# The tropical symplectic Grassmannian

George Balla and Jorge Alberto Olarte  Algebra and Representation Theory, RWTH
Aachen University, Pontdriesch 10-16, 52062 Aachen, Germany<EMAIL_ADDRESS>aachen.de Institut für Mathematik, Technische Universität Berlin, Straße des
17. Juni 135, 10623 Berlin, Germany<EMAIL_ADDRESS>
###### Abstract.

We launch the study of the tropicalization of the symplectic Grassmannian,
that is, the space of all linear subspaces isotropic with respect to a fixed
symplectic form. We formulate tropical analogues of several equivalent
characterizations of the symplectic Grassmannian and determine all
implications between them. In the process, we show that the Plücker and
symplectic relations form a tropical basis if and only if the rank is at most
2. We provide plenty of examples that show that several features of the
symplectic Grassmannian do not hold after tropicalizing. We show exactly when
do conormal fans of matroids satisfy these characterizations, as well as doing
the same for a valuated generalization. Finally, we propose several directions
to extend the study of the tropical symplectic Grassmannian.

##  1\. Introduction

Given a field $\mathbb{K}$, the Grassmannian, denoted by
$\textsc{G}\mathsf{r}_{\mathbb{K}}(k,n)$, is the space of all linear subspaces
of $\mathbb{K}^{n}$ of dimension $k$. Let $\omega$ denote the standard
symplectic form on $\mathbb{K}^{2n}$. The symplectic Grassmannian, which will
be denoted by $\textsc{Sp}\textsc{G}\mathsf{r}_{\mathbb{K}}(k,2n)$, is the
subset of the Grassmannian consisting of all isotropic subspaces of
$\mathbb{K}^{2n}$, that is, spaces whose elements are orthogonal to each other
with respect to $\omega$. We consider the Grassmannian under its Plücker
embedding and the symplectic Grassmannian as a subset under the same
embedding. The defining ideal of the latter variety with respect to this
embedding is generated by Plücker and certain linear relations that we term
symplectic relations [Dec79]. We will therefore refer to this ideal as the
Plücker-Symplectic ideal. For a given valuation
$\operatorname{val}:\mathbb{K}\to\mathbb{T}:=\mathbb{R}\cup\\{\infty\\}$, the
tropicalization of the Grassmannian is the space of tropicalizations of linear
subspaces [SS04]. We define the tropical symplectic Grassmannian as the
tropicalization of the symplectic Grassmannian with respect to the Plücker-
Symplectic ideal. Therefore, it is a subset of the tropical Grassmannian
consisting of tropicalizations of isotropic linear subspaces. We will denote
it by $\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(k,2n)$, where $p$ is the
characteristic of the field $\mathbb{K}$.

It is known that the geometry of the tropical Grassmannian is governed by the
combinatorics of matroids [Spe08]. Coxeter matroids are generalizations of
matroids to different root systems, where the type A Coxeter matroids are the
usual matroids [BGW03]. In [Rin12], Rincón studied the tropicalization of the
type D Grassmannian, known as the Spinor variety, using the type D Coxeter
matroids, also known as $\Delta$-matroids, to obtain similar results as those
in [Spe08] for type A. The symplectic Grassmannian can be interpreted as the
type C Grassmannian. However, very little is known about the type C Coxeter
matroids, known as symplectic matroids (see [BGW98] and [BGW03, Chapter 3] for
what is known about them). In particular, there is no good characterization of
symplectic matroids in terms of axioms for their bases. This makes it more
challenging to build a theory of their valuated counterparts as tropical
Plücker vectors similar to the ones for types A and D. So, we take a different
approach, relying on the already rich theory of type A valuated matroids.

There are several equivalent ways of saying that a linear space belongs to the
symplectic Grassmannian, among them:

  1. (1)

Its Plücker vector satisfies all polynomials in the Plücker-Symplectic ideal.

  2. (2)

Its Plücker vector satisfies the symplectic relations.

  3. (3)

It is isotropic.

  4. (4)

It has a basis of pairwise orthogonal vectors (with respect to $\omega$).

  5. (5)

It is the row span of a matrix $(A|B)$ such that $A\cdot B^{T}$ is symmetric,
where $A,B\in\mathbb{K}^{k\times n}$.

There is a natural way of obtaining a tropical analogue for each of the
statements above, by imposing a condition on a given tropical linear space.
The first is the most obvious one, as it just asks to be in the
tropicalization of the symplectic Grassmannian. Just like the Dressian is the
tropical prevariety consisting of vectors satisfying the tropical Plücker
relations, we define the symplectic Dressian, which will be denoted by
$\textsc{Sp}\textsc{D}\mathsf{r}(k,2n)$, to be the prevariety consisting of
vectors satisfying the tropical symplectic relations as well as the tropical
Plücker relations.

A tropical notion of isotropic linear spaces was already studied in [Rin12].
Rincón’s motivation was in type D, that is, in the context of isotropic spaces
with respect to a symmetric bilinear relation. However, the sign that
distinguishes that context to ours vanishes after tropicalizing, so his
definition is the relevant one for us too. For the last two characterizations,
we use the tropical Stiefel map from [FR15] as an analogue of being the span
of a collection of vectors. However, we recognize the limitations of this last
analogy, since it forces us to restrict ourselves to tropical linear spaces of
this kind. Thus we obtain the following conditions on a given tropical linear
space that can be considered tropical analogues of the characterizations of
the symplectic Grassmannian above.

  1. (1)

It is in the tropical symplectic Grassmannian.

  2. (2)

It is in the symplectic Dressian.

  3. (3)

It is tropically isotropic.

  4. (4)

It is the image under the tropical Stiefel map of a matrix with (tropically)
orthogonal rows.

  5. (5)

It is the image under the tropical Stiefel map of a matrix $(A|B)$ such that
$A\odot B^{T}$ is symmetric, where $A,B\in\mathbb{T}^{k\times n}$.

Our main result (Theorem 4.16), determines for each $k$ (dimension of the
tropical linear space) and $n$ (dimension of the ambient space), which of the
above statements implies another. It is fully described in Figure 1 below.

Figure 1. Map of the Matroid Zoo.

The five boxes represent the five tropical analogues. Black full arrows show
implications, which are containment of sets. For example, the left most
downward arrow means that
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(k,2n)\subseteq\textsc{Sp}\textsc{D}\mathsf{r}(k,2n)$.
Dashed arrows represent implications for specific cases. They come with a
label indicating the conditions on $k$ and $n$ for them to hold. Black arrows
come labeled with the corresponding statement in Section 4 when not trivial.
Red arrows represent counterexamples that show that such implications fail to
hold. They come with the letters that label the corresponding counterexample
in Section 7. The dashed box titled ‘Stiefel image’ is there to clarify that
the arrows which interact with the two right boxes are taken under the
assumption that we are restricted to the Stiefel image. It is not there to say
that only these valuated matroids are in the Stiefel image; for example, we
have that the matroid of B is also in the Stiefel image.

A particularly interesting case covered in our main theorem is the rank two
case. We see that in this case, the tropical symplectic Grassmannian
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(2,2n)$ and the symplectic
Dressian $\textsc{Sp}\textsc{D}\mathsf{r}(2,2n)$ coincide (Theorem 4.8). This
result also implies that the Plücker and symplectic relations form a tropical
basis for the Plücker-Symplectic ideal in this case. This is analogous to the
type A result by Maclagan and Sturmfels [MS15]. Furthermore, Speyer and
Sturmfels [SS04] showed that the tropical Grassmannian in rank two coincides
with the space of phylogenetic trees. We show that this coincidence holds in
the symplectic case as well, after quotienting
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(2,2n)$ by the tropical
hyperplane corresponding to the single tropical symplectic relation
(Proposition 5.1). We then go on to obtain enumerative information on this
fan, namely, the number of rays and facets (Corollary 5.2) and the Betti
numbers for the corresponding simplicial complex (Corollary 5.3).

Another case of particular interest is the Lagrangian case, that is, when
$k=n$. In [ADH20], the authors proved that the $h$-vector of the independence
complex and the broken circuit complex of a matroid is log-concave, solving
conjectures that were standing since the 80’s. One of their main tools is what
they call the conormal fan of a matroid $M$, which has the same support as the
product of the Bergman fan of $M$ and its dual. The conormal fan has a natural
generalization to any valuated matroid, forming a class of linear spaces which
make sense to study under our context. We show that indeed, they are
tropically isotropic, they satisfy the tropical symplectic relations, and
provided the involved valuated matroids are realizable, they are in the
tropical symplectic Grassmannian (Theorem 8.2).

The paper is organized as follows: Section 2 provides background on the
classical Grassmannian as well as the symplectic Grassmannian while Section 3
provides background on the tropical Grassmannian and its connection to
valuated matroids. In Section 4, we formulate each of the tropical analogues
of characterizations of the symplectic Grassmannian and show the implications
between them. Section 5 deals with the rank two case and in particular, we
show that for this case, the symplectic and Plücker relations form a tropical
basis for the symplectic Grassmannian. Section 7 contains the other side of
the main result, by exhibiting all counterexamples needed to finish the proof,
plus some other interesting examples that exhibit some pathologies that can
occur. Before that, Section 6 explains why the list given in Section 7 is
complete to prove Theorem 4.16. In Section 8, we answer the question: where do
conormal bundles fit in Figure 1? Finally, Section 9 suggests several
directions for future work, including the connection to symplectic matroids
and applications to toric degenerations of flag varieties.

### Acknowledgments

The authors are grateful to Ghislain Fourier for suggesting this project and
for insightful discussions. The second author would also like to thank Felipe
Rincón for useful discussions. Great thanks to Xin Fang, Christian Steinert
and an anonymous referee for reading an earlier version, and for pointing out
misprints. The first author is funded by the Deutscher Akademischer
Austauschdienst (DAAD, German Academic Exchange Service) scholarship program:
Research Grants - Doctoral Programs in Germany. The second author is funded by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through
Project-ID 286237555 – TRR 195 and under Germany’s Excellence Strategy; The
Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID 390685689,
sub-project AA/EFx-y).

##  2\. The Symplectic Grassmannian

We begin by fixing some notation that we will be using. Let
$[n]:=\\{1,\ldots,n\\}$ and
$[2n]:=\\{1,\ldots,n,\overline{1},\ldots,\overline{n}\\}$ ordered as
$1<\cdots<n<\overline{1}<\cdots<\overline{n}$. Adding a bar is a dual
operation, so that $\bar{\bar{i}}=i$. For a set $S$, we write
$Si:=S\cup\\{i\\}$ and $\overline{S}:=\\{\bar{i}\mid i\in S\\}$. Let
$\mathbb{K}$ be an algebraically closed field of characteristic $p$. For a
matrix $M\in\mathbb{K}^{k\times n}$ we write $M^{T}$ for the transpose and for
a sub-sequence $\textsc{J}\subseteq[n]$ let $M_{\textsc{J}}$ denote the sub-
matrix of $M$ with columns indexed by J.

The _Grassmannian_ $\textsc{G}\mathsf{r}_{\mathbb{K}}(k,n)$ is the space of
all $k$-dimensional linear subspaces of $\mathbb{K}^{n}$. Whenever the field
$\mathbb{K}$ is not relevant, we omit it from the subscript and write just
$\textsc{G}\mathsf{r}(k,n)$. A subspace $L\subseteq\mathbb{K}^{n}$ can be
identified by its _Plücker coordinates_ $\textsc{X}_{\textsc{J}}$, which are
determinants of the sub-matrices $M_{\textsc{J}}$ for every
$\textsc{J}\in\binom{[n]}{k}$, where $M$ is a $(k\times n)$-matrix whose rows
span $L$. A Plücker vector $[\textsc{X}]\in\mathbb{P}^{\binom{n}{k}-1}$ is a
vector whose coordinates are the $\binom{n}{k}$ Plücker coordinates. The
Grassmannian $\textsc{G}\mathsf{r}(k,n)$ can be identified with the set of all
Plücker vectors in $\mathbb{P}^{\binom{n}{k}-1}$ via the _Plücker embedding_.
It is a projective variety of dimension $k(n-k)$ that satisfies the _Plücker
relations_ :

$\forall\,\,S\in\binom{[n]}{k-1},\quad\forall\,\,T\in\binom{[n]}{k+1},\quad\sum_{i\in
T\backslash S}\operatorname{sign}(i,T\backslash S)\textsc{X}_{T\setminus
i}\cdot\textsc{X}_{Si}=0,$ (2.1)

where the sign $\operatorname{sign}(i,T\backslash S)\in\\{-1,1\\}$ alternates
with respect to the ordering of $T\backslash S$. We write
$\mathcal{P}_{k,n}(\mathbb{K})$ for the ideal generated by the above
relations, called the _Plücker ideal_.

The subspace $L\subseteq\mathbb{K}^{n}$ can be recovered from the Plücker
vector $\textsc{X}\in\mathbb{P}^{\binom{n}{k}-1}$ as follows:

$L=\left\\{x\in\mathbb{K}^{n}\mid\forall\,\,T\in\binom{[n]}{k+1},\quad\sum_{i\in
T}\operatorname{sign}(i,T)\textsc{X}_{T\setminus i}\cdot x_{i}=0\right\\}.$
(2.2)

A symplectic vector space $W$ is a vector space with a symplectic form, which
is a bilinear form $\omega:W\times W\to\mathbb{K}$ satisfying:

  * •

$\forall\,\,v\in W\enspace\omega(v,v)=0$ (alternating).

  * •

if $\omega(u,v)=0$ for every $u\in W$, then $v=0$ (non-degenerate).

The first condition implies that
$\omega(u,v)=-\omega(v,u)\,\,\forall\,\,u,v\in W$, however the converse only
holds if $p\neq 2$.

It is well known that any symplectic vector space has even dimension and that
there is a basis $\\{e_{1},\dots,e_{n},e_{\bar{1}},\dots,e_{\bar{n}}\\}$ such
that $\omega$ is given by $\omega(e_{i},e_{\bar{i}})=1$ (for $i<\bar{i}$) and
$\omega(e_{i},e_{j})=0$ for $j\neq\bar{i}$. In other words, $\omega$ can be
written as

$\begin{pmatrix}0&\operatorname{Id}_{n}\\\
-\operatorname{Id}_{n}&0\end{pmatrix},$ (2.3)

with respect to this basis, where $\operatorname{Id}_{n}$ is the $n\times n$
matrix with $1$’s on the diagonal and zeros elsewhere.

We say that two vectors $u$ and $v$ are _orthogonal_ if $\omega(u,v)=0$. A
$k$-dimensional linear subspace $L\subseteq\mathbb{K}^{2n}$ is called
_isotropic_ if every two vectors in $L$ are orthogonal. By bilinearity, it is
enough that $L$ has a basis consisting of pairwise orthogonal vectors. If we
write such a basis as rows of a matrix $(A|B)$ with $A$ and $B$ two $k\times
n$ matrices, it is straightforward to see that the basis is orthogonal if and
only if $A\cdot B^{T}$ is symmetric. If $L$ is isotropic then it follows that
$k\leq n$. When $\dim(L)=n$, we also call $L$ _Lagrangian_.

The _symplectic Grassmannian_
$\textsc{Sp}\textsc{G}\mathsf{r}_{\mathbb{K}}(k,2n)$ is the sub-variety of
$\textsc{G}\mathsf{r}_{\mathbb{K}}(k,2n)$ consisting of all isotropic linear
subspaces of $\mathbb{K}^{2n}$. Again, we omit the subscript $\mathbb{K}$ when
the field does not play a role and write
$\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)$. It is an irreducible projective
variety of dimension $\frac{k(4n-3k+1)}{2}$ (see for example, [MWZ20]). It
follows from the work of De Concini [Dec79] that the defining ideal of
$\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)$, which we denote by
$S_{k,2n}(\mathbb{K})$, is generated by the Plücker relations together with
the _symplectic relations_ :

$\forall\,\,S\in\binom{[2n]}{k-2},\quad\sum_{i}\textsc{X}_{Si\overline{i}}=0,$
(2.4)

with respect to the Plücker embedding, where the sum goes over all $i\in[n]$
such that $\\{i,\overline{i}\\}\cap S=\emptyset$. We call
$S_{k,2n}(\mathbb{K})$ the _Plücker-Symplectic ideal_. We include the
following short proof that the symplectic relations really define the
symplectic Grassmannian, for completeness. It was originally written by
Carrillo-Pacheco and Zaldivar [CPZ11] for the Lagrangian case $k=n$, but it
works for any $k$.

######  Proposition 2.1.

The defining ideal of $\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)$ with respect to
the Plücker embedding is the ideal $S_{k,2n}(\mathbb{K})$.

###### Proof.

For $1\leq k\leq n$, let $\bigwedge^{k}\mathbb{K}^{2n}$ be the $k$-th exterior
power over $\mathbb{K}^{2n}$. Define a linear map:

$\displaystyle f:\bigwedge^{k}\mathbb{K}^{2n}$
$\displaystyle\longrightarrow\bigwedge^{k-2}\mathbb{K}^{2n},$ $\displaystyle
v_{1}\wedge\cdots\wedge v_{k}$ $\displaystyle\longmapsto\sum_{1\leq r<s\leq
k}(-1)^{r+s}\omega(v_{r},v_{s})v_{1}\wedge\cdots\wedge\hat{v}_{r}\wedge\cdots\wedge\hat{v}_{s}\wedge\cdots\wedge
v_{k},$

where $\hat{v}$ means that the corresponding term is omitted. Notice that the
sign $(-1)^{r+s}$ is necessary for $f$ to be well defined with respect to
commuting the terms in the wedge product (this sign is missing in [CPZ11]).
Let $\mathcal{K}$ denote the kernel of $f$ and denote by
$\mathbb{P}(\mathcal{K})$ the projectivization of $\mathcal{K}$. Then
$\mathbb{P}(\mathcal{K})$ is a closed irreducible subset of
$\mathbb{P}\left(\bigwedge^{k}\mathbb{K}^{2n}\right)$ under the Plücker
embedding.

Claim 1:
$\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)=\textsc{G}\mathsf{r}(k,2n)\cap\mathbb{P}(\mathcal{K})$.
It is clear that
$\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)\subseteq\textsc{G}\mathsf{r}(k,2n)\cap\mathbb{P}(\mathcal{K})$.
For the opposite inclusion, if
$v\in\textsc{G}\mathsf{r}(k,2n)\cap\mathbb{P}(\mathcal{K})$, then $v$ is the
equivalence class of $v_{1}\wedge\cdots\wedge v_{k}$ with $v_{1},\ldots,v_{k}$
linearly independent and $f(v)=0$, that is:

$f(v)=\sum_{1\leq r<s\leq
k}(-1)^{r+s}\omega(v_{r},v_{s})v_{1}\wedge\cdots\wedge\hat{v}_{r}\wedge\cdots\wedge\hat{v}_{s}\wedge\cdots\wedge
v_{k}=0,$

and thus $\omega(v_{r},v_{s})=0$ for all $1\leq r<s\leq k$, and hence the
claim follows.

Claim 2: $v\in\mathcal{K}$ if and only if the linear relations (2.4) vanish.
For this, we use the linearity of $f$. Write $v$ with its Plücker coordinates,
that is,

$v=\sum\limits_{I\in\binom{[2n]}{k}}\textsc{X}_{I}e_{i_{1}}\wedge\dots\wedge
e_{i_{k}}$

where $I=\\{i_{1}\dots i_{k}\\}$. So

$f(v)=\sum\limits_{I\in\binom{[2n]}{k}}\textsc{X}_{I}f(e_{i_{1}}\wedge\dots\wedge
e_{i_{k}}).$

Notice that $f(e_{i_{1}}\wedge\dots\wedge e_{i_{k}})$ vanishes, unless
$\\{i,\bar{i}\\}\subseteq I$ for some $i$. Let $S=\\{s_{1}\dots
s_{k-2}\\}\in\binom{[2n]}{k-2}$. The coefficient of
$e_{s_{1}}\wedge\dots\wedge e_{s_{k}}$ in $f(v)$ is

$\sum_{\\{i,\bar{i}\\}\cap S=\emptyset}\textsc{X}_{Si\bar{i}}(-1)^{2k-1}.$

So $f(v)=0$ if and only if the coefficient above is $0$ for every $S$, which
are precisely the linear relations (2.4). ∎

To summarize, we have established five equivalent conditions for a linear
space $L$ to be in the symplectic Grassmannian
$\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)$:

  1. (1)

All polynomials in the Plücker-Symplectic ideal $S_{k,2n}(\mathbb{K})$ vanish
on the Plücker coordinates of $L$.

  2. (2)

The Plücker coordinates of $L$ satisfy the symplectic relations (Equation
2.4).

  3. (3)

For every pair $u,v\in L$, we have $\omega(u,v)=0$.

  4. (4)

There is a basis of $L$, $u_{1},\dots,u_{k}$, such that
$\forall\,\,i,j,\enspace\omega(u_{i},u_{j})=0$.

  5. (5)

If $L$ is the row-span of a matrix $(A|B)$, then $A\cdot B^{T}$ is symmetric,
where $A,B\in\mathbb{K}^{k\times n}$.

##  3\. The Tropical Grassmannian

In this section, we recall notions related to the tropical Grassmannian, which
was first introduced and studied by Speyer and Sturmfels [SS04]. We refer the
reader to Maclagan and Sturmfels [MS15] for a comprehensive introduction to
tropical geometry.

To get started, we recall a couple of key tropical notions and fix some
conventions. Let $\mathbb{T}$ be the semi-field of tropical numbers
$\mathbb{T}=\mathbb{R}\cup\\{\infty\\}$ with operations $\oplus:=\min$ and
$\odot:=+$. Let $\operatorname{val}:\mathbb{K}\to\mathbb{T}$ be a non-
Archimidean valuation. Given an ideal
$\textsc{I}\subseteq\mathbb{K}[\textsc{X}_{1},\ldots,\textsc{X}_{n}]$, let
$V(\textsc{I})$ denote the algebraic variety corresponding to the ideal I. The
tropical variety $\operatorname{trop}(V(\textsc{I}))$ is the intersection of
the tropical hypersurfaces $\operatorname{trop}(V(f))$ for all
$f\in\textsc{I}$.

On the other hand, a tropical prevariety is the intersection of finitely many
tropical hypersurfaces, which is not necessarily a tropical variety. A finite
set of polynomials is said to be a tropical basis of the ideal it generates if
the tropical prevariety it defines is equal to the tropical variety of that
ideal.

###  3.1. The Tropical Grassmannian and the Dressian

First, we recall the _tropical projective space_

$\mathbb{T}\mathbb{P}^{\binom{n}{k}-1}:=\left(\mathbb{T}^{\binom{n}{k}}\setminus(\infty,\dots,\infty)\right)/\mathbb{R}(1,\dots,1).$

The _tropical Grassmannian_ , which we denote by
$\textsc{T}\textsc{G}\mathsf{r}_{p}(k,n)$, is the tropicalization of
$\textsc{G}\mathsf{r}(k,n)$ i.e.,
$\textsc{T}\textsc{G}\mathsf{r}_{p}(k,n)=\operatorname{trop}(\textsc{G}\mathsf{r}(k,n))$.
We want to emphasize that we use the notation
$\textsc{T}\textsc{G}\mathsf{r}_{p}(k,n)$ because the tropicalization of the
Plücker ideal only depends on the characteristic $p$ of $\mathbb{K}$. However,
similarly as before, whenever we say something about the tropical Grassmannian
that holds for every characteristic, we omit $p$ from the subscript and write
$\textsc{T}\textsc{G}\mathsf{r}(k,n)$.

The _Dressian_
$\textsc{D}\mathsf{r}(k,n)\supseteq\textsc{T}\textsc{G}\mathsf{r}(k,n)$ is the
prevariety given by all the Plücker relations, i.e., all vectors in
$\mathbb{T}\mathbb{P}^{\binom{n}{k}-1}$ that satisfy the _tropical Plücker
relations_ :

$\forall\,\,S\in\binom{[n]}{k-1},\enspace\,\,\forall\,\,T\in\binom{[n]}{k+1},\,\,\text{the
minimum in}\bigoplus_{i\in T\setminus S}(x_{T\setminus i}\odot x_{Si})$

is achieved at least twice (or is equal to $\infty$).

