}i\\\ w(e)-r&\text{if }e\text{ is an ingoing edge to }i\\\
w(e)&\text{otherwise. }\end{cases}$

This is sometimes known as the _gauge action_ on $w$. Equipping $\Gamma$ with
these new edge weights has the effect that every path starting from $i$ is
increased by $r$, but the weight of every other path remains the same, even if
they use the vertex $i$ somewhere that is not the source. The resulting flag
matroid ${\bm{\mu}}^{\prime}(\Gamma,w^{\prime},\textbf{S})$ is the translation
of ${\bm{\mu}}$ we wanted.

If $i$ is in the the flag of sinks S, then just create a new vertex $v$ and an
edge from $i$ to $v$ of weight $0$, replace $i$ by $v$ in S and then apply the
gauge transformation described above. ∎

When the vertices of $\Gamma$ are exactly $[n]$ with no other additional
vertices, we can all the resulting (valuated) flag matroid a _(valuated)
strict flag gammoid_ , extending from the definitions in [30]. Their duals can
be called _(valuated) transversal flag matroid_ [18, Proposition 7.3]. It is
easy to prove that every polytope in the subdivision in a valuated (strict)
flag gammoid is a (strict) gammoid using a suitable gauge transformation as
described in Proposition 7.3 and removing edges with positive weight. However
the converse is far from trivial. It was proven in [18, Theorem 6.20] that a
regular subdivision consisting of strict gammoids corresponds to a valuated
strict gammoid. We ask if we can generalize this in the flag setting, and
moreover, to the non-strict setting:

######  Question 1.

If we have a flag valuated matroid ${\bm{\mu}}$ such that every polytope in
the subdivision is a (strict) gammoid, does it follow that ${\bm{\mu}}$ is a
valuated (strict) flag gammoid?

Notice that this question is similar in the local vs global spirit of some of
the positivity questions we deal with in this paper. Example 5.2 shows that a
subdivision of tnn-flag matroids does not imply that the valuated matroid is
tnn. Whether a subdivision of flag positroids is a valuated flag positroid is
still unknown to us; this is part of 1.5. Theorem 1.4 provides a weaker
version of this statement.

Just like valuated positroids can be parametrized using valuated gammoids
[24], we can use valuated flag gammoids to parametrize flag matroids in tnn
tropical flag varieties. The following is an adaptation of a construction
given in [13], which itself is an interpretation of the parametrization of
Richardson varieties given in [31]. Our construction is an ad-hoc
simplification for hollow flags using the language of flag gammoids and using
Theorem 6.6 to use as a starting point the flag matroid instead of the Bruhat
interval.

Let $\textbf{M}=(M,N)$ be a matroid of rank $(1,n-1)$ such that
$P_{\textbf{M}}$ is a Bruhat polytope.Let $a_{1}<\dots<a_{\alpha}$ be the non-
loops of $M$. By the second part of Theorem 6.6, if there is an element $i$
which is neither a loop of $M$ nor a coloop in $N$, then either $i-1$ or $i+1$
satisfy the same. So the set $C$, consistinig of the elements which are not
loops of $M$ nor coloops of $N$ consists of consecutive segments of size at
least $2$. Let $b_{1}<\dots<b_{\beta}$ be the set of non-coloops of $N$ except
for the last element of each consecutive segment in $C$.

We construct a weighted directed graph which we call $\Gamma_{\textbf{M}}$ in
the following way. The set of vertices is $[n]\sqcup[\overline{n}]$ where
$[\overline{n}]=\\{\overline{1},\dots,\overline{n}\\}$. There are three kinds
of edges:

  * •

For every $i$, there is an edge from $i$ to $\overline{i}$ with weight
$w(i,\overline{i})=0$.

  * •

For every $i<\alpha$, there is an edge from $a_{i}$ to $a_{i+1}$ with
arbitrary weight $w(a_{i},a_{i+1})$.

  * •

For every $i<\beta$, There is an edge from $b_{i}$ to $\overline{b_{i+1}}$
with arbitrary weight $w(b_{i},\overline{b_{i+1}})$.

Let $S_{1}=\\{\overline{a_{\alpha}}\\}$ and
$S_{2}=[\overline{n}]\backslash\\{\overline{b_{1}}\\}$ and consider the
valuated flag matroid ${\bm{\mu}}(\Gamma_{\textbf{M}},w,\textbf{S})$ for the
flag of sources $\textbf{S}=(S_{1},S_{2})$.

######  Example 7.4.

Consider the flag matroid $\textbf{M}=(M,N)$ on $[8]$ where the loops of $M$
are $3$ and $8$ and the only coloop of $N$ is $7$. This matroid satisfies
(1.5) so $P_{\textbf{M}}$ is a Bruhat polytope. The graph
$\Gamma_{\textbf{M}}$ is pictured in Figure 5.

Figure 5. The graph $\Gamma_{\textbf{M}}$ of Example 7.4. The sink $\bar{7}$
is in red as it is the only element in $S_{1}$. The element $\bar{1}$ is
removed as it is the only element of $[\bar{n}]$ not in $S_{2}$, so it is
irrelevant.

######  Remark 7.5.

The construction in [13] has several convention differences to this one.
First, the elements of the matroid are placed as sinks as and look at linkings
onto them from marked sources. However in the classical construction of
gammoids [14, 30] the arrows go the other way, looking instead at likings from
a subset of elements of a matroid onto marked sinks, which is what we use
here. Moreover, the twisted convention of Bruhat polytopes is used instead of
the untwisted. From the proof of Theorem 4.6 we have that to change
conventions one takes a Bruhat interval for the dual and then takes inverse of
the permutations in the interval. Notice that a word for the inverse of a
permutation can be obtained by reversing the order of the letters. So together
the change of these conventions amounts to doing the same construction in [13,
Definition 3.11] for $[u^{\prime},v^{\prime}]$ where
$P_{\textbf{M}^{*}}=Q_{u^{\prime},v}$ while reversing all arrows and placing
the arrows from right to left in the figures therein.

######  Proposition 7.6.

A valuated flag matroid ${\bm{\mu}}$ with support M is in
$\operatorname{TFl}^{\textsl{tnn}}(1,n-1;n)$ if and only if
${\bm{\mu}}=\mu(\Gamma_{\textbf{M}},w,\textbf{S})$ for some edge weights $w$.

###### Proof.

If $P_{\textbf{M}^{*}}=Q_{u,v}$ for some Bruhat interval $[u,v]$ of rank
$(1,n-1)$, then $v$ has a word which consists of an ascending subword
$\tau_{1}\tau_{2}\dots$ followed by a descending subword
$\tau_{n-1}\tau_{n-2}\dots$. Following the construction in [13, Definition
3.11], we have that the ascending subword corresponds to some ascending path
$a_{1}\rightarrow\dots\rightarrow a_{\alpha}$ in the vertices labeled $[n]$
and the descending part corresponds to edges $(b_{i},\overline{b_{i+1}})$ for
some sequence $b_{1}<\dots<b_{\beta}$. We need to show that the sets
$A=\\{a_{1},\dots,a_{\alpha}\\}$ and $B=\\{b_{1},\dots b_{\beta}\\}$ are the
same as the ones constructed above for $\Gamma_{\textbf{M}}$.

First, it is clear that the only vertices with a path to
$S_{1}=\\{\overline{a_{\alpha}}\\}$ are those in $A$. So necessarily $A$ must
be the set of bases i.e. non-loops of $M$. Then for every element in $b_{i}$
we can obtain a linking from $[n]\backslash b_{i}$ onto $S_{2}$ by taking the
edges $(b_{j-1},\overline{b_{j}})$ for $j\leq i$ and $(j,\overline{j})$ for
all other remaining sinks in $S_{2}$. However, if $b_{i}$ is also in $A$, say
$b_{i}=a_{j}$, then we can also add the edge $(a_{j},a_{j+1})$ and obtain a
linking from $[n]\backslash a_{j+1}$ onto $S_{2}$, so $b_{i}$ must not be a
coloop in $N$. By Theorem 6.6, there is always at least two consecutive
elements which are both non-loops of $M$ and non-coloops of $N$. Hence we can
and must omit the last element in every consecutive segment of non-coloops of
$N$ which are also not loops of $M$. By taking a minimal $u$ we can assume
every other non-coloop of $N$ is in $B$, showing that indeed
$\Gamma_{\textbf{M}}$ is the end result. The statement follows by [13, Theorem
3.13]. ∎

##  8\. Positivity for hollow flag matroids

In this section we will prove part 1. (Theorem 8.2) and part 3. (Theorem 8.4)
of Theorem 1.2. We begin with the following lemma:

######  Lemma 8.1.

A sequence of real numbers $\lambda_{1},\dots,\lambda_{n}$ satisfies (1.4) if
and only if there exists $x_{1},\dots,x_{n-1}$ such that
$\lambda_{i}=x_{i}\oplus x_{i-1}$ for every $i$ (where $x_{0}=x_{n}=\infty$).

###### Proof.

First, if such $x_{1},\dots x_{n-1}$ exist, then

$\lambda_{i-1}\oplus\lambda_{i}=(x_{i-1}\oplus x_{i-2})\oplus(x_{i+1}\oplus
x_{i})\leq x_{i-1}\oplus x_{i}=\lambda_{i}.$

Now suppose that $\lambda_{1},\dots,\lambda_{n}$ satisfy (1.4). Let
$x_{i}:=\max(\lambda_{i},\lambda_{i+1})$. We have that $x_{i}\oplus
x_{i-1}=\max(\lambda_{i},\lambda_{i+1})\oplus\max(\lambda_{i+1},\lambda_{i})$
is larger than $\lambda_{i}$ because both terms are larger than $x_{i}$. But
for both terms to be larger than $x_{i}$, we would have that
$\lambda_{i+1}\oplus\lambda_{i-1}>\lambda_{i}$, which contradicts (1.4). ∎

######  Theorem 8.2.

A valuated flag matroid of hollow rank is tnn if and only if its lambda-values
satisfy (1.4).

###### Proof.

Let ${\bm{\mu}}=(\mu,\nu)$ be a valuated matroid satisfying (1.4). Then its
support $\textbf{M}=(M,N)$ satisfies (1.5). By Proposition 7.6, ${\bm{\mu}}$
is in $\operatorname{TFl}(1,n-1;n)$ if and only if it can be represented as a
valuated flag gammoid by $\Gamma_{\textbf{M}}$ as constructed in Section 7.

First, as total non-negativity is invariant under translation, we can
translate ${\bm{\mu}}$ so that $\mu$ is a $(0,\infty)$-vector. This has the
effect that the edges of the form $(a_{i},a_{i+1})$ must be all of weight $0$,
since $\mu_{a_{j}}$ is the sum of all these edges for $i\geq j$. Furthermore,
by scaling $\nu\in\mathbb{TP}^{n-1}$ we can assume that $\nu_{b_{1}}=0$. This
way $\lambda_{i}=\nu_{[n]\backslash i}$ is the minimum weight of all linkings
from $[n]\backslash i$ to $S_{2}$.

Let $y_{i}$ be the weight of the edge $(b_{i},\overline{b_{i+1}})$. Let
$x_{i}=y_{1}+\dots+y_{i-1}$, so in particular $x_{1}=0$. Clearly we can
recover the edge weights $y_{i}$ from the $x_{i}$ and viceversa. Let us
consider the possible linkings of $[n]\backslash i$ into $S_{2}$.

  1. I.

If $i=b_{j}$ for some $j$, then we can have the path
$(b_{k},\overline{b_{k+1}})$ for every $k<j$ and $(m,\overline{m})$ for every
sink $\overline{m}\in S_{2}$ not yet used. This linking has weight $x_{j}$.

  2. II.

If $i=a_{l}$ for some $l>2$ and $a_{l-1}=b_{j}$ for every some $j$, then we
can have the path $(a_{l-1},a_{l},\overline{a_{l}})$, the path
$(b_{k},\overline{b_{k+1}})$ for every $k<j$ and $(m,\overline{m})$ for every
sink $\overline{m}\in S_{2}$ not yet used. Again, this linking has weight
$x_{j}$.

No other linking exists. We call these linkings type I and type II
respectively.

If $b_{i}$ is a loop of $M$, then we have that $[n]\backslash b_{i}$ has only
linking onto $S_{2}$ of type I and weight $x_{i}$. Now consider a consecutive
segment $b_{i},b_{i}+1\dots,b_{i}+k$ of $C$, the set of non-loops of $M$ and
non-coloops of $N$. In other words $\lambda_{b_{i}},\dots,\lambda_{b_{i}+k}$
are finite but $\lambda_{j-1}=\lambda_{k+1}=\infty$. Equivalently,
$b_{i}+j=b_{i+j}$ for $0\leq j<k$ but not for $j=k$ or $j=-1$. We compute
$\lambda_{b_{i}+j}$ considering the following cases

  * •

Because $b_{i}$ is the first element of a consecutive segment of $C$, the
largest non-loop of $M$ before $b_{i}$ is either a coloop in $N$ or the last
element in another consecutive segment of $C$. Either way, we have that
$b_{i}$ does not have a linking of type II, it only has a linking of type I
with weight $x_{i}$, so $\lambda_{b_{i}}=x_{i}$.

