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\newcommand{\diag}{\mathrm{diag}}
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\title[Pl\"ucker inequalities]{Pl\"ucker inequalities for weakly separated coordinates in TNN Grassmannians}

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\author[D. Soskin and P.K. Vishwakarma]{\texorpdfstring{Daniel Soskin\thanks{\href{mailto:dsoskin@ias.edu}{dsoskin@ias.edu}. D.S. was supported by the National Science Foundation under Grant No. DMS-1926686.}\addressmark{1} \and Prateek Kumar Vishwakarma\thanks{\href{mailto:prateek-kumar.vishwakarma.1@ulaval.ca}{prateek-kumar.vishwakarma.1@ulaval.ca}. P.K.V. is supported by the Centre de recherches math\'ematiques-Laval (CRM-Laval) Postdoctoral Fellowship. He also acknowledges the Pacific Institute for the Mathematical Sciences (PIMS), a SwarnaJayanti Fellowship from DST and SERB, Govt. of India (PI: Apoorva Khare), the SPARC travel support (Scheme for Promotion of Academic and Research Collaboration, MHRD, Govt. of India; PI: Tirthankar Bhattacharyya), and the University of Plymouth (UK).}\addressmark{2}}{Daniel Soskin and Prateek Kumar Vishwakarma}}

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\address{\addressmark{1}Institute for Advanced Study, Princeton, NJ, USA \\ 
\addressmark{2}Universit\'e Laval, Qu\'ebec, QC, Canada}

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\abstract{
The fundamental connections of the Grassmannian with both weak separability and Pl\"ucker relations are well known. In this work we focus on the totally nonnegative (TNN) part of the Grassmannian, and discover the intrinsic connection between weak separability and Pl\"ucker relations. In particular, we show that certain natural sums of terms in a long Pl\"ucker relation for pairs of weakly separated Pl\"ucker coordinates oscillate around $0$ over the TNN Grassmannian. This generalizes the classical oscillating inequalities by Gantmacher--Krein (1941) and recent results on TNN matrix inequalities by Fallat--Vishwakarma (2024). In fact we obtain a characterization of weak separability, by showing that no other pairs of Pl\"ucker coordinates satisfy this property. Moreover, our work uncovers the natural connections between weak separability, Pl\"ucker relations, as well as Temperley--Lieb immanants, and provides a general and natural class of additive inequalities in Pl\"ucker coordinates on the totally nonnegative part of the Grassmannian.}

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\keywords{Determinantal inequalities, total nonnegativity, Pl\"ucker relations, oscillating inequalities, weak separability, totally nonnegative Grassmannian, cluster algebras}

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\begin{document}

\maketitle
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\section{Introduction and main results}
A real matrix is referred to as {\textit{totally nonnegative (TNN)}} if the determinants of all its square submatrices are {nonnegative.} Total {nonnegativity} arose initially in a few different areas: by Gantmacher--Krein \cite{GantKreinOsc} in oscillations of vibrating systems, by Fekete--P\'olya \cite{FP1912} (following Laguerre) in understanding the variation diminishing property of linear operators, and by Schoenberg and coauthors \cite{ASW} in applications to the analysis of real roots of polynomials and spline functions. These matrices play significant roles in algebraic and enumerative combinatorics, integrable systems, probability, classical mechanics, and many other areas \cite{ando,FJTNNMatrices,gas2013total,K2}. 
%The area of total positivity continues to be studied by many researchers, see for instance \cite{Chepuri-Sherman-Bennett-CJM, Khare-Chou-Kannan, Galashin-Karp-Lam, Galashin-Pylyavskyy, Parisi-Sherman-Bennett-Williams, Serhiyenko-Sherman-Bennett-Williams}. 
Lusztig extended the notion of total positivity to reductive Lie groups \emph{G} \cite{LusztigTP}, where the totally nonnegative part $\emph{G}^{\geq0}$ of \emph{G} is a semialgebraic submonoid of $\emph{G}$ generated by Chevalley generators. $\emph{G}^{\geq0}$ is the subset of $\emph{G}$ where all elements of the dual canonical basis are nonnegative \cite{lusztig1998introduction}. This concept could be generalized even further to varieties \emph{V}. The totally nonnegative subvariety is defined as the subset of \emph{V} where certain regular functions on \emph{V} have nonnegative values \cite{berenstein1997total,fomin1999db,gekhtman2010cluster}. Lusztig proved that specializations of elements of the dual canonical basis in the representation theory of quantum groups at $q=1$ are totally nonnegative polynomials. Thus, it is important to investigate classes of functions on matrices that are nonnegative on totally nonnegative matrices. We will discuss a source of such functions that are closely related to the \emph{(long) Pl\"ucker relations}.

This project traces its roots to the classical work by Gantmacher--Krein \cite{GantKreinOsc} where they modify the Laplace formula (1772) for the determinant expansion along the first row of the matrices to derive a sequence of inequalities oscillating about 0 and holding for all totally nonnegative matrices. These were expanded into new systems of inequalities by Fallat--Vishwakarma \cite{fallat2024inequalities}, who similarly derived an oscillating set of inequalities from the generalized Laplace identity. To state this result more precisely we introduce some notations: for a matrix $A,$ define $A_{P,Q}$ as the submatrix with rows indexed by $P$ and columns indexed by $Q,$ $[m,n]:=\{m,\dots,n\}$ and $[n]:=[1,n]$ for integers $0\leq m\leq n.$ 

\begin{theorem}[Fallat--Vishwakarma \cite{fallat2024inequalities}]\label{FVthm}
Let $1\leq d < n$ be integers. Suppose $P_d:=[d],$ and $Q_{dk} := [n-d,n]\setminus\{n-d+k\},$ for all $k\in [0,d].$ Then
\begin{align*}
\sum_{k=0}^{l} (-1)^{l+k} \det A_{P_{d},Q_{dk}} \det A_{[n]\setminus P_{d},[n]\setminus Q_{dk}} \geq 0, \mbox{ for all }l\in [0,d],
\end{align*}
for all $n\times n$ totally nonnegative matrices $A.$
\end{theorem}
In this extended abstract we explain why these inequalities refine certain long Pl\"ucker relations, via the Temperley--Lieb immanants introduced by Rhoades--Skandera \cite{RSkanTLImmp}. Another classical identity, which can also be viewed as one of the Pl\"ucker relations, is the well-known Karlin's identity \cite{K2}. Fallat and Vishwakarma provided a complete refinement of this identity for TNN matrices (see Theorem B in \cite{fallat2024inequalities}).
%Another classical identity that can also be seen as one of the Pl\"ucker relations is the famous Karlin's identity \cite{K2}, and Fallat--Vishwakarma completely refined this identity for TNN matrices (see Theorem~B in \cite{fallat2024inequalities}). 
%These two instances suggest exploring the possible existence of a larger set of ``{Pl\"ucker inequalities}'' over the TNN (totally nonnegative) Grassmannian. 
These two examples suggest investigating the potential existence of a broader set of ``Pl\"ucker inequalities'' over the totally nonnegative Grassmannian. This is exactly the focus of our main theorem; to present it, we establish some classical facts.

Throughout this article we assume $1\leq m\leq n$ are integers. The Grassmannian $\Gr(m,m+n)$ is the manifold of $m$-dimensional subspaces of $\R^{m+n}.$ Such subspaces can be represented by matrices $X\in \R^{(m+n) \times m}$ of full rank. Let $X_I$ denote the submatrix corresponding to rows indexed by $m$ element subsets $I \subseteq [m+n]$ and columns indexed by $[m].$ Here $\Delta_I(X):=\det X_{I}$ denotes the corresponding maximal minor of $X.$ Note that right multiplication by an invertible $m\times m$ matrix $B$ rescales each $\Delta_I(X)$ by $\det B.$ Thus, the minors $\Delta_I(X)$ together form projective coordinates of $\Gr(m,m+n),$ known as the corresponding \textit{Pl\"ucker coordinates.} These are related via \textit{Pl\"ucker relations}: defined for $m$-element ordered subsets $I:=(i_1,\dots,i_m), J:=(j_1,\dots,j_m)$ of $[m+n],$ and $I_{k,r}:=(i_{1},\dots,i_{r-1},j_{k},i_{r+1},\dots,i_{m})$ and $J_{k,r}:=(j_{1},\dots,j_{k-1},i_{r},j_{k+1},\dots,j_{m})$ for $k,r\in [1,m]$ \cite{gekhtman2010cluster,postnikov2006total}:
\begin{align}\label{Plucker-rel-2}
\Delta_I\Delta_J =  \sum_{k\in [m]} \Delta_{I_{k,r}}\Delta_{J_{k,r}} \quad \mbox{at each point in $\Gr(m,m+n)$\quad (and conversely).} 
\end{align}
Here $\Delta_{(i_{1},...,i_{m})}$ is labelled by an ordered sequence, and equals $\Delta_{\{i_{1},...,i_{m}\}}$ if $i_{1} < \dots < i_{m}.$ Also, $\Delta_{(i_{1},...,i_{m})}=(-1)^{\sgn(w)}\Delta_{(i_{w(1)},...,i_{w(m)})}$ for all permutations $w \in \sm.$ The totally nonnegative (TNN) Grassmannian $\Gr^{\geq 0}(m,m+n)\subset \Gr(m,m+n)$ contains those vector subspaces which have a representative matrix $X\in \R^{(m+n)\times m}$ with all $\Delta_{I} \geq 0$ \cite{postnikov2006total}.

The main idea in this paper involves two steps: (1) restricting the Pl\"ucker relations to the TNN Grassmannian, and (2) showing that natural families of inequalities can be ``extracted'' from these relations. That is, we show that for \textit{weakly separated} $I,J$ (\Cref{DIJlr}), one can extract inequalities out of \eqref{Plucker-rel-2} that look like
\begin{align}\label{Plucker-rel-3}
\Delta_I\Delta_J \leq  \sum_{k\in \mathcal{M}} \Delta_{I_{k,r}}\Delta_{J_{k,r}}, \quad \mbox{ and hold for each point in $\Gr^{\geq 0}(m,m+n),$} 
\end{align}
for ``nice'' subsets $\mathcal{M} \subseteq [m].$ In fact we describe \textit{all} $I,J$ for which such natural inequalities can be obtained.
% \begin{align*}
% \Delta_{(i_{1},\dots,i_{m})}\cdot \Delta_{(j_{1},\dots,j_{m})}= \sum_{k=1}^{m} \Delta_{(j_{k},i_{2},\dots,i_{m})}\cdot\Delta_{(j_{1},\dots,j_{k-1},i_{1},j_{k+1},\dots,j_{m})} 
% \end{align*}
% for all $i_1,\dots,i_{m},j_1,\dots,j_{m}\in [m+n].$ 
%{(To be precise, the TNN Grassmannian refers to those subspaces that have a representative matrix $X\in \R^{(m+n)\times m}$ with all nonzero $\Delta_{I}$ of the same sign.) Similarly, the totally positive (TP) Grassmannian $\Gr^{> 0}(m,m+n)\subset \Gr^{\geq 0}(m,m+n)$ is the collection of those vector subspaces which have a representative matrix $X\in \R^{(m+n)\times m}$ with all $\Delta_{I} > 0.$ (For precision, the TP Grassmannian refers to those subspaces that have a representative matrix $X\in \R^{(m+n)\times m}$ with all $\Delta_{I}$ nonzero and of the same sign.)} 
%In this paper we are concerned with certain inequalities in Pl\"ucker coordinates over the entire TNN Grassmannian. 
We now discuss the required notations. For $c_{I,J}\in \R$ and Pl\"ucker coordinates $\Delta_I$ and $\Delta_J,$ we say 
\begin{align*}
\sum_{I,J} c_{I,J}\Delta_{I}\Delta_{J} \mbox{ is nonnegative } \big{(}\geq 0\big{)} \quad \mbox{over}\quad \Gr^{\geq 0}(m,m+n)
\end{align*}
if for all subspaces $V\in \Gr^{\geq 0}(m,m+n)$ and all representative matrices $X \in \R^{(m+n)\times m}$ of $V,$ $\sum_{I,J} c_{I,J}\Delta_{I}(X)\Delta_{J}(X) \geq 0.$ The main result in this paper unifies the classical \cite{GantKreinOsc} and recent \cite{fallat2024inequalities} oscillating inequalities mentioned above for TNN matrices, by refining the Pl\"ucker relations for $\Gr^{\geq 0}(m,m+n).$ To state it, we require some notations:

\begin{defn}\label{DIJlr}
Fix integers $m,n\geq 1,$ and let $I$ and $J$ be ordered $m$-element subsets of $[m+n].$ Imagine that the elements of $I$ and $J$ are on the circle, among the points $1,2,\dots,m+n$ which are marked in clockwise order. 
\begin{enumerate}
\item We call $I,J$ \textit{weakly separated} if the sets $I\setminus J$ and $J\setminus I$ can be separated by a chord in the circle.
\item Suppose $\eta=|I\setminus J|=|J\setminus I|.$ Denote $I\setminus J := \{i_1,\dots,i_{\eta}\}$ and $J \setminus I := \{j_1,\dots,j_{\eta}\},$ such that $i_1~{<}_{c}~\cdots~{<}_{c}~i_{\eta}~{<}_{c}~ j_{\eta}$ and $i_1~{<}_{c}~j_1~{<}_{c}~\cdots~{<}_{c}~j_{\eta}$ in the clockwise order ${<}_{c},$ where each order is decided by moving in the clockwise direction on the circle starting from $i_1.$ (For example: if $m=n=6,$ $I=(1,5,3,4,10,11)$ and $J=(2,6,7,8,9,11),$ then $(i_1,i_2,i_3,i_4,i_5)=(10,1,3,4,5)$ and $(j_1,j_2,j_3,j_4,j_5)=(2,6,7,8,9)$. In particular, we choose $i_1$ and $j_{\eta}$ such that all other $i_k,j_k$ lie in between them while we traverse in the clockwise order starting from $i_1.$) Now define the following pair of tuples for $k,r\in [1,\eta]$: 
\begin{equation}\label{Eq:IJlr}
I_{k,r}:=(\dots,j_k,\dots), \quad \mbox{and} \quad 
J_{k,r}:=(\dots,i_r,\dots),
\end{equation}
where $j_k$ and $i_r$ replace each other in $I$ and $J,$ respectively.
\end{enumerate}
\end{defn}
%We begin by describing our main result for weakly separated $I,J.$ In the next inequalities, denote by $I^{\uparrow}$ the re-ordering of an $m$ element ordered subset $I$ of $[m+n]$ into increasing (in $\mathbb{Z}$) coordinates. