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\title{On Two Notions of Total Positivity for Generalized Partial Flag Varieties of Classical Lie Types}

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\author{\texorpdfstring{Grant Barkley\thanks{\href{mailto:gbarkley@math.harvard.edu}{gbarkley@math.harvard.edu}. Grant Barkley was supported in part by NSF grant DMS-2152991.}\addressmark{1}, Jonathan Boretsky\thanks{\href{mailto:jonathan.boretsky@mis.mpg.de}{jonathan.boretsky@mis.mpg.de}. Jonathan Boretsky was supported in part by NSF grant DMS-2152991. Jonathan Boretsky was
also partially supported by NSERC.
Jonathan Boretsky a été aussi partiellement financé par CRSNG [Ref. no. 557353-2021]. }\addressmark{2},  Chris Eur\thanks{\href{mailto:ceur@andrew.cmu.edu}{ceur@andrew.cmu.edu}. Chris Eur was supported by NSF grant
DMS-2246518.}\addressmark{3}, \and Jiyang Gao\thanks{\href{mailto:jgao@math.harvard.edu}{jgao@math.harvard.edu}. Jiyang Gao was supported in part by NSF grant DMS-2152991.}\addressmark{1}}{Grant Barkley, Jonathan Boretsky,  Chris Eur, and Jiyang Gao}}

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\address{\addressmark{1}Mathematics Department, Harvard University, Cambridge, MA \\ \addressmark{2} Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany \\
\addressmark{3}Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania}

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\abstract{For Grassmannians, Lusztig's notion of total positivity coincides with positivity of the Pl\"ucker coordinates.  This coincidence underpins the rich interaction between matroid theory, tropical geometry, and the theory of total positivity.  Bloch and Karp furthermore characterized the (type A) partial flag varieties for which the two notions of positivity similarly coincide.  We characterize the symplectic (type C) and odd-orthogonal (type B) partial flag varieties for which Lusztig's total positivity coincides with Pl\"ucker positivity.
}

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\keywords{total positivity, flag varieties, symplectic group, orthogonal group}

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\begin{document}

\maketitle

\section{Introduction}
Let $n$ be a positive integer, and denote $[n] = \{1, \dotsc, n\}$.
The \emph{totally positive} (resp.\ \emph{nonnegative}) \emph{part} $GL_n^{>0}$ (resp.\ $GL_n^{\geq 0}$) of the general linear group $GL_n$ consists of the real invertible matrices whose minors are all positive (resp.\ nonnegative).
The study of these spaces traces back to \cite{Loewnerpositive, Whitneypositivity}.
Lusztig generalized this notion of total positivity to an arbitrary connected reductive ($\RR$-split) algebraic group $G$ and its partial flag varieties $G/P$ \cite{lusztig1994positivity,lusztig1998positivity}.
The study of total positivity has since been a nexus for fruitful interactions between algebraic geometry, representation theory, combinatorics, and physics \cite{physicsbook,FZsurvey,GWpos}.

 
Underpinning such fruitful interactions is the interplay between ``parametric'' and ``implicit'' descriptions of total positivity.
The original definitions are ``parametric'' in nature: Lusztig defined the \emph{Lusztig positive} (resp.\ \emph{Lusztig nonnegative}) part $G^{>0}$ (resp.\ $G^{\geq 0}$) of $G$ as a semigroup in $G$ generated by certain elements (see \Cref{subsect:Luspositivity}).
For a parabolic subgroup $P\subset G$, the Lusztig positive (resp.\ Lusztig nonnegative) part $(G/P)^{>0}$ (resp.\ $(G/P)^{\geq 0}$) of the partial flag variety $G/P$ is then defined as the image of (resp.\ the closure of the image of) $G^{>0}$ under the projection map $G\to G/P$.  Marsh and Rietsch gave a combinatorial parametrization of $(G/P)^{\geq 0}$ in terms of its Deodhar cells \cite{marshrietsch2004}. A challenge of studying Lusztig nonnegative flag varieties is that, from these parametric descriptions alone, it is difficult to recognize when a partial flag is Lusztig nonnegative. 

 
To address this difficulty, one may seek an ``implicit'' description of Lusztig positivity for $G/P$ in terms of positivity of suitably natural coordinates on $G/P$. Our main result, \Cref{thm:main}, gives a simple implicit description for certain flag varieties when $G$ is the symplectic group on $2n$ elements or the orthogonal group on $2n+1$ elements.
As a motivating example, consider the \emph{Grassmannian} $\operatorname{Gr}_{k;n}$ of $k$-dimensional subspaces in $\RR^n$.
Its Pl\"ucker coordinates allow one to consider the \emph{Pl\"ucker positive} (resp.\ \emph{nonnegative}) \emph{part} of $\operatorname{Gr}_{k;n}$, defined as
\[
\text{$\operatorname{Gr}_{k;n}^{\Delta > 0}$ (resp.\ $\operatorname{Gr}_{k;n}^{\Delta \geq 0}$)} := \left\{L \subseteq \RR^n \ \middle| \ \begin{matrix}\text{$L$ is the row-span of a real $k\times n$ matrix with all}\\  \text{maximal minors positive (resp.\ nonnegative)}\end{matrix}\right\}.
\]
Lam \cite{Lam2014} and, independently, Talaska and Williams \cite{talaska2013network} showed that $\operatorname{Gr}_{k;n}^{> 0} = \operatorname{Gr}_{k;n}^{\Delta > 0}$ and $\operatorname{Gr}_{k;n}^{\geq 0} = \operatorname{Gr}_{k;n}^{\Delta \geq 0}$.
More generally, for a subset $K = \{k_1 < \dotsb < k_j\} \subseteq [n-1]$, one may consider the (type A) partial flag variety
\[
\operatorname{Fl}_{K ; n} := \{\text{flags of subspaces $L_\bullet = (L_1 \subseteq \dotsb \subseteq L_j)$ with $\dim L_i = k_i$ for all $i = 1, \dots, j$}\}.
\]
Under the natural embedding $\operatorname{Fl}_{K ; n} \hookrightarrow \prod_{i = 1}^j \operatorname{Gr}_{k_i;n}$,
its \emph{Pl\"ucker positive} (resp.\ \emph{nonnegative}) part is defined as the intersection
\[
\text{$\operatorname{Fl}_{K ; n}^{\Delta>0}$ (resp.\ $\operatorname{Fl}_{K ; n}^{\Delta \geq 0}$)} := \operatorname{Fl}_{K ; n} \cap  \prod_{i = 1}^j \operatorname{Gr}_{k_i;n}^{\Delta >0} \text{ (resp.\ $\prod_{i = 1}^j \operatorname{Gr}_{k_i;n}^{\Delta \geq 0}$)}.
\]
Bloch and Karp \cite{blochkarp2023} showed the following.  The second author independently showed a similar result in the case of $K =[n-1]$ \cite{boretsky2023totally}.

\begin{thm}[{\cite[Theorem 1.1]{blochkarp2023}}]\label{thm:BK}
The following are equivalent for a subset $K\subseteq[n-1]$:
\begin{enumerate}[label = (\arabic*)]
\item $\operatorname{Fl}_{K ; n}^{>0} = \operatorname{Fl}_{K; n}^{\Delta>0}$,
\item $\operatorname{Fl}_{K; n}^{\geq 0} = \operatorname{Fl}_{K; n}^{\Delta\geq 0}$, and
\item $K$ consists of consecutive integers.
\end{enumerate}
\end{thm}

In summary, these results establish the coincidence of Lusztig's positivity and Pl\"ucker positivity for partial flag varieties $\operatorname{Fl}_{K;n}$ with consecutive $K$.
This coincidence supports the rich interaction between matroid theory, tropical geometry, and total positivity \cite{postnikov2006, SpeyerWilliams2005, JLLO,boretsky2023totally}. For instance, it is used to prove that rank $K$ positively oriented flag matroids are flag positroids and that the positive tropical flag variety equals the positive flag Dressian \cite{boretskyeurwilliams}. %\cite{ardilarinconwilliams, postnikov2006, SpeyerWilliams2005, SpeyerWilliams2021, boretskyeurwilliams, JLLO, PSBW23}.
Here, with a view towards the theory of Coxeter matroids \cite{BGW2003}, we characterize partial flag varieties of the symplectic group $\operatorname{Sp}_{2n}$ (type C) and the odd-orthogonal group $\operatorname{SO}_{2n+1}$ (type B) for which Lusztig's positivity coincides with Pl\"ucker positivity. This yields an explicit test for membership in the Lusztig positive part of those partial flag varieties: A partial flag is Lusztig positive if and only if all of its Pl\"ucker coordinates are positive.  

 
Let $\be_i$ denote the $i$-th standard basis vector in a coordinate space, and $\be_i^*$ its dual.
For type C, endow $\RR^{2n}$ with the symplectic bilinear form $\omega = \sum_{i = 1}^{n} (-1)^i \be_i^* \wedge \be_{2n+1 - i}^*$.
For type B, endow $\RR^{2n+1}$ with the symmetric bilinear form $Q = \sum_{i = 1}^{n+1} (-1)^i \be_i^* \cdot \be_{2n+2 - i}^*$.
Let $\operatorname{Sp}_{2n}$ and $\operatorname{SO}_{2n+1}$ be the linear groups preserving the bilinear forms $\omega$ and $Q$, respectively.
% Explicitly, we have
% \begin{align*}
% \mathrm{Sp}_{2n}(\RR) & :=\{A\in \mathrm{SL}_{2n}(\RR)| A^tEA = E\},\quad\text{where\quad}E=\left[\begin{smallmatrix}
%     &&&&1\\&&&-1&\\&&\iddots&&\\&1&&&\\-1&&&&
% \end{smallmatrix}\right], \quad\text{and}\\
% \operatorname{SO}_{2n+1}(\RR) &:= \{A\in \mathrm{SL}_{2n+1}(\RR) | A^tE'A = E'\},\quad\text{where}\quad E'= \begin{bmatrix} & -1 \\ E & \end{bmatrix} = \left[\begin{smallmatrix}
%     &&&&-1\\&&&1&\\&&\iddots&&\\&1&&&\\-1&&&&
% \end{smallmatrix}\right].
% \end{align*}
Recall that a subspace of a vector space with a symmetric or alternating form is \emph{isotropic} if the restriction of the form to the subspace is trivial. The partial flag varieties of these groups have the following description in terms of isotropic subspaces: For $K\subseteq [n]$,
\begin{align*}
\operatorname{SpFl}_{K;2n} &:= \{ L_\bullet \in \operatorname{Fl}_{K;2n} : \text{each $L_i$ in the flag $L_\bullet$ is isotropic with respect to $\omega$}\}, \quad\text{and}\\
\operatorname{SOFl}_{K;2n+1} &:= \{ L_\bullet \in \operatorname{Fl}_{K;2n+1} : \text{each $L_i$ in the flag $L_\bullet$ is isotropic with respect to $Q$}\}.
\end{align*}
We define their \emph{Pl\"ucker positive} (resp.\ \emph{nonnegative}) parts as the intersections
\begin{align*}
\operatorname{SpFl}_{K;2n}^{\Delta >0}\text{ (resp.\ $\operatorname{SpFl}_{K; 2n}^{\Delta \geq 0}$)} &:= \operatorname{SpFl}_{K;2n} \cap \operatorname{Fl}_{K; 2n}^{\Delta > 0} \text{ (resp.\ $\operatorname{Fl}_{K; 2n}^{\Delta \geq 0}$)}, \quad\text{and}\\
\operatorname{SOFl}_{K;2n+1}^{\Delta >0}\text{ (resp.\ $\operatorname{SOFl}_{K; 2n+1}^{\Delta \geq 0}$)} &:= \operatorname{SOFl}_{K;2n+1} \cap \operatorname{Fl}_{K; 2n+1}^{\Delta > 0} \text{ (resp.\ $\operatorname{Fl}_{K; 2n+1}^{\Delta \geq 0}$)}.
\end{align*}
Our main theorem is as follows.


\begin{maintheorem}\label{thm:main}
In type C, for $n \geq 2$ and a subset $K\subseteq [n]$, the following are equivalent:
\begin{enumerate}[label = (\arabic*)]
\item\label{mainthm:pos} $\operatorname{SpFl}_{K ; 2n}^{>0} = \operatorname{SpFl}_{K; 2n}^{\Delta>0}$,
\item\label{mainthm:nonneg} $\operatorname{SpFl}_{K; 2n}^{\geq 0} = \operatorname{SpFl}_{K; 2n}^{\Delta\geq 0}$, and
\item\label{mainthm:consec} $K = \{k, k +1, \dotsc, n\}$ for some $1\leq k\leq n$.
\end{enumerate}
In type B, for $n \geq 3$ and a subset $K\subseteq [n]$, the following are equivalent:
\begin{enumerate}[label = (\arabic*)]
\item $\operatorname{SOFl}_{K ; 2n+1}^{>0} = \operatorname{SOFl}_{K; 2n+1}^{\Delta>0}$,
\item $\operatorname{SOFl}_{K; 2n+1}^{\geq 0} = \operatorname{SOFl}_{K; 2n+1}^{\Delta\geq 0}$, and
\item $K = \{k, k +1, \dotsc, n\}$ for some $1\leq k\leq n$.
\end{enumerate}
In type B, when $n=2$, the statements (1) and (2) hold for all partial flag varieties $\operatorname{SOFl}_{K;5}$.
\end{maintheorem}

The proof of \ref{mainthm:consec}$\implies$\ref{mainthm:pos}$\implies$ \ref{mainthm:nonneg} uses the fact that, with the forms $\omega$ and $Q$, $\operatorname{SpFl}_{K;2n}$ and $\operatorname{SOFl_{K;2n+1}}$ are compatible with type A flag varieties in a way we make precise in condition \hyperref[condition1]{($\dagger$)} of \Cref{condition1}. As a result, we can invoke \Cref{thm:BK}. In fact, any flag variety satisfying condition \hyperref[condition1]{($\dagger$)} also has the appropriate analogue of \ref{mainthm:consec}$\implies$\ref{mainthm:pos}$\implies$ \ref{mainthm:nonneg}. To prove \ref{mainthm:nonneg}$\implies$\ref{mainthm:consec}, we explicitly construct examples of flags where \ref{mainthm:nonneg} fails whenever $K$ is not of the form $\{k, k +1, \dotsc, n\}$. 

We briefly discuss type D. We show that the methods employed in types B and C cannot be extended to type D. However, forthcoming work of the second author \cite{forthcoming} gives a straightforward test for positivity in the type D Grassmannian $\operatorname{OGr}(n,2n)$. 

\subsection*{Previous works} Karpman showed that statements (1) and (2) of \Cref{thm:main} hold for Lagrangian Grassmannians, i.e.\ $\operatorname{SpFl}_{n;2n}$ \cite{karpman2018}. For a general reductive ($\RR$-split) algebraic group $G$ of simply-laced type, Lusztig showed that Lusztig positivity for a partial flag variety $G/P$ coincides with positivity of the coordinates from the canonical basis of a sufficiently large irreducible representation of $G$ \cite{lusztig1998positivity}.
However, due to the ``sufficiently large'' condition, this does not recover any of the aforementioned results of Lam, Talaska--Williams, Bloch--Karp, or Karpman. This abstract is based on a preprint \cite{BBEG}, which contains further details.


% Chevalier \cite[Example 10.2]{chevaliercounterexample} gave an example showing that the ``sufficiently large'' condition cannot be removed; we verify the example explicitly in \Cref{sect:typeD}.

\section{Pinnings}\label{sect:pinning}



Let $G$ be a connected, reductive, $\RR$-split linear algebraic group.  We often identify $G$ with its $\RR$-valued points.
A \emph{pinning} of $G$ is an additional set of choices for $G$ that is part of the input data for the definition of Lusztig positivity for $G$.

Fix a split maximal torus $T$ in $G$, and let $X$ be the character lattice of $T$.
Let $\Phi \subset X$ be the set of roots of the corresponding root system.
Fix a system of positive roots $\Phi^+$, and let $B_+$ be the corresponding Borel subgroup of $G$.
Let $B_-$ be the opposite Borel subgroup such that $B_+\cap B_{-} =T$.
Let $U_+$ and $U_-$ be the unipotent radicals of $B_+$ and of $B_-$, respectively.
Let $I$ be an indexing set for the set $\{\alpha_i : i\in I\}$ of simple roots in $\Phi^+$.
For every $i\in I$, fix a homomorphism $\phi_i:SL_2\rightarrow G$ such that in the induced map $\mathfrak{sl}_2 \to \mathfrak g$ of Lie algebras, the element $\left[\begin{smallmatrix} 0 & 1\\ 0 & 0\end{smallmatrix}\right]\in \mathfrak{sl}_2$ maps to a generator of the root space in $\mathfrak g$ of weight $\alpha_i$.
We then define homomorphisms $x_i: \RR \to U_+$, $y_i: \RR \to U_-$, and $\chi_i: \RR^* \to T$ by 
\[
x_i(m) :=\phi_i\left(\begin{bmatrix}
        1 & m\\
        0 & 1
    \end{bmatrix}\right),
\quad
y_i(m) :=\phi_i\left(\begin{bmatrix}
        1 & 0\\
        m & 1
    \end{bmatrix}\right),
\quad\text{and}\quad
\chi_i(t) := \phi_i\left(\begin{bmatrix}
        t & 0\\
        0 & t^{-1}
    \end{bmatrix}\right).
\]

\begin{defn}
The data $(T, B_+, B_-, \{x_i\}_{i\in I}, \{y_i\}_{i\in I})$ is called a \emph{pinning} for $G$.
\end{defn}

When multiple groups are in play, we write superscripts of the root system name, for example $T^{\Phi}$, $s_i^\Phi$, and $y_i^\Phi$, to distinguish between the pinnings of different groups.

 A pinning of $G$ identifies the reflection group $W$ of the root system $\Phi$ with the Weyl group $N_G(T)/T$, as follows.
For each $i\in I$, the simple reflection $s_i\in W$ is identified with $\dot{s}_iT$ where
\[
\dot{s}_i := \phi_i\left(\left[\begin{matrix}
    0 & -1 \\
    1 & 0
\end{matrix}\right]\right).
\]


 
In type $A_{n-1}$, when $G=GL_n$, we use the \emph{standard pinning} $(T^{A}, B_+^{A}, B_-^{A}, \{x_i^{A}\}_{i\in [n-1]}, \allowbreak \{y_i^{A}\}_{i\in [n-1]})$, defined as follows.
The torus $T^{A}$ consists of diagonal matrices with non-zero entries on the diagonal. The Borels $B_+^{A}$ and $B_-^{A}$ consist of upper and lower triangular invertible matrices, respectively. The set of simple roots is $\{\be_1-\be_2, \dotsc, \be_{n-1} -\be_n\}$. Accordingly, for each $i\in [n-1]$, the maps $\phi_i^{A}$ are given by
\begin{equation*}
    \phi_i^{A}\left(\begin{pmatrix}
        a & b\\
        c&d
    \end{pmatrix}\right) :=\begin{blockarray}{ccccccc}
& &  & i & i+1 & & \\
\begin{block}{c(cccccc)}
  & 1 &  &  & & & \\
  & & \ddots &  & &  & \\
  i& &  & a & b &  &  \\
  i+1& &  &c  & d &  &  \\
  & &  &  &  & \ddots  & \\
  &&&&&&1\\ 
\end{block} 
\end{blockarray}\hspace{5pt}, 
\end{equation*}
where unmarked off-diagonal matrix entries are $0$.  The Weyl group is the permutation group $\mathfrak S_n$ on $[n]$ with $s_i^A$ the transposition $(i\ i+1)$.

We now remark on notations we will use in subsequent sections. 
For a sequence $\mathbf i = (i_1, \dotsc, i_\ell)$ with entries in the indexing set $I$ of the simple roots, we denote by $\mathbf s_{\mathbf i}$ the element
\[
\mathbf s_{\mathbf i} := s_{i_1}\dotsm s_{i_\ell} \in W.
\]
When clear from context, we use $\mathbf s_{\mathbf i}$ to denote also the word $(s_{i_1}, \dotsc, s_{i_\ell})$.
Define the function $\mathbf y_{\mathbf i}: \RR^{\ell} \to G$ by
\[
\mathbf y_{\mathbf i}(a_1, \dotsc, a_\ell) := y_{i_1}(a_1) \dotsm y_{i_\ell}(a_\ell),
\]
and similarly define $\mathbf x_{\mathbf i}$, $\boldsymbol \chi_{\mathbf i}$, and $\dot{\mathbf s}_{\mathbf i}$.
The length $\ell$ of the sequence $\mathbf i$ is denoted $|\mathbf i|$.


% \subsection{Pinnings of \texorpdfstring{$\operatorname{Sp}_{2n}$}{Sp2n} and \texorpdfstring{$\operatorname{SO}_{2n+1}$}{SO2n+1}}
% We provide explicit descriptions of the pinnings of $\operatorname{Sp}_{2n}$ and $\operatorname{SO}_{2n+1}$ in \S\ref{sssec:Cpinning} and \S\ref{sssec:Bpinning}, respectively.
% One may verify that they are indeed valid pinnings from \cite{BilleyLakshmibai}, which provides explicit descriptions of Chevalley generators of the Lie algebras $\mathfrak{sp}_{2n}$ and $\mathfrak{so}_{2n+1}$.
% We record the key properties of these pinnings that we will use in \Cref{sec:compatibility}.

% \subsubsection{Type C pinning}\label{sssec:Cpinning}
% The pinning $(T^{C_{n}}, B_+^{C_{n}}, B_-^{C_{n}}, \{x_i^{C_{n}}\}_{i\in [n]}, \{y_i^{C_{n}}\}_{i\in [n]})$ of $\operatorname{Sp}_{2n}$ is defined as follows.
% The torus $T^{C_{n}}$ consists of matrices 

% \begin{equation*}
% \begin{pmatrix}
%     t_1&&&&&&&\\
%     &t_2&&&&&&\\
%     &&\ddots&&&&&\\
%     &&&t_n&&&&\\
%     &&&&t_n^{-1}&&&\\
%     &&&&&\ddots&&\\
%     &&&&&&t_2^{-1}&\\
%     &&&&&&& t_1^{-1}
% \end{pmatrix},
% \end{equation*}
% where $t_i\in \mathbb{R}^*$ for $i\in [n]$ and all off-diagonal entries are $0$.
% The Borels $B_+^{C_{n}}$ and $B_-^{C_{n}}$ consist of upper and lower triangular matrices in $\operatorname{Sp}_{2n}$, respectively, with nonzero entries on the diagonal. The set of simple roots is $\{\be_1-\be_2, \dotsc, \be_{n-1}-\be_n, 2\be_n\}$. Accordingly, for $i\in [n-1]$, the map $\phi_i^{C_{n}}$ is given by
% \begin{equation*}
%     \phi_i^{C_{n}}\left(\begin{pmatrix}
%         a&b\\
%         c&d
%     \end{pmatrix}\right)=\begin{blockarray}{cccccccccc}
% & &  & i & i+1 & & 2n-i& 2n-i+1 & &\\
% \begin{block}{c(ccccccccc)}
%   & 1 &  &  & & & & & &\\
%   & & \ddots &  & &  & & & & \\
%   i& &  & a & b &  & & & & \\
%   i+1& &  &c  & d &  &  & & & \\
%   & &  &  &  & \ddots  & & & &  \\
%   2n-i&&&&&&a&b&&\\
%   2n-i+1&&&&&&c&d&&\\
%  &&&&&&&&\ddots&\\
%  &&&&&&&&&1\\
% \end{block} 
% \end{blockarray}\hspace{5pt}, 
% \end{equation*}
% where all unmarked off-diagonal entries are $0$.
% For $i=n$, the map $\phi_n^{C_n}$ is given by
% \begin{equation*}
%     \phi_n^{C_{n}}\left(\begin{pmatrix}
%         a&b\\
%         c&d
%     \end{pmatrix}\right)=\begin{blockarray}{ccccccc}
% & &  & n & n+1 & & \\
% \begin{block}{c(cccccc)}
%   & 1 &  &  & & & \\
%   & & \ddots &  & &  & \\
%   n& &  & a & b &  &  \\
%   n+1& &  &c  & d &  &  \\
%   & &  &  &  & \ddots  & \\
%   &&&&&&1\\ 
% \end{block} 
% \end{blockarray}\hspace{5pt}, 
% \end{equation*}
% where all unmarked off-diagonal entries are $0$.