  * •

As $b_{i}+k$ is the last element of the cosecutive segment, it is not equal to
some $b_{m}$, so it can not have a linking of type I. It only has a linking of
type II which has weight $x_{i+k-1}$, so $\lambda_{b_{i}+k}=x_{i+k-1}$.

  * •

If $1\leq j<k$, then $b_{i}+j$ has a linking of type I of weight $x_{i}$ and
linking of weight $x_{i+j}$ and a linking of type II of weight $x_{i+j-1}$, so
$\lambda_{b_{i}+k}=x_{i+j}\oplus x_{i+k-1}$.

By Lemma 8.1, $\lambda_{b_{i}},\dots,\lambda_{b_{i}+k}$ must satisfy (1.4). If
every consecutive segment of finite lambda values satisfy (1.4), then all of
them satisfy (1.4). Conversely, if all $\lambda_{1},\dots,\lambda_{n}$ satisfy
(1.4), then again we can choose appropriate $x_{i}$ on each consecutive
segment as in the proof of Lemma 8.1. Namely,
$x_{i}=\max(\lambda_{b_{i}},\lambda_{b_{i}+1})$ if both $\lambda_{b_{i}}$ and
$\lambda_{b_{i}+1}$ are finite and $b_{i}+1$ is not the last element of the
consecutive segment in $C$. If $b_{i}$ is the last element of a consecutive
segment of $C$ or it is a loop of $M$, then $x_{i}=\nu_{[n]\backslash
b_{i+1}}$. This way we can recover $x_{i}$ and hence recover the edge weights
$y_{i}$ which will give us the valuated matroid ${\bm{\mu}}$ by Lemma 8.1. ∎

######  Example 8.3.

Recall the flag matroid M from Example 7.4. Consider ${\bm{\mu}}$ be the
valuated flag matroid with support in M where the valuation on the bases of
$M$ are all equal to $0$ and

$\displaystyle\mu_{[n]\backslash 1}=0\quad$ $\displaystyle\mu_{[n]\backslash
2}=0\quad\mu_{[n]\backslash 3}=2\quad\mu_{[n]\backslash 4}=1$
$\displaystyle\mu_{[n]\backslash 5}=1\quad$ $\displaystyle\mu_{[n]\backslash
6}=3\quad\mu_{[n]\backslash 7}=\infty\quad\mu_{[n]\backslash 8}=-1.$

Hence we have that
$\lambda=(0:0:\infty:1:1:3:\infty:\infty)\in\mathbb{T}\mathbb{P}^{7}$
satisfies (1.4). Recall the graph $\Gamma_{\textbf{M}}$ pictured in Figure 5.
As the valuation on $M$ is trivial, we can assume all horizontal edges have
weight 0 (as well as the vertical weights). Thus we just need to determine the
weights of the diagonal edges. Using the procedure described above, we get the
edge weights

$y_{1,\bar{3}}=2\quad y_{3,\bar{4}}=-1\quad y_{4,\bar{5}}=2\quad
y_{5,\bar{8}}=-4.$

If we had a valuated flag matroid ${\bm{\mu}}$ where the valuation on $M$ is
not trivial, then we can translate to a valuated flag matroid
${\bm{\mu}}^{\prime}$ where the valuation in $M$ is trivial to obtain edge
weights as above and then use the gauge action as described in Proposition 7.3
to obtain a graphical presentation of ${\bm{\mu}}$ as a totally positive
valuated flag matroid.

######  Theorem 8.4.

A valuated flag matroid ${\bm{\mu}}$ of hollow rank is a flag positroid if and
only if ${\bm{\mu}}\in\operatorname{FlDr}^{\geq 0}(1,n-1;n).$

###### Proof.

One of the directions is immediate for arbitrary rank, so we do the other
direction. Suppose $\lambda$ satisfies (1.6) for hollow valuated flag matroid
${\bm{\mu}}(\mu,\nu)$. Let $i$ be an even index where $\lambda$ achieves the
minimum and $j$ an odd index where $\lambda$ achieves the minimum. As the
lambda-values, as well as realizability by positive linear spaces is invariant
under translation, we suppose again that $\mu$ is the trivial valuation on its
support. Moreover, we can take a representative of $\nu$ that has non-negative
entries and that $\nu([n]\backslash i)=\nu([n]\backslash j)=0$.

Let us build a matrix $A\in\mathbb{R}\\{\\!\\!\\{t\\}\\!\\!\\}^{(n-1)\times
n}$ that realizes ${\bm{\mu}}$ as a flag positroid. For the first row of $A$
let

$A_{1,k}=\begin{cases}n\quad\text{if $k=i$ or $k=j$}\\\ 0\quad\text{if $k$ is
a loop of $M$}\\\ 1\quad\text{otherwise.}\end{cases}$

Let $A_{f(k),k}=1$ for every $k\in[n]\backslash\\{i,j\\}$ where $f$ is the
order preserving function $[n]\backslash\\{i,j\\}\to[n-1]\backslash 1$. Let
$A_{k,i}=\alpha t^{\nu([n]\backslash k)}$ for every even $k$ and
$A_{k,j}=\alpha t^{\nu([n]\backslash k)}$ for every odd $k$ where

$\alpha=\begin{cases}1\quad\text{if $j<i$}\\\ -1\quad\text{if
$i<j$}\end{cases}$

(where $t^{\infty}:=0$). Finally, let every other entry of $A$ equal to $0$.
See (8.1) for an example of such a matrix.

Let us show that the row spans of $A$ realizes ${\bm{\mu}}$ as flag positroid.
Clearly the first row has non-negative coordinates and realizes $\mu$. Now let
us look at the $(n-1)\times(n-1)$ minors. Consider the minor given by deleting
the column $k$ with $k$ different from $i$ or $j$. Suppose $k$ is even. Any
function $\sigma:[n]\backslash k\to[n-1]$ on non-zero entries must have that
$\sigma(i)=k$ as it is the only column with a non-zero entry in that row. Now
$\sigma(j)$ could be $1$ or any odd value. If $\sigma(j)=1$, then for rest of
the coordinates we have $\sigma(l)=f(l)$. Because of how $\alpha$ was defined,
the resulting sign for $\sigma$ is going to be positive and the product of the
corresponding entries is going to be equal to $nt^{\nu([n]\backslash k)}$. If
$\sigma(j)\neq 1$, the valuation of the resulting product is at least
$\nu([n]\backslash k)$. This is because we would need $\sigma(j)$ not to be a
loop of $M$ for the entry $A_{1,\sigma(j)}$ to be different to $0$ and then
$\nu([n]\backslash\sigma(j))=\lambda_{\sigma(j)}$ is non-negative. However,
the absolute value of the leading coefficient of such term would be $1$ So the
term $nt^{\nu([n]\backslash k)}$ dominates all others. Hence this minor is
positive and has valuation $\nu([n]\backslash k)$ as desired.

A similar reasoning shows that this is also the case for $k$ odd. If $k=i$,
then again the dominating term in the minor corresponds to $\sigma(j)=1$ and
$\sigma(l)=f(l)$ and this term is equal to $n$. Similarly for $k=j$, so this
matrix realizes ${\bm{\mu}}$ as a flag positroid. ∎

Notice that valuated flag positroids always induce flag positroids
subdivisions. On the other hand if a hollow valuated flag matroid ${\bm{\mu}}$
does not satisfy (1.6), then clearly none of maximal cells in the subdivision
induced by ${\bm{\mu}}$ are going to be flag positroid polytopes. Therefor
Theorem 8.4 implies part 3. of Theorem 1.2. With this we complete our proof of
Theorem 1.2 and we have now completely determined all positivity notions with
explicit conditions on the Plücker coordinates for the hollow case.

######  Example 8.5.

Recall the flag valuated matroid from Example 8.3. In this case we have $i=2$
and $j=1$ and the matrix

$A=\begin{pmatrix}8&8&0&1&1&1&1&0\\\ 0&t^{2}&1&0&0&0&0&0\\\
t^{1}&0&0&1&0&0&0&0\\\ 0&t^{1}&0&0&1&0&0&0\\\ t^{3}&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&1&0\\\ t^{-1}&0&0&0&0&0&0&1\end{pmatrix}.$ (8.1)

realizes it as a flag positroid. This matrix may not have all minors positive,
though it is possible to construct such a matrix by recovering the
parametrization of [31] from the weighted graph constructed in Example 8.3.

##  9\. Arbitrary rank

In this section we turn our attention to arbitrary rank d and use Theorem 1.2
to prove Theorem 1.4. Before we continue to the proof, we want to comment on
why we have to include Plücker pairs $(S,T)$ corresponding to Plücker
relations, i.e. when $S\in{[n]\choose d_{i}-1}$ and $T\in{[n]\choose
d_{i}+1}$.