A vector $\mu\in\textsc{D}\mathsf{r}(k,n)$ is called a _valuated matroid_ or a
_tropical Plücker vector_. A _matroid_ can be defined to be a tropical Plücker
vector consisting of just $0$ and $\infty$ as entries. The set of subsets
whose corresponding coordinate is $0$ is the usual set of _bases_ of the
matroid (see Oxley [Oxl06] for more on the theory of ordinary matroids).

The difference between the tropical Grassmannian and the Dressian is an
extension of representable matroids vs non-representable matroids. When the
bases of a matroid are the set of non-zero coordinates of a point in the
Grassmannian $\textsc{G}\mathsf{r}_{\mathbb{K}}(k,n)$, we say that the matroid
is _representable_ or _realizable_ over the field $\mathbb{K}$. The existence
of non-realizable matroids for fields of a given characteristic, such as the
Fano matroid (for characteristic different than 2) and non-Fano matroid (for
characteristic equal to 2), implies that the tropical Grassmannian indeed
depends on the characteristic and that the Plücker relations (2.1) do not form
a tropical basis for the Plücker ideal $\mathcal{P}_{k,n}$. In other words,
the tropical Grassmannian $\textsc{T}\textsc{G}\mathsf{r}_{p}(k,n)$ and the
Dressian $\textsc{D}\mathsf{r}(k,n)$ do not coincide in general, except in
some cases:

######  Theorem 3.1 (Maclagan-Sturmfels, [MS15]).

The Dressian $\textsc{D}\mathsf{r}(2,n)$ and the tropical Grassmannian
$\textsc{T}\textsc{G}\mathsf{r}(2,n)$ coincide.

In an analogous fashion as before, a _tropical linear space_ can be recovered
from a tropical Plücker vector $\mu$:

$L_{\mu}=\left\\{x\in\mathbb{T}^{n}\mid\forall\,\,T\in\binom{[n]}{k+1},\text{
the minimum in}\,\,\bigoplus\limits_{i\in T}\mu_{T\setminus i}\odot
x_{i}\,\,\text{is A.A.T or equals }\infty\right\\},$ (3.1)

where A.A.T means “achieved at least twice”. If
$\mu\in\textsc{T}\textsc{G}\mathsf{r}_{p}(k,n)$ has rational coefficients and
$\mathbb{K}$ is algebraically closed, then by the Fundamental Theorem of
Tropical Algebraic Geometry [MS15, Theorem 3.2.3],
$\mu=\operatorname{val}(\textsc{X})$ for some Plücker vector X over
$\mathbb{K}$.

The tropical linear space corresponding to $\mu$ coincides with the
tropicalization of the linear space corresponding to X, i.e.,
$\operatorname{trop}(L_{\textsc{X}})=L_{\mu}$ [SS04, Theorem 3.8]. If $\mu$ is
a valuated matroid which is not in the tropical Grassmannian, $L_{\mu}$ is not
the tropicalization of a linear space, however there exist reasons to still
call such $L_{\mu}$ a tropical linear space (see, for example, [Fin13, Theorem
6.5]).

The study of tropical linear spaces equals the study of the valuated matroids,
which relies heavily on a polyhedral view of matroids. Valuated matroids are
characterized by being vectors which induce a regular subdivision of a matroid
polytope into matroid polytopes (Speyer [Spe08]). The tropical linear space of
an unvaluated matroid (such as the case where the valuation is trivial) is a
fan whose support is the same as a fan known as the _Bergman fan_.

The condition that a linear subspace contains another can be read off from
their Plücker vectors via the so called incidence relations. In [Haq12], Haque
shows that the incidence relations tropicalize well:

######  Proposition 3.2 ([Haq12]).

Let $\mu_{1}$ and $\mu_{2}$ be two valuated matroids on $[n]$ of rank
$k_{1}\leq k_{2}$. Then $L_{\mu_{1}}\subseteq L_{\mu_{2}}$ if and only if

$\forall S\in\binom{[n]}{k_{1}-1},\enspace T\in\binom{[n]}{k_{2}+1}\text{ the
minimum in }\bigoplus_{i\in T\setminus
S}(\mu_{1})_{Si}\odot(\mu_{2})_{T\backslash i}$

is achieved at least twice.

###  3.2. The Tropical Stiefel Map

The classical Stiefel map sends full rank $k\times n$ matrices to their row
span. In [FR15], Fink and Rincón introduced a tropical analogue. Let
$A\in\mathbb{T}^{k\times n}$ be a matrix with tropical entries. For
$\textsc{J}\in\binom{[n]}{k}$, let $A_{\textsc{J}}$ be the sub-matrix with
columns indexed by J. Then:

######  Definition 3.3 ([FR15]).

The _tropical Stiefel map_ $\pi:\mathbb{T}^{k\times
n}\dashrightarrow\mathbb{T}^{\binom{[n]}{k}}$ is given by

$\pi(A)_{\textsc{J}}:=\operatorname{tdet}(A_{\textsc{J}}),$

where $\operatorname{tdet}$ denotes the tropical determinant, that is, the
usual determinant computed with tropical operations. It is only defined for
matrices such that at least one of such determinants is different from
$\infty$, which is one notion of tropical full rank [DSS05].

Not all valuated matroids arise this way. The matroids that arise this way are
known as _transversal matroids_ and the corresponding valuated matroids are a
natural generalization so they are also called _transversal_. For a general
introduction to (unvaluated) transversal matroids, we recommend [Bru87].

For some valuated matroid $\mu$, a matrix $A$ such that $\pi(A)=\mu$ is called
a _presentation_ of $\mu$. We naturally have
$\pi(A)\in\textsc{T}\textsc{G}\mathsf{r}_{0}(k,n)$ but not all valuated
matroids that are representable over characteristic $0$ are of this form (see,
for example, [FR15, Figure 1]). The space of transversal valuated matroids in
$\textsc{T}\textsc{G}\mathsf{r}_{0}(k,n)$ is called the _Stiefel image_.

The fibers of the tropical Stiefel map were explicitly described by Fink and
the second author [FO19, Theorem 6.6]. This generalizes the characterization
of presentations of valuated matroids by Brualdi and Dinolt [BD72, Theorem
4.7]. It is easy to see that any row of $A$ is a point in $L_{\pi(A)}$. In
general, the fiber $\pi^{-1}(\mu)$ is the orbit of a certain fan in
$L_{\mu}^{k}$ under the action of permuting rows.

##  4\. Tropical Analogies

In this section, we describe the tropical analogues of the five equivalent
classical realizations of isotropic subspaces discussed in Section 2 following
section 3, and then we go on to discuss the analogies between them. The key
point to catch is that these five realizations are no longer tropically
equivalent in general.

###  4.1. Tropical Analogues of the Five Equivalent Characterizations of
Classical Isotropic Subspaces

We start by introducing the analogue of the tropical Grassmannian; the
_tropical symplectic Grassmannian_ which will be denoted by
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)$.

######  Definition 4.1.

The _tropical symplectic Grassmannian_
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(k,2n)$ is the tropicalization
of the symplectic Grassmannian $\textsc{Sp}\textsc{G}\mathsf{r}_{\mathbb{K}}$.
That is

$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(k,2n)=\\{w\in\mathbb{T}\mathbb{P}^{\binom{2n}{k}-1}\mid
w\in\operatorname{trop}(V(f))\quad\forall\,\,f\in S_{k,2n}(\mathbb{K})\\}.$

Again, the tropical symplectic Grassmannian depends on the characteristic $p$
of the field $\mathbb{K}$, but we omit the subscript when it is not relevant.

The following is a consequence of the Structure Theorem [MS15, Theorem 3.3.5]:

######  Corollary 4.2.

The tropical symplectic Grassmannian
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)$ is a pure polyhedral fan in
$\mathbb{T}\mathbb{P}^{\binom{2n}{k}-1}$ of dimension $\frac{k(4n-3k+1)}{2}$.

Below we provide explicit examples of the fan
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)$ for small $k$ and $n$ that
we compute using the _Tropical_ package (Améndola et al [AKL+17]) based on
_Gfan_ (Jensen [Jen08]) in _Macaulay2_ (Grayson and Stillman [GS97]). [We use
the same software for all other computational examples in this paper.]

######  Example 4.3 (k=2, n=3).

The ideal $S_{2,6}(\mathbb{K})$ of $\textsc{Sp}\textsc{G}\mathsf{r}(2,6)$ is
generated by the corresponding Plücker relations and the symplectic relation:
$\textsc{X}_{1,\overline{1}}+\textsc{X}_{2,\overline{2}}+\textsc{X}_{3,\overline{3}}$.
Here $\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,6)$ is a pure
$7$-dimensional fan in $\mathbb{T}\mathbb{P}^{14}$ with a lineality space of
dimension 3. Its f vector modulo the lineality space is $(28,180,420,315)$, so
it has 28 rays and 315 maximal cones.

######  Example 4.4 (k=3, n=3).

For $\textsc{Sp}\textsc{G}\mathsf{r}_{0}(3,6)$, the ideal
$S_{3,6}(\mathbb{C})$ is generated by the respective Plücker relations and the
following set of symplectic relations:

$\small\textsc{X}_{2,\overline{1},\overline{2}}+\textsc{X}_{3,\overline{1},\overline{3}},\,\,\textsc{X}_{1,3,\overline{1}}+\textsc{X}_{2,3,\overline{2}},\,\,-\textsc{X}_{1,\overline{1},\overline{2}}+\textsc{X}_{3,\overline{2},\overline{3}},$
$\textsc{X}_{1,\overline{1},\overline{3}}+\textsc{X}_{2,\overline{2},\overline{3}},\,\,\textsc{X}_{1,2,\overline{1}}+\textsc{X}_{2,3,\overline{3}},\,\,\textsc{X}_{1,2,\overline{1}}-\textsc{X}_{2,3,\overline{3}}.$

The fan $\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{0}(3,6)$ is pure of
dimension 6 in $\mathbb{T}\mathbb{P}^{19}$, with a 3-dimensional lineality
space. Its f vector modulo the lineality space is $(35,157,159)$.

We now define the symplectic counterpart of the usual Dressian
$\textsc{D}\mathsf{r}(k,2n)$; the _symplectic Dressian_ which we denote by
$\textsc{Sp}\textsc{D}\mathsf{r}(k,2n)$:

######  Definition 4.5.

The _symplectic Dressian_
$\textsc{Sp}\textsc{D}\mathsf{r}(k,2n)\supseteq\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)$
is the prevariety corresponding to the ideal $S_{k,2n}(\mathbb{K})$, i.e., the
set of valuated matroids (tropical Plücker vectors) $\mu$ satisfying:

$\forall\,\,S\in\binom{[2n]}{k-2},\text{ the minimum in
}\quad\bigoplus_{i}\mu_{Si\overline{i}}\quad\text{is achieved at least
twice.}$ (4.1)

We call the elements of $\textsc{Sp}\textsc{D}\mathsf{r}(k,2n)$, _tropical
symplectic Plücker vectors_.

We have seen the notion of a tropical linear space $L_{\mu}$ associated to a
valuated matroid $\mu\in\textsc{D}\mathsf{r}(k,2n)$ in Section 3. The
following definition is a tropical analogue of being isotropic, which was
first introduced by Rincón in [Rin12]. Under the scope of Coxeter matroids
(Borovik et al [BGW03]), Rincón [Rin12] studies valuated matroids
corresponding to type D, while here we are considering type C. However,
orthogonality and isotropicity in type D vs type C differ by just a sign,
which becomes irrelevant under tropicalization, so his definition remains the
relevant one for us:

######  Definition 4.6.

We say that two points $x,y\in\mathbb{T}^{2n}$ are _(tropically) orthogonal_
if the minimum is achieved at least twice in

$\bigoplus_{i\in[2n]}x_{i}\odot y_{\overline{i}}.$

A tropical linear space $L\subseteq\mathbb{T}^{2n}$ is _isotropic_ if all of
its points are pairwise _orthogonal_.

Recall the tropical Stiefel map $\pi$ from Definition 3.3. In the following
definition, we establish two more analogies for transversal valuated matroids:

######  Definition 4.7.

Let $\mu\in\textsc{D}\mathsf{r}(k,2n)$ and suppose $\mu=\pi(A|B)$, for
$A,B\in\mathbb{T}^{k\times n}$.

  1. (1)

If the rows of $(A|B)$ are (tropically) orthogonal, we say $\mu$ has a _row-
orthogonal_ presentation.

  2. (2)

If $A\odot B^{T}$ is symmetric, we say $\mu$ has a _symmetric_ presentation.

To summarize, we have the following tropical analogues of the five equivalent
characterizations of isotropic linear subspaces with respect to a symplectic
form:

  1. (1)

$\mu\in\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(k,2n)$.

  2. (2)

$\mu\in\textsc{Sp}\textsc{D}\mathsf{r}(k,2n)$.

  3. (3)

$L_{\mu}$ is isotropic.

  4. (4)

$\mu$ has a row-orthogonal presentation.

  5. (5)

$\mu$ has a symmetric presentation.

Since it turns out they do not agree in general, it is convenient to give a
name to each of the sets they define, respectively, the:

  1. (1)

_tropical symplectic Grassmannian._

  2. (2)

_symplectic Dressian._

  3. (3)

_isotropic region._

  4. (4)

_row-orthogonal Stiefel image._

  5. (5)

_symmetric Stiefel image._

###  4.2. Implications between the Five Tropical Analogues

We are now set to examine when does one of the analogues above imply another.

To begin with, we examine the cases when
$\textsc{Sp}\textsc{D}\mathsf{r}(k,2n)$ is equal to
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)$. It turns out that the two
fans are equal when $k=2$, as stated in the following theorem. We keep the
proof for Section 5, where we study the rank two case in detail.

######  Theorem 4.8.

The set of Plücker and symplectic relations forms a tropical basis for the
ideal $S_{2,2n}(\mathbb{K})$, i.e., the symplectic Dressian
$\textsc{Sp}\textsc{D}\mathsf{r}(2,2n)$ and the tropical symplectic
Grassmannian $\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(2,2n)$ coincide.
In particular, $\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(2,2n)$ does not
depend on $p$.

In A of the matroid zoo (Section 7), we describe an explicit point that is in
$\textsc{Sp}\textsc{D}\mathsf{r}(3,6)$ but not in
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(3,6)$, hence showing that these two
fans are not equal. This is in contrast with type ${\tt A}$, where
$\textsc{D}\mathsf{r}(3,6)=\textsc{T}\textsc{G}\mathsf{r}(3,6)$. With the help
of this example, we also find the missing relation to obtain a tropical basis
for $\textsc{Sp}\textsc{G}\mathsf{r}_{2}(3,6)$. The computational information
of the fan $\textsc{Sp}\textsc{D}\mathsf{r}(3,6)$ is shown below:

######  Example 4.9.

The symplectic Dressian $\textsc{Sp}\textsc{D}\mathsf{r}(3,6)$ is a non-pure
non-simplicial 7-dimensional polyhedral fan in $\mathbb{T}\mathbb{P}^{19}$
with f vector $\\{35,153,155,2\\}$. It has 149 maximal cones; of these, 2 are
of dimension 7 while 147 are of dimension 6. However, the rays of
$\textsc{Sp}\textsc{D}\mathsf{r}(3,6)$ are the rays of
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(3,6)$.

We will show in Theorem 4.16, that the non-equality of the two fans for
$k=n=3$ discussed above holds for all $k\geq 3,n\geq 3$.

We now go on to study connections between the tropical symplectic
Grassmannian, the symplectic Dressian and isotropic tropical linear spaces. We
first show that a tropical linear space corresponding to any valuated matroid
in $\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(k,n)$ is isotropic:

######  Proposition 4.10.

Let $\mu\in\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(k,2n)$ be a valuated
matroid in the tropical symplectic Grassmannian. Then $L_{\mu}$ is isotropic.

###### Proof.

Consider $\mathbb{K}$ an algebraically closed field of characteristic $p$ such
that $\operatorname{val}:\mathbb{K}\to\mathbb{T}$ is surjective, for example
the field of formal Hahn series (see for example [JS21]). Since
$\mu\in\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(k,2n)$, then there exists
an isotropic linear space $L\subseteq\mathbb{K}^{2n}$ realizing $\mu$, that
is, its Plücker vector X satisfies $\operatorname{val}(\textsc{X})=\mu$ (here
we are using that $\mathbb{K}$ has a surjective valuation). Since $L$ is given
by Equation 2.2 and $L_{\mu}$ is given by Equation 3.1, then by [MS15, Theorem
3.2.5 and Proposition 4.1.6], we have that $L_{\mu}=\operatorname{val}(L)$.
Since any two points $x,y\in L$ satisfy $\omega(x,y)=0$, we have that
$\operatorname{val}(x)$ and $\operatorname{val}(y)$ are orthogonal, so
$L_{\mu}$ is isotropic. ∎

The following is a result due to Rincón [Rin12]. It gives a sufficient
condition for a tropical linear space to be isotropic in the Lagrangian case.

######  Proposition 4.11 ([Rin12]).

Let $\mu\in\textsc{D}\mathsf{r}(n,2n)$ be a valuated matroid. Then $L_{\mu}$
is isotropic if and only if
$\mu_{\overline{\textsc{J}}}=\mu_{[2n]\setminus\textsc{J}}$ for all
$\textsc{J}\in\binom{[2n]}{n}$.

Notice that for $k=n\leq 3$, the relations
$\mu_{\overline{\textsc{J}}}=\mu_{[2n]\setminus\textsc{J}}$ are equivalent to
the symplectic relations (in other words, these relations are trivially true),
so we have the following result:

######  Proposition 4.12.

For $k=n\leq 3$, $L_{\mu}$ is isotropic if and only if
$\mu\in\textsc{Sp}\textsc{D}\mathsf{r}(n,2n)$.

######  Remark 4.13.

The equations from Proposition 4.11 are also known to cut out the Lagrangian
Grassmannian in the non-tropical case for any $n$. That is, a Plücker vector X
satisfies
$\textsc{X}_{\overline{\textsc{J}}}=\textsc{X}_{[2n]\setminus\textsc{J}}$ if
and only if $\textsc{X}\in\textsc{Sp}\textsc{G}\mathsf{r}(n,2n)$ [Kar18,
Theorem 5.16].

The proof of Proposition 4.11 is based on the observation that the dual linear
space $L_{\mu^{*}}$ (where
$\mu^{*}_{\textsc{J}}:=\mu_{[2n]\setminus\textsc{J}}$) consists of all points
that are orthogonal to $L_{\mu}$ with respect to the tropical dot product. So
being isotropic means that $L_{\overline{\mu}}\subseteq L_{\mu^{*}}$, where
$\overline{\mu}_{\textsc{J}}:=\mu_{\overline{\textsc{J}}}$. For the Lagrangian
case, $\dim(L_{\overline{\mu}})=\dim(L_{\mu^{*}})=n$ so they must be the same,
hence Proposition 4.11. However, for $k<n$, we can combine this approach with
the Proposition 3.2 to obtain a characterization of isotropic tropical linear
spaces in terms of the Plücker coordinates, although not as clean as
Proposition 4.11.

######  Proposition 4.14.

A tropical linear space $L_{\mu}$ is isotropic if and only if

$\forall\,\,S_{1},S_{2}\in\binom{[2n]}{k-1},\text{ the minimum in
}\bigoplus_{i}\mu_{S_{1}i}\odot\mu_{S_{2}\bar{i}}$

is achieved twice (if $i\in S_{1}$ or $\bar{i}\in S_{2}$ we assume
$\mu_{S_{1}i}=\infty$ or $\mu_{S_{2}\bar{i}}=\infty$ respectively).

Finally, we show that the symmetric Stiefel image is contained in the row-
orthogonal Stiefel image:

######  Proposition 4.15.

Let $A,B\in\mathbb{T}^{k\times n}$ such that $A\odot B^{T}$ is symmetric. Then
the rows of $(A|B)$ are (tropically) orthogonal.

###### Proof.

Since $A\odot B^{T}$ is a $k\times k$ symmetric matrix, we have that for any
$i,j$:

$\bigoplus_{l\in[n]}A_{il}\odot B_{jl}=\bigoplus_{l\in[n]}A_{jl}\odot B_{il}.$

This implies that rows $i$ and $j$ of $(A|B)$ are orthogonal, since

$\bigoplus_{l\in[2n]}(A|B)_{il}\odot(A|B)_{j\bar{l}}=\left(\bigoplus_{l\in[n]}A_{il}\odot
B_{jl}\right)\oplus\left(\bigoplus_{l\in[n]}B_{il}\odot A_{jl}\right).$

∎

The following is our main theorem. It combines the results from this section
and Sections 5, 6, 7 to give a full description of all the implications.

######  Theorem 4.16.

None of the statements (1)-(5) are equivalent in general. The following is a
complete list of the implications between them for each $k$ and $n$:

  * (a)

In general, $(1)\Rightarrow(2)$ and $(1)\Rightarrow(3)$.

  * (b)

For transversal valuated matroids, $(3)\Rightarrow(4)$ and
$(5)\Rightarrow(4)$.

  * (c)

For $k=1$, they are all equivalent.

  * (d)

For $k=2$, $(1)\Leftrightarrow(2)$.

  * (e)

For $k=n\leq 3$, $(2)\Leftrightarrow(3)$.

There is an example showing that any other implication fails (even when
restricting to transversal valuated matroids, for the cases involving (4) and
(5)).

Notice that set $(1)$, the tropical symplectic Grassmannian, is special in
that it depends on the characteristic $p$. However, Theorem 4.16 remains true
regardless of the characteristic chosen.

###### Proof.

The proof is summarized in Figure 1, the map of the matroid zoo.

  * (a)

$(1)\Rightarrow(2)$ is trivial and $(1)\Rightarrow(3)$ is Proposition 4.10.

  * (b)

For transversal valuated matroids, $(3)\Rightarrow(4)$ is immediate, since the
rows of $A$ are always points in $L_{\pi(A)}$ and $(5)\Rightarrow(4)$ is
Proposition 4.15.

  * (c)

For $k=1$ all spaces are equal to the usual Dressian
$\textsc{D}\mathsf{r}(1,2n)$.

  * (d)

This is Theorem 4.8.

  * (e)

This is Proposition 4.12.

Now to show that there are no other implications, we provide a complete list
of _minimal_ counterexamples in Section 7. We call them minimal in the
following sense: there is no example of rank $k^{\prime}\leq k$ on
$2n^{\prime}<2n$ elements with $n^{\prime}-k^{\prime}\leq n-k$ showing that
the same implication fails. The reason for this notion of minimality is that
we can use Lemma 6.1 to extend any counterexample to a counterexample with
more elements, without decreasing the rank nor the co-rank. So it is enough to
consider the following minimal counterexamples:

  * •

A shows that neither (2) nor (3) imply (1) for $k=n=3$.

  * •

Examples B and C show that (3) does not imply (2) for $k=2$, $n=3$ and
$k=n=4$, respectively. B is also a minimal example for (3) not implying (1).

  * •

D shows that none imply (4), for $k=n=2$.

  * •

E shows that neither (4) nor (5) imply (1), (2) or (3), for $k=n=2$.

  * •

Examples F and G show that (2) does not imply (3) or (4), for $k=3$, $n=4$ and
$k=n=4$, respectively.

∎

##  5\. Rank Two Case

In this section, we prove Theorem 4.8 and deduce from it enumerative
information for $\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$.

###  5.1. Coincidence of the Tropical Symplectic Grassmannian with the
Symplectic Dressian

Notice that the ideal $\textsc{I}_{2,2n}$ for
$\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$ is generated by the following
$\binom{2n}{4}$ Plücker relations

$\textsc{X}_{i,j}\textsc{X}_{k,l}-\textsc{X}_{i,k}\textsc{X}_{j,l}+\textsc{X}_{i,l}\textsc{X}_{j,k}=0\quad\text{for}\quad
1\leq i<j<k<l\leq\overline{n},$ (5.1)

and a single linear relation

$\textsc{X}_{1,\overline{1}}+\cdots+\textsc{X}_{n,\overline{n}}=0.$ (5.2)

### Proof of Theorem 4.8

We want to prove that
$\textsc{Sp}\textsc{D}\mathsf{r}(2,2n)=\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(2,2n)$.
We know that isotropic linear spaces satisfy a single linear relation (5.2)
when $k=2$.

Obviously,
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(2,2n)\subseteq\textsc{Sp}\textsc{D}\mathsf{r}(2,2n)$,
so let $\mu\in\textsc{Sp}\textsc{D}\mathsf{r}(2,2n)$. We have that the minimum
in $\mu_{1,\overline{1}}\oplus\dots\oplus\mu_{n,\overline{n}}$ is achieved
twice. By Theorem 3.1 we know that
$\mu\in\textsc{T}\textsc{G}\mathsf{r}(2,2n)$. Let $\mathbb{K}$ be an
algebraically closed field of characteristic $p$ and
$\mathbb{K}\\{\\!\\!\\{t\\}\\!\\!\\}$ be the field of Puiseux series with
coefficients in $\mathbb{K}$. Suppose $\mu$ has rational entries. Then there
exists a linear subspace
$L\subseteq(\mathbb{K}\\{\\!\\!\\{t\\}\\!\\!\\})^{2n}$ whose Plücker vector X
satisfies $\operatorname{val}(\textsc{X}_{I})=\mu_{I}$ for every
$I\in\binom{[2n]}{2}$.

It may be that $L$ is not isotropic, i.e., X does not satisfy Equation 5.2.
However, we have that the minimum of

$\bigoplus\limits_{i\in[n]}\operatorname{val}(\textsc{X}_{i,\overline{i}})$

is achieved twice. Let $i$ be such that
$\operatorname{val}(\textsc{X}_{i,\overline{i}})$ achieves the minimum and
assume it is not infinite (otherwise we would already have that $L$ is
isotropic). Consider a matrix
$A\subseteq\mathbb{K}\\{\\!\\!\\{t\\}\\!\\!\\}^{2\times 2n}$ such that its
rows form a basis of $L$.

Notice that multiplying column $i$ by a factor
$\alpha\in\mathbb{K}\\{\\!\\!\\{t\\}\\!\\!\\}$ with valuation $0$, changes
$\textsc{X}_{I}$ by a factor of $\alpha$ if $i\in I$, otherwise it remains
unchanged. However, since $\operatorname{val}(\alpha)=0$ we observe that
$\operatorname{val}(\textsc{X}_{I})$ remains unchanged anyway. We are going to
show that there exists such $\alpha$ such that the tropical linear space
$L^{\prime}$, generated by the rows of the matrix $A^{\prime}$ which is the
result of multiplying column $i$ of $A$ by $\alpha$, is isotropic. By the
discussion above, the valuation of the Plücker vector of $L^{\prime}$ is $\mu$
and this will show that $\mu$ is in the tropical symplectic Grassmannian.

Let
$a=\operatorname{val}(\textsc{X}_{1,\overline{1}}+\dots+\textsc{X}_{n,\overline{n}})$.
If $a=\infty$ we have already that $L$ is isotropic. So suppose $a$ is finite.
By the axioms of valuation,
$a\geq\operatorname{val}(\textsc{X}_{i,\overline{i}})$. If
$a=\operatorname{val}(\textsc{X}_{i,\overline{i}})$, consider

$b=\frac{\overline{-t^{-a}\sum\limits_{j\neq
i}^{n}\textsc{X}_{j,\overline{j}}}}{\overline{t^{-a}\textsc{X}_{i,\overline{i}}}}$

where $\overline{x}$ denotes the value of $x$ in the residue field (which in
this case is $\mathbb{K}$). Since the minimum is achieved twice for
$\operatorname{val}(\textsc{X}_{j,\overline{j}})$, without loss of generality
we can assume that $i$ is such that $b\neq 0$. We have now that

$\operatorname{val}(\textsc{X}_{1,\overline{1}}+\cdots+b\textsc{X}_{i,\overline{i}}+\cdots+\textsc{X}_{n,\overline{n}})>\operatorname{val}(\textsc{X}_{i,\overline{i}}).$

Let $m$ be least common denominator of all the exponents of $t$ in all entries
of $A$. We are going to construct $\alpha$ recursively as follows:

  1. (1)

Start with $\alpha_{1}=b$ where $b=1$ if
$a>\operatorname{val}(\textsc{X}_{i,\overline{i}})$ and let
$S_{1}=\textsc{X}_{1,\overline{1}}+\cdots+\alpha_{1}\textsc{X}_{i,\overline{i}}+\cdots+\textsc{X}_{n,\overline{n}}$.

  2. (2)

Let
$\lambda_{j}=\operatorname{val}(S_{j})-\operatorname{val}(\textsc{X}_{i,\overline{i}})$
which is a positive rational and let
$c_{j}=\overline{-t^{-\operatorname{val}(S_{j})}S_{j}}$.

  3. (3)

Define $\alpha_{j+1}=\alpha_{j}+t^{\lambda_{j}}c_{j}$ and
$S_{j+1}=\textsc{X}_{1,\overline{1}}+\cdots+\alpha_{j+1}\textsc{X}_{i,\overline{i}}+\cdots+\textsc{X}_{n,\overline{n}}$.

  4. (4)

Repeat the two previous steps infinitely (or until $S_{j}=0$).

  5. (5)

Let $\alpha$ be the resulting series, that is

$\alpha:=b+\sum_{j=1}^{\infty}c_{j}t^{\lambda_{j}}.$

We have that $\alpha$ is a Puiseux series since all $\lambda_{j}$ share the
denominator $m$. Furthermore $\operatorname{val}(\alpha)=0$. Notice that
$\operatorname{val}(S_{j+1})\geq\operatorname{val}(S_{j})+\dfrac{1}{m}$, so we
must have that
$\operatorname{val}(\textsc{X}_{1,\overline{1}}+\cdots+\alpha\textsc{X}_{i,\overline{i}}+\cdots+\textsc{X}_{n,\overline{n}})=\infty$
as we wanted. ∎

###  5.2. Coincidence with the Space of Phylogenetic Trees

We consider the space of phylogenetic trees with $2n$ labeled leaves (studied
by Billera et al [BHV01]) and its underlying simplicial complex which we
denote by $\mathbf{T}_{2n}$; first introduced by Buneman [Bun74] and later
studied by Robinson and Whitehouse [RW96] and Vogtmann [Vog90].

We prove the following proposition which relates
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$ to the simplicial complex
$\mathbf{T}_{2n}$.

######  Proposition 5.1.

The tropical symplectic Grassmannian
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$ is isomorphic as a fan to
$\mathbb{R}^{n}\times H_{n-2}\times\mathbf{T}_{2n}$ where $H_{n-2}$ denotes a
generic hyperplane in $\mathbb{T}\mathbb{P}^{n-1}$.

###### Proof.

Let $Y$ denote the tropical hyperplane given by
$\mu_{1,\overline{1}}\oplus\dots\oplus\mu_{n,\overline{n}}$, which is the
tropicalization of the linear relation (5.2). Recall that the lineality space
$L$ of $\textsc{T}\textsc{G}\mathsf{r}(2,2n)$ is the largest linear space
contained in every cone of $\textsc{T}\textsc{G}\mathsf{r}(2,2n)$. It is the
$(2n-1)$-dimensional vector space spanned by the vectors:

$u_{i}=\sum\limits_{B\ni i}e_{B}$

for every $i\in[2n]$ (the span of this vectors in $\mathbb{R}^{\binom{2n}{2}}$
contains the line $\mathbb{R}(1,\dots,1)$ which is collapsed in the tropical
projective plane).

Theorem 4.8 tells us that $\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$ is
the intersection of $\textsc{T}\textsc{G}\mathsf{r}(2,2n)$ with the tropical
hyperplane $Y$. The intersection $Y\cap L$ is a fan with a $n$-dimensional
lineality space $L^{\prime}$ spanned by the vectors:

$u_{i}-u_{\overline{i}}$

Every other cone of $L\cap Y$ is of the form
$L^{\prime}+\operatorname{cone}\\{u_{i}\mid i\in I\\}$ for any subset
$I\subseteq[n]$ of size at most $n-2$. So $L\cap Y$ is isomorphic as a fan to
$\mathbb{R}^{n}\times H_{n-2}$.

For any point $\mu\in\mathbb{T}\mathbb{P}^{{\binom{2n}{2}}-1}$, it is
straightforward to see that there is a point $\mu^{\prime}\in\mu+L$ where the
minimum in Equation 5.2 is achieved at every term. From this we can deduce
that $(\mu+L)\cap Y=\mu^{\prime}+(L\cap Y)$. One way to interpret the last
equation is that $Y$ intersects $\textsc{T}\textsc{G}\mathsf{r}(2,2n)$ through
its lineality space. As a fan, $\textsc{T}\textsc{G}\mathsf{r}(2,2n)$ is
isomorphic to $L\times\mathbf{T}_{2n}$ [SS04, Theorem 3.4]. So we conclude
that

$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)\cong(L\cap
Y)\times\mathbf{T}_{2n}\cong\mathbb{R}^{n}\times
H_{n-2}\times\mathbf{T}_{2n}.$

∎

We can now extract enumerative information of
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$:

######  Corollary 5.2.

The fan $\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$ has $2^{2n-1}-n-1$
rays and $\binom{n}{2}[(4n-5)!!]$ facets.

###### Proof.

The number of rays of $H_{n-2}$ is $n$ and the number of rays in
$\mathbf{T}_{2n}$ is $2^{2n-1}-2n-1$ [BHV01], so the number of rays of
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$ is
$n+2^{2n-1}-2n-1=2^{2n-1}-n-1$.

The number of facets of $H_{n-2}$ is $\binom{n}{2}$ and the number of facets
in $\mathbf{T}_{2n}$ is $(4n-5)!!=1\times 3\times 5\times\cdots\times(4n-5)$
[RW96, Vog90], so the number of facets of
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$ is $\binom{n}{2}[(4n-5)!!]$.
∎

One can turn any simplicial fan into a simplicial complex by quotienting out
the lineality space and intersecting with the sphere of radius 1. We say that
the homology of the simplicial fan is the simplicial homology of this complex.
The following corollary specifies some topological information of
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$; its homotopy type in
particular:

######  Corollary 5.3.

The Betti numbers for the simplicial complex corresponding to the fan
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$ are as follows: for $n\geq
4$ the only positive Betti numbers are $b_{0}=1$, $b_{n-3}=n-1$,
$b_{2n-4}=(2n-2)!$ and $b_{3n-7}=(n-1)(2n-2)!$; for $n=3$ they are $b_{0}=3$
and $b_{2}=72$ and for $n=2$ we only have $b_{0}=3$.

###### Proof.

The simplicial complex $\mathbf{T}_{2n}$ has the homotopy type of a bouquet of
$(2n-2)!$ spheres of dimension $2n-4$ [RW96, Vog90] while the complex
corresponding to $H_{n-2}$ has the homotopy type of a bouquet of $n-1$ spheres
of dimension $n-3$. Now we use Künneth formula for product of spaces [Kün23]
to compute the Betti numbers of the simplicial complex of
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(2,2n)$ as stated in the corollary.
∎

##  6\. Direct Sums

Let $\mu_{1}\in\textsc{D}\mathsf{r}(k_{1},n_{1})$ and
$\mu_{2}\in\textsc{D}\mathsf{r}(k_{2},n_{2})$ be two valuated matroids. The
_direct sum_ between them
$\mu_{1}\oplus\mu_{2}\in\textsc{D}\mathsf{r}(k_{1}+k_{2},n_{1}+n_{2})$ is
defined as

$(\mu_{1}\oplus\mu_{2})_{\textsc{J}_{1}\sqcup\textsc{J}_{2}}:={\mu_{1}}_{\textsc{J}_{1}}+{\mu_{2}}_{\textsc{J}_{2}}$

where $\textsc{J}_{1}\subset[n_{1}]$ and $\textsc{J}_{2}\subset[n_{2}]$.
Notice that this implies that $(\mu_{1}\oplus\mu_{2})_{\textsc{J}}=\infty$ if
$|\textsc{J}\cap n_{1}|\neq k_{1}$.

