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\title[The positive orthogonal Grassmannian]{The positive orthogonal Grassmannian}

\author[Y.~El Maazouz and Y.~Mandelshtam]{\texorpdfstring{
Yassine El Maazouz \thanks{\href{mailto:maazouz@caltech.edu}{maazouz@caltech.edu}. 
YEM is supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) SFB-TRR 195 “Symbolic Tools in Mathematics and their Application.”}\addressmark{1} \and 
Yelena Mandelshtam \thanks{\href{mailto:yelenam@ias.edu}{yelenam@ias.edu}.
YM is supported by the National Science Foundation under Award No. DMS-2402069 and the Bob Moses Fund of the Institute for Advanced Study.}
\addressmark{2}}{
Yassine El Maazouz  and 
Yelena Mandelshtam 
}}

\address{\addressmark{1} California Institute of Technology \\ \addressmark{2} Institute for Advanced Study}

\received{\today}

\abstract{The Pl\"ucker positive region $\OGr_+(k,2k)$ of the orthogonal Grassmannian  emerged as the positive geometry behind the ABJM scattering amplitudes. In this paper we initiate the study of the positive orthogonal Grassmannian $\OGr_+(k,n)$ for general values of $k,n$. We determine the boundary structure of the quadric $\OGr_+(1,n)$ in $\PP^{n-1}_{+}$ and show that it is a positive geometry. We show that $\OGr_+(k,2k+1)$ is isomorphic to  $\OGr_+(k+1, 2k+2)$ and connect its combinatorial structure to matchings on $[2k+2]$. Finally, we show that in the case $n>2k+1$, the \emph{positroid cells} of $\Gr_+(k,n)$ no longer suffice to induce a CW cell decomposition of $\OGr_+(k,n)$.}

\keywords{Orthogonal Grassmannian, Flag variety, Positive geometry.}


\begin{document}

\maketitle

	\section{Introduction}
	
	Let $n \geq k$ be positive integers and denote by $\Gr(k,n)$ the \emph{Grassmannian} of $k$-dimensional subspaces of $\CC^n$. The \emph{positive Grassmannian} $\Gr_+(k,n)$ is the semialgebraic set in $\Gr(k,n)$ where all Pl\"ucker coordinates are real and nonnegative. The matroid stratification of the Grassmannian \cite{GGMS} induces a natural decomposition of $\Gr_+(k,n)$ into the so-called \emph{positroid cells}, see \cite{postnikov06}.
	
	After Postnikov's landmark paper \cite{postnikov06}, the positive Grassmannian became a rich object of research in algebraic combinatorics \cite{PostnikovICM,GKL22,W21}. Its study accelerated, in recent years, largely due to its unexpected and profound connection to Physics, in particular shallow water waves \cite{KodamaWilliams, AG18} and scattering amplitudes in quantum field theory \cite{TheAmplituhedron,ABCGPT,PositiveGeometries}. The object of study in this article is the \emph{positive orthogonal Grassmannian} $\OGr_+^{\omega}(k,n)$.
    
        \begin{definition}\label{def:OGr}
        Let $\omega: \RR^{n} \times \RR^{n} \to \RR$ be a non-degenerate symmetric bilinear form. We denote by $\OGr^{\omega}(k,n)$ the algebraic variety of \emph{isotropic} $k$-dimensional subspaces $V$ of $\CC^n$ with respect to $\omega$ i.e. $\omega(x,y) = 0$ for any $x,y \in V$. The positive orthogonal Grassmannian $\OGr^\omega_+(k,n)$ is the semi-algebraic subset of $\OGr^\omega(k,n)$ where the Pl\"ucker coordinates are all real and have the same sign.
    \end{definition}
	In the special case $n = 2k$ and $\omega(x,y) = \sum_{i=1}^{2k} (-1)^{i-1} x_iy_i$, this object was first studied in the context of ABJM scattering amplitudes \cite{huang2014abjm} and later connected to the Ising model in \cite{galashin2020ising}. In this paper we initiate the study of $\OGr^{\omega}_+(k,n)$ for general values of $k,n$ with respect to the quadratic form
	\begin{equation}\label{eq:altForm}
		\omega_0(x,y) = x_1y_1 - x_2y_2 + \cdots + (-1)^{n-1}x_ny_n.
	\end{equation}
    The choice of the quadratic form $\omega$ is extremely important.
    For certain quadratic forms the variety $\OGr^{\omega}(k,n)$ has no real points, or its positive part is not full dimensional.
    
	This article is organized as follows. 
        In \Cref{sec:2} we collect some facts on the geometry of the orthogonal Grassmannian $\OGr^{\omega}(k,n)$. 
        In particular we determine the ideal of quadrics that cut out $\OGr(k,n)$ in $\PP(\wedge^k \CC^n)$, and determine a Gr\"obner basis for this ideal. 
        In \Cref{sec3} we investigate $\OGr_+^{\omega_0}(1,n)$ with respect to the alternating form \eqref{eq:altForm}. Namely, we describe its face structure and show that it is a positive geometry. 
        \Cref{sec4} is devoted to $\OGr^{\omega_0}_+(k,2k+1)$. In this section we show that $\OGr^{\omega_0}_+(k,2k+1)$ is isomorphic to $\OGr^{\omega_0}_+(k+1,2k+2)$ and we relate the face structure of $\OGr^{\omega_0}_+(k,2k+1)$ to matchings on $[2k+2]$. In \Cref{sec5} we initiate the study of $\OGr^{\omega_0}_+(k,n)$ starting with the case $k=2$. Already in this specific case, we show that the positroid cell decomposition of $\Gr_+(2,n)$ is no longer sufficient to induce a CW cell decomposition of $\OGr_+^{\omega_0}(2,n)$.
	