% \subsubsection{Type B pinning}\label{sssec:Bpinning}

% The pinning $(T^{B_{n}}, B_+^{B_{n}}, B_-^{B_{n}}, \{x_i^{B_{n}}\}_{i\in [n]}, \{y_i^{B_{n}}\}_{i\in [n]})$ of $\operatorname{SO}_{2n+1}$ is defined as follows.
% The torus $T^{B_{n}}$ consists of matrices 
% \begin{equation*}
% \begin{pmatrix}
%     t_1&&&&&&&&\\
%     &t_2&&&&&&&\\
%     &&\ddots&&&&&&\\
%     &&&t_n&&&&&\\
%     &&&&1&&&&\\
%     &&&&&t_n^{-1}&&&\\
%     &&&&&&\ddots&&\\
%     &&&&&&&t_2^{-1}&\\
%     &&&&&&&& t_1^{-1}
% \end{pmatrix},
% \end{equation*}
% \noindent where $t_i\in \mathbb{R}^*$ for $i\in [n]$ and all off-diagonal entries are $0$.
% The Borels $B_+^{B_{n}}$ and $B_-^{B_{n}}$ consist of upper and lower triangular matrices in $\operatorname{SO}_{2n+1}$, respectively, with nonzero entries on the diagonal. The set of simple roots is $\{\be_1-\be_2, \dotsc, \be_{n-1}-\be_n, \be_n\}$.
% Accordingly, for $i\in [n-1]$, the map $\phi_i^{B_{n}}$ is given by

% \begin{equation*}
%     \phi_i^{B_{n}}\left(\begin{pmatrix}
%         a&b\\
%         c&d
%     \end{pmatrix}\right)=\begin{blockarray}{cccccccccc}
% & &  & i & i+1 & & 2n-i+1& 2n-i+2 & &\\
% \begin{block}{c(ccccccccc)}
%   & 1 &  &  & & & & & &\\
%   & & \ddots &  & &  & & & & \\
%   i& &  & a & b &  & & & & \\
%   i+1& &  &c  & d &  &  & & & \\
%   & &  &  &  & \ddots  & & & &  \\
%   2n-i+1&&&&&&a&b&&\\
%   2n-i+2&&&&&&c&d&&\\
%  &&&&&&&&\ddots&\\
%  &&&&&&&&&1\\
% \end{block} 
% \end{blockarray}\hspace{5pt}, 
% \end{equation*}
% where all unmarked off-diagonal entries are $0$.
% For $i=n$, the map $\phi_n^{B_n}$ is determined by 

% \begin{equation*}
%     \phi_n^{B_{n}}\left(\begin{pmatrix}
%         t&0\\
%         0&t^{-1}
%     \end{pmatrix}\right)=\begin{blockarray}{cccccccc}
% & &  & n & n+1 & n+2 & & \\
% \begin{block}{c(ccccccc)}
%   & 1 &  &  & & & & \\
%   & & \ddots &  & & &  & \\
%   n& &  & t^2 &  & &  &  \\
%   n+1& &  &  & 1 &  & &  \\
%   n+2 & & & & & t^{-2} & \\
%   & &  &  & & & \ddots  & \\
%   &&&&&&&1\\ 
% \end{block} 
% \end{blockarray}\hspace{5pt},
% \end{equation*}

% \begin{equation*}
%     \phi_n^{B_{n}}\left(\begin{pmatrix}
%         1&m\\
%         0&1
%     \end{pmatrix}\right)=\begin{blockarray}{cccccccc}
% & &  & n & n+1 & n+2 & & \\
% \begin{block}{c(ccccccc)}
%   & 1 &  &  & & & & \\
%   & & \ddots &  & & &  & \\
%   n& &  & 1 & \sqrt{2}m & m^2&  &  \\
%   n+1& &  &  & 1 & \sqrt{2}m & &  \\
%   n+2 & & & & & 1 & \\
%   & &  &  & & & \ddots  & \\
%   &&&&&&&1\\ 
% \end{block} 
% \end{blockarray}\hspace{5pt},
% \end{equation*}
% and

% \begin{equation*}
%     \phi_n^{B_{n}}\left(\begin{pmatrix}
%         1&0\\
%         m&1
%     \end{pmatrix}\right)=\begin{blockarray}{cccccccc}
% & &  & n & n+1 & n+2 & & \\
% \begin{block}{c(ccccccc)}
%   & 1 &  &  & & & & \\
%   & & \ddots &  & & &  & \\
%   n& &  & 1 &  & &  &  \\
%   n+1& &  & \sqrt{2}m & 1 & & &  \\
%   n+2 & & & m^2 & \sqrt{2}m & 1 & \\
%   & &  &  & & & \ddots  & \\
%   &&&&&&&1\\ 
% \end{block} 
% \end{blockarray}\hspace{5pt}, 
% \end{equation*}
% where all unmarked off-diagonal entries are $0$.  

% \subsection{Pinnings of \texorpdfstring{$\operatorname{Sp}_{2n}$}{Sp2n} and \texorpdfstring{$\operatorname{SO}_{2n+1}$}{SO2n+1}}
% \label{sec:compatibility}

To avoid repeated arguments, we shall often use the following general setup. The function $\psi$ is an extra piece of combinatorial data that plays a crucial role in \Cref{condition1}.

\begin{setup}\label{setup}
Let $G$ be a connected, reductive, $\RR$-split linear algebraic group with a fixed pinning $(T^\Phi,B_+^\Phi,B_-^\Phi,\{x_i^\Phi\}_{i\in I}, \{y_i^\Phi\}_{i\in I})$ with simple roots indexed by $I$ in the root system $\Phi$.
Let $\iota: G \hookrightarrow GL_N$ be an embedding, and fix a function $\psi: I \to \{\text{sequences in $[N-1]$}\}$.  We write $\psi$ also for the function $\{\text{sequences in $I$}\} \to \{\text{sequences in $[N-1]$}\}$ defined by
\[
(i_1, \dotsc, i_\ell) \mapsto \big(\text{the concatenation of }\psi(i_1),\dotsc, \psi(i_\ell)\big).
\]
\end{setup}







\section{Lusztig positivity and the proof of (3)\texorpdfstring{$\implies$}{=>}(1)\texorpdfstring{$\implies$}{=>}(2)}\label{sect:Luspositivity}

\subsection{Lusztig's total positivity}\label{subsect:Luspositivity}
Let $G$ be a connected, reductive, $\RR$-split linear algebraic group with a fixed pinning $(T,B_+,B_-,\{x_i\}_{i\in I}, \{y_i\}_{i\in I})$.
We recall Lusztig's definition of total positivity for $G$.

\begin{defn}\label{defn:LusztigPos}
For a sequence $\mathbf i$ in $I$ such that $\mathbf s_{\mathbf i}$ is a reduced expression for the longest element $w_0\in W$,
define $U_-^{>0}$ (resp.\ $U_-^{\geq 0}$) to be the image $\mathbf y_{\mathbf i}(\RR^{|\mathbf i|}_{>0})$ (resp.\ $\mathbf y_{\mathbf i}(\RR^{|\mathbf i|}_{\geq0})$),
and similarly define $U_+^{>0}$ and $U_+^{\geq 0}$ in terms of $\mathbf{x}_{\mathbf i}$.
Define $T^{>0}$ to be the subgroup of the $\RR$-split torus $T$ generated by the elements $\chi(t)$ for $t\in \RR_{>0}$ and $\chi: \RR^* \to T$ a cocharacter of $T$.
Define the \emph{positive} (resp.\ \emph{nonnegative}) \emph{part} of $G$ to be
\[
G^{>0} := U_-^{>0}T^{>0}U_+^{>0} \quad\text{(resp.\ $G^{\geq 0} := U_-^{\geq0}T^{> 0}U_+^{\geq0}$)}.
\]
\end{defn}

The sets $\mathbf y_{\mathbf i}(\RR^{|\mathbf i|}_{>0})$ and $\mathbf y_{\mathbf i}(\RR^{|\mathbf i|}_{\geq0})$ depend only on the element $\mathbf s_{\mathbf i}\in W$ as long as $\mathbf{s}_\mathbf i$ is a reduced expression \cite{lusztig1994positivity}.
When $G = GL_n$ with the standard pinning, it is a classical result \cite{Cryer73, Cryer76}
that $GL_n^{>0}$ (resp.\ $GL_n^{\geq 0}$) as defined here is the space of invertible matrices with all positive (resp.\ nonnegative) minors.


 
For a parabolic subgroup $P \subset G$ containing $B_+$, let $\pi: G \to G/P$ be the projection map.
For $S\subseteq G/P$, we denote by $\overline S$ its closure in the Euclidean topology on $(G/P)(\RR)$.

\begin{defn} Define the \emph{positive} (resp.\ \emph{nonnegative}) \emph{part} of the partial flag $G/P$ to be
\[
(G/P)^{>0} := \pi(G^{>0}) \quad\text{(resp.\ $(G/P)^{\geq0} := \overline{\pi(G^{>0})}$)}.
\]
\end{defn}

We caution that although $G^{\geq 0}$ is the closure of $G^{>0}$ \cite{lusztig1994positivity}, the image $\pi(G^{\geq 0})$ may be strictly contained in $(G/B_+)^{\geq 0}$, since $\pi: G \to G/{B_+}$ may not be proper and so we may not in general conclude that $(G/P)^{\geq 0}=\pi(G^{\geq 0})$.
However, note that the projection map $G/{B_+} \to G/P$ is proper, and hence $(G/P)^{\geq 0}$ is the image of $(G/B_+)^{\geq 0}$. For a linear subspace $L\subset \mathbb{R}^{N}$, let $L^{\perp,\omega}$ denote the orthogonal complement of $L$ with respect to the symplectic form $\omega$, and similarly for the symmetric form $Q$. One can show the following: 

\begin{lem}\label{cor:pinning3}
For a subset $K \subseteq [n]$, we have
\begin{align*}
\operatorname{Sp}_{2n}/P_J^C &= \{L_\bullet \in \operatorname{Fl}_{K\cup (2n-K);2n} : L_{i} = L_j^{\perp,\omega} \text{ if $\dim L_i + \dim L_j = 2n$}\} \\
&\simeq  \operatorname{SpFl}_{K;2n}, \quad\text{and}\\
\operatorname{SO}_{2n+1}/P_J^B &= \{L_\bullet \in \operatorname{Fl}_{K\cup (2n+1-K);2n+1} :  L_{i} = L_j^{\perp,Q}  \text{ if $\dim L_i + \dim L_j = 2n+1$}\} \\ &\simeq  \operatorname{SOFl}_{K;2n+1}.
\end{align*}
Consequently, we have $(\operatorname{Sp}_{2n}/P_J^C) \cap \operatorname{Fl}_{K\cup (2n-K);2n}^{\Delta >0} \simeq \operatorname{SpFl}_{K;2n}^{\Delta>0}$ and $(\operatorname{SO}_{2n+1}/P_J^B) \cap \operatorname{Fl}_{K\cup (2n+1-K);2n+1}^{\Delta >0} \simeq \operatorname{SOFl}_{K;2n+1}^{\Delta>0}$, and similarly with $\Delta \geq 0$ in place of $\Delta >0$.
\end{lem}


\subsection{Proof of (1)\texorpdfstring{$\implies$}{=>}(2)}
We work now in the general setting of \Cref{setup}, where $\iota: G \hookrightarrow GL_N$ is an embedding and $\psi$ maps sequences in $I$ to sequences in $[N-1]$. Let $G$ have root system $\Phi$. 

\begin{defn}\label{condition1}
We say that $(\iota,\psi)$ has \emph{property ($\dagger$)} if the following are satisfied.
\begin{enumerate}
\item[($\dagger1$)] For every $i\in I$, we have $\dot s_i^\Phi = \dot{\mathbf s}_{\psi(i)}^A$, and we have $y_i^\Phi(a) = \mathbf y_{\psi(i)}^A(f_1(a), \dotsc, f_{|\psi(i)|}(a))$ for some sequence $(f_1, \dotsc, f_{|\psi(i)|})$ of differentiable functions $f_j: \RR \to \RR$ such that:

\begin{itemize}
    \item [$\bullet$] $f_j(\RR_{>0}) \subseteq \RR_{>0}$,
and similarly for $x_i^\Phi$ and $\chi_i^\Phi$, 
    \item [$\bullet$] $f_j(\RR_{< 0}) \subseteq \RR_{< 0}$ (so $f_j(\RR^*) \subset \RR^*$),
        \item[$\bullet$] $\lim\limits_{a\to+\infty}f_j(a)=+\infty$ for at least one $j\in\psi(i)$, and
        \item[$\bullet$] $\lim\limits_{a\to-\infty}f_{j}(a)=-\infty$ for at least one $j\in\psi(i)$.
    \end{itemize}
\item[($\dagger2$)] For some sequence $\mathbf i$ in $I$ such that ${\mathbf s}^{\Phi}_\mathbf i$ is a reduced word for the longest element $w_0\in W^\Phi$, the word ${\mathbf s}_{\psi(\mathbf i)}^A$ is a reduced word for the longest element of $W^A$. 
\end{enumerate}
\end{defn}

Condition \hyperref[condition1]{($\dagger$)} asserts that $(\iota,\psi)$ is compatible with the type A flag variety in that $B_+^\Phi \subseteq B_+^A$. In particular, if $(\iota, \psi)$ satisfies \hyperref[condition1]{($\dagger$)} and $P$ is a parabolic subgroup of $GL_N$ containing $B_+^A$, then $P \cap G$ is a parabolic subgroup of $G$ containing $B_+^\Phi$. Note that in the definition above, $y_i^A$ denotes a map into $GL_{N}$, where $N$ is implicit from the context. We always take $N=2n$ for type C and $N=2n+1$ for type B.

Whenever we use $\iota: \operatorname{Sp}_{2n} \hookrightarrow GL_{2n}$, we fix $\psi$ to be the function $\psi(i) = (i,2n-i)$ for $i\in [n-1]$, and $\psi(n) = n$. Similarly, whenever we use $\iota: \operatorname{SO}_{2n+1} \hookrightarrow GL_{2n+1}$, we fix $\psi$ to be the function $\psi(i) = (i,2n+1-i)$ for $i\in [n-1]$, and $\psi(n) = (n,n+1,n)$. 



\begin{lem}\label{lem:pinning1}

There exist pinnings $(T^C, B_+^C, B_-^C, \{x_i^C\}_{i\in [n]}, \{y_i^C\}_{i\in [n]})$ of $\operatorname{Sp}_{2n}$ and $(T^B, B_+^B, \allowbreak B_-^B, \{x_i^B\}_{i\in [n]}, \allowbreak \{y_i^B\}_{i\in [n]})$ of $\operatorname{SO}_{2n+1}$, respectively, which satisfy \hyperref[condition1]{($\dagger$)}. 

% such that, for $i\in[n-1]$, $\chi_i^C(t) = \chi_i^A(t)\chi_{2n-i}^A(t)$, $y_i^C(m) =y_i^A(m)y_{2n-i}^A(m)$, and $\dot s_i^C =\dot s_i^A \dot s_{2n-i}^A$, and moreover, $\chi_n^C(t) = \chi_n^A(t)$, $y_n^C(m) = y_n^A(m)$, and $\dot s_n^C = \dot s_n^A$.

% Similarly, there exists a pinning $(T^B, B_+^B, B_-^B, \{x_i^B\}_{i\in [n]}, \{y_i^B\}_{i\in [n]})$ of $\operatorname{SO}_{2n+1}$ such that, for $i\in[n-1]$, $\chi_i^B(t) = \chi_i^A(t)\chi_{2n+1-i}^A(t)$, $y_i^B(m) =y_i^A(m)y_{2n+1-i}^A(m)$, and $\dot s_i^B =\dot s_i^A \dot s_{2n+1-i}^A$, and moreover, $\chi_n^B(t) = \chi_n^A(t^2)\chi_{n+1}^A(t^2)$, $y_n^B(m) = y_n^A(\frac{m}{\sqrt{2}})y_{n+1}^A(\sqrt{2}m)y_n^A(\frac{m}{\sqrt{2}})$, and $\dot s_n^B = \dot s_n^A \dot s_{n+1}^A \dot s_n^A$.



\end{lem}

\begin{proof}[Proof sketch]
We can choose $y_i^C(m) =y_i^A(m)y_{2n-i}^A(m)$ and $y_i^B(m) =y_i^A(m)y_{2n+1-i}^A(m)$ for $i\in [n-1]$, and $y_n^C(m) = y_n^A(m)$ and $y_n^B(m) = y_n^A(\frac{m}{\sqrt{2}})y_{n+1}^A(\sqrt{2}m)y_n^A(\frac{m}{\sqrt{2}})$. These choices satisfy condition \hyperref[condition1]{($\dagger1$)}. Condition \hyperref[condition1]{($\dagger2$)} can be verified directly on any choice of reduced word for $w_0^C$ and for $w_0^B$. (Since $W^C=W^B$, we actually have $w_0^C=w_0^B$. However, due to the different $\psi$ for $\operatorname{Sp}_{2n} \hookrightarrow GL_{2n+1}$ and $\operatorname{SO}_{2n+1} \hookrightarrow GL_{2n+1}$, there are still two statements to verify here.) 
\end{proof}

% One can directly verify the following proposition for any fixed choice of expression for $w_0^\Phi$, which suffices for the proof of \Cref{thm:main}. 
% \begin{prop}
% \label{cor:preserveslongestword}
%     If $\bv$ is a reduced expression for $w_0^\Phi\in W^\Phi$, then $\psi(\bv)$ is a reduced expression for $w_0^A \in W^A$. 
% \end{prop}


% In the lemma above, the notation $y_i^A$ denoted a map into either $GL_{2n}$ or $GL_{2n+1}$, depending on context, and likewise for $\chi_i^A$ and $\dot s_i^A$.  We will continue this abuse of notation, as we trust that this ambiguity will cause no confusion.




\begin{prop}\label{prop:dagger}
Suppose $(\iota, \psi)$ has property \hyperref[condition1]{($\dagger$)}.
Then, we have the following.
\begin{enumerate}[label = (\alph*)]
\item $G^{>0} \subseteq GL_N^{>0}$.
\item For $J\subseteq [N-1]$, let $P^A_J \supseteq B_+^A$ be the parabolic subgroup of $GL_N$, so that $GL_N/P^A_J = \operatorname{Fl}_{K;N}$ where $K = [N-1]\setminus J$. Let $P = P^A_J \cap G$.  Then, we have that
\[
(G/P) \cap \operatorname{Fl}_{K;N}^{\Delta \geq 0} 
= \overline{\Big( (G/P) \cap \operatorname{Fl}_{K;N}^{\Delta >0} \Big)}.
\]
\end{enumerate}
\end{prop}

% \begin{proof}
% The property \hyperref[condition1]{($\dagger$)} implies that $(U_-^\Phi)^{>0}\subseteq(U_-^A)^{>0}$, $(T^\Phi)^{>0}\subseteq(T^A)^{>0}$, and $(U_+^\Phi)^{>0}\subseteq(U_+^A)^{>0}$, from which the statement (a) follows.  
% For the statement (b), let $p$ be a point in $(G/P) \cap \operatorname{Fl}_{K;N}^{\Delta \geq 0}$.
% We exhibit $p$ as a limit of points in $(G/P) \cap \operatorname{Fl}_{K;N}^{\Delta > 0}$, as follows.  Let $K = \{k_1, \ldots, k_j\}$.

% For an $N\times N$ matrix $M$, let $L_\bullet(M)$ denote the flag of subspaces whose $i$-th subspace is the span of the first $k_i$ columns of $M$.
% By definition, there is an $N \times N$ matrix $M$ in $\iota(G)$ such that $p= L_\bullet(M)$, and such that for every $1\leq i \leq j$, the first $k_i$ columns of $M$ have all nonnegative maximal minors, at least one of which is positive by invertibility of $M$. 
% By the Cauchy--Binet formula, for any $A\in GL_N^{>0}$ and $1\leq i\leq j$, the first $k_i$ columns of $AM$ have all positive maximal minors, i.e.\ $L_\bullet(AM) \in \operatorname{Fl}_{K;N}^{\Delta > 0}$.  Thus, for any curve $A(t): \RR_{>0} \to G^{>0} \subseteq GL_N^{>0}$ such that $\lim_{t\to 0} A(t)$ equals the identity, the curve $L_\bullet(A(t)M)$ lies in $(G/P) \cap \operatorname{Fl}_{K;N}^{\Delta >0}$ and limits to $p$ as $t\to 0$.
% Such a curve exists since $G^{\geq 0}$ is the closure of $G^{>0}$, or explicitly, one may take $A(t) = \mathbf y_{\mathbf i}^{\Phi}(t,\dotsc, t) \boldsymbol \chi_{\mathbf i}^{\Phi}(1, \dotsc, 1) \mathbf x_{\mathbf i}^{\Phi}(t,\dotsc, t)$ where $\mathbf s_{\mathbf i} = w_0^\Phi$.
% \end{proof}

Applying this proposition to $\operatorname{Sp}_{2n}$ and $\operatorname{SO}_{2n+1}$ yields \Cref{thm:main} \ref{mainthm:pos}$\implies$\ref{mainthm:nonneg}:

\begin{proof}[Proof of \Cref{thm:main} \ref{mainthm:pos}$\implies$\ref{mainthm:nonneg}]
Since $(G/P)^{\geq 0}$ is the closure of $(G/P)^{>0}$, by \Cref{cor:pinning3}, it suffices to show that the Pl\"ucker nonnegative part is the closure of the Pl\"ucker positive part.
By \Cref{prop:dagger}(b), this follows from \Cref{lem:pinning1}.
 
\end{proof}

\subsection{Proof of (3)\texorpdfstring{$\implies$}{=>}(1)}
% We again work first in the general setting as stated in Setup~\ref{setup} involving an embedding $\iota: G \hookrightarrow GL_N$ and a map $\psi$.
% We prepare by recalling the following fact.

% \begin{lem}[{\cite[Proposition 5.2, Theorem 11.3]{marshrietsch2004}}]\label{lem:positiveparam}
% Suppose $\mathbf{s}_{\mathbf i}$ is a reduced expression for the longest element $w_0 \in W$ of length $\ell=\ell(w_0)$, and let $\mathring{\mathbf y}_{\mathbf i}: (\RR^*)^\ell \to G/B_+$ be the composition of the restriction of the map $\mathbf y_{\mathbf i}: \RR^\ell \to G$ to the torus $(\RR^*)^\ell$ with the projection map $G\to G/B_+$.  The map $\mathring{\mathbf y}_{\mathbf i}$ is an isomorphism onto a Zariski open subset of $G$.  In particular, the map $\mathring{\mathbf y}_{\mathbf i}$ has a dense image in $G/B_+$, and induces a bijection $(\RR_{>0})^\ell \overset\sim\to (G/B_+)^{>0}$.
% \end{lem}

Property \hyperref[condition1]{($\dagger$)}, together with {\cite[Proposition 5.2, Theorem 11.3]{marshrietsch2004}}, implies the following: 

\begin{prop}\label{prop:strongdagger}
Suppose the pair $(\iota, \psi)$ satisfies the property \hyperref[condition1]{($\dagger$)}. Then, one has
\[
(G/B_+)^{>0} = (G/B_+) \cap \operatorname{Fl}_{[N-1];N}^{\Delta >0}.
\]
\end{prop}

% \begin{proof}
% By \Cref{thm:BK} (also \cite[Theorem 4.11]{boretsky2023totally}), we may prove the statement with $\operatorname{Fl}_{[N-1];N}^{\Delta >0}$ replaced by $\operatorname{Fl}_{[N-1];N}^{>0}$.
% Fix a sequence $\mathbf i$ such that $\mathbf{s}_\mathbf{i}$ is a reduced expression for the longest element of $W^\Phi$, as in \hyperref[condition1]{($\dagger2$)}, and denote $U = \operatorname{image}(\mathring{\mathbf y}_{\mathbf i}^\Phi)$ and $V = \operatorname{image}(\mathring{\mathbf y}_{\psi(\mathbf i)}^A)$.
% The strengthening \hyperref[condition1strong]{($\dagger1'$)} implies that
% % the image of the map $\mathbb{R}^*\to(\mathbb{R}^*)^{|\psi(i)|}$ given by $a\mapsto(f_j(a))_{j\in \psi(i)}$ is closed for all $i\in I$. Therefore, we have that
% $U$ is a closed subset of $V$ under the embedding $G/B_+ \hookrightarrow \operatorname{Fl}_{[N-1];N}$ induced by $\iota: G \hookrightarrow GL_N$.
% As $U$ is dense in $G/B_+$ by \Cref{lem:positiveparam}, and as $G/B_+ \hookrightarrow \operatorname{Fl}_{[N-1];N}$ is a closed embedding, we find that $U$ is dense and closed in $(G/B_+) \cap V$, and hence $U = (G/B_+) \cap V$.
% The desired equality for the Lusztig positive part now follows from \Cref{lem:positiveparam}, since \hyperref[condition1]{($\dagger1$)} and \hyperref[condition1strong]{($\dagger1'$)} imply that $f_j^{-1}(\RR_{>0}) = \RR_{>0}$.
% \end{proof}

We now prove \Cref{thm:main} \ref{mainthm:consec}$\implies$\ref{mainthm:pos}.

\begin{proof}[Proof of \Cref{thm:main} \ref{mainthm:consec}$\implies$\ref{mainthm:pos}]
Let $n\geq 2$.
As the embeddings $\iota:\operatorname{Sp}_{2n} \hookrightarrow GL_{2n}$ and $\iota:\operatorname{SO}_{2n+1} \hookrightarrow GL_{2n+1}$ satisfy \hyperref[condition1]{($\dagger$)}, \Cref{defn:LusztigPos} and \Cref{prop:dagger}(a) together imply that $\operatorname{SpFl}_{K ; 2n}^{>0} \subseteq \operatorname{SpFl}_{K; 2n}^{\Delta>0}$ and $\operatorname{SOFl}_{K ; 2n+1}^{>0} \subseteq \operatorname{SOFl}_{K; 2n+1}^{\Delta>0}$ for any $K\subseteq [n]$.
It remains to show the reverse inclusions when $K = \{k, k +1, \dotsc, n\}$ for some $k\in [n]$.

We first reduce to the case $K = [n]$ as follows.
We show this reduction for the $\operatorname{Sp}_{2n}$ case; the case of $\operatorname{SO}_{2n+1}$ is similar.
By \Cref{cor:pinning3}, a point in $\operatorname{SpFl}_{K; 2n}^{\Delta>0}$ is a point $L_\bullet$ in $(\operatorname{Sp}_{2n}/P_K^C)\cap \operatorname{Fl}_{K\cup (2n-K); 2n}^{\Delta >0}$. By our assumption on $K$, \Cref{thm:BK} implies $ \operatorname{Fl}_{K\cup (2n-K); 2n}^{\Delta >0} =  \operatorname{Fl}_{K\cup (2n-K); 2n}^{>0}$.  
 Since by definition $Fl^{>0}_{
K\cup(2n-K);2n}$ is the
projection of $Fl^{>0}_{
[2n-1];2n}$, we may extend the flag $L_\bullet$ to a flag $\widetilde L_\bullet$ in $Fl^{>0}_{
[2n-1];2n} = Fl^{\Delta>0}_{
[2n-1];2n}$. One can show that there exists such an extension with $\widetilde L_\bullet$ in $(\operatorname{Sp}_{2n}/B_+) \cap \operatorname{Fl}_{[2n-1]; 2n}^{\Delta>0}$. 
The projection of $\widetilde L_\bullet$ to $\operatorname{Fl}_{[n];2n}$ is a point in $\operatorname{SpFl}_{[n];2n}^{\Delta >0}$. Hence, if Lusztig positivity and Pl\"ucker positivity agrees for the case of $K = [n]$, then $\widetilde L_\bullet \in (\operatorname{Sp}_{2n}/B)^{>0}$ so that its projection $L_\bullet$ is Lusztig positive also.

The case of $K = [n]$ follows from \Cref{prop:strongdagger} and \Cref{cor:pinning3} since, by \Cref{lem:pinning1}, the embeddings $\operatorname{Sp}_{2n} \hookrightarrow GL_{2n}$ and $\operatorname{SO}_{2n+1} \hookrightarrow GL_{2n+1}$ satisfy the property \hyperref[condition1]{$(\dagger$)}.
\end{proof}

% \begin{rem}
% For situations where the functions $f_j$ appearing in \hyperref[condition1]{($\dagger1$)} are not simply linear functions, one condition that implies \hyperref[condition1strong]{($\dagger1'$)} is as follows.
% \begin{itemize}
% \item[($\dagger1''$)]\label{condition1strongstrong}
% For each $i\in I$, the functions $f_j: \RR \to \RR$ appearing in \hyperref[condition1]{($\dagger1$)} further satisfy the property that:
%     \begin{itemize}
%         \item[$\bullet$] $f_j(\RR_{< 0}) \subseteq \RR_{< 0}$ (so $f_j(\RR^*) \subset \RR^*$),
%         \item[$\bullet$] $\lim\limits_{a\to+\infty}f_j(a)=+\infty$ for at least one $j\in\psi(i)$, and
%         \item[$\bullet$] $\lim\limits_{a\to-\infty}f_{j}(a)=-\infty$ for at least one $j\in\psi(i)$.
%     \end{itemize}
% \end{itemize}
% One can check that \hyperref[condition1strongstrong]{($\dagger1''$)} implies \hyperref[condition1strong]{($\dagger1'$)} through a simple application of the Bolzano--Weierstrass Theorem. The second and third bullet points cannot be dropped, as shown by counterexamples such as $f_j(a)=\arctan(a)$, $\forall j\in\psi(i)$.
% \end{rem}



\section{Examples, and the proof of (2)\texorpdfstring{$\implies$}{=>}(3)}\label{sect:constructions}


\subsection{Proof of (2)\texorpdfstring{$\implies$}{=>}(3)}



Let $G$ be either $\operatorname{Sp}_{2n}$ or $\operatorname{SO}_{2n+1}$.
As we have shown  \ref{mainthm:consec}$\implies$\ref{mainthm:pos} $\implies$ \ref{mainthm:nonneg}, in particular for $K = [n]$, we have that $(G/B_+)^{\geq 0} = (G/B_+)^{\Delta \geq 0}$.
We now provide examples to demonstrate the contrapositive of \ref{mainthm:nonneg}$\implies$\ref{mainthm:consec} for $\operatorname{Sp}_{2n}$ (type C). Similar examples can be constructed for $\operatorname{SO}_{2n+1}$ (type B).
For each relevant $K\subseteq [n]$ and $J = [n]\setminus K$, we will find a Pl\"ucker nonnegative point in $G/P_J$ that does not extend to a Pl\"ucker nonnegative point in $G/B_+$.
Such a point cannot be in the Lusztig nonnegative part $(G/P_J)^{\geq 0}$, since $(G/P_J)^{\geq 0}$ is the projection of $(G/B_+)^{\geq 0} = (G/B_+)^{\Delta \geq 0}$. We begin from the following observation:

\begin{lem}\label{lem:smallcases}
For $n\geq 2$, define $[L_{1;2n}] = \begin{bmatrix} 1 & 0 &  \dotsc & 0 & 1\end{bmatrix}\in \RR^{2n}$.
Then, the span of $[L_{1;2n}]$ is a point in $\mathrm{SpFl}_{1;2n}^{\Delta \geq 0}$ that does not extend to a point in $\mathrm{SpFl}_{1,2;2n}^{\Delta \geq 0}$.
\end{lem}


 
We now demonstrate how to construct a point in $\mathrm{SpFl}_{K;2n}^{\Delta \geq 0}\setminus  \mathrm{SpFl}_{K;2n}^{\geq 0}$ for all $n\geq 2$ and $K$ not of the form $\{k, k+1, \dotsc, n\}$ for some $1 \leq k \leq n$. 
Fix such a subset $K$, and denote $g$ to be the integer satisfying $g\notin K$ and $\{g+1, g+2, \dotsc, n\} \subset K$, i.e.\ ``the first gap from the right.'' Denote $f = \max\{i \mid i\in K \text{ and } i < g\}$. We consider three cases.


\noindent \textbf{Case (i).} Suppose $g=n$. The $f\times 2n$ matrix
\[
M = \begin{bmatrix}
I_{f-1} & 0 & 0 \\
0 & [L_{1;2(n-f+1)}] & 0
\end{bmatrix},
\]
where $I_{f-1}$ is the $(f-1)\times (f-1)$ identity matrix,
represents a flag in $\operatorname{SpFl}_{K;2n}^{\Delta\geq 0}$. If it could be extended to a flag in $\operatorname{SpFl}_{K\cup \{f+1\};2n}$, we would have a contradiction to \Cref{lem:smallcases}. Thus, $M$ provides the desired counterexample.

\noindent \textbf{Case (ii).} Suppose $g=n-1$. Let $\ell = n-f-2$.  The $n\times 2n$ matrix
\[
M = \begin{bmatrix}
I_{f-1} & \vline & 0 & \vline & 0& \vline & 0\quad \\
\hline
0 & \vline & 0 & \vline & \begin{matrix} 1 & 0 & 0 & 0 & 0 & 1 \end{matrix}& \vline & 0\quad\\
\hline
0 & \vline & (-1)^{\ell}I_{\ell}  & \vline&\begin{matrix} 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}& \vline & 0\quad \\
\hline
0 & \vline & 0  & \vline & \begin{matrix} 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\end{matrix} & \vline & 0&\quad 
\end{bmatrix}
\]
represents a flag in $\operatorname{SpFl}_{K;2n}^{\Delta\geq 0}$. Suppose it could be extended to a flag in $\operatorname{SpFl}_{[n];2n}$. Then, restricting our flag to the coordinates in the third block of columns of $M$ would yield a contradiction to \Cref{lem:smallcases} with $n=3$. Thus, $M$ provides the desired counterexample.



\noindent \textbf{Case (iii).} We use a construction from Bloch and Karp appearing in \cite[Proof of Theorem 1.1 (ii)$\implies$(iii)]{blochkarp2023}. Suppose $g<n-1$. The $n\times 2n$ matrix
\[
M = \begin{bmatrix}
0 & I_{f-1} & 0 &0 \\
B & 0 & 0 &0\\
0 & 0 & I_{n-f-3} &0\\
C & 0 & 0 & 0
\end{bmatrix}
\]
where
\[B = (-1)^{f-1}\begin{bmatrix}
1 & 0 & 0 & 1\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0
\end{bmatrix}
\quad\text{and}\quad  C = (-1)^{n-f-3}\begin{bmatrix} 0 &0 &0& 1\end{bmatrix},
\]
represents a flag in $\operatorname{SpFl}_{K;2n}^{\Delta \geq 0}$.  Considered as a point in $\operatorname{Fl}_{K;2n}^{\Delta \geq 0}$, it cannot be extended to a point in $\operatorname{Fl}_{[2n-1];2n}^{\Delta \geq 0}$. 


% \subsubsection{Type B examples}
% Since \ref{mainthm:nonneg}$\implies$\ref{mainthm:consec} fails for $B_2$ (see Remark~\ref{rem:B2}), we assume $n\geq 3$ throughout.
% Our examples will be constructed from the following observation.

% \begin{lem}\label{lem:Bsmallcases}
% For $n\geq 3$, let the subspace $L_{1;B_n} \subset \RR^{2n+1}$ be defined by
% \[
% L_{1;B_n} := 
% \begin{cases}
% \operatorname{span}(\be_1 + \sqrt{2}\be_{n+1}+\be_{2n+1}) & \text{if $n$ is odd}\\
% \operatorname{span}(\be_2 + \sqrt{2}\be_{n+1}+\be_{2n}) & \text{if $n$ is even}.
% \end{cases}
% \]
% Then, $L_{1;B_n}$ is a point in $\mathrm{SOFl}_{1;2n+1}^{\Delta \geq 0}$ that does not extend to a point in $\mathrm{SOFl}_{[n];2n+1}^{\Delta \geq 0}$.
% \end{lem}

% \begin{proof}
% In both cases (odd and even $n$), one verifies the isotropicity with respect to the symmetric form $Q$ defined in the introduction by $2\cdot1\cdot1 - \sqrt{2}^2 = 0$.
% Pl\"ucker nonnegativity is clear.
% We show more strongly that $L_{1;2n}$ does not extend to a point $L_\bullet$ in $\mathrm{SOFl}_{1,2,3;2n+1}^{\Delta \geq 0}$.
% Suppose for a contradiction otherwise.
% We treat the two cases, odd and even $n$, separately.

% When $n$ is odd, the 2-dimensional subspace $L_2$ in the flag $L_\bullet$ is the row-span of a matrix of the form 
% \[
% \begin{bmatrix}
% 1 & 0 &  \dotsc &0 & \sqrt{2}& 0 & \dotsc & 0 & 1\\
% 0 & x_2 &  \dotsc & x_n & x_{n+1}& x_{n+2} & \dotsc & x_{2n} & x_{2n+1}
% \end{bmatrix}.
% \]
% In order for $2\times 2$ minors to be nonnegative, we find $x_2 = \dotsb = x_{n} = x_{n+2} = \dotsb = x_{2n} = 0$, but then the isotropicity of the second row of the matrix demands $x_{n+1} = 0$, after which the isotropicity of the two rows with each other demands $x_{2n+1} = 0$.  This contradicts that $L_2$ is 2-dimensional.

% When $n$ is even, the 2-dimensional subspace $L_2$ in the flag $L_\bullet$ is the row-span of a matrix of the form
% \[
% \begin{bmatrix}
% 0 & 1 & 0 &  \dotsc &0 & \sqrt{2}& 0 & \dotsc & 0 & 1 & 0\\
% -a & 0 & 0 &  \dotsc &0 & b& 0 & \dotsc & 0 & c & d
% \end{bmatrix}.
% \]
% Here, several entries in the second row have been forced to be 0 by row-reduction or the nonnegativity of the 2$\times$2 minors.  Because the isotropicity of the two rows implies $2ad = b^2$ and $c = \sqrt{2}b$, we find that either $a$ or $d$ must be nonzero, since otherwise $a = b= c= d= 0$.  Without loss of generality, let us set $a = 1$, since the argument that follows is similar for the case of $d =1$ by symmetry.  Now, the 3-dimensional subspace $L_3$ in the flag $L_\bullet$ is the row-span of a matrix of the form
% \[
% \begin{bmatrix}
% 0 & 1 & 0 &  \dotsc &0 & \sqrt{2}& 0 & \dotsc & 0 & 1 & 0\\
% -1 & 0 & 0 &  \dotsc &0 & b& 0 & \dotsc & 0 & c & d\\
% 0 & 0 & x_3 &  \dotsc &x_n & x_{n+1}& x_{n+2} & \dotsc & x_{2n-1} & x_{2n} & x_{2n+1}\\
% \end{bmatrix}.
% \]
% For each $i \in \{3, \dotsc, n\} \cup \{n+2, \dotsc, 2n-1\}$, we have that $x_i= P_{12i} = -P_{1i(2n)}$, so that Pl\"ucker nonnegativity implies $x_i = 0$.  Then, the isotropicity of the third row implies $x_{n+1} = 0$, after which the isotropicity of the third row with the first and the second row implies $x_{2n} = 0$ and $x_{2n+1} = 0$, respectively.  This contradicts that $L_3$ is 3-dimensional.
% \end{proof}

% Let us write $[L_{1;B_n}]$ for the nonnegative matrix whose row-span is $L_{1;B_n}$.
% We make one more observation in preparation.

% \begin{lem}\label{lem:Bcornercase}
% The 2-dimensional subspace defined as the row-span of
% \[
% \begin{bmatrix}
% 1 & 2 & 2 & 2 & 2 & 1 & 0\\
% 0 & 1 & 2 & 2 & 2 & 2 & 1
% \end{bmatrix}
% \]
% is a point in $\mathrm{SOFl}_{1,2;7}^{\Delta\geq 0}$ that does not extend to a point in $\mathrm{SOFl}_{[3];7}^{\Delta\geq 0}$.
% \end{lem}

% \begin{proof}
% Suppose it extends, so that the 3-dimensional subspace in the extension is the row-span of a matrix of the form
% \[
% \begin{bmatrix}
% 1 & 1 & 0 & 0 & 0 & -1 & -1\\
% 0 & 1 & 2 & 2 & 2 & 2 & 1\\
% 0 & 0 & a & b & c & d & e
% \end{bmatrix}.
% \]
% The nonnegativity of the Pl\"ucker coordinates $P_{134}$ and $P_{145}$ implies $a\leq b \leq c$, whereas the nonnegativity of $P_{346}$ and $P_{456}$ implies $a \geq b \geq c$, so that $a= b = c$.
% Then, the isotropicity of the bottom row implies $2ac =b^2$, which implies $a = b= c = 0$.  Lastly, the isotropicity of the bottom row with the first and the second row implies $d = e$ and $d = 0$, respectively.
% \end{proof}

% We now demonstrate how to construct a point in $\mathrm{SOFl}_{J;2n+1}^{\Delta \geq 0}\setminus  \mathrm{SOFl}_{J;2n+1}^{\geq 0}$ for all $n\geq 3$ and $K$ not of the form $\{k, k+1, \dotsc, n\}$ for some $1 \leq k \leq n$.
% Given the Lemmas~\ref{lem:Bsmallcases} and \ref{lem:Bcornercase}, the arguments are nearly identical to the type C case, so we will omit details.

% Fix such a subset $K$, and denote $g$ to be the integer satisfying $g\notin K$ and $\{g+1, g+2, \dotsc, n\} \subset K$, i.e.\ ``the first gap from the right.''
% We have three cases:
% (i) $g = n$, (ii) $g = n-1$, and (iii) $g \leq n-2$.
% For all cases, denote $f = \max\{i \mid i\in K \text{ and } i < g\}$.  Note that $f+1 \notin K$.

% \smallskip
% \noindent \textbf{Case (i).}
% By assumption $f\leq n-1$.
% We further divide into two cases.  When $f = n-1$, consider the $k\times (2n+1)$ matrix
% \[
% M = \begin{bmatrix}
% I_{f-2} & 0 & 0 \\
% 0 & A & 0
% \end{bmatrix} \quad\text{where}\quad A= \begin{bmatrix}
% 1 & 2 & 2 & 2 & 2 & 1 & 0\\
% 0 & 1 & 2 & 2 & 2 & 2 & 1
% \end{bmatrix}.
% \]
% When $f \leq n-2$, so that $n-f+1 \geq 3$, consider the $f\times (2n+1)$ matrix
% \[
% M = \begin{bmatrix}
% I_{f-1} & 0 & 0 \\
% 0 & [L_{1;B_{n-f+1}}] & 0
% \end{bmatrix}.
% \]
% In both cases, the appropriate row-spans define a point in $\mathrm{SOFl}_{K;2n+1}^{\Delta\geq 0}$ that does not extend to a point in $\mathrm{SOFl}_{[n];2n+1}^{\Delta\geq 0}$.


% \smallskip
% \noindent \textbf{Case (ii).}
% By assumption $f \leq n-2$.  Let $\ell = n-f-2$.  The $n\times (2n+1)$ matrix
% \[
% M = \begin{bmatrix}
% I_{f-1} & \vline & 0 & \vline & 0& \vline & \quad0\quad  &\vline & 0\quad \\
% \hline
% 0 & \vline & 0 & \vline & \begin{matrix} 1 & 0 & 0 & \sqrt{2} & 0 & 0 & 1 \end{matrix}& \vline & \quad0\quad&\vline & 0\quad \\
% \hline
% 0 & \vline & (-1)^{\ell}I_{\ell}  & \vline& 0 & \vline & \quad0\quad&\vline & 0\quad \\
% \hline
% 0 & \vline & 0  & \vline & \begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0\end{matrix} & \vline & \quad0\quad &\vline & 0\quad 
% \end{bmatrix}
% \]
% provides the desired example.

% \smallskip
% \noindent \textbf{Case (iii).} In this case, the construction from Bloch and Karp, identical to the one recalled here in the type C case, provides the desired example.

% \begin{rem}\label{rem:B2}
% Let us treat the $B_2$ case.  We claim that every point in $\operatorname{SOFl}_{1;5}^{\Delta> 0}$ extends to a point in $\operatorname{SOFl}_{1,2;5}^{\Delta> 0}$.
% Let $L$ be the row-span of a positive matrix $[a \ b \ c \ d \ e]$.  For it to be isotropic, we have $2ae+c^2 = 2bd$.
% We may assume $a = 1$, and thus $e = bd - \frac{c^2}{2}$.
% Consider then the matrix
% \[
% \begin{bmatrix}
% 1 & b & c & d & bd-\frac{c^2}{2}\\
% 0 & 1 & 2x & 2x^2 & 2bx^2+d-{2}cx
% \end{bmatrix}.
% \]
% One verifies the isotropicity, and notes that every minor is a polynomial in $x$ whose leading term is positive.
% Hence, for a sufficiently large $x>0$, the matrix represents a point in $\operatorname{SOFl}_{1,2;5}^{\Delta> 0}$.
% \end{rem}




\section{Brief Comments on Type D}\label{sect:typeD}

In this section, we discuss why the methods used above in type B and C do not apply in type D. We take the type D Lie group $\operatorname{SO}_{2n}$ to be defined as the linear subgroup of $\operatorname{SL}_{2n}$ that preserves a non-degenerate symmetric bilinear form $Q$ on $\mathbb{R}^{2n}$.  Specifically, we have:
\[\operatorname{SO}_{2n}:=\{A\in \operatorname{SL}_{2n}(\mathbb{R})|A^tEA=E\},\]
where $E$ is the symmetric matrix associated with $Q$. 

\begin{prop}\label{prop:typeDpinning}
    There does not exist a choice of $E$ (which determines the embedding $\iota:\operatorname{SO}_{2n}\hookrightarrow\operatorname{SL}_{2n}$), a pinning $(T^D, B_+^D,B_-^D,\{x_i^D\},\{y_i^D\})$ of $\operatorname{SO}_{2n}$, and a map $\psi$ satisfying \hyperref[condition1]{($\dagger1$)} in \Cref{condition1}.
 
\end{prop}

Despite \Cref{prop:typeDpinning}, there are other methods for working with type D flag varieties. Upcoming work of the second author \cite{forthcoming} describes a set $\mathcal{I}$ of Pl\"ucker coordinates such that a point in the type D Grassmannian $\operatorname{OGr}(n,2n)$ (with a suitable form $E$) is positive if and only if the Pl\"ucker coordinates in $\mathcal{I}$ have a particular sign pattern; as one would expect from \Cref{prop:typeDpinning}, some Pl\"uckers must be negative.
% We first prove the following lemma.

% \begin{lem}\label{lem:typeDpinning}
%     Given a symmetric matrix $E$, which determines an embedding $\iota:\operatorname{SO}_{2n}\hookrightarrow\operatorname{SL}_{2n}$. If the maximal torus $T^D$ of $\operatorname{SO}_{2n}$ satisfies $\iota(T^D)\subseteq T^A$, then $E$ must be a monomial matrix.
% \end{lem}
% \begin{proof}
%     It is easier to work with the Lie algebra
%     \[\mathfrak{so}_{2n}=\{A\in \mathfrak{sl}_{2n}\mid AE+EA^t=0\}.\]
%     Let $t=\text{diag}(t_1,t_2,\dots,t_{2n})$ be a diagonal matrix in $\mathfrak{sl}_{2n}$. Then $t\in \mathfrak{so}_{2n}$ implies $tE+Et=0$, or $(t_i+t_j)E(i,j)=0$ for any $i,j$ (here $E(i,j)$ is the $(i,j)^\text{th}$ entry of $E$) Therefore, $t_i+t_j=0$ if $E(i,j)\neq 0$. We create a graph $G$ on $2n$ vertices, where we connect $i,j$ if the entry $E(i,j)\neq 0$. Note we add a self-loop to vertex $i$ in this graph if $E(i,i)\neq 0$. Note that $\dim(T^D)$ is at most the number of connected components of $G$ since all $t_i$'s in that connected component are the same up to a sign. Also, since $E$ is not degenerate, all vertices in $G$ have degree at least one. As a result, any connected component of size one must be a self loop, at some vertex $i$. This forces $t_i=0$. Assume $G$ has $a$ connected components with size $1$ and $b$ connected components with size $\geq 2$. Then $a+2b\leq 2n$, and $n=\dim(T^D)\leq b$. Both inequalities hold simultaneously if and only if $a=0$, $b=n$ and all connected components have size $2$. This implies $E$ is a monomial matrix that represents a permutation $w\in S_{2n}$, where $w$ is an involution with no fixed points.
% \end{proof}

% \begin{proof}[Proof of \Cref{prop:typeDpinning}]
%     Let $e_i^A\in \mathfrak{sl}_{2n}$ and $e_i^D\in \mathfrak{so}_{2n}$ be the Chevalley generators of the Lie algebras. If \hyperref[condition1]{($\dagger1$)} holds, where $y_i^D(a)=\mathbf{y}^A_{\psi(i)}(f_1(a), \dots, f_{|\psi(i)|}(a))=\prod_{j\in \psi(i)}y_{i_j}^A(f_j(a))$, then:
%     \[e_i^D=\lim_{a\to 0}\frac{y_i^D(a)-I}{a}=\lim_{a\to 0}\frac{\prod_{j\in \psi(i)}y_{i_j}^A(f_j(a))-I}{a}=\lim_{a\to 0}\frac{\prod_{j\in \psi(i)}(I+f_j(a)\cdot e^A_{i_j})-I}{a}=\sum_{j\in I}f'_j(0)e_{i_j}^A.\]
%     % Note that since $y_i^{D}$ gives a smooth map from $\RR$ to $\operatorname{SO}_{2n}\subseteq \operatorname{GL}_{2n}$ and $y_i^D(0) = I$, it follows that the sum on the right-hand side is well-defined, even if $f_j'(0)$ is not defined for some $j$.
%     Therefore, $e_i^D$ is a sub-diagonal matrix, meaning all non-zero entries lies on the diagonal directly above the main diagonal. Let the support of $e_i^D$, denoted as $\text{supp}(e_i^D)$, be the set of entries in $e_i^D$ that are non-zero. The contradiction arises as follows:
%     \begin{enumerate}
%         \item Note that $e_i^DE$ is a skew-symmetric matrix since $e_i^D\in \mathfrak{so}_{2n}$. If $|\text{supp}(e_i^D)|=1$, then $e_i^DE$ would have exactly one non-zero entry,
%         given that $E$ is a monomial matrix by \Cref{lem:typeDpinning}. This is impossible for a skew-symmetric matrix, as it would violate the property of having an even number of non-zero entries. Therefore, we conclude that $|\text{supp}(e_i^D)|\geq 2$.
%         \item For any $i\neq j$, $\text{supp}(e_i^D)\cap\text{supp}(e_j^D)=\emptyset$. To see this, let $t=\text{diag}(t_1,\dots,t_{2n})$ be a generic element in $T^D$. If they did share entry $(k,k+1)$, then since $e_i^D$ and $e_j^D$ are both eigenvectors of $T^D$, they both must belong to the $\lambda=(t_k-t_{k+1})$ eigenspace in order to satisfy $[t,e_i^D]=\lambda e_i^D$ in position $(k,k+1)$ and the analogous equation for $e_j^D$. This contradicts the fact that $e_i^D$ and $e_j^D$ belong to different $T^D$-eigenspaces. 
        
        
%         \item Now, all $\text{supp}(e_i^D)$ for $i\in [n]$ are disjoint and each has size at least two. However, there are only $2n-1$ entries on the sub-diagonal. This is a contradiction.\qedhere
%     \end{enumerate}
    
%     \end{proof}

% \begin{rem}
%     We conjecture that \Cref{prop:typeDpinning} should hold for any embedding $\iota: \operatorname{SO}_{2n}\hookrightarrow\operatorname{SL}_N$.
% \end{rem}



% \subsection{Pfaffian positivity}
% We have so far compared Pl\"ucker positivity with Lusztig positivity. Pl\"ucker coordinates may be thought of as arising from the action of $G$ on exterior powers of the standard representation of $G$. Types B and D also have spin representations, not arising from tensor powers of the standard representation, which give coordinates resembling matrix Pfaffians. One might wonder whether these coordinates also may be used for positivity tests. 
% In this section, we will show that these Pfaffian coordinates fail to detect Lusztig positivity for an orthogonal Grassmannian of type D$_4$. This also gives a counterexample to \cite[Chapter 4, Proposition 5.1]{Rietsch} and the note at the end of \cite{lusztig1998positivity}, which (as a special case) asserted that the canonical basis for a fundamental representation detects positivity for its associated Grassmannian.

% \subsubsection{Pinning for $\operatorname{SO}_{2n}$}

% Set $E_0=\left[\begin{smallmatrix}
%          & &(-1)^n \\ &\iddots &\\ -1& &
%     \end{smallmatrix}\right]$. We take $\mathfrak{so}_{2n}$ to be the $2n\times 2n$ matrices $A$ so that $A^tE+EA=0$, where
% \[ E = \begin{bmatrix}
%     & E_0^t \\ E_0 &
% \end{bmatrix}=\left[\begin{smallmatrix}
%     &&&&&&&-1\\
%     &&&&&&1&\\
%     &&&&&\iddots&&\\
%     &&&&(-1)^n&&&\\
%     &&&(-1)^n&&&&\\
%     &&\iddots&&&&&\\
%     &1&&&&&&\\
%     -1&&&&&&&
% \end{smallmatrix}\right]. \]
% Then $\operatorname{O}_{2n}$ is the invertible $2n\times 2n$ matrices $A$ so that $A^tEA=E$, and $\operatorname{SO}_{2n}$ is its subgroup of determinant 1 matrices. We take the maximal torus to be $\mathrm{diag}(t_1,\ldots,t_n,t^{-1}_n,\ldots,t^{-1}_1)$. 

% \begin{rem}
%     We pick the embedding of $\operatorname{SO}_{2n}$ into $\operatorname{GL}_{2n}$ so that the matrix $E$ defining $\operatorname{SO}_{2n}$ is antidiagonal, to maintain consistency with the other groups considered in this paper. One could instead use  $E = \begin{bmatrix}
%     0 & I_n \\ I_n & 0
% \end{bmatrix},$
% which would make the
% canonical basis coordinates for the orthogonal Grassmannian (equivalently, generalized minors for the spin representation) coincide with matrix Pfaffians on the nose. The two embeddings are related via conjugation by $\begin{bmatrix}
%     I_n & \\
%     & E_0
% \end{bmatrix}$. See \cite[Remarks 5.2 and 5.5]{GalPyl} for more discussion.
% \end{rem}




% The simple roots are $\{\be_1-\be_2,\ldots, \be_{n-1}-\be_{n}, \be_{n-1}+\be_n\}$. We write the corresponding fundamental weights as $\varpi_1,\ldots, \varpi_n$, and denote the irreducible representation of highest weight $\lambda$ by $V_{\lambda}$. For $i\in [n-1]$, the map $\phi_i^{D_{n}}$ is given by
% \begin{equation*}
%     \phi_i^{D_{n}}\left(\begin{pmatrix}
%         a&b\\
%         c&d
%     \end{pmatrix}\right)=\begin{blockarray}{cccccccccc}
% & &  & i & i+1 & & 2n-i& 2n-i+1 & &\\
% \begin{block}{c(ccccccccc)}
%   & 1 &  &  & & & & & &\\
%   & & \ddots &  & &  & & & & \\
%   i& &  & a & b &  & & & & \\
%   i+1& &  &c  & d &  &  & & & \\
%   & &  &  &  & \ddots  & & & &  \\
%   2n-i&&&&&&a&b&&\\
%   2n-i+1&&&&&&c&d&&\\
%  &&&&&&&&\ddots&\\
%  &&&&&&&&&1\\
% \end{block} 
% \end{blockarray}\hspace{5pt}, 
% \end{equation*}
% where all unmarked off-diagonal entries are $0$.
% For $i=n$, the map $\phi_n^{D_n}$ is given by

% \begin{equation*}
%     \phi_n^{D_{n}}\left(\begin{pmatrix}
%         a&b\\
%         c&d
%     \end{pmatrix}\right)=\begin{blockarray}{cccccccccc}
% & &  & n-1 & n & n+1& n+2 & & &\\
% \begin{block}{c(ccccccccc)}
%   & 1 &  &  &  & & & & &\\
%   & & \ddots &   &  & & & & &\\
%   n-1& &  & a &    & b& & && \\
%   n& &  &  & a &   & b& & &\\
%   n+1&&&c&&d&&&&\\
%   n+2&&&&c&&d&&&\\
%  &&&&&&&&\ddots&\\
%  &&&&&&&&&1\\
% \end{block} 
% \end{blockarray}\hspace{5pt}, 
% \end{equation*}

% Set $\overline{i} = 2n-i+1$. The Weyl group $\mathfrak{D}_n$ acts via permutations of $\{1,\ldots,n, \overline{n},\ldots, \overline{1}\}$ so that $\sigma(\overline{i})=\overline{\sigma(i)}$ and so that an even number of un-barred elements are sent to barred elements. 

% \subsubsection{Spin representations}
% Let $Q(\cdot,\cdot)$ be the symmetric form on $\CC^{2n}$ determined by $E$ and let $\{\be_i\}$ be the standard basis of $\CC^{2n}$. Consider $\mathcal{C}$, the tensor algebra $T(\CC^{2n})$ modulo the relations 
% \[\mathbf v\otimes\mathbf w + \mathbf w\otimes \mathbf v - Q(\mathbf v,\mathbf w).\] 
% The resulting algebra $\mathcal{C}$ is called the \emph{Clifford algebra}.
% The underlying vector space of $\mathcal{C}$ is graded by $\mathcal{C} = \bigoplus_{i=0}^{2n} \mathcal{C}^i$, where $\mathcal{C}^i$ is spanned by products of $i$ pairwise orthogonal vectors. In particular, the degree 1 part $\mathcal{C}^1$ is $\CC^{2n}$. The degree 2 part $\mathcal{C}^2$ is a Lie algebra under commutator bracket. It has a basis of the form $\frac{1}{2}(\be_i\be_j-\be_j\be_i)_{i<j}$ and acts on $\mathcal{C}^1$ via commutator bracket. This action identifies $\mathcal{C}^2$ with $\mathfrak{so}_{2n}$. Explicitly, if $E_{ij}$ is the matrix with $1$ at position $(i,j)$ and $0$s elsewhere, then the vector $\frac{1}{2}(\be_i\be_j-\be_j\be_i)$ is sent to $E_{i\overline{j}}-E_{j\overline{i}}$.


% Let $\CC^{2n}=W\oplus W^*$ be the decomposition of $\CC^{2n}$ into the span of $\be_1,\ldots, \be_n$ and the span of $\be_{n+1},\ldots, \be_{2n}$. We can identify $W^*$ as the dual space of $W$ using the bilinear form $Q$. The spin representation $S = S^-\oplus S^+ = V_{\varpi_{n-1}}\oplus V_{\varpi_n}$ has underlying vector space $\bigwedge^\bullet W = \bigwedge^{\mathrm{odd}}W\oplus \bigwedge^{\mathrm{even}}W$. This space is a module for $\mathcal{C}$, with action induced by $\mathbf w\cdot \omega = \mathbf w\wedge \omega$ for $\mathbf w\in W$ and $\mathbf w^*\cdot\omega = \iota_{\mathbf w^*}\omega$ for $\mathbf w^*\in W^*$. Here $\iota_{\mathbf w^*}$ is the interior product, determined recursively by $\iota_{\mathbf w^*}(1) = 0$ and $\iota_{\mathbf w^*}(\mathbf w\wedge \omega') = (\mathbf w^*,\mathbf w)\omega' - \mathbf w\wedge \iota_{\mathbf w^*}\omega'$ for $\mathbf w\in W$. 


% If $I = \{i_1 < \cdots < i_r\}$, then write $\be_I = \be_{i_1}\wedge \cdots \wedge \be_{i_n}$. 
% We see that $E_{ij}-E_{\overline{ji}}$ in $\mathfrak{so}_{2n}$ acts via $\frac{1}{2}(\be_i\be_{\overline{j}}-\be_{\overline{j}}\be_i)$. 
% Hence $\be_I$ is a weight vector of weight $\sum_{i\in I} \frac{1}{2}\be_i - \sum_{i\not\in I}\frac{1}{2}\be_i$, and $\{\be_I \mid I\subseteq [n]\}$ is a weight basis of $S$. The weight spaces associated to $V_{\varpi_{n-1}}$ are those with $|I|$ odd, and the weight spaces associated to $V_{\varpi_n}$ are those with $|I|$ even. Since the basis $\{\be_I\mid I\subseteq [n],~|I|\text{ even}\}$ of $V_{\varpi_n}$ (resp. the basis $\{\be_I\mid I\subseteq [n],~|I|\text{ odd}\}$ of $V_{\varpi_{n-1}}$) is a single $W^{D_n}$-orbit containing a highest weight vector, it coincides with the Lusztig canonical basis.

% \subsubsection{The Grassmannian $\operatorname{SOGr}(n,2n)$}
% The orthogonal Grassmannian $\operatorname{SOGr}(n,2n)_+$ is (one of two components of) the space of $n$-dimensional isotropic subspaces of $\CC^{2n}$ under the symmetric form defined by $E$. It is the maximal parabolic quotient associated to the simple root $\alpha_{n}$, and hence embeds in $\PP V_{\varpi_{n}}$.  A representative element is the isotropic subspace spanned by $\be_1,\ldots,\be_{n-1},\be_{n}$. We can represent an element of $\operatorname{SOGr}(n,2n)_+$ by a full-rank $2n\times n$ matrix $\begin{bmatrix}A\\B\end{bmatrix}$ satisfying $A^tE_0B+B^tE_0^tA=0$, up to right multiplication by an invertible matrix. We remark that the isotropic subspace spanned by $\be_1,\ldots,\be_{n-1},\be_{\overline{n}}$ is \emph{not} an element of $\operatorname{SOGr}(n,2n)_+$; it is instead in $\operatorname{SOGr}(n,2n)_-$, which we identify as the maximal parabolic quotient associated to the simple root $\alpha_{n-1}$.



% \subsubsection{Coordinates on $\operatorname{SOGr}(n,2n)_+$}
% \newcommand{\Pf}{\mathrm{Pf}}
% \newcommand{\Spin}{\mathrm{Spin}}
% The weight vector coordinates on $V_{\varpi_n}=S^+$ induce homogeneous coordinates on $\operatorname{SOGr}(n,2n)_+$. Let $\Pf_I$ denote the coordinate $g\mapsto \langle \be_{[n]\setminus I}, g \be_{[n]}\rangle$, for $I\subseteq [n]$ with $|I|$ even. (Here $\langle \be_{[n]\setminus I},g\be_{[n]}\rangle$ means the $\be_{[n]\setminus I}$-coefficient of $g\be_{[n]}$ in the basis $\{\be_{I}\}$.) If $|I|$ is odd, then set $\Pf_I=0$. If $X=\begin{bmatrix}A\\B\end{bmatrix}$ represents a point in $\operatorname{SOGr}(n,2n)_+$, then the vector $(\Pf_I(X)^2)_{I\subseteq [n]}$ coincides with the vector $(\Delta_{\widetilde I})_{I\subseteq [n]}$ up to rescaling. Here $\Delta_{\widetilde I}$ is the minor of $X$ with rows $\left([n]\setminus I \right)\sqcup \overline{I}$, and $\Pf_I(X)$ is the value of $\Pf_I$ on an element of
% (the universal cover $\Spin_{2n}$ lifting an element of) $\operatorname{SO}_{2n}$ of the form $\begin{bmatrix} A & ? \\ B & ? \end{bmatrix}$. These vectors coincide because the highest weight of $V_{\varpi_n}\otimes V_{\varpi_n}$ is the same as the highest weight of $\bigwedge^n V_{\varpi_1}$, so the coordinates with respect to the $W^{D_n}$ orbit of their highest weight vectors are the same. We normalize $\Pf_I(X)$ so that when $X= \begin{bmatrix} I_n \\ B \end{bmatrix}$, then $\mathrm{sgn}(I,[n]\setminus I)\Pf_{I}(X)$ is the Pfaffian of the (antisymmetric) submatrix of $E_0B$ given by selecting the rows and columns indicated by $I$. (Points of this form make up the dense Schubert cell in $\operatorname{SOGr}(n,2n)_+$.)

% \subsubsection{Counterexample to $\operatorname{SOGr}(n,2n)_+^{\geq 0} = \operatorname{SOGr}(n,2n)_+^{\mathrm{Pf}\geq 0}$}
% \label{sec:typeDcounterexample}
% The spin representation $S^+$ is miniscule, so its canonical basis coincides with its weight basis. 
% Hence \cite[Chapter 4, Proposition 5.1]{Rietsch} would imply that $\operatorname{SOGr}(n,2n)_+^{\geq 0}$ is the locus of non-negativity of $\{\mathrm{Pf}_I \mid I\subseteq [n],~|I|\text{ even}\}$ in $\operatorname{SOGr}(n,2n)_+$. However, this is not the case: there is an element of $\operatorname{SOGr}(n,2n)_+$ with strictly positive values of its Pfaffians which is not in $\operatorname{SOGr}(n,2n)_+^{\geq 0}$. Indeed, consider the reduced word
% \[ w_0^{\{1,\ldots,n-1\}} = s_4s_2s_3s_1s_2s_4,  \]
% and the element $y_4(t_1)y_2(t_2)y_3(t_3)y_1(t_4)y_2(t_5)y_4(t_6)P$ in $\operatorname{SOGr}(n,2n)_+$. This element of $\operatorname{SOGr}(n,2n)_+$ is represented by the matrix
% \[
% \begin{bmatrix}
%     1 & 0 & 0 & 0 \\
%     0 & 1 & 0 & 0 \\
%     0 & 0 & 1 & 0 \\
%     0 & 0 & 0 & 1 \\
%     t_4t_5t_6 & -(t_2+t_5)t_6 & t_1+t_6 & 0 \\
%      t_3t_4t_5t_6 & -t_3t_5t_6 & 0 & t_1+t_6 \\
%      t_2t_3t_4t_5t_6 & 0 & -t_3t_5t_6 & (t_2+t_5)t_6 \\
%      0 & t_2t_3t_4t_5t_6 & -t_3t_4t_5t_6 & t_4t_5t_6
% \end{bmatrix}.
% \]
% The associated antisymmetric matrix is
% \[ E_0B = \begin{bmatrix}
%     0 & -t_2t_3t_4t_5t_6 & t_3t_4t_5t_6 & -t_4t_5t_6 \\
%     t_2t_3t_4t_5t_6 & 0 & -t_3t_5t_6 & (t_2+t_5)t_6 \\
%     -t_3t_4t_5t_6 & t_3t_5t_6 & 0 & -t_1-t_6 \\
%     t_4t_5t_6 & -(t_2+t_5)t_6 & t_1+t_6 & 0
% \end{bmatrix}. \]
% The Pfaffians are $\Pf_{\varnothing}=1$, $\Pf_{\{1,2,3,4\}}=t_1t_2t_3t_4t_5t_6$, and the Pfaffians with index of size $2$, which are (up to sign) the entries of the matrix,
% \[t_2t_3t_4t_5t_6,~t_3t_4t_5t_6,~t_4t_5t_6,~t_3t_5t_6,(t_2+t_5)t_6,t_1+t_6. \]
% We see that if $t_1,t_2,t_3,t_4$ are positive, $t_5,t_6$ are negative, and $|t_6|\ll t_1$ and $|t_5|\ll t_2$, then all of the canonical basis coordinates are positive. However, this point of $\operatorname{SOGr}(n,2n)_+$ is Lusztig non-negative if and only if $t_i\geq 0$ for all $i$.

\acknowledgements{The authors would like to thank Pavel Galashin, Steven Karp, Konstanze Rietsch, and Lauren Williams for helpful communications regarding this work.}

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