The main motivation behind this construction is that if $\mu_{1}$ and
$\mu_{2}$ are representable valuated matroids, represented by $L_{1}$ and
$L_{2}$, then $\mu_{1}\oplus\mu_{2}$ is representable and represented by
$L_{1}\times L_{2}$. It is straightforward to see that

$L_{\mu_{1}\oplus\mu_{2}}=L_{\mu_{1}}\times L_{\mu_{2}}$ (6.1)

and that if $\mu_{1}=\pi(A_{1})$ and $\mu_{2}=\pi(A_{2})$ then

$\mu_{1}\oplus\mu_{2}=\begin{pmatrix}\pi(A_{1})&\infty\\\
\infty&\pi(A_{2})\end{pmatrix}.$ (6.2)

Moreover, the converse is also true: every presentation of
$\mu_{1}\oplus\mu_{2}$ is of this form. This follows from Equation 6.1 and
[FO19, Theorem 6.6].

From the observations above we can prove the following statements which help
us to extend the counter-examples from Section 7 to counter-examples for all
suitable $k$ and $n$.

######  Lemma 6.1.

Let $X$ be any of the five sets considered in Theorem 4.16 for all $k$ and $n$
and let $\mu$ be a valuated matroid. Then the following are equivalent:

  1. (1)

$\mu\in X$,

  2. (2)

$\mu\oplus U_{0,2}\in X$,

  3. (3)

$\mu\oplus U_{1,2}\in X$,

where $U_{0,2}=(0)\in\textsc{T}\textsc{G}\mathsf{r}(0,2)$ and
$U_{1,2}=(0,0)\in\textsc{T}\textsc{G}\mathsf{r}(1,2)$ are the uniform
matroids.

###### Proof.

We are going to examine each set that we are considering one by one:

  * •

The tropical symplectic Grassmannian: If
$\mu\in\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)$ then there is an
isotropic linear space $L\subset\mathbb{K}^{2n}$ such that
$\mu=\operatorname{val}(L)$. Then the space
$L^{\prime}=L\times\\{(0,0)\\}\subset\mathbb{K}^{2n+2}$ is obviously isotropic
(when pairing the last two coordinates) and represents $L\oplus U_{0,2}$. Any
representation of $L\oplus U_{0,2}$ is of this form so the other direction
holds.

Similarly, $L^{\prime\prime}=L\times\\{(x,x)\mid x\in\mathbb{K}\\}$ is a
$(k+1)$-dimensional vector space representing $\mu\oplus U_{1,2}$ which is
isotropic if and only if $L$ is isotropic.

  * •

The symplectic Dressian: Since $(\mu\oplus U_{0,2})_{\textsc{J}}=\infty$ if
$\textsc{J}\cap\\{2n+1,2n+2\\}\neq\emptyset$, the only non-trivial symplectic
relations occur when $S$ in Equation 4.1 does not contain $2n+1$ or $2n+2$ and
they are equivalent to the symplectic relations on $\mu$.

For $\mu\oplus U_{1,2}$, notice that $(\mu\oplus U_{1,2})_{\textsc{J}}=\infty$
unless $|\textsc{J}\cap\\{2n+1,2n+2\\}|=1$. So again we have that the only
non-trivial symplectic relation occurs when $|S\cap\\{2n+1,2n+2\\}|=1$ and
they are again equivalent to those for $\mu$ (since $(\mu\oplus
U_{1,2})_{\textsc{J}}=\mu_{\textsc{J}\setminus k}$ for $k\in\\{2n+1,2n+2\\}$.

  * •

The isotropic region: By Equation 6.1, $L_{\mu\oplus
U_{0,2}}=L_{\mu}\times\\{(\infty,\infty)\\}$ and $L_{\mu\oplus
U_{1,2}}=L_{\mu}\times\\{(x,x)\mid x\in\mathbb{T}\\}$, which are clearly
isotropic if and only if $L_{\mu}$ is isotropic.

  * •

Row-orthogonal Stiefel image: Suppose $\mu$ is a transversal valuated matroid.
We have that $\mu=\pi(A)$ for a tropical matrix $A\in\mathbb{K}^{k\times 2n}$
if and only if the matrix $A^{\prime}$ which is obtained by adding two columns
to $A$ with all entries equal to $\infty$ satisfies $\pi(A^{\prime})=\mu\oplus
U_{0,2}$. Moreover, all presentations of $\mu\oplus U_{0,2}$ are of this form.

Similarly, consider the matrix $A^{\prime\prime}$ that consists of adding a
row to $A^{\prime}$ whose first $2n$ entries are equal to $\infty$ and the
last two are equal to $0$. We have that
$\pi(A)=\mu\Leftrightarrow\pi(A^{\prime\prime})=\mu\oplus U_{1,2}$ and all
presentations are of this form.

Now it is clear that the rows of $A$ are orthogonal if and only if the rows of
$A^{\prime}$ are orthogonal and if and only if the rows of $A^{\prime\prime}$
are orthogonal.

  * •

Symmetric Stiefel image: Let us write a presentation of $\mu$ as $(A|B)$ and
the corresponding presentations of $\mu\oplus U_{0,2}$ and $\mu\oplus U_{1,2}$
as $(A^{\prime}|B^{\prime})$ and $(A^{\prime\prime}|B^{\prime\prime})$, so for
example

$A^{\prime}=\begin{pmatrix}A&\infty\end{pmatrix}\in\mathbb{T}^{k\times(n+1)}$

and

$A^{\prime\prime}=\begin{pmatrix}A&\infty\\\
\infty&0\end{pmatrix}\in\mathbb{T}^{(k+1)\times(n+1)}.$

Then $A^{\prime}\odot B^{\prime T}=A\odot B$ and

$A^{\prime\prime}\odot B^{\prime\prime T}=\begin{pmatrix}A\odot
B^{T}&\infty\\\ \infty&0\end{pmatrix}\in\mathbb{T}^{(k+1)\times(k+1)},$

from which the desired statement follows.

∎

##  7\. The Matroid Zoo

Welcome to the matroid zoo, a section devoted to show examples that exhibit
different pathologies. Examples A through G are used in the proof of Theorem
4.16. Additional highlights of the zoo include:

  * •

A shows that the tropical symplectic Grassmannian depends on the
characteristic.

  * •

B shows that neither being in the tropical symplectic Grassmannian nor in the
Dressian is closed under subspaces (as opposed to the classical symplectic
Grassmannian).

  * •

H shows a non realizable flag with symplectically realizable components.

  * •

I shows that (non-realizable) symplectic matroids do not always satisfy the
tropical symplectic relations.

###  7.1. In the Symplectic Dressian but not in the Tropical Symplectic
Grassmannian

######  Example A.

Let $M_{K_{4}}$ be the matroid whose affine representation is depicted in
Figure 2.

Figure 2. The matroid $M_{K_{4}}$

We call it $M_{K_{4}}$ because it is the graphical matroid that corresponds to
the complete graph $K_{4}$ with edges $i$ and $\overline{i}$ opposite to each
other. It is easy to see that this matroid satisfies all the symplectic
relations, since all of its non-bases are admissible sets. However, notice
that the polynomial

$\textsc{X}_{123}\textsc{X}_{\bar{3}\bar{2}\bar{1}}-\textsc{X}_{12\bar{3}}\textsc{X}_{3\bar{2}\bar{1}}-\textsc{X}_{13\bar{2}}\textsc{X}_{2\bar{3}\bar{1}}+\textsc{X}_{1\bar{3}\bar{2}}\textsc{X}_{23\bar{1}}+2\textsc{X}_{12\bar{2}}\textsc{X}_{3\bar{3}\bar{1}}$

is in the ideal $S_{3,6}(\mathbb{K})$. However, unless the characteristic is
2, the tropicalization of the polynomial attains the minimum only in
$x_{12\bar{2}}\odot x_{3\bar{3}\bar{1}}$, so
$M_{K_{4}}\notin\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(3,6)$ for $p\neq
2$. However, $M_{K_{4}}\in\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(3,6)$,
so already with 6 elements the tropical symplectic Grassmannian depends on the
characteristic. This is remarkable, since for type ${\tt A}$ the smallest non
realizable matroid has 7 elements (the Fano and non-Fano matroids) and for
type ${\tt D}$ it is known that the smallest non-realizable tropical Wick
vector has at least 12 elements and at most 14 elements.

We checked computationally that for
$\textsc{Sp}\textsc{G}\mathsf{r}_{2}(3,6)$, the Plücker relations, the
symplectic relations and the relation

$\textsc{X}_{123}\textsc{X}_{\bar{3}\bar{2}\bar{1}}-\textsc{X}_{12\bar{3}}\textsc{X}_{3\bar{2}\bar{1}}-\textsc{X}_{13\bar{2}}\textsc{X}_{2\bar{3}\bar{1}}+\textsc{X}_{1\bar{3}\bar{2}}\textsc{X}_{23\bar{1}}$

form a tropical basis. The relation above is interesting as it consists of
only admissible sets. Perhaps this can shine a light into the relation to
symplectic matroids (see Section 9.1).

###  7.2. Isotropic but not in the Symplectic Dressian.

######  Example B.

Consider the matroid $M$ with $2n=6$ elements and rank $2$ with parallelism
classes given by $1\bar{1}$, $2\bar{2}$, $3$ and $\bar{3}$ (see Figure 3).
This matroid clearly does not satisfy the only symplectic relation
$x_{1\bar{1}}\oplus x_{2\bar{2}}\oplus x_{3\bar{3}}$. However, the Bergman fan
of $L_{M}$ consists of four rays: $\operatorname{cone}(e_{1\bar{1}})$,
$\operatorname{cone}(e_{2\bar{2}})$, $\operatorname{cone}(e_{3})$ and
$\operatorname{cone}(e_{\bar{3}})$. It is easy to see that any two points on
these 4 rays are orthogonal to each other, since they all have at least 4
coordinates achieving the minimum, so this space is isotropic.

Figure 3. The matroid in B

Moreover, notice that $L_{M}$ is a subspace of the Bergman fan of the uniform
matroid $U_{3,6}$, which does not lie in the tropical symplectic Grassmannian.
So this example also shows that being a subspace of a tropical linear space in
the tropical symplectic Grassmannian (or symplectic Dressian) does not
guarantee being in tropical symplectic Grassmannian (respectively symplectic
Dressian). This is somewhat anti-intuitive, since any subspace of a linear
space in the classical symplectic Grassmannian is obviously also in the
symplectic Grassmannian (of the corresponding rank).

One could think this would extend over to the tropical case, by taking any
isotropic linear space $H$ that realizes $U_{3,6}$ and intersecting it with
the pre-image under $\operatorname{val}$ of $L_{M}$. However, any linear
subspace of rank two of $H$ that has points mapped to
$\operatorname{cone}(e_{1\bar{1}})$ and $\operatorname{cone}(e_{2\bar{2}})$
via $\operatorname{val}$, must also have points mapped to
$\operatorname{cone}(e_{3\bar{3}})$ (this follows from $H$ being isotropic).
So the tropicalization of any such subspace would have to be $L_{U_{1,2}\oplus
U_{1,2}\oplus U_{1,2}}$.

The previous example shows that if $k<n$, then there are tropical isotropic
linear spaces that are not in the symplectic Dressian. However, for $k=n\leq
3$ a tropical linear space is isotropic if and only if it is in the Dressian.
The remaining cases are $k=n\geq 4$. The following are the tropical isotropic
relations that are not tropical symplectic relations for $k=n=4$:

  1. (1)

$x_{12\bar{1}\bar{2}}\oplus x_{34\bar{3}\bar{4}}$,

  2. (2)

$x_{13\bar{1}\bar{3}}\oplus x_{24\bar{2}\bar{4}}$,

  3. (3)

$x_{14\bar{1}\bar{4}}\oplus x_{23\bar{2}\bar{3}}$,

while the following are the tropical symplectic relations which are not
isotropic:

  1. (1)

$x_{12\bar{1}\bar{2}}\oplus x_{13\bar{1}\bar{3}}\oplus x_{14\bar{1}\bar{4}}$,

  2. (2)

$x_{12\bar{1}\bar{2}}\oplus x_{23\bar{2}\bar{3}}\oplus x_{24\bar{2}\bar{4}}$,

  3. (3)

$x_{13\bar{1}\bar{3}}\oplus x_{23\bar{2}\bar{3}}\oplus x_{34\bar{3}\bar{4}}$,

  4. (4)

$x_{14\bar{1}\bar{4}}\oplus x_{24\bar{2}\bar{4}}\oplus x_{34\bar{3}\bar{4}}$.

######  Example C.

Consider the matrix

$A=\begin{pmatrix}1&1&1&1&1&1&1&1\\\ 0&1&1&0&0&1&1&0\\\ 0&0&2&2&0&0&2&2\\\
0&0&0&0&1&1&1&1\end{pmatrix},$

and let $M$ be the matroid given by the columns of $A$ as vectors (see Figure
4). We have that $13\bar{1}\bar{3}$ and $24\bar{2}\bar{4}$ are bases while
$12\bar{1}\bar{2},23\bar{2}\bar{3},34\bar{3}\bar{4}$ and $14\bar{1}\bar{4}$
are not. So $M$ does not satisfy any of the four symplectic relations listed
above, but it does satisfy all isotropic relations, (to see that it satisfies
the relations that are both isotropic and symplectic, notice all sets of the
form $ijk\overline{i}$ are bases).

Figure 4. Affine representation of C.

###  7.3. With Row-orthogonal Presentation but without Symmetric Presentation

######  Example D.

Consider the the matrix $(A|B)=\left(\begin{smallmatrix}0&0&1&1\\\
0&0&0&0\end{smallmatrix}\right)$ and the valuated matroid $\mu=\pi(A|B)$. The
rows of $(A|B)$ are orthogonal, however, $A\odot
B^{T}=\left(\begin{smallmatrix}1&0\\\ 1&0\end{smallmatrix}\right)$ is not
symmetric. Moreover, every presentation $(A^{\prime}|B^{\prime})$ of $\mu$ is
the result of replacing one of the 1’s in $(A|B)$ by some $x\geq 1$ and
replacing either $A_{2,1}$ or $A_{2,2}$ by some $y\geq 0$ [FO19, Example 3.3].
So in any case $A^{\prime}\odot B^{\prime T}=\left(\begin{smallmatrix}1&0\\\
1&0\end{smallmatrix}\right)$.

###  7.4. With Symmetric Presentation but not Isotropic

######  Example E.

Consider the the matrix $(A|B)=\left(\begin{smallmatrix}0&1&0&1\\\
0&0&0&0\end{smallmatrix}\right)$ and the valuated matroid $\mu=\pi(A|B)$. We
have that $A\odot B^{T}=\left(\begin{smallmatrix}0&0\\\
0&0\end{smallmatrix}\right)$ is symmetric. However, $\mu_{1\overline{1}}=0$
and $\mu_{2\overline{2}}=1$ so it does not satisfy the isotropic relations
from Proposition 4.11.

###  7.5. In the Symplectic Dressian but without Row-orthogonal Presentation

Notice that by Proposition 4.12, the smallest example of a transversal
valuated matroid in the symplectic Dressian that does not have a presentation
with orthogonal rows must have at least 8 elements. To completely deal with
all other cases, we need an example of rank 3 and another of rank 4.

######  Example F.

Consider the rank 3 matroid $M$ whose affine representation is given by Figure
5. We have that $M$ is transversal (by [BD72, Theorem 4.6], for example) and,
by [FO19, Proposition 4.3], any valuated presentation $A\in\pi^{-1}(M)$ of $M$
must have a row equal to $R_{1}=(0,\infty,\infty,0,0,\infty,\infty,0)$ (modulo
tropical scalar multiplication, i.e., adding a constant) and another row equal
to $R_{2}=(0,0,0,\infty,\infty,0,0,\infty)$, which are not orthogonal to each
other.

Figure 5. The matroid in F

######  Example G.

Now consider the rank 4 matroid $M$, whose affine representation is almost as
Figure 5, but with $23\bar{2}\bar{3}$ being co-planar instead of co-linear. In
other words, $M$ is the matroid whose only cyclic flats are $23\bar{2}\bar{3}$
of rank 3 and $4\bar{1}\bar{4}$ of rank two. Again, this matroid is
transversal and, by [FO19, Proposition 4.3], any valuated presentation
$A\in\pi^{-1}(M)$ of $M$ must have a row equal to $R_{1}$. However, now there
are two rows which should have $\infty$ entries at $4,\bar{1}$ and $\bar{4}$,
but are not necessarily equal to $R_{2}$. One of them could be equal to
$(\infty,0,0,\infty,\infty,0,0,\infty)$ and be orthogonal with $R_{1}$, but
not both at the same time (other wise it forces $1$ to be in the
$4\bar{1}\bar{4}$ line). So there is always a row which is not orthogonal to
$R_{1}$.

###  7.6. Non-realizable Flag with Symplectically Realizable Components

######  Example H.

Consider the non-Pappus matroid $P$ depicted in Figure 6.

Figure 6. The non-Pappus matroid.

Let $M_{1}=P\backslash x$ and $M_{2}=P/x$. The non realizability of the non-
Pappus matroid results in the non-realizability of the flag
$\operatorname{Be}(M_{1})\subseteq\operatorname{Be}(M_{2})$. This is Example 7
from [BGW03], but we have added a labeling i.e., a pairing of the points so
that $M_{1}$ and $M_{2}$ are symplectically realizable. We checked
computationally that
$M_{2}\in\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{0}(3,8)$ while
$M_{1}\in\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{0}(2,8)$ by Theorem 4.8.

###  7.7. Symplectic Matroid not in the Symplectic Dressian

######  Example I.

Consider the symplectic matroid depicted in Figure 7 (the definition of
symplectic matroid can be found below in Section 9). Explicitly, we have the
symplectic matroid with non bases $\\{12,1\bar{2},\bar{1}3,\bar{1}\bar{3}\\}$.
This example appeared in [BGW03, §3.4.3] as an example of a non-realizable
symplectic matroid.

Figure 7. The polytope of a non-realizable symplectic matroid that does not
satisfy the symplectic relations.

If we were to complete this symplectic matroid to a usual matroid (i.e.,
determining its bases for non-admissible sets), we would necessarily have that
$13\bar{3}$ and $\bar{1}2\bar{2}$ are the two parallel classes. But then the
symplectic relation $x_{1\bar{1}}\oplus x_{2\bar{2}}\oplus x_{3\bar{3}}$
achieves its minimum once, so such matroid is not in the symplectic Dressian.

##  8\. Conormal Bundles

Recall that the dual of a valuated matroid $\mu$ on $[n]$ of rank $k$ is a
valuated matroid $\mu^{*}$ on $[n]$ of rank $n-k$ where
$\mu^{*}_{B}=\mu_{[n]\backslash B}$.

In [ADH20], Ardila, Denham and Huh studied the _conormal fan_ of a matroid,
$\Sigma_{M,M^{*}}$, whose support equals $L_{M\oplus M^{*}}$. The latter is a
tropical linear space of dimension $n$ on $[2n]$. The conormal fan serves as a
tropical analogue of conic Lagrangian subvarieties. Therefore, a natural
question is: where does this kind of tropical linear space fit in our picture
(that is, in Figure 1)?

We answer this question for a more general class of tropical linear spaces,
namely for the following natural generalization to valuated matroids:

######  Definition 8.1.

Let $\mu$ be a valuated matroid. The _conormal bundle_ of $\mu$ is the
tropical linear space $L_{\mu\oplus\mu^{*}}$.

We remark that this is a generalization of conormal fans only in the same way
as tropical linear spaces generalize Bergman fans: for unvaluated matroids the
conormal bundle coincides with the support of the conormal fan, but the
conormal fan may have a finer subdivision. As the name suggests, for valuated
matroids the conormal bundle may not be a fan.

######  Theorem 8.2.

Let $\mu$ be a valuated matroid. Then:

  1. (1)

if $\mu$ is realizable over a field $\mathbb{K}$ of characteristic $p$, then
$\mu\oplus\mu^{*}\in\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(n,2n)$.

  2. (2)

for any $\mu$, $\mu\oplus\mu^{*}\in\textsc{Sp}\textsc{D}\mathsf{r}(n,2n)$,
i.e., $\mu\oplus\mu^{*}$ satisfies the tropical symplectic relations.

  3. (3)

the conormal bundle $L_{\mu\oplus\mu^{*}}$ is isotropic.

  4. (4)

if $\mu$ and $\mu^{*}$ are transversal, $\mu\oplus\mu^{*}$ has a symmetric
presentation if and only if $\mu$ has a single basis.

###### Proof.

For the first statement, if $\mu=\operatorname{trop}(V)$ for a linear subspace
$V\subseteq\mathbb{K}^{n}$, then the orthogonal complement $V^{\perp}$
satisfies that $\operatorname{trop}(V^{\perp})=\mu^{*}$ and $V\times
V^{\perp}\subseteq\mathbb{K}^{2n}$ is isotropic with respect to the canonical
symplectic form given by Equation 2.3. So
$\mu\oplus\mu^{*}=\operatorname{trop}(V\times
V^{\perp})\in\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}_{p}(n,2n)$.

Next, let $\mu$ be any valuated matroid (not necessarily realizable). Notice
that the tropical symplectic relations for $\mu\oplus\mu^{*}$ are exactly the
same as the tropical Plücker relations for $\mu$:

$\bigoplus_{i}(\mu\oplus\mu^{*})_{Si\bar{i}}=\bigoplus_{i}\mu_{(S\cap[n])\cup
i}\odot\mu^{*}_{S\backslash[n]\cup\bar{i}}=\bigoplus_{i}\mu_{S^{\prime}i}\odot\mu_{T\backslash
i}$

where $S^{\prime}=S\cap[n]$ and $T=[n]\backslash\bar{S}$.

For the third statement, we use Proposition 4.11. Let $J=B_{1}\sqcup
B_{2}\in\binom{2n}{n}$ where $B_{1}=J\cap[n]$. Then

$(\mu\oplus\mu^{*})_{[2n]\backslash J}=\mu_{[n]\backslash
B_{1}}\odot\mu^{*}_{[n]\backslash
B_{2}}=\mu_{B_{2}}\odot\mu^{*}_{B_{1}}=(\mu\oplus\mu^{*})_{\bar{J}}.$

Finally, to relate conormal bundles with the Stiefel image, we need to work
under the assumption that both $\mu$ and $\mu^{*}$ are transversal
(transversality is closed under direct sums but not under duality). If such is
the case, then by part (2) we have that $\mu\oplus\mu^{*}$ has row-orthogonal
presentation.

Now, if $\mu$ has a single basis, it means it is the direct sum of loops and
coloops (an element $i$ is a loop of $\mu$ if it is not in any basis and it is
a coloop if it is a loop of $\mu^{*}$). Therefore, $\mu^{*}$ and
$\mu\oplus\mu^{*}$ are also of this form. Then, if column $i$ of a
presentation $(A|B)$ of $\mu\oplus\mu^{*}$ has a finite entry, the column
$\bar{i}$ must have all entries equal to $\infty$. From this it follows that
$A\odot B^{T}$ has all infinity entries for a presentation $(A|B)$ and in
particular it is symmetric.

If $\mu$ is not of this form, then there is an element $x\in[n]$ which is not
a loop in $\mu$ or in $\mu^{*}$. Therefore, there are rows $i$ and $j$ in
$(A|B)$ with a finite entry at $x$ and $\bar{x}$ respectively. This implies
that the $(i,j)$-entry of $A\odot B^{T}$ is finite. However, the $(j,i)$ entry
of $A\odot B^{T}$ is given by the tropical dot product of two all $\infty$
vectors, so it is equal to $\infty$. Therefore $A\odot B^{T}$ is not
symmetric. ∎

##  9\. Future Bridges

In this section, we discuss several notions including the tropical symplectic
flag variety, symplectic flag Dressian, flag Stiefel image, degenerations of
symplectic flag varieties and total positivity and pose several questions
about them.

###  9.1. Symplectic Matroids

We follow Borovik et al [BGW98, BGW03] and we refer the reader there for
details on symplectic matroids.

We begin by recalling that we call a subset $\textsc{J}\subset[2n]$
_admissible_ if for every $i\in\textsc{J}$, $\bar{i}\notin\textsc{J}$. Let
$e_{\bar{i}}:=-e_{i}$ and for an admissible set $\textsc{J}\subseteq[2n]$
consider
$e_{\textsc{J}}:=\sum\limits_{i\in\textsc{J}}e_{i}\in\mathbb{R}^{[n]}$.

######  Definition 9.1.

A _symplectic matroid_ is a collection $\mathcal{B}\subseteq\binom{[2n]}{k}$
of admissible sets such that every edge of the polytope
$\operatorname{conv}\\{e_{\textsc{J}}\mid\textsc{J}\in\mathcal{B}\\}$ is
either a translation of $e_{i}-e_{j}$ for some $i,j$ or of $2e_{i}$ for some
$i$.

The main motivation behind this definition is the following:

######  Theorem 9.2 ([BGW98]).

Let $L\subseteq\mathbb{K}^{[2n]}$ be an isotropic linear space. Then the
collection of admissible bases of $L$ is a symplectic matroid.

A symplectic matroid constructed as above is called _representable_. From
Theorem 9.2, it follows that any realizable symplectic matroid can be
completed to a matroid which is in the tropical symplectic Grassmannian (as
opposed to I).

One possible direction would be to find axioms for which one can enhance a
symplectic matroid with a valuation on the tropical numbers, which includes
every vector in the tropical symplectic Grassmannian
$\textsc{T}\textsc{Sp}\textsc{G}\mathsf{r}(k,2n)$ and such that symplectic
matroids are exactly the valuations that take values in $\\{0,\infty\\}$.
However, even the $\\{0,\infty\\}$ symplectic matroids do not always satisfy
the tropical symplectic relations (see Section 7 above), so this would not be
contained in the symplectic Dressian $\textsc{Sp}\textsc{D}\mathsf{r}(k,2n)$.

In general, the connection of symplectic matroids with the tropical symplectic
Grassmannian remains largely not understood. One reason for this is that the
symplectic relations (Equation 4.1) are not in terms of variables
corresponding to admissible sets. However, there are equations in the ideal
generated by them which are purely in terms of variables corresponding to
admissible sets (at least in characteristic 2, see A). Finding more of these
relations and studying their combinatorics might lead to a better
understanding of this connection.

###  9.2. Characterization in terms of Matroid Subdivisions

It would be desirable to obtain a characterization of tropical symplectic
Plücker vectors in terms of polyhedral subdivisions, similar to Speyer’s
theorem for type A [Spe08, Proposition 2.2], generalized by Rincón for any
matroid polytope in type A as well as type D [Rin12, Theorem 5.14] and in
[BEZ20, Theorem A] to tropical flags.

However, such a characterization is, in principle, not possible, since
satisfying the tropical symplectic relations is not closed under translations.
This implies that being a tropical symplectic Plücker vector can not be
determined from the matroid subdivision it induces. But one could still ask
which matroid subdivisions are induced by vectors from
$\textsc{Sp}\textsc{D}\mathsf{r}(k,2n)$.

###  9.3. The Tropical Symplectic Flag Variety and Symplectic Flag Dressian

The complete symplectic flag variety which will be denoted by
$\textsc{SpF}_{2n}$, is the set of all full flags
$\\{\textsc{U}_{1}\subset\cdots\subset\textsc{U}_{n},\,\,\dim\,\,\textsc{U}_{k}=k\\}$,
with the extra condition that each $\textsc{U}_{k}$ is isotropic with respect
to the symplectic form $\omega$. We consider the Plücker embedding of
$\textsc{SpF}_{2n}$ into the product of projective spaces
$\mathbb{P}\Big{(}\bigwedge^{k}\mathbb{C}^{2n}\Big{)}$, for $1\leq k\leq n$.
It is an irreducible projective variety of dimension $n^{2}$. With respect to
this embedding, its defining ideal which we denote by $S_{2n}$, is seen to be
generated by the corresponding Plücker and symplectic relations following the
work of De Concini [Dec79]. See also [Mak20] and [Bal20] for a further
description and some examples of this ideal. Let $G_{2n}$ denote the set of
the generators of the ideal $S_{2n}$.

Let $\textsc{TF}_{n}$ denote the usual tropical flag variety, i.e., the
tropicalization of the (type An-1) complete flag variety with respect to the
Plücker embedding. Let $N=\sum_{i=1}^{n}\binom{2n}{i}$ denote the number of
Plücker coordinates on $\textsc{SpF}_{2n}$ with respect to the Plücker
embedding as discussed above. We introduce the symplectic counterpart of
$\textsc{TF}_{n}$; the _tropical symplectic flag variety_ , which will be
denoted by $\textsc{T}\textsc{SpF}_{2n}$.

######  Definition 9.3.

The _tropical symplectic flag variety_ $\textsc{T}\textsc{SpF}_{2n}$, is the
tropicalization of the symplectic flag variety $\textsc{SpF}_{2n}$. That is

$\textsc{T}\textsc{SpF}_{2n}=\\{\mu\in\mathbb{T}\mathbb{P}^{N-1}\mid\mu\in\operatorname{trop}(V(f))\quad\forall\,\,f\in
S_{2n}\\}.$

######  Example 9.4 (n=2).

The tropical symplectic flag variety $\textsc{T}\textsc{SpF}_{4}$ is a pure
4-dimensional fan in $\mathbb{R}^{8}$ with a 2-dimensional lineality space. It
has 10 rays and 15 maximal cones.

In the following, we define the _symplectic flag Dressian_ which we denote by
$\textsc{SpF}\textsc{D}\mathsf{r}_{2n}$. It is the analogue of the usual flag
Dressian; introduced and studied by Brandt, Eur and Zhang [BEZ20].

######  Definition 9.5.

The _symplectic flag Dressian_ $\textsc{SpF}\textsc{D}\mathsf{r}_{2n}$ is
given by:

$\textsc{SpF}\textsc{D}\mathsf{r}_{2n}=\\{\mu\in\mathbb{T}\mathbb{P}^{N-1}\mid\mu\in\operatorname{trop}(V(f))\quad\forall\,\,f\in
G_{2n}\\}.$

As it would be expected, the tropical symplectic flag variety
$\textsc{T}\textsc{SpF}_{2n}$ is not equal to
$\textsc{SpF}\textsc{D}\mathsf{r}_{2n}$ in general. However, the two fans
coincide computationally for $n=2$. It is important to note that the issue of
representability of symplectic flags is subtle, in that there are flags with
symplectically representable components but the flags themselves are not
symplectically representable. Such a scenario is illustrated in Example H of
the matroid zoo.

As in the classical setting, we can also obtain a flag of tropical linear
spaces out of an $n\times n$ matrix, by considering all minors (not
necessarily maximal) whose rows are indexed by $[k]$ for some $k\leq n$.

######  Definition 9.6.

The _tropical flag Stiefel map_ for type ${\tt A}$,

$\pi_{\mathcal{F}}^{\tt A}:\mathbb{T}^{n\times
n}\dashrightarrow\textsc{TF}_{n},$

sends a matrix to the vector where for every non-empty proper subset
$S\subset[n]$, $\pi_{\mathcal{F}}(A)_{S}$ is the tropical determinant of the
submatrix of $A$ with columns indexed by $S$ and the first $|S|$ rows.

Similarly, we have a _tropical flag Stiefel map_ for type ${\tt C}$,

$\pi_{\mathcal{F}}^{\tt C}:\mathbb{T}^{n\times
2n}\dashrightarrow\textsc{T}\textsc{SpF}_{2n},$

where we consider the minors indexed by non-empty subsets $S\subset[2n]$ of
size at most $n$.

The image under the above maps we call _flag Stiefel image_.

###  9.4. Degenerations of Symplectic Grassmann/Flag Varieties and Tropical
Geometry

A cone of a tropical variety is said to be prime if the corresponding initial
ideal is a prime ideal. Prime maximal cones of tropicalizations of algebraic
varieties give rise to a nice class of toric degenerations, namely, those that
are irreducible. According to [SS04], all maximal cones of
$\textsc{T}\textsc{G}\mathsf{r}(2,n)$ are prime, but not all those of
$\textsc{T}\textsc{G}\mathsf{r}(3,6)$ are. On the other hand, Bossinger,
Lambogila, Mincheva, and Mohammadi in [BLMM17], computed toric degenerations
of the type A flag variety corresponding to both prime and non-prime maximal
cones in the tropicalization. So, it is natural to ask which of the maximal
cones of the tropical symplectic Grassmann or flag variety are prime cones.

Now, an explicit computation of tropicalizations of flag varieties is very
cumbersome, in that, it is only possible for small $n$ (see again [BLMM17]).
However, a full facet description of a maximal prime cone with nice features
in the type A tropical flag variety has been given in [FFFM19], and
connections made to various degenerations of flag varieties. For example, some
of its facets correspond to linear degenerations [CFF+19].

We point out that such a description can be explained through the tropical
flag Stiefel map discussed above. Consider the cone
$\mathcal{A}\subseteq\mathbb{T}^{n\times n}$ of upper triangular matrices $A$
such that:

  * •

$A_{i,i}=0$,

  * •

$A_{i,j}=\infty,\qquad\qquad\qquad\qquad\qquad\quad$ for $j<i$,

  * •

$A_{i,j}+A_{i+1,j+1}\geq A_{i+1,j}+A_{i,j+1},\quad\,$ for $1\leq i<j\leq n-2$.

The original description of the cone in [FFFM19] was stated by setting the
tropical Plücker coordinates as

$\mu_{B}=\sum_{i=1}^{l}A_{p_{i},q_{i}}$

where $p_{1}>\dots>p_{l}$ are the elements of $B\setminus[k]$,
$q_{1}<\dots<q_{l}$ are the elements of $[k]\setminus B$ and $k=|B|$. However,
we notice that this corresponds precisely to the minimal term in the tropical
determinant of the submatrix with rows $[k]$ and columns $B$ got by adding the
0-terms $A_{i,i}$ for $i\in[k]\cap B$. To see that, notice that this is the
lexicographic maximal permutation using finite coordinates and the third
condition ensures that this is the minimal term. So we conclude that the
maximal prime cone described in [FFFM19] is actually $\pi_{\mathcal{F}}^{\tt
A}(\mathcal{A})$, where $\pi_{\mathcal{F}}^{\tt A}$ is the type A tropical
flag Stiefel map (Definition 9.6). We see therefore that the valuated matroids
within this cone are transversal.

A symplectic counterpart of this cone is under construction in [BF21] and we
conjecture that it also lives in the tropical flag Stiefel image for type C.

###  9.5. Total Positivity

A major object of study has been the non-negative Grassmannian
$\textsc{G}\mathsf{r}_{\geq 0}(k,n)$, that is, the set of all Plücker vectors
in $\textsc{G}\mathsf{r}_{\mathbb{R}}(k,n)$ with non-negative entries.
Extensive research has been conducted on its positroid decomposition,
specially motivated by its connections to quantum physics, such as computing
Feynman integrals in the supersymmetric-Yang-Mills model (see, for example,
[ABC+16] for an overview). The cells of the positroid decomposition are
indexed by a variety of combinatorial gadgets and can be parametrized using
networks through the boundary measurement map [Pos06]. Adaptations of the
combinatorics of the non-negative Grassmannian have been constructed both for
type C, more precisely, for the non-negative Lagrangian Grassmannian
$\textsc{Sp}\textsc{G}\mathsf{r}_{\geq 0}(n,2n):=\textsc{G}\mathsf{r}_{\geq
0}(n,2n)\cap\textsc{Sp}\textsc{G}\mathsf{r}(n,2n)$ [Kar18]. On the other hand,
the study of the totally positive tropical Grassmannian as defined in [SW05]
has gained a lot of momentum recently (see for example [ALS21, LPW20, SW21])
where the combinatorics of positroids play again a protagonist role. Therefore
we believe it is likely that the combinatorial objects used in [Kar18], such
as symmetric plabic graphs, can be used in a tropical setting to provide
information about the non-negative part of the tropical symplectic
Grassmannian.

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