	\section{Commutative algebra and geometry of \texorpdfstring{$\OGr(k,n)$}{}}
	\label{sec:2}
	
	In this section we collect some facts on the algebraic variety $\OGr^{\omega}(k,n)$ over $\CC$. Since all non-degenerate symmetric bilinear forms over $\CC$ are equivalent to the standard inner product $(\cdot, \cdot)$, the varieties $\OGr^{\omega}(k,n)$ for different $\omega$ are isomorphic. So, in this section we may assume that $\omega$ is $(\cdot, \cdot)$, and we suppress $\omega$ and write $\OGr(k,n)$. Recall that the Grassmannian $\Gr(k,n)$ comes with $\binom{n}{k}$  \emph{Pl\"ucker} coordinates, which we denote by $p_I$ for any subset $I = \{i_1< i_2 < \dots < i_k\}$ of $[n]$. 
    
    \begin{theorem}\label{thm:OGr_Eq}
		The orthogonal Grassmannian $\OGr(k,n)$ is cut out in $\PP(\wedge^k\CC^n)$ by the Pl\"ucker relations and the following $ \frac{1}{2}\binom{n}{k-1} \Big(\binom{n}{k-1}+1\Big)$ equations:
		\begin{equation}\label{eq:OrthogonalEquations}
			\sum_{\ell=1}^{n} \varepsilon(I\ell) \varepsilon(J\ell) \ p_{I\ell} p_{J\ell} = 0, \quad \text{for } I,J \in \binom{[n]}{k-1}.
		\end{equation}
		where $\epsilon(I\ell) = (-1)^{|\{i \in I \colon i > \ell \}|}$ denotes the sign of the permutation that sorts $I\ell$.	
    	\end{theorem}
        
	
	 \begin{remark}
         %        For a quadratic form $\omega$ with matrix $\Omega$ in the standard basis of $\CC^n$, we obtain the equations for $\OGr^{\omega}(k,n)$ by setting $P \Omega P^T = 0$ instead where $P$ is the matrix:
    	% 	 \[
    	% 	 P_{I \ell} = \begin{cases} \epsilon(I \ell) \ p_{I \ell}, \quad \text{if } \ell \not \in I \\
    	% 	 	0, \quad \quad \quad \quad \text{otherwise } \ell \in I \end{cases}
    	% 	 ,\quad \text{for} \quad I \in \binom{[n]}{k-1} \text{ and } 1 \leq \ell \leq n.
    	% 	 \]
       In the case of the bilinear form \eqref{eq:altForm}, the equations \eqref{eq:OrthogonalEquations} become:
	 	\begin{equation}\label{eq:orthRelationsAlt}
	 		\sum_{\ell=1}^{n} (-1)^{\ell - 1} \epsilon(I\ell) \epsilon(J\ell) p_{I\ell}p_{J\ell} = 0 ,\quad \text{for} \quad I, J \in \binom{[n]}{k-1}.
	 	\end{equation}
	 \end{remark}
	
	\begin{proposition}\label{prop:irred} The variety $\OGr(k,n)$ is empty if $n < 2k$. When $n = 2k$ it splits into two irreducible connected components, and it is irreducible when $n > 2k$. Moreover we have:
		\[
		\dim(\OGr(k,n)) = k(n-k) - \binom{k+1}{2} \quad \text{for } n \geq 2k.
		\]
	\end{proposition}
    
	Following \cite{EPS24}, let $Y_{k,n}$ denote \emph{Young's lattice}. This is a poset whose elements are subsets of size $k$ in $[n]$ and the order relation in $Y_{k,n}$ is:
	\[\resizebox{\textwidth}{!}{$
	\langle i_1 < \dots < i_k \rangle \leq \langle j_1 < \dots < j_k \rangle \quad \text{if} \quad i_1 \leq j_1, \ i_2\leq j_2, \ \dots, \ i_{k-1} \leq j_{k-1} \text{ and } i_k \leq j_k.$}
	\]
	We denote by $\widetilde{Y}_{k,n}$ another copy of Young's lattice. As a set $\widetilde{Y}_{k,n} = \binom{[n]}{n-k}$ and the order relation is given by:
	\[
	[ i'_1 < \dots < i'_{n-k} ] \leq [j'_1 < \dots < j'_{n-k} ] \quad \text{if} \quad i'_1 \geq j'_1, \ \dots, \ i'_{n-k} \geq j'_{n-k}.
	\]
	Finally we denote by $\mcP_{k,n}$ the poset which, as a set, is the disjoint union of $Y_{k,n}$ and $\widetilde{Y}_{k,n}$. All order relations in $Y_{k,n}$ and $\widetilde{Y}_{k,n}$ remain order relations in $\mcP_{k,n}$ and in addition to these relations we add $\binom{2k}{k}$ covering relations:
	\[
	   [j'_1< \dots <j'_{n-k}] \ \ <  \ \ \langle i_1< \dots <i_k\rangle 
	\]
	whenever $\{1,2,3, \dots, 2k\} = \{i_1, \dots, i_k\} \sqcup \{j_1, \dots, j_k\}$ is a partition where $\{j_1, \dots ,j_k\}$ is the complement $[n] \setminus \{j'_1, \dots ,j'_{n-k}\}$. See \Cref{fig:P26} for an example.
    
    \begin{figure}[ht]
        \centering
        \scalebox{0.6}{
         \begin{tikzpicture}
            
            \draw (2,4) node  {$\langle 56 \rangle$};
            \draw (1,3) node  {$\langle 46 \rangle$};
            \draw (0,2) node  {$\langle 36 \rangle$};
            \draw (-1,1) node {$\langle 26 \rangle$};
            \draw (-2,0) node {$\langle 16 \rangle$};
            
            \draw (1.25,3.25)--(1.75,3.75);
            \draw (0.25,2.25)--(0.75,2.75);
            \draw (-0.75,1.25)--(-0.25,1.75);
            \draw (-1.75,0.25)--(-1.25,0.75);
    
    
            \draw (1.25,2.75)--(1.75,2.25);
            \draw (0.25,1.75)--(0.75,1.25);
            \draw (-0.75,0.75)--(-0.25,0.25);
            \draw (-1.75,-0.25)--(-1.25,-0.75);
    
            \draw (2,2) node {$\langle 45 \rangle$};
            \draw (1,1) node {$\langle 35 \rangle$};
            \draw (0,0) node {$\langle 25 \rangle$};
            \draw (-1,-1) node {$\langle 15 \rangle$};
    
            
            \draw (1.25,1.25)--(1.75,1.75);
            \draw (0.25,0.25)--(0.75,0.75);
            \draw (-0.75,-0.75)--(-0.25,-0.25);
    
    
            
            \draw (1.25,0.75)--(1.75,0.25);
            \draw (0.25,-0.25)--(0.75,-0.75);
            \draw (-0.75,-1.25)--(-0.25,-1.75);
    
    
        
            
            \draw (2,0) node {$\langle 34 \rangle$};
            \draw (1,-1) node {$\langle 24 \rangle$};
            \draw (0,-2) node {$\langle 14 \rangle$};
    
    
            \draw (1.25,-0.75)--(1.75,-0.25);
            \draw (0.75,-1.25)--(0.25,-1.75);
            
    
            \draw (1.25,-1.25)--(1.75,-1.75);
            \draw (0.25,-2.25)--(0.75,-2.75);
            
            
            \draw (2,-2) node {$\langle 23 \rangle$};
            \draw (1,-3) node {$\langle 13 \rangle$};
    
            
            \draw (1.75,-2.25)--(1.25,-2.75);
    
            \draw (1.25,-3.25)--(1.75,-3.75);
             
            \draw (2,-4) node {$\langle 12 \rangle$};
    
    
            %%%%%%%% Boxed variables
            \begin{scope}[shift={(7.2,-4)}, rotate=180]
        
    
            \draw (2,4) node  {[1234]};
            \draw (1,3) node  {[1235]};
            \draw (0,2) node  {[1245]};
            \draw (-1,1) node {[1345]};
            \draw (-2,0) node {[2345]};
            
    
            \draw (1.25,3.25)--(1.75,3.75);
            \draw (0.25,2.25)--(0.75,2.75);
            \draw (-0.75,1.25)--(-0.25,1.75);
            \draw (-1.75,0.25)--(-1.25,0.75);
    
    
            \draw (1.25,2.75)--(1.75,2.25);
            \draw (0.25,1.75)--(0.75,1.25);
            \draw (-0.75,0.75)--(-0.25,0.25);
            \draw (-1.75,-0.25)--(-1.25,-0.75);
    
    
    
    
            \draw (2,2) node {[1236]};
            \draw (1,1) node {[1246]};
            \draw (0,0) node {[1346]};
            \draw (-1,-1) node {[2346]};
    
            
            \draw (1.25,1.25)--(1.75,1.75);
            \draw (0.25,0.25)--(0.75,0.75);
            \draw (-0.75,-0.75)--(-0.25,-0.25);
    
    
            
            \draw (1.25,0.75)--(1.75,0.25);
            \draw (0.25,-0.25)--(0.75,-0.75);
            \draw (-0.75,-1.25)--(-0.25,-1.75);
    
    
        
            
            \draw (2,0) node {[1256]};
            \draw (1,-1) node {[1356]};
            \draw (0,-2) node {[2356]};
    
    
            \draw (1.25,-0.75)--(1.75,-0.25);
            \draw (0.75,-1.25)--(0.25,-1.75);
            
    
            \draw (1.25,-1.25)--(1.75,-1.75);
            \draw (0.25,-2.25)--(0.75,-2.75);
            
            
            \draw (2,-2) node {[1456]};
            \draw (1,-3) node {[2456]};
    
            
            \draw (1.75,-2.25)--(1.25,-2.75);
    
            \draw (1.25,-3.25)--(1.75,-3.75);
             
            \draw (2,-4) node {[3456]};
    
            \end{scope}
    
            %% RED lines 
    
            \draw[red] (2.35,0)--(4.65,0);
            \draw[red] (1.35,-1)--(5.65,-1);
    
            \draw[red] (2.35,-2) .. controls (4,-1.5) and (5,-1.5) .. (6.65,-2);
            \draw[red] (0.35,-2) .. controls (2,-2.5) and (3,-2.5) .. (4.65,-2);
            
            
            \draw[red] (1.35,-3) -- (5.65,-3);
            \draw[red] (2.35,-4) -- (4.65,-4);
            
        \end{tikzpicture}}
        \caption{The poset $\mcP_{2,6}$ glued from
            $Y_{2,6}$ and $\widetilde{Y}_{2,6}$ using the covering relations in red.}
        \label{fig:P26}
    \end{figure}
    
	An incomparable pair $\big(\langle i_1, \dots, i_k \rangle , \langle j_1, \dots, j_{k}\rangle\big)$ or $\big(\langle i_1, \dots, i_k \rangle , [j'_1, \dots, j'_{n-k}]\big)$ in $\mcP_{k,n}$ yields a non-semistandard Young tableau $\mu$ of shape $(k,k)$ or $\lambda$ of shape $(n-k,k)$:
	
	\setcounter{MaxMatrixCols}{11}
	\begin{equation}\label{eq:skewSymTableau}
        \resizebox{\textwidth}{!}{$
        \mu \ = 
		\ \begin{bmatrix} 
			j_1 & \cdots &j_{\ell -1}& \bm{j}_{\ell} &\bm{j}_{\ell+1} & \cdots  & \bm{j}_{k}\\
			\bm{i}_1  & \cdots  &\bm{i}_{\ell-1}& \bm{i}_{\ell}  &i_{\ell+1}  & \cdots & i_{k}  \\
		\end{bmatrix}, \quad
		\lambda \ = 
		\ \begin{bmatrix} 
			j_1' & \cdots &j'_{\ell -1}& \bm{j}'_{\ell} &\bm{j}'_{\ell+1} & \cdots  &\bm{j}'_{k} &  \cdots & \bm{j}'_{n-k}\\
			\bm{i}_1  & \cdots  &\bm{i}_{\ell-1}& \bm{i}_{\ell}  &i_{\ell+1}  & \cdots & i_{k} & &  \\
		\end{bmatrix}$.}
	\end{equation}
    The tableau $\mu$ or $\lambda$ being non-semistandard means that there exists $\ell$ in $[k]$ such that:
	\begin{equation}\label{eq:snake}
		i_1 < \dots < i_{\ell} < j_{\ell} < \dots < j_{k} \quad \text{or} \quad i_1 < \dots < i_{\ell} < j'_{\ell} < \dots < j'_{n-k}.
	\end{equation} 
	We pick $\ell$ to be the smallest index with this property. The strictly increasing sequences of integers in \eqref{eq:snake} are highlighted in bold in \eqref{eq:skewSymTableau}. Now consider the permutations $\pi$ of the sequence $i_1 < \dots< i_\ell < j_\ell < \dots < j_k$ which make the first $\ell$ entries and the last $k - \ell + 1$ entries separately increasing, and similarly, the permutations $\sigma$ of the sequence $i_1 < \dots< i_\ell < j'_\ell < \dots < j'_{n-k}$ which make the first $\ell$ entries and the last $n- k - \ell + 1$ entries separately increasing. Such permutations permute the bold entries in the tableaux $\mu$ and $\lambda$ in \eqref{eq:skewSymTableau} and yield
    \begin{equation*}
		\begin{aligned}
        \pi(\mu) \ &= 
		\ \begin{bmatrix} 
			j_1 & \cdots &j_{\ell -1}& \pi(\bm{j}_{\ell}) & \pi(\bm{j}_{\ell+1}) & \cdots  & \pi(\bm{j}_{k})\\
			\pi(\bm{i}_1)  & \cdots  &\pi(\bm{i}_{\ell-1})& \pi(\bm{i}_{\ell})  &i_{\ell+1}  & \cdots & i_{k}  \\
		\end{bmatrix},\\
	\sigma(\lambda) \ &= 
		\ \begin{bmatrix} 
			j_1' & \cdots &j'_{\ell -1}& \pi(\bm{j}'_{\ell}) &\pi(\bm{j}'_{\ell+1}) & \cdots  &\pi(\bm{j}'_{k}) &  \cdots & \pi(\bm{j}'_{n-k})\\
			\pi(\bm{i}_1)  & \cdots  &\pi(\bm{i}_{\ell-1})& \pi(\bm{i}_{\ell})  &i_{\ell+1}  & \cdots & i_{k} & &  \\
		\end{bmatrix}.
		\end{aligned}
	\end{equation*}
    Summing over these permutations, the tableaux $\mu$ and $\lambda$ yield quadrics
	\begin{equation} \label{eq:StrLaws} 
        \begin{aligned}
	    f_{\mu} := \sum\limits_{\pi} \sign(\pi) \ \langle \pi(i_1),\dots, \pi(i_\ell), i_{\ell+1}, \dots i_k\rangle \ \langle j_1,\dots,j_{\ell-1}, \pi(j_{\ell}), \dots, \pi(j_k)\rangle,\\
         f_{\lambda} := \sum\limits_{\pi} \sign(\pi) \ \langle \pi(i_1),\dots, \pi(i_\ell), i_{\ell+1}, \dots i_k\rangle \ [j'_1,\dots,j'_{\ell-1}, \pi(j'_{\ell}), \dots, \pi(j'_k)].
	\end{aligned} \quad 
	\end{equation}
    Here, whenever $J' = \{j'_1<\dots<j'_{n-k}\}$ and $[n]\setminus J' = \{ \bar{j}_1 < \dots < \bar{j}_k \}$ we set
    \[
        [j'_1, \dots, j'_{n-k}] := (-1)^{\sum_{r=1}^{n-k} j'_{r}} \langle \bar{j}_1, \dots, \bar{j}_{k}\rangle.
    \]
 
	
	
	\begin{theorem}\label{thm:Groebner}
		The quadrics in \eqref{eq:StrLaws} form a Gr\"obner basis for the ideal ${\rm I}_{k,n}$ in $\CC[p_{I}]$ generated by the Pl\"ucker relations and the quadratic equations in \eqref{eq:OrthogonalEquations} with respect to any monomial ordering given by a linear extension of the poset $\mcP_{k,n}$.
	\end{theorem}

        \begin{proposition}\label{prop:degree}
           Let $n > 2k$, $m := \lfloor n/2 \rfloor$, and set $D := k(n-k) - \binom{k+1}{2}$. The degree of $\OGr(k,n)$ in the Pl\"ucker embedding is
           \begin{equation}\label{eq:degreeFormula}
            \resizebox{.915\hsize}{!}{$\begin{aligned}
                  & D! \cdot \left(\prod_{\substack{1 \leq i \leq k \\ k< j \leq m }}  \frac{1}{(2m - i - j)(j-i)}\right)
                \left(\prod_{1 \leq i < j \leq k }  \frac{2}{2m - i - j}\right),  && \text{if } n = 2m, \\
                  & D! \cdot  \left(\prod\limits_{1 \leq i \leq k }  \frac{2}{2m - 2i + 1}\right)  \left(\prod\limits_{\substack{1 \leq i \leq k \\ k< j \leq m }}  \frac{1}{(2m - i - j)(j-i)}\right)
                \left(\prod\limits_{1 \leq i < j \leq k }  \frac{2}{2m - i - j+1}\right),  && \text{if } n = 2m+1.
            \end{aligned}$}
        \end{equation}    
    \end{proposition}

        \begin{theorem}\label{thm:Primality}
		 When $n > 2k$, the ideal ${\rm I}_{k,n}$ in $\CC\left[p_{I} : I \in \binom{[n]}{k} \right]$ generated by the Pl\"ucker relations and the quadrics in \eqref{eq:OrthogonalEquations} is the prime ideal of $\OGr(k,n)$. In particular, the degree of $I_{k,n}$ is given by  \eqref{eq:degreeFormula}.
	\end{theorem}
    
         \begin{remark}
         The ideal ${\rm I}_{k,2k}$ is clearly not prime since $\OGr(k,2k)$ has two irreducible connected components and we know that $I_{k,2k}$ cuts out $\OGr(k,2k)$ in $\PP^{\binom{2k}{k}-1}$. Moreover, if $\omega = \omega_0$ is the sign alternating quadratic form in \eqref{eq:altForm}, then for any $p \in \Gr(k,2k)$ we have $p \in \OGr^{\omega_0}(k,2k)$ if and only if
        \begin{equation}\label{eq:LinearRelationOGr(k,2k)}
            p_{I} = p_{I^c} \quad \text{for all } I \in \binom{[2k]}{k} \quad \text{ or } p_{I} = - p_{I^c} \quad \text{for all } I \in \binom{[2k]}{k}.
        \end{equation}		
        \end{remark}
        
        We define the \emph{standard component} of $\OGr^{\omega_0}(k,2k)$ to be the connected component where $p_I = p_{I^c}$ for all $I \in \binom{[2k]}{k}$. The semialgebraic set in the standard component where all Pl\"ucker coordinates are real and have the same sign is denoted by $\OGr^{\omega_0}_+(k,2k)$.
    
\section{The positive orthogonal Grassmannian \texorpdfstring{$\mathrm{OGr}_{+}(1,n)$}{}}
	\label{sec3}
    In this section we study the positive geometry, in the sense of \cite{PositiveGeometries}, of $ \OGr(1,n)$ with  the quadratic form $\omega_0$ given by \eqref{eq:altForm}. From now on, unless specifically mentioned, we always work with $\omega_0$. We denote by $(p,q)$ its signature where $p = \lceil \frac{n}{2} \rceil$ and $q = \lfloor \frac{n}{2} \rfloor$.

    We think of the elements of $[n]$ as vertices of a regular $n$-gon ordered clockwise from $1$ to $n$. For each pair of non-empty subsets $A \subset [n]\cap (2\ZZ+1)$ and $B \subset [n] \cap 2\ZZ$, there exists a unique cycle $\sigma(A,B)$ in the symmetric group $S_n$ such that $\sigma(A,B)$ has exactly one excedance and the support of $\sigma(A,B)$ is $A \sqcup B$. The set of such permutations\footnote{These are decorated permutations with all fixed points having a ``$+$'' decoration.} $\sigma(A,B)$ is denoted $\mathfrak{S}_{1,n}$. The set $\mathfrak{S}_{1,n}$ is endowed with a partial order given by:
    \[
        \sigma(C,D) \preceq \sigma(A,B) \iff C \subseteq A \text{ and } D \subseteq B.
    \]
    For $\sigma(A,B) \in \mathfrak{S}_{1,n}$, we denote by $\Pi_{\sigma(A,B)}$ the subset of $\PP^{n-1}_{+}$ where $x_i = 0$ if and only if $i$ is a fixed point of $\sigma(A,B)$ i.e. $i \not \in A \sqcup B$. Here, $\PP^{n-1}_{+}$ is simply $\Gr_+(1,n)$.
    
	\begin{theorem}\label{thm:OGr(1,n)}
		The positive orthogonal Grassmannian $\OGr_{+}(1,n)$ is combinatorially isomorphic to the product of simplices $\Delta_{p-1} \times \Delta_{q-1}$. More precisely, the following hold:

        \begin{enumerate}[leftmargin = 20pt]
            \item $\OGr_+(1,n) = \bigsqcup\limits_{ \sigma \in \mathfrak{S}_{1,n} } \OGr_+(1,n) \cap \Pi_{\sigma}$.
            \item $\overline{\OGr_+(1,n) \cap \Pi_{\sigma}} =  \bigsqcup\limits_{\tau \preceq \sigma} \OGr_+(1,n) \cap \Pi_{\tau}$.
            \item If $A = \{i_1 < \dots < i_r\}$ and $B = \{ j_1 < \dots < j_m\}$, $\sigma = \sigma(A,B)$ the cell $\OGr_+(1,n) \cap \Pi_{\sigma(A,B)}$ can be parameterized as follows. For each $t_{1},\dots, t_{r-1}$ and $s_{1},\dots, s_{m-1}$ in $\RR_{>0}$ we get a point $x \in \OGr_+(1,n) \cap \Pi_{\sigma(A,B)} $ by setting $x_i = 0$ whenever $i \not \in (A \cup B) $ and:
        \end{enumerate}
        \begin{equation}\label{eq:parametrization}
            \resizebox{.915\hsize}{!}{$\begin{aligned}
                x_{i_1} &= \frac{e^{t_1} - e^{-t_1}}{e^{t_1}+e^{-t_1}}, \ &x_{i_2} &= \frac{2}{e^{t_1} + e^{-t_1}} \frac{e^{t_2} - e^{-t_2}}{e^{t_2}+e^{-t_2}}, \ \dots, \ &x_{i_{r-1}} &= \frac{2}{e^{t_{r-1}} + e^{-t_{r-1}}}  \prod\limits_{\ell=1}^{r-1} \frac{e^{t_\ell} - e^{t_\ell}}{e^{t_{\ell}} + e^{-t_{\ell}}},\\
                x_{j_1} &= \frac{e^{s_1} - e^{-s_1}}{e^{s_1}+e^{-s_1}}, \ &x_{j_2} &= \frac{2}{e^{s_1} + e^{-s_1}} \frac{e^{s_2} - e^{-s_2}}{e^{s_2}+e^{-s_2}}, \ \dots, \ &x_{j_{m-1}} &= \frac{2}{e^{s_{m-1}} + e^{-s_{m-1}}}  \prod\limits_{\ell=1}^{m-1} \frac{e^{s_\ell} - e^{s_\ell}}{e^{s_{\ell}} + e^{-s_{\ell}}}.
            \end{aligned}$}
        \end{equation}
        
	\end{theorem}


    \begin{example} The orthogonal Grassmannian $\OGr_+(1,5)$ has the same combinatorial structure as $\Delta_{1} \times \Delta_{2}$. The poset of the boundaries of $\OGr_+(1,n)$ is depicted in \Cref{fig:OGr15}.
    \begin{figure}[ht]
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        \draw (202.74,382.19) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$5$};
        \draw (387.6,416.4) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$1$};
        \draw (344.58,403.75) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$2$};
        \draw (343.8,367.35) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$3$};
        \draw (387,353.93) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$4$};
        \draw (406.74,383.19) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$5$};
        \draw (486.6,419.4) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$1$};
        % Text Node
        \draw (443.58,406.75) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$2$};
        \draw (442.8,370.35) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$3$};
        \draw (484,353.93) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$4$};
        \draw (505.74,386.19) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$5$};
        \draw (286.2,417.4) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$1$};
        \draw (243.39,404.75) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$2$};
        \draw (243.2,368.35) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$3$};
        \draw (284.67,354.93) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$4$};
        \draw (303.74,385.19) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$5$};
        \draw (575.6,419.4) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$1$};
        \draw (532.58,406.75) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$2$};
        \draw (531.8,370.35) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$3$};
        \draw (573,353.93) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$4$};
        \draw (594.74,386.19) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$5$};
        \end{tikzpicture}
        }
        \caption{The Hasse diagram of the poset structure on $\mathfrak{S}_{1,5}$.}
        \label{fig:OGr15}
    \end{figure}
    \end{example}

The next theorem shows that $\OGr_+(1,n)$ is a positive geometry. For convenience, we permute\footnote{Here, since $k=1$, permuting the coordinates does not change the signs of the "minors".} the coordinates of $\PP^{n-1}$ and write:
\[
    \OGr_+(1,n) = \big\{(y_1: \dots: y_n) \in \PP_{+}^{n-1} \colon y_1^2 + \dots + y_p^2 - y_{p+1}^2 - \dots - y_n^{2} = 0 \big\}.
\]

 \begin{theorem}
         The semi-algebraic set $\OGr_+(1,n)$ is a positive geometry. Its canonical form is:
		\[
		\Omega = (1+u_{2,1}^2 + u_{3,1}^2 + \cdots + u_{p,1}^2) \  \frac{ du_{2,1} \wedge du_{3,1} \wedge \dots \wedge du_{n-1,1} }{ u_{2,1} \ u_{3,1} \ \ \cdots \ u_{n-1,1} \ u_{n,1}^2},
		\]
		where $u_{i,j}= x_i/x_1$ in the projective coordinates $(x_1 : \dots : x_n)$ of $\PP^{n-1}$.
    \end{theorem}
    

% 	\begin{example}[$\OGr_{+}(1, 4)$]
%     The region $\OGr_{+}(1,4)$ is a curvy quadrilateral inside the $3$-simplex in $\PP^3_{+}$ as depicted in \Cref{fig:OGr+1,4}. It is a positive geometry and its canonical form has the following expression in the coordinates $u_{i,j} = x_i/x_j$
% 		\begin{align*}
% 			\Omega &= (1 + u_{3,1}^2) \ \frac{du_{2,1} \wedge du_{3,1}}{u_{3,1} \ u_{2,1} \ u_{4,1}^2 } =  (1+u_{1,3}^2) \ \frac{du_{2,1} \wedge du_{0,1}}{ u_{0,1} \ u_{2,1} \ u_{3,1}^2}.
% 		\end{align*}
        
		
% 		\begin{figure}[ht]
% 			\centering
% 			\begin{center}
% 				\begin{tikzpicture}
% 					% coordinates
% 					\coordinate (a) at (-2,0);
% 					\coordinate (b) at (2,0);
% 					\coordinate (c) at (0,1.5);
% 					\coordinate (d) at (0,-2);
					
% 					\draw[draw=black, line width=1pt, dashed] (c) -- (d);
					
% 					\draw[black] (2.7,0) node  [xscale = 0.6, yscale = 0.6] {$[1:0:0:0]$};
% 					\draw[black] (-2.7,0) node  [xscale = 0.6, yscale = 0.6] {$[0:1:0:0]$};
% 					\draw[black] (0,1.7) node  [xscale = 0.6, yscale = 0.6] {$[0:0:1:0]$};
% 					\draw[black] (0,-2.3) node  [xscale = 0.6, yscale = 0.6] {$[0:0:0:1]$};
					
					
% 					\filldraw[draw=black, fill=red, opacity = 0.8] 
% 					(1,0.7) .. controls (0.5,0.5) and (-0.5,0.5) .. (-1,0.7)  
% 					.. controls (-0.75,0.5) and (-0.75,-0.5) .. (-1,-1)
% 					.. controls (-0.5,-0.7) and (0.5,-0.7) .. (1,-1)
% 					.. controls (0.75,-0.5) and (0.75,0.5) .. (1,0.7);
% 					\filldraw[draw=black, fill=gray, opacity = 0.75, line width=0pt] (a) -- (b) -- (c) -- cycle;
% 					\draw[draw=black, fill=gray, opacity = 0.4, line width=0pt] (a) -- (b) -- (d) -- cycle;        
% 				\end{tikzpicture}
% 			\end{center}
% 			\caption{The red region in the tetrahedron $\PP^3_{+}$ is $\OGr_{+}{(1,4)}$. The four facet boundaries of $\OGr_{+}(1,4)$ lie on the four facets of $\PP^3_{+}$.}
% 			\label{fig:OGr+1,4}
% 		\end{figure}		
% 	\end{example}


\section{The positive orthogonal Grassmannian \texorpdfstring{$\OGr_+(k,2k+1)$}{}} \label{sec4}
    
	We recall that we are working with the sign alternating form \eqref{eq:altForm}. The positroid cells of $\Gr_+(k,2k)$ induce a cell decomposition on the nonnegative orthogonal Grassmannian $\OGr_+(k,2k)$, and the cells of this decomposition are indexed by fixed-point-free involutions of $[2k]$. The face structure of $\OGr_+(k,2k)$ and the parametrization of its cells are studied in detail in \cite[Section 5]{galashin2020ising}.
    
    One of the reasons positroid cells induce a cell decomposition of the nonnegative orthogonal Grassmannian $\OGr_+(k,2k)$ is that, per \eqref{eq:LinearRelationOGr(k,2k)}, the latter is obtained by slicing $\Gr_+(k,2k)$ by a linear space. In general, one can obtain $\OGr_+(k,n)$ by slicing the positive flag variety with a linear space. For a subspace $V$ in $\CC^n$ of dimension $k$, we denote by $V^\perp$ its orthogonal complement with respect to the form \eqref{eq:altForm}.
    
	\begin{lem}\label{lem:hodgeStarMap}
		The Hodge star map $\Gr(k,n) \to \Gr(n-k,n), \ V \to V^\perp$ is given in Pl\"ucker coordinates by:
		\[
		q_J = p_{J^c}, \quad \text{for any } J \in \binom{[n]}{n-k},
		\]
		where $p_I$ and $q_J$ are Pl\"ucker coordinates in $\Gr(k,n)$ and $\Gr(n-k,n)$ respectively. In particular it restricts to an isomorphism of positive geometries between $\Gr_+(k,n)$ and $\Gr_+(n-k,n)$.
	\end{lem}
	
    Let $\mcF(k,n)$ be the $2$-step flag variety of partial flags $V \subset W \subset \CC^n$ where $\dim(V) = k$ and $\dim(W) = n-k$. The nonnegative part $\mcF_+(k,n)$ of $\mcF(k,n)$ is the semi-algebraic set of points $(V,W) \in \Gr_+(k,n) \times \Gr_+(n-k, n)$ such that $(V,W) \in \mcF(k,n)$. We denote by $\mcD$ the \emph{diagonal} subset of $\PP^{\binom{n}{k}} \times \PP^{\binom{n}{n-k}}$ i.e.
	\[
	\mcD := \left\{(p, q) \colon p_I = q_{I^c} \quad \text{for any } I \in \binom{[n]}{k} \right\}.
	\]
    
    \begin{proposition}
    The positive orthogonal Grassmannian $\OGr_+(k,n)$ is the intersection of the positive flag variety $\mcF_+(k,n)$ with $\mcD$ i.e.:
    	\begin{equation}\label{eq:DiagSlice}
    		\OGr_+(k,n) = \mcF_+(k,n) \cap \mcD.
    	\end{equation}
    \end{proposition}
 
	This motivates the choice of the sign alternating form \eqref{eq:altForm} in \cite{galashin2020ising,huang2014abjm}. However, unlike $\mcF_+(k,2k) \cong \Gr_+(k,2k)$, the nonnegative region $\mcF_+(k,n)$ is not well understood\footnote{The Lusztig positive part of $\mcF(k,n)$ is well understood but it can be shown that the Pl\"ucker positive region $\mcF_+(k,n)$ strictly contains the Lusztig positive region when $n > 2k + 1$, see \cite{BBEG24}.} for general $k$. This motivates the following question:
     
     \begin{problem}
       Study the face structure of $\mcF_+(k,n)$ and find a parametrization of its cells.
     \end{problem}

	\begin{proposition}
		The homogeneous coordinate rings of the $2$-step flag variety $\mcF(k,2k+1)$ and the Grassmannian $\Gr(k+1, 2k+2)$ are isomorphic.
	\end{proposition}
 
	\begin{theorem}\label{thm:LinIsom}
		The orthogonal Grassmannians $\OGr(k,2k+1)$ and $\OGr(k+1, 2k+2)$ can be identified through a linear isomorphism $(q_J) \mapsto p_I = q_{I \cup \{2k+2\}}$. This isomorphism restricts to an isomorphism of the positive regions $\OGr_+(k,2k+1)$ and $\OGr_+(k,2k+1)$.
	\end{theorem}
	\begin{remark}
		The equations that cut out $\OGr(k,2k+1)$ in $\Gr(k,2k+1)$ are all quadrics. It is remarkable that we can still describe the face structure of $\OGr_+(k,2k+1)$ from our understanding of the face structure of $\OGr_+(k+1, 2k+2)$.
	\end{remark}
	
	\begin{example}[$\OGr_+(2,5)$]
		The orthogonal Grassmannian $\OGr_+(2,5)$ is isomorphic to $\OGr_+(3,6)$. The Hasse diagram of the face poset of the latter is in \cite[Figure 7]{galashin2020ising}. \Cref{fig:OGr25} gives the same Hasse diagram in the realizable permutations in $\OGr_+(2,5)$. These cells can be parameterized using the isomorphism in \Cref{thm:LinIsom} and \cite[Theorem 5.17 (i)]{galashin2020ising}.
	\end{example}
 
     \begin{figure}[ht]
         \centering
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         \caption{The face poset of $\OGr_+(2,5)$ matches that of $\OGr_+(3,6)$. See \cite[Figure 7]{galashin2020ising}. \label{fig:OGr25}}
     \end{figure}

    

     We finish this section by explaining how one goes from matchings $\tau$ on $[2k+2]$ to the realizable permutations in $[2k+1]$ i.e. permutations $\sigma$ of $[2k+1]$ with corresponding positroid cell $\Pi_{\sigma}$ such that $\Pi_{\sigma} \cap \OGr_+(k,2k+1)$ is nonempty. Let $c$ denote the chord in $\tau$ attached to the vertex $2k+2$ and, starting from the vertex $2k+2$, consider the largest sequence $c = c_1, c_2, \dots, c_r$ of pairwise intersecting chords of $\tau$. Denote the $2r$ vertices of these chords by $i_1 < \dots < i_{2r-1} < 2k+2$. Then the cell $\Pi_{\tau} \cap \OGr_+(k+1, 2k+2)$ is isomorphic to the cell $\Pi_{\sigma} \cap \OGr_+(k, 2k+1)$ where $\sigma$  is the permutation of $[2k+1]$ obtained by replacing the chords $c_1, \dots, c_r$ with the unique cycle with support $\{i_1, \dots, i_{2r-1}\}$ and $r$ excedances. See \Cref{fig:matchingCorrespondence} for an example.
     
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    \end{tikzpicture}
    }
         \caption{A matching $\tau$ of $[2k+2]$ and the corresponding permutation $\sigma$ of $[2k+1]$ for $k = 7$. On the left, starting vertex $16$ (in blue), the chords in red are longest sequence of chords $c_1,\dots, c_r$ that intersect pairwise. On the right, vertex $16$ is deleted and the red chords turn into the unique cycle with support $\{1,2,4,8,11,13,15\}$ and 4 excedances.}
         \label{fig:matchingCorrespondence}
     \end{figure}

    \section{What goes wrong when \texorpdfstring{$n > 2k+1$}{} and \texorpdfstring{$k>1$}{}?}
	\label{sec5}
    In this section we show why positroid cells fail to induce a cell decomposition  of the orthogonal Grassmannian $\OGr_{+}(k, n)$ as soon as $n > 2k + 1$ and $k>1$. Let us start with the following:
    
	\begin{definition}\label{def:ortho-Positroids}
		For any positroid $\scrM$ of type $(k,n)$ and for any pair of subsets $I,J$ of $[n]$ of size $k-1$ we define the following two subsets of $[n]$:
		\[
		A_{IJ}^{\pm}(\scrM) = \big\{ \ell \in [n] \colon I\ell, J\ell \in \scrM \text{ and } (-1)^{\ell - 1} \epsilon_{I\ell} \epsilon_{J\ell} =  \pm 1 \big\}.
		\]
		We say that $\scrM$ is an orthopositroid if for any $I,J \in \binom{[n]}{k-1}$ we have:
		\[
		A_{IJ}^{+}(\scrM) = \emptyset \quad \iff \quad    A_{IJ}^{-}(\scrM) = \emptyset. 
		\]
	\end{definition}
 
	\begin{example}Let $n = 5$ and consider the two following positroids:
		\[
		\scrM_1 = \big\{\{1, 2\}, \{1, 4\}, \{2, 5\}, \{4, 5\} \big\} \ \text{and} \ \scrM_2 = \big\{ \{1, 2\}, \{1, 3\}, \{2, 4\}, \{3, 4\}\big\}.
		\]
		We then have $A^+_{24}(\scrM_1)  = \emptyset  \quad \text{and} \quad A^-_{24}(\scrM_1)  =  \{2\}$. So $\scrM_1$ is \textbf{\emph{not}} an orthopositroid. One can check that $\scrM_2$ is an orthopositroid.
	\end{example}

    The motivation behind this definition is that if $X$ is a point in $\OGr_+(k,n)$ and $\mathscr{M}_X$ is its associated positroid then $\mathscr{M}_X$ is necessarily an orthopositroid in the sense of \Cref{def:ortho-Positroids}. This is because the Pl\"ucker coordinates of $X$ satisfy the equations \eqref{eq:orthRelationsAlt}.

    \begin{conjecture}
        For each orthopositroid $\mathscr{M}$ of type $(k,n)$, there exists $X$ in $\OGr_+(k,n)$ such that $\mathscr{M}_X = \mathscr{M}$.
    \end{conjecture}
    
    Since we will show that positroid cells do not give a cell decomposition of $\OGr_+(k,n)$, we refrain from elaborating on the realizability of orthopositroids for general $k$.
    
        \begin{table}[ht]
                \centering
                \input{tableO26}
                \caption{The 99 realizable permutations in $\OGr_+(2, 6)$, organized by dimension.}
                \label{tab:O26}
        \end{table}
    
    Let us start with $\OGr_+(2,6)$. An exhaustive computation shows that, out of all the positroids $\mathscr{M}$ (or decorated permutations $\sigma$) of type $(2,6)$, there are exactly $99$ orthopositroids (or admissible permutations). We list them in \Cref{tab:O26}. Let us focus on the following two orthopositroid cells in $\OGr_+(2,6)$:

    \begin{center}
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        }
    \end{center}
    
    Let $C_\sigma := \Pi_{\sigma} \cap \OGr_+(2,6)$ and $C_\tau :=  \Pi_{\tau} \cap \OGr_+(2,6)$ be the two positroid cells in $\OGr_+(2,6)$ corresponding to $\sigma$ and $\tau$ respectively. We start by giving generic matrices $M_\sigma, M_\tau$ that parametrize the points of $C_\sigma, C_\tau$ respectively:
    %\begin{align*}
    %    M_\sigma &= \begin{bmatrix} 1 & 1 & 0 & 0 & -x& -x\\ 0 & 0 & 1 & 1 &  y & y \end{bmatrix}, \quad &\text{for } &x,y > 0,\\
    %    M_\tau &= \begin{bmatrix} 1 & 1 & 0 & 0 & 0& 0\\ 0 & 0 & 1 & a &  b & c \end{bmatrix}, \quad &\text{for } &\begin{cases}
    %                              a,b,c>0 \\
    %                              1 + b^2  = a^2 +  c^2
    %                          \end{cases}.
    %\end{align*}
     \[
       M_\sigma = \begin{bmatrix} 1 & 1 & 0 & 0 & -x& -x\\ 0 & 0 & 1 & 1 &  y & y \end{bmatrix} 
        \quad \text{and} \quad M_\tau = \begin{bmatrix} 1 & 1 & 0 & 0 & 0& 0\\ 0 & 0 & 1 & a &  b & c \end{bmatrix}
     \]
      for $x,y > 0$ and $ a,b,c > 0$ such that $1 + b^2  = a^2 +  c^2$.
      
      The closure $\overline{C_\tau}$ of $C_\tau$ has the combinatorial type of a square and the closure $\overline{C_\sigma}$ of the cell $C_\sigma$ has the combinatorial type of a triangle. The edges of the latter are given by:
    \[
   \resizebox{.99\hsize}{!}{$
    e_1 = \begin{bmatrix} 1 & 1 & 0 & 0& b& b\\ 0 & 0 & 1 & 1 &  0& 0 \end{bmatrix}, \quad
    e_2 = \begin{bmatrix} 1 & 1 & b & b & 0& 0\\ 0 & 0 & 0 & 0 & 1& 1 \end{bmatrix}, \quad e_3 = \begin{bmatrix} 1 & 1 & 0 & 0& 0& 0\\ 0 & 0 & 1 & 1 &  b& b \end{bmatrix} \quad b \geq 0$}.
    \]
    The edge $e_1$ is one of the diagonals of the ``square'' $C_\tau$. So the cell $C_\sigma$ glues with the cell $C_\tau$ as in \Cref{fig:gluing}. 
    \begin{figure}[ht]
        \centering
        \begin{center}
                \scalebox{0.8}{
				\begin{tikzpicture}
					\filldraw[draw=black, fill=red, opacity = 0.8] 
					(1,0.7) .. controls (0.5,0.5) and (-0.5,0.5) .. (-1,0.7)  
					.. controls (-0.75,0.5) and (-0.75,-0.5) .. (-1,-1)
					.. controls (-0.5,-0.7) and (0.5,-0.7) .. (1,-1)
					.. controls (0.75,-0.5) and (0.75,0.5) .. (1,0.7);
                    \filldraw[draw=black, fill=green, opacity = 0.6] 
					(0.75,1.7) .. controls (0.5,0.5) and (-0.5,0.5) .. (-1,-1)
					.. controls (-0.5,0.2) and (0.1,-0.4) .. (1,0.71)
					.. controls (0.75,0.4) and (0.75,0.8) .. (0.75,1.7);
				\end{tikzpicture}
                }
    \end{center}
        \caption{A cartoon of the cell $C_\sigma$ (in green) glued to the cell $C_\tau$ (in red).}
        \label{fig:gluing}
    \end{figure}
    
    This shows that the positroid cells are not enough to induce a CW cell decomposition on $\OGr_+(2,6)$. In general this problem arises as soon as $n > 2k+1$.  This is because whenever $n > 2k+1$ we have $n-6 \geq  2(k-2)$, so we can extend a $2 \times 6$ matrix in $\OGr_+(2,6)$ by a $(k-2)\times(n-6)$ as follows
    \setcounter{MaxMatrixCols}{20}
    \[
        \resizebox{0.6\hsize}{!}{$\begin{bmatrix}
    1 & 1 &  0  & \cdots  & \cdots  & \cdots  & \cdots  & \cdots  & \cdots  & \cdots  & \cdots  & 0   &  &  &  &  &  & \\
    0 & 0 &  1  & 1       &       0 & \cdots  & \cdots  & \cdots  & \cdots  & \cdots  & \cdots  & 0   &  &  &  &  &  & \\
    0 & \cdots  & \cdots  &  & \ddots  & \cdots  & \cdots  & \cdots  & \cdots  & \cdots  & \cdots  & \vdots  &  &  &  & (0) &  & \\
    \vdots  & \cdots  & \cdots  & \cdots  & 0 & 1 & 1 & \cdots  & \cdots  & \cdots  & \cdots  & 0 &  &  &  &  &  & \\
    0 & \cdots  & \cdots  & \cdots  & \cdots  & \cdots  & 0 & 1 & 1 & 0 & \cdots  & 0 &  &  &  &  &  & \\
     &  &  &  &  &  &  &  &  &  &  &  & \ast  & \ast  & \ast  & \ast  & \ast  & \ast \\
     &  &  &  &  & (0) &  &  &  &  &  &  & \ast  & \ast  & \ast  & \ast  & \ast  & \ast 
    \end{bmatrix}$.
    }
    \]    
    We can then realize each positroid cell in $\OGr_+(2,6)$ as some positroid cell of $\OGr_+(k,n)$ and the same problem as above arises again. This highlights the need for new combinatorics to give a CW cell decomposition of $\OGr_+(k,n)$ when $n > 2k+1$ and $k > 1$.    

    \begin{problem}
        Find a cell decomposition for $\OGr_+(k,n)$ when~$n > 2k+1$ and describe the combinatorics behind its face poset.
    \end{problem}
    
\printbibliography


\end{document}
