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%%%%% Auteur
\author{\firstname{Fatemeh} \lastname{Mohammadi}}%%%1
\address{University of Bristol\\
Bristol\\
BS8 1TW, UK}
\email{fatemeh.mohammadi@bristol.ac.uk}
\author{\firstname{Kristin} \lastname{Shaw}}%%%2
\address{University of Oslo\\
P.O. box 1053\\
Blindern\\
0316 OSLO, Norway}
\email{krisshaw@math.uio.no}
\thanks{The first author's research was partially
supported by the EPSRC Early Career Fellowship EP/R023379/1. The research of the second author was partially supported by the MPI Leipzig and the Bergen Research Foundation project grant ``Algebraic and topological cycles in tropical and complex geometry''. }
%%%%% Sujet
%\subjclass{10X99, 14A12, 11L05}
\subjclass{14M15, 14M25, 14T05}
\keywords{toric degenerations, SAGBI and Khovanskii bases, Grassmannians, tropical geometry}
%%%%% Gestion
%\DOI{10.5802/alco.77}
%\datereceived{2018-09-18}
%\daterevised{2019-03-04}
%\dateaccepted{2019-03-21}
%%%%% Titre et résumé
\title[Toric degenerations of Grassmannians from matching fields]
{Toric degenerations of Grassmannians from matching fields}
\begin{abstract}
We study the algebraic combinatorics of monomial degenerations of Pl{\"u}cker forms which is governed by matching fields in the sense of Sturmfels and Zelevinsky.
We provide a necessary condition for a matching field to yield a SAGBI basis of the Pl{\"u}cker algebra for $3$-planes in $n$-space. When the ideal associated to the matching field is quadratically generated this condition is both necessary and sufficient.
Finally, we describe a family of matching fields, called $2$-block diagonal, whose ideals are quadratically generated. These matching fields produce
a new family of toric degenerations of $\text{\upshape Gr}(3, n)$.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
\section{Introduction}
In this note we offer a new family of toric degenerations
of $\Gr(3, n)$ arising from monomial degenerations of the Pl{\"u}cker forms.
Toric degenerations provide a useful tool to study algebraic varieties. This is mainly because toric geometry is inextricably linked to the theory of polytopes and polyhedral fans.
Combinatorial invariants of polytopes provide geometric information about toric varieties, and many of these invariants
are preserved under degeneration. Here, a toric degeneration is a Gr{\"o}bner degeneration such that the corresponding initial ideal is binomial and prime, see Definition~\ref{def:toric}.
For general Grassmannians and flag varieties there are prototypic examples of toric degenerations which are related to Young tableaux, Gelfand--Cetlin integrable systems, and their polytopes~\cite{ FvectorGC, KOGAN}.
In the case of the Grassmannian $\Gr(2,n)$, there are many other toric degenerations generalizing this primary example. Namely, any trivalent tree with $n$ number of labelled leaves gives rise to a toric degeneration of $\Gr(2, n)$.
The toric variety is governed by the isomorphism type of the trivalent tree~\cite{SpeyerSturmfels, Witaszek}.
These degenerations are related to bending systems on polygon spaces and integrable systems~\cite{Kapovich, NISHINOU}.
The Gelfand--Cetlin degenerations arise from monomial initial degenerations of the Pl{\"u}cker forms. These degenerations arise from the general theory of Khovanskii bases, and in fact come from SAGBI bases for the Pl{\"u}cker algebra, see Definition~\ref{def:Khovanskii}.
The leading term of each Pl{\"u}cker form in this case is the monomial of the determinant corresponding to the identity permutation. The new degenerations of $\Gr(3, n)$ provided here depend only on the underlying \emph{coherent matching field}. A matching field is a choice of permutation for each Pl{\"u}cker form, see Definitions~\ref{def:matching} and~\ref{def:coherent}. A coherent matching field provides a monomial degeneration of the Pl{\"u}cker forms and are therefore candidates for SAGBI bases.
To a general matching field, we associate a toric ideal in the polynomial ring $\mathbb{K}[x_{ij}]$ where $\mathbb{K}$ is a field. These matching field ideals are most conveniently represented by matching field tableaux, which we also introduce here. These tableaux generalize Young tableaux which are usually strictly increasing in the columns. Columns of a matching field tableau are filled according to the permutation chosen by the matching field.
A matching field ideal is the kernel of a monomial map, and hence toric, see Equation~\eqref{eq:binomials}. Moreover, it is generated by binomials which come from pairs of matching field tableaux whose contents are row-wise equal.
The Pl{\"u}cker forms are a SAGBI basis of the Pl{\"u}cker algebra with respect to a matching field if and only if its matching field ideal is equal to the initial ideal of the corresponding degeneration of the Pl{\"u}cker ideal, see Theorem~\ref{Theorem:Stu} or~\cite[Theorem~11.4]{sturmfels1996grobner}. Therefore, obtaining a SAGBI basis from a matching field is equivalent to obtaining a toric degeneration of the Grassmannian.
From a weight matrix that produces a monomial degeneration of the Pl{\"u}cker forms, we can produce a tropical hyperplane arrangement, and the matching field can be determined from this geometric picture~\cite{fink2015stiefel}.
Using the associated tropical hyperplane arrangement, we introduce the notion of hexagonal matching fields of size $3 \times 6$ and non-hexagonal matching fields, see Definition~\ref{def:hex}. This leads to our first theorem.
\begin{theo}\label{Theorem:necess}
If a $3 \times n$ matching field produces a toric degeneration of $\Gr(3,n)$, then it is non-hexgonal.
\end{theo}
We also define submatching fields by using natural maps between Grassmannians of different sizes, see Definition~\ref{def:submatching}.
This allows us to extend Theorem~\ref{Theorem:necess} to higher Grassmannians.
\begin{theo}\label{Theorem:subhex}
A $k \times n$ matching field that has a hexagonal submatching field does not produce a toric degeneration of the Grassmannian $\Gr(k, n)$.
\end{theo}
If a $3 \times n$ matching field ideal is quadratically generated, then the necessary condition from Theorem~\ref{Theorem:necess} is also sufficient.
\begin{theo}\label{Theorem:iff}
A $3 \times n$ matching field whose ideal is quadratically generated provides a toric degeneration of $\Gr(3, n)$ if and only if it is non-hexagonal.
\end{theo}
Describing a generating set of toric ideals is a well-studied and difficult problem. In particular, proving that an ideal is quadratically generated is quite a difficult task. There are some combinatorial criteria for the toric ideals arising from graphs, matroids and simplicial complexes to be generated by quadratics, see \eg \cite{White, Ohsugi, Cone, Mateusz}. Such a criterion guarantees that the associated Koszul algebra is normal.
Not all coherent matching field ideals are quadratically generated. The first examples that we know of are of size $3 \times 8$.
However, we introduce a class of matching fields of size $3 \times n$, called \emph{block diagonal}, which are quadratically generated when they have two blocks.
\begin{theo}\label{Theorem:2block}
The ideal of a $2$-block diagonal matching field of size $3 \times n$ is quadratically generated.
\end{theo}
\begin{coro}\label{cor:blockdiag}
A $2$-block diagonal matching field produces a toric degeneration of the Grassmannian $\Gr(3, n)$.
Equivalently, when $k= 3$ the Pl{\"u}cker forms are a Khovanskii basis with respect to any weight matrix arising from a $2$-block diagonal matching field.
\end{coro}
\looseness-1
Before reviewing the contents of the paper we would like to comment on some related works. Rietsch and Williams describe families of toric degenerations of Grassmannians arising from plabic graphs~\cite{RietschWilliams}. In~\cite{Bossinger}, Bossinger et.~al.~show that already in the case of $\Gr(3, 6)$ there is a discrepancy between the toric degenerations arising on one hand from plabic graphs and on the other hand from top dimensional cones of the tropical Grassmannian. The toric degenerations studied here are a subset of the latter type coming from top dimensional cones of the tropical Grassmannian associated to Stiefel tropical linear spaces, see~\cite{fink2015stiefel} for more details. In general, the combinatorial connection between matching fields and plabic graphs is still unknown.
Kaveh and Manon provide a general connection between tropical geometry and Khovanskii bases (and hence toric degenerations) in~\cite{Kaveh}. This question has been studied in~\cite{bossinger2017computing} for small flag varieties. Here, we are interested in determining when the Pl{\"u}cker forms are a SAGBI basis, equivalently when the associated initial degeneration of the Pl{\"u}cker ideal is toric.
The results presented here offer a family of examples
that fit into the general framework of Kaveh and Manon and which are linked to combinatorics. Lastly, we would like to remark that toric degenerations of flag varieties and Schubert varieties arising from matching fields is another open direction of research at the present time.
\smallskip
We finish the introduction with an outline of the paper. Section~\ref{prelim} fixes notations for the Grassmannians and introduces matching fields. In Section~\ref{sec:arr}, we review the connection between matching fields and tropical hyperplane arrangements. Here we introduce the notion of hexagonal matching fields and prove Theorems~\ref{Theorem:necess}, \ref{Theorem:subhex}, and~\ref{Theorem:iff}. Block diagonal matching fields are introduced in Section~\ref{sec:block} and the proof of Theorem~\ref{Theorem:2block} is also given here. The final section defines matching field polytopes and provides some examples as well as remarks about their combinatorics.
\section{Preliminaries}\label{prelim}
Throughout we set $[n]\coloneqq \{1, \ldots , n\}$ and we use $\mathbf{I}_{k, n}$ to denote the collection of subsets of $[n]$ of size $k$. The symmetric group on $k$ elements is denoted by $S_k$. We also fix a field $\mathbb{K}$.
\smallskip
The Grassmannian $\Gr(k, n)$ is the space of all $k$ dimensional linear subspaces of $\mathbb{K}^n$. A point in $\Gr(k, n)$ can be represented by a $k\times n$ matrix with entries in $\mathbb{K}$.
Let $X=(x_{ij})$ be a $k\times n$ matrix of indeterminates. For a subset $I = \{i_1,\ldots,i_k\} \in \mathbf{I}_{k, n}$, let $X_I$ denote the $k\times k$ submatrix with the column indices $i_1,\ldots,i_k$.
The \emph{Pl{\"u}cker forms (or Pl{\"u}cker coordinates)} are $P_I = \det (X_I)$ for $I \in\mathbf{I}_{k, n}$. These forms determine the \emph{Pl{\"u}cker embedding} from
$ \Gr (k,n)$ into $\mathbb{P}^{\binom{n}{k}-1}$.
In the following, we consider the polynomial ring $\mathbb{K}[x_{ij}]$ on the variables $x_{ij}$ with $1\leq i\leq k$ and $1\leq j\leq n$ and the polynomial ring $\mathbb{K}[P_I]$ on the Pl{\"u}cker variables with $|I|=k$.
\begin{Definition}\label{def:ideal}The Pl{\"u}cker ideal $\I_{k,n}$ is defined as the kernel of the map
\begin{equation}\label{eqn:pluckermap}
\begin{aligned}
\psi \colon \mathbb{K}[P_I] &\rightarrow \mathbb{K}[x_{ij}]
\\
P_{I} &\mapsto \det (X_I).
\end{aligned}
\end{equation}
The \emph{Pl{\"u}cker algebra} is the finitely generated algebra $\mathbb{K}[P_I ]/{\mathcal{I}_{k,n}}$
denoted by $\mathcal{A}_{k,n}$.
\end{Definition}
\begin{Definition}\label{def:matching}
A $k\times n$ matching field is a map $\Lambda: \mathbf{I}_{k,n} \to S_k$.
\end{Definition}
Given a $k\times n$ matching field $\Lambda$ and a subset $I = \{i_1, \ldots , i_k\} \in \mathbf{I}_{k, n}$
we consider the set to be ordered by $i_1 < \cdots < i_{k}$. We think of the permutation {$\sigma=\Lambda(I)$} as inducing a new ordering on the elements of $I$
where the position of $i_s$ is determined by the value of $\sigma(s)$.
It is convenient to represent the variable $P_I$ as a $k \times 1$ tableau
where $(\sigma(r), 1)$ contains $i_{r}$.
\begin{Definition}
Let $\Lambda$ be a size $k \times n$ matching field.
A $\Lambda$-tableau is a tableau of size $k \times d$ for any $d\geq 1$ with entries in $[n]$,
so that the entries in each column are pairwise distinct and filled according to the order determined by $\Lambda$.
\end{Definition}
\begin{exam}\label{ex:diagonal1}
The \emph{diagonal matching field} assigns to
each subset $I \in \mathbf{I}_{k,n}$ the identity permutation~\cite[Example~1.3]{sturmfels1993maximal}.
Therefore, a $\Lambda$-tableau is a rectangular tableau of size $k \times d$ filled with entries in $[n]$ such that the columns are strictly increasing.
\end{exam}
\begin{exam}\label{ex:pointed1}
A $k \times n$ matching field is called \emph{pointed} if there exists $i_1, \ldots, i_k \in [n]$ such that if $i_s \in I$ for some
$1 \leq s \leq k$ then $\Lambda(I)(s) = s$~\cite[Example~1.4]{sturmfels1993maximal}. In other words, if $i_s \in I$ for some $1\leq s \leq k$ then the column corresponding to $P_I$ contains $i_s$ in row $s$. Below are the tableaux representing $P_I$ for a pointed matching field of size $3 \times 5$ which is otherwise filled diagonally:
\[
\begin{array}{c}1 \\2 \\ 3 \end{array} , \quad
\begin{array}{c} 1 \\2 \\4 \end{array},\quad
\begin{array}{c}1 \\2 \\ 5 \end{array} , \quad
\begin{array}{c}1 \\ 4 \\ 3 \end{array} ,\quad
\begin{array}{c}1 \\5 \\ 3\end{array} ,\quad
\begin{array}{c}4 \\ 2\\ 3 \end{array} ,\quad
\begin{array}{c}5 \\2 \\ 3\end{array} ,\quad
\begin{array}{c} 4 \\ 2 \\ 5 \end{array}.
\]
Generalising this, we say that a size $k \times n$ matching field $\Lambda$ is pointed on $S \subset [n]$ if for all $i \in S$ there exists a $j_i$ such that $i$ always appears in row $j_i$ of any $\Lambda$ matching field tableau.
Here $S$ need not have size equal to $k$.
\end{exam}
A monomial $\Pi_{I \in A} P_I $ can be represented by a $\Lambda$-tableau of size $k \times |A|$ given by the concatenation of the columns with content $I$ filled according to the matching field.
To each monomial $\Pi_{I \in A} P_I $ we associate a sign $\epsilon_{A} = \pm 1$ determined by the signature of the permutations $\Lambda(I)$ for all $I \in A$. More precisely,
\[
\epsilon_{A} = \Pi_{I \in A} \sgn (\Lambda(I)).
\]
\begin{Definition}
Given a matching field $\Lambda$, the \emph{matching field ideal} $\mathcal{I}_{\Lambda} \subset \mathbb{K}[x_{ij}]$ is generated by the
binomial relations
\begin{equation}\label{eq:binomials}
\epsilon_A \Pi_{I \in A} P_I - \epsilon_B \Pi_{J \in B} P_J
\end{equation}
if and only if the contents of the corresponding $\Lambda$-tableau of size $k \times |A|$ are row-wise equal.
\end{Definition}
To $I \in \mathbf{I}_{k, n}$ {with $\sigma=\Lambda(I)$} we associate the monomial
\[
\mathbf{x}_{\Lambda(I)}\coloneqq x_{\sigma(1) i_{1}}x_{\sigma(2)i_2}\cdots x_{{\sigma(k)i_k}}.
\]
A $k\times n$ matching field $\Lambda$,
gives a map of polynomial rings
\begin{equation}\label{eqn:monomialmap}
\begin{aligned}
\phi_{\Lambda} \colon\ \mathbb{K}[P_I] &\rightarrow \mathbb{K}[x_{ij}]
\\
P_{I} &\mapsto \sgn(\Lambda(I)) \mathbf{x}_{\Lambda(I)}.
\end{aligned}
\end{equation}
\smallskip
\begin{Proposition}
Given a matching field $\Lambda$, the matching field ideal $\mathcal{I}_{\Lambda}$ is the kernel of the monomial map
$\phi_{\Lambda}$ from Equation~\eqref{eqn:monomialmap}.
\end{Proposition}
\begin{Definition}\label{def:coherent}
A $k \times n$ matching field $\Lambda$ is \emph{coherent} if there exists a $k \times n$ matrix $M$ with entries in $\mathbb{R}$
such that
for every $I \in \mathbf{I}_{k, n}$
the initial form of the Pl{\"u}cker form $P_I \in \mathbb{K}[x_{ij}]$, the sum of all terms in $M_I$ of lowest weight, is $\ini_M (P_I) = \sgn (\Lambda(I)) \mathbf{x}_{\Lambda(I)}$.
In this case, we say that the matrix $M$ \emph{induces the matching field} $\Lambda$.
\end{Definition}
\begin{exam}
When $k = 2$ all coherent matching fields are induced by a total ordering on the set $[n]$, see~\cite[Proposition~1.11]{sturmfels1993maximal}.
\end{exam}
\begin{exam}\label{ex:diagonal2}
The diagonal matching field of size $k \times n$ is coherent
\cite[Example~1.3]{sturmfels1993maximal}.
For example, this matching field is induced by a $k\times n$ weight matrix $M$ whose $i,j$-th entry
is $(i-1)(n-j)$. Therefore, we have,
\[
M=\begin{bmatrix}
0 & 0 & 0 & \cdots & 0 \\
n-1 &\cdots & 2 & 1 & 0 \\
2(n-1) &\cdots & 4 & 2 & 0 \\
\cdots & \cdots & \cdots & \cdots \\
(k-1)(n-1)& \cdots & 2(k-1) & k-1 & 0 \\
\end{bmatrix}.
\]
For any size $k$ subset $I=\{i_1,i_2,\ldots,i_k\}$ with $i_1\!\!>0 $ except for in row $s$ where the entry can be chosen to be $0$.
\end{exam}
\begin{Definition}\label{def:toric}
Let $\I$ be an ideal in the polynomial ring $S = \mathbb{K}[y_1, \ldots, y_m]$ and let $w \in \mathbb{R}^m$.
The initial degeneration with respect to $w$ is called \emph{toric} if the initial ideal $\ini_w(\I)$ is prime and binomial.
\end{Definition}
\begin{Definition}[\cite{robbiano1990subalgebra}]\label{def:Khovanskii}
The set of Pl{\"u}cker forms $\{P_I\}_{I\in \mathbf{I}_{k,n}}\subset \mathbb{K}[x_{ij}]$ is a \emph{SAGBI basis} for the Pl{\"u}cker algebra $\mathcal{A}_{k,n}$ with respect to a weight matrix $M$
if for all $I \in \mathbf{I}_{k,n} $ the initial form $\ini_M(P_I) $ is a monomial and
$\ini_{w_M}(\mathcal{A}_{k,n})=\mathbb{K}[\ini_M(P_I)]_{I \in \mathbf{I}_{k,n}}$. Here $w_M$ is the weight vector on the variables $P_I$ induced by the weight matrix $M\in\mathbb{R}^{k\times n}$ on the variables $x_{ij}$.
\end{Definition}
The following theorem intimately relates SAGBI bases and toric initial degenerations. It is phrased in the context of matching fields and Grassmannians.
\begin{theo}[{\cite[Theorem~11.4]{sturmfels1996grobner}}]\label{Theorem:Stu}
The set of Pl{\"u}cker forms $\{P_I\}_{I\in \mathbf{I}_{k,n}}\subset \mathbb{K}[x_{ij}]$
is a SAGBI basis with respect to a weight matrix $M$ if and only if $\ini_{w_M}(\I_{k, n}) = \I_\Lambda$, where $w_M$ is the weight vector on the variables $P_I$ induced by $M$ and ${\Lambda}$ is the matching field induced by $M$.
\end{theo}
\section{Coherent matching fields and tropical hyperplane arrangements}\label{sec:arr}
In~\cite{fink2015stiefel}, Fink and Rinc\'on provide a link between tropical hyperplane arrangements and coherent matching fields (and more generally multi-matching fields). We will summarize the facts needed here and refer the reader to~\cite{fink2015stiefel} for more details. In~\cite{fink2015stiefel}, tropical hyperplane arrangements are described in tropical projective space. We do not require this level of generality here, therefore we simplify our considerations in the following summary.
Let $M=(a_{ij}) \in \mathbb{R}^{k \times n}$ be a weight matrix. For each $1 \leq j \leq n$ consider the piecewise linear function $F_j: \mathbb{R}^{k-1} \to \mathbb{R}$, given by
\begin{equation}\label{eqn:linearform}
F_j (x) = \max \{ a_{1j}, a_{2j} + x_2, \ldots, a_{kj}+ x_k\}.
\end{equation}
In $\mathbb{R}^{k-1}$ let $\Sigma$ be the $k-2$ dimensional polyhedral fan whose top dimensional cones
are spanned by subsets of size $k-2$ of the vectors $v_1, \ldots, v_{k}$, where $v_1 = (1, \ldots, 1)$ and
$v_i = - e_{i-1}$ otherwise. For $k=3$ this amounts to three rays in the directions $(1, 1), (-1, 0)$ and $(0, -1)$ emanating from the origin.
If $a_{1j} = 0$, then the non-differentiability locus of $F_j$ is the fan $\Sigma \subset \mathbb{R}^{k-1}$ translated by the vector $(-a_{2j} , \ldots, -a_{kj}) \in \mathbb{R}^{k-1}$. Any coherent matching field is induced by a weight matrix $M$ whose first row consists of zeros.
So we may assume that $a_{1j} = 0$ for all $j$.
\begin{Remark}\label{rem:maxmin}
In this paper we purposely use the minimum conventions for tropical arithmetic and the maximum conventions for tropical geometry. We do this to avoid the appearance of many minus signs when passing from the algebra of weight matrices to the geometry of tropical hyperplane arrangements.
\end{Remark}
\begin{figure}[htb]\centering
\begin{tikzpicture} [scale = .4, very thick = 1mm, every node/.style={inner sep=0,outer sep=0}]
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\node (i2) at (2,0 ) [Cwhite] {};
\foreach \from/\to in {i/i0,i/i1,i/i2}
\draw[black] (\from) -- (\to);
\node (j) at (4,2) [Cblack3] {};
\node (j0) at (9,7) [Cwhite]{} ;
\node (j1) at (0,2) [Cwhite] {\huge{$1$}};
\node (j2) at (4,0 ) [Cwhite] {};
\foreach \from/\to in {j/j0,j/j1,j/j2}
\draw[black] (\from) -- (\to);
\node (j) at (8,4) [Cblack3] {};
\node (j0) at (10,6) [Cwhite]{} ;
\node (j1) at (0,4) [Cwhite] {\huge{$3$}};
\node (j2) at (8,0 ) [Cwhite] {};
\foreach \from/\to in {j/j0,j/j1,j/j2}
\draw[black] (\from) -- (\to);
\node (j) at (18+ 4,6) [Cblack3] {};
\node (j0) at (18+6,8) [Cwhite]{} ;
\node (j1) at (18+0,6) [Cwhite] {\huge{$2$}};
\node (j2) at (18+4,0 ) [Cwhite] {};
\foreach \from/\to in {j/j0,j/j1,j/j2}
\draw[black] (\from) -- (\to);
\node (j) at (18+ 6,4) [Cblack3] {};
\node (j0) at (18+8, 6) [Cwhite]{} ;
\node (j1) at (18+0,4) [Cwhite] {\huge{$1$}};
\node (j2) at (18+6,0 ) [Cwhite] {};
\foreach \from/\to in {j/j0,j/j1,j/j2}
\draw[black] (\from) -- (\to);
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\node (j1) at (18+0,2) [Cwhite] {\huge{$3$}};
\node (j2) at (18+2,0 ) [Cwhite] {};
\foreach \from/\to in {j/j0,j/j1,j/j2}
\draw[black] (\from) -- (\to);
\end{tikzpicture}\caption{Two arrangements of three tropical lines in $\mathbb{R}^2$. On the left the configuration intersects properly on the right the three lines are concurrent.}\label{concurrent}
\end{figure}
A $k \times n$ weight matrix $M \in \mathbb{R}^{k \times n}$, whose first row consists of zeros, produces an arrangement of tropical hyperplanes $\mathcal{A} = \{ H_1, \ldots , H_n\}$ defined by the functions $F_1, \ldots , F_n$, whose coefficients come from $M$.
To each $k-1$ dimensional cell $\tau$ of the complement of the arrangement $\mathcal{A}$ in $\mathbb{R}^{k-1}$ there is an associated \emph{covector} $c_{\tau} \in \mathcal{P}[n]^k$, where $\mathcal{P}[n]$ denotes the power set of $n$.
The $i$-th entry of the covector $c_{\tau}$ is a subset $S_i \subset [n]$ corresponding to the collection of hyperplanes in $\mathcal{A}$ which intersect the ray $x + tv_i$ for $x \in \tau^{\circ}$ and $t \geq 0$. Here the vectors $v_i = -e_{i-1}$ for $i = 2, \ldots, k-1$ and $v_1 = (1, \ldots, 1)$. The \emph{coarse covector}
of a cell is simply the vector which records the sizes of the subsets of the covector.
\begin{Definition}
A collection of $k$ hyperplanes $H_1, \ldots, H_k$ in $\mathbb{R}^{k-1}$ is said to \emph{intersect properly} if $\cap_{i = 1}^k H_i = \emptyset$.
Equivalently, a collection of $k$ hyperplanes $H_1, \ldots, H_k$ in $\mathbb{R}^{k-1}$ intersects properly if and only if there is a $k-1$ dimensional cell in the complement of $\cup_{i = 1}^k H_i$
whose coarse covector is $(1, \ldots, 1)$.
\end{Definition}
\begin{exam}
Consider the two $3 \times 3$ weight matrices,
\[
M_1=\begin{bmatrix}
0 & 0 & 0 \\
4& 2 & 8 \\
2& 3 & 4 \\
\end{bmatrix},
\qquad
M_2=\begin{bmatrix}
0 & 0 & 0 \\
6 & 4 & 2 \\
4 & 6 & 2 \\
\end{bmatrix}.
\]
The matrix $M_1$ corresponds to tropical lines intersecting properly in Figure~\ref{concurrent} (left) and the matrix $M_2$ corresponds to concurrent tropical lines in Figure~\ref{concurrent} (right).
\end{exam}
The following proposition can be extracted from~\cite[Proposition~2.4]{DevSturm} and~\cite[Propositions~5.11 and~5.12]{fink2015stiefel}.
\begin{Proposition}\label{prop:initialPlucker}
Let $M$ be a $k \times n$ weight matrix such that for any size $k$ subset $I \subset [n]$, the collection of hyperplanes $\{H_i\}_{i\in I}$ intersects properly.
Then
\[
\ini_{M} (P_{I}) =\sgn ({\Lambda(I)}) x_{1c_1}x_{2c_2}\ldots x_{kc_{k}}
\]
where $(c_1, \ldots, c_{k})$ is the covector of the unique cell with coarse covector $(1, 1, \ldots, 1)$ and
$\sgn ({\Lambda(I)})$ is the sign of the permutation $i \mapsto \sigma(i)$, where $c_{\sigma(1)}<\cdots3$ since in these cases quadratic generation of the initial ideals does not directly imply that the initial degeneration is toric.
\end{Remark}
In general there is a $\mathbb{Z}^4$-grading given by the number of elements of type $I_1$ in different rows of a tableau. A $3\times d$ tableau $T$ is of degree $(\alpha, \beta,\gamma, d-\alpha-\beta-\gamma)$
where
\begin{enumerate}
\item $\alpha=|\{\text{Content of row 3 of } T\}\cap I_1| = |\{I\in T: \ |I \cap I_1| = 3\}|$
\item $\beta=|\{\text{Content of row 1 of } T\}\cap I_1|-\alpha = |\{I\in T: \ |I \cap I_1| = 2\}|$
\item $\gamma=|\{\text{Content of row 2 of } T\}\cap I_1|-\alpha-\beta = |\{I\in T: \ |I \cap I_1| = 1\}|$
\end{enumerate}
For two $\Lambda$-tableaux $T,T'$ which are row-wise equal, these numbers are equal. This implies the following lemma.
% \medskip
\begin{lemm}\label{lem:homo}
The ideal of a block diagonal matching field
has a $\mathbb{Z}^4$-grading given by $(\alpha, \beta,\gamma, d-\alpha-\beta-\gamma)$
from above.
\end{lemm}
\begin{proof}[Proof of Theorem~\ref{Theorem:2block}]
Consider a binomial relation obtained from two $\Lambda$-tableaux $T$ and $T'$ of size $3 \times d$ where $d >2$ whose contents are row-wise equal. By applying quadratic changes to the tableaux (changes involving only two columns) we will reduce the degree of this relation thus proving that the matching field ideal is quadratically generated.
Given a $\Lambda$-tableau $T$, arrange the columns so that the first columns are those for which the matching field assigns the identity permutation and to the last column the matching field assigns the transposition $(12)$. Let $C$ denote the subtableau formed by the first columns and let $D$ denote the subtableau formed by the last columns.
The tableaux $C$ and $D$ can each be put into semi-standard format.
In other words, we can rearrange both $C$ and $D$ so that all rows are in weakly increasing order, and the columns of $C$ are strictly increasing, whereas the columns of $D$ are arranged so that the first and second entries are permuted from the diagonal order.
Now given a binomial relation obtained from two $\Lambda$-tableaux $T$ and $T'$. We assume that $T$ (respectively $T'$) is organized as a pair of subtableaux $C$, $D$ (respectively $C'$, $D'$) satisfying the requirements described above.
If the first columns of $T$ and $T'$ are equal, then we can cancel them from the binomial relation and it is not a minimal generator.
We let $I$ and $I'$ denote the first columns of $T$ and $T'$, respectively.
Otherwise by Lemma~\ref{lem:homo} the matching field relations are homogeneous with respect to the $\mathbb{Z}^4$-grading and so $\lvert I \cap I_1 \rvert = \lvert I' \cap I_1\rvert$.
\let\oldthecdrthm\thecdrthm
\setcounter{cdrthm}{0}
\def\thecdrthm{\arabic{cdrthm}}
\begin{enonce}[remark]{Case}\label{Case1}%\refstepcounter{cdrthm}
Suppose that $\lvert I \cap I_1 \rvert = \lvert I' \cap I_1\rvert = 3$. In this case, the columns $I$ and $I'$ could only differ in the second row. Suppose the entries of the second row of $I$ and $I'$ are $j $ and $j'$, respectively. We can also assume that $j