In [22, Theorem 3.10] it is shown that the incidence relations on some
functions $\mu:{[n]\choose d}\to\mathbb{R}$ and $\nu:{[n]\choose
d+1}\to\mathbb{R}$ imply the Plücker relations on $\mu$ and $\nu$. We could
ask whether the same holds for valuated matroids whose rank differ by more
than $1$. However, the answer is no, as the following example shows:

######  Example 9.1.

Consider the $\mu:{[5]\choose 2}:\mathbb{R}$ where $\mu(ij)=0$ for
$ij=12,23,34,45,15$, and an arbitrary positive number in the rest of the
cases. This function satisfies all incidence relations with
$\nu=(0,0,0,0,0)\in\operatorname{Dr}(4,5)$. However $\mu$ is not a valuated
matroid; for example, the minimum in
$\mu(12)\odot\mu(34)\oplus\mu(13)\odot\mu(24)\oplus\mu(14)\odot\mu(24)$ is
achieved only in $\mu(12)\odot\mu(34)$.

Furthermore the positive version of the three-term incidence relations imply
the positive Plücker relations for the complete flag variety of full support
[22, Lemma 6.7]. We could ask whether this extends to partial flags, adding
the assumption that the functions are actually valuated matroids, because of
the previous example. Again, the answer is no.

######  Example 9.2.

Let ${\bm{\mu}}\in\operatorname{Dr}(2,4;5)$ where ${\bm{\mu}}(13)=1$ and
${\bm{\mu}}(I)=0$ for every other set $I\in{5\choose 2}\cup{5\choose 4}$. This
valuated flag matroid satisfies the positive relations ((1.10) for
$S\in{5\choose 1}$ and $T=[5]$), but $\mu_{1}$ does not satisfy the positive
Plücker relations as
${\bm{\mu}}(13)\odot{\bm{\mu}}(24)>{\bm{\mu}}(12)\odot{\bm{\mu}}(34)\oplus{\bm{\mu}}(14)\odot{\bm{\mu}}(23)$.

Because the positive three-term Plücker relations are equivalent to a valuated
matroid being a valuated positroid [1, 41], all of the positivity concepts
lisited in the introduction imply that the constituents of ${\bm{\mu}}$ are
valuated positroids. So from now on we will assume that ${\bm{\mu}}$ has
constituents that are valuated positroids and we will only consider Plücker
pairs $(S,T)$ that correspond to incidence relations (i.e. $T\in{[n]\choose
d_{i+1}+1}$)

######  Definition 9.3.

Let $(S,T)$ be a Plücker pair of ${\bm{\mu}}$. Then the _hollow projection_ of
${\bm{\mu}}$ induced by $(S,T)$ is the valuated flag matroid

$\eta_{S,T}({\bm{\mu}}):=(\mu_{i}/S|_{T},\mu_{i+1}/(S\cap T)|_{T})$

The adjective hollow is to emphasize that the result is a matroid over
$T\backslash S$ of hollow rank. To see that it is actually a flag matroid,
notice that $(\mu_{i}/S|_{T},\mu_{i+1}/S|_{T},\mu_{i+1}/(S\cap T)|_{T})$ is a
flag matroid, the first quotient is due to the fact that taking minors
preserves valuated matroid quotient [8, Corollary 4.3.2] and the second
quotient is due to Theorem 3.2.

The reason to consider hollow projections is the following:

######  Lemma 9.4.

The lambda values of $\eta_{S,T}({\bm{\mu}})$ equal the lambda values of
${\bm{\mu}}$ for the Plücker pair $(S,T)$.

###### Proof.

Let $(\mu^{\prime},\nu^{\prime})=\eta_{S,T}({\bm{\mu}})$. The lemma simply
follows from the fact that for all $j\in T\backslash S$ we have that
$\mu^{\prime}(j)={\bm{\mu}}(Sj)$ and $\nu^{\prime}(T\backslash
Sj)={\bm{\mu}}(T\backslash j)$. ∎

Theorem 1.4 follows directly from Theorem 1.2, Lemma 9.4 and the following
theorem:

######  Theorem 9.5.

Let $(S,T)$ be a Plücker pair of flag valuated matroid ${\bm{\mu}}$ where
$|S|=d_{i}-1$ and $|T|=d_{i+1}+1$.

  1. (1)

If ${\bm{\mu}}$ is tnn, then $\eta_{S,T}({\bm{\mu}})$ is tnn.

  2. (2)

If ${\bm{\mu}}$ induces a Bruhat subdivision, then $\eta_{S,T}({\bm{\mu}})$
induces a Bruhat subdivision.

  3. (3)

If ${\bm{\mu}}$ induces a positroid subdivision, then $\eta_{S,T}({\bm{\mu}})$
induces a positroid subdivision.

###### Proof.

  1. (1)

Suppose ${\bm{\mu}}$ is totally non negative. So there is a flag of linear
spaces
$\textbf{L}=(L_{1},\dots,L_{n})\in\operatorname{Fl}^{\textsl{tnn}}(\textbf{n},C^{n})$
which restricted to d realizes ${\bm{\mu}}$ as a tnn valuated flag matroid.
Consider $\pi_{T}$ the projection to the $T$ coordinates and

$L_{j}^{\prime}=\pi_{T}(L_{j}\cap\\{x_{k}=0\mid k\in S\\}).$

We have that $(L_{d_{i}}^{\prime},\dots,L_{d_{i+1}}^{\prime})$ is a flag of
linear spaces of consecutive rank $(1,\dots,|T|-|S|-1)$ with positive Plücker
coordinates. Now order the elements of $S\backslash T$, say
$a_{1},\dots,a_{k}$ and let

$L_{j}^{\prime\prime}=\pi_{T}(L_{d_{i+1}}\cap\\{x_{k}=0\mid k\in
S\backslash\\{a_{l}\mid l<j\\}\\}).$

We obtain another flag of linear spaces
$(L_{1}^{\prime\prime},\dots,L_{k+1}^{\prime\prime})$ of linear spaces of
consecutive rank $(|T|-|S|-1,\dots,|T\backslash S|-1)$. Notice that
$L_{1}^{\prime\prime}=L_{d_{i+1}}^{\prime}$, $L_{1}^{\prime}$ is a realization
of $\mu^{\prime}$ and $L_{k+1}^{\prime\prime}$ is a realization of
$\nu^{\prime}$. So

$(L_{d_{i}}^{\prime},\dots,L_{d_{i+1}}^{\prime}=L_{1}^{\prime\prime},\dots,L_{k+1}^{\prime\prime})$

is a complete flag with all of its Plücker coordinates positive showing that
$\eta_{S,T}({\bm{\mu}})$ is totally non-negative.

  2. (2)

Suppose ${\bm{\mu}}$ induces a Bruhat subdivision. Let
$x\in\mathbb{R}^{T\backslash S}$ and $(M,N):=\eta_{S,T}({\bm{\mu}})^{x}$.

Let $\tilde{x}\in\mathbb{R}^{n}$ be a vector such that
$\tilde{x}|_{T\backslash S}=x$, $x_{i}$ is sufficiently large for $i\in S$ and
sufficiently small for $i\in[n]\backslash(S\cup T)$. For such $\tilde{x}$, we
have that $\mu_{d_{i}}^{\tilde{x}}/S|_{T}=M$ and
$\mu_{d_{i}+1}^{\tilde{x}}/S|_{T}=(\mu_{d_{i}+1}/S|_{T})^{x}$. As
${\bm{\mu}}^{\tilde{x}}$ is a Bruhat polytope, so is
${\bm{\mu}}^{\tilde{x}}/S|_{T}$. In particular we have that
$(M,(\mu_{d_{i}+1}/S|_{T})^{x})$ can be completed to flag positroid
$\textbf{M}^{\prime}=(M_{1}^{\prime},\dots,M_{|T|-|S|-1}^{\prime})$ of
consecutive rank. In particular, $\textbf{M}^{\prime}$ satisfies the 3-term
incidence relations.

Let $\mu_{j}^{\prime\prime}=\mu_{d+1}/(S\backslash\\{a_{l}\mid l<j\\})|_{T}$
where again $S\backslash T=\\{a_{1},\dots a_{k}\\}$. If ${\bm{\mu}}$ induces a
Bruhat subdivision then $\mu_{d+1}$ induces a positroid subdivision and hence
satisfies the 3-term positive Plücker relations. This implies that
${\bm{\mu}}^{\prime\prime}:=(\mu_{1}^{\prime\prime},\dots,\mu_{k+1}^{\prime\prime})$
and therefor $\textbf{M}^{\prime\prime}={\bm{\mu}}^{\prime\prime
x}=(\mu_{1}^{\prime\prime x},\dots,\mu_{k+1}^{\prime\prime x})$ satisfy the
3-term positive incidence relations.

So the flag matroid
$(M_{1}^{\prime},\dots,M_{|T|-|S|-1}^{\prime}=\mu_{1}^{\prime\prime
x},\dots,\mu_{k+1}^{\prime\prime x})$ satisfies all the 3-term positive
incidence relations and by [22, Theorem 6.10] it has a Bruhat polyotpe. But
$M_{1}^{\prime}=M$ and $\mu_{k+1}^{\prime\prime x}=N$, so the restriction of
this flag matroid to hollow rank is $(M,N)$. So $P_{(M,N)}$ is a Bruhat
polytope. As $x\in\mathbb{R}^{T\backslash S}$ was arbitrary, we have that
$\eta_{S,T}({\bm{\mu}})$ induces a Bruhat subdivision.

  3. (3)

This proof is similar to the one above. Suppose ${\bm{\mu}}$ induces a
positroid subdivision. Let $x\in\mathbb{R}^{T\backslash S}$ and
$(M,N):=\eta_{S,T}({\bm{\mu}})^{x}$. Let $\tilde{x}$ be as before. We have
that
$(\mu_{d_{i}}^{\tilde{x}},\mu_{d_{i}+1}^{\tilde{x}})=(M,\mu_{d_{i}+1}^{\tilde{x}}/S|_{T})$
is a flag positroid and it satisfies the positive incidence relations (1.10).
Let $\textbf{M}^{\prime\prime}$ be as before. We have that
$\textbf{M}^{\prime\prime}$ is a flag positroid, so it also satisfies the
positive incidence relations. So we have that
$(M,\mu_{d_{i}+1}^{\tilde{x}}/S|_{T}=\mu_{1}^{\prime\prime
x},\dots,\mu_{k+1}^{\prime\prime x}=N)$ is the non negative flag Dressian by
[21, Theorem 2.16], (see [7, Remark 3.11]). This implies that
$(M,N)\in\operatorname{FlDr}^{\geq 0}(1,|T\backslash S|-1,n)$ and, by Theorem
8.4, it is a flag positroid. As $x$ was arbitrary, $\eta_{S,T}({\bm{\mu}})$
induces a positroid subdivision.

∎

Now we discuss some information that should be taken into account regarding
1.5.

######  Proposition 9.6.

  1. (1)

If part 2. of 1.5 holds for unvaluated matroids, then it holds for all
valuated matroids.

  2. (2)

If 1.3 implies d for unvaluated matroids, then 1.3 implies d holds for all
valuated matroids.

###### Proof.

Recall that the lambda values are invariant under translation. So for every
$x\in\mathbb{R}^{n}$ and for each Plücker pair $(S,T)$, the lambda values of
the flag matroid ${\bm{\mu}}^{x}$ for $(S,T)$ is either the result of
replacinig by $0$ every term that achieves the minimum and everything else is
replaced by $\infty$, or they all become $\infty$. Either way, if ${\bm{\mu}}$
satisfies (1.9) then ${\bm{\mu}}^{x}$ satisfies (1.9) and if ${\bm{\mu}}$
satisfies (1.10) then ${\bm{\mu}}^{x}$ satisfies (1.10). ∎

In other words, part 2. of 1.5 follows from a generalization of Theorem 6.4,
while 1.3 $\leftrightarrow$ d is equivalent to [7, Question 1.6]. Notice that
parts 1. and 2. of 1.5 are the same at the level of matroids. Hence we have
the following

######  Corollary 9.7.

Part 1. of 1.5 implies part 2.

######  Remark 9.8.

It may be tempting to believe that to prove 1.5 it is enough for flags with 2
constituents, as one could then fill the gaps independently. However, if one
we have that $(\mu_{1},\mu_{2})$ and $(\mu_{2},\mu_{3})$ can be realized by
flags $(L_{1},L_{2})$ and $(L_{2}^{\prime},L_{3})$ with positive Plücker
coordinates, they might not satisfy that $L_{2}=L_{2}^{\prime}$.

We finish with the following questions:

######  Question 2.

If we have a regular Bruhat subdivision, does there always exist a tnn flag
valuated matroid that induces it?

######  Question 3.

Is the (positive) flag Dressian generated by (positive) Plücker-incidence
relations of minmal length (i.e. three term Plücker relations plus incidence
relations with $S\subseteq T$)?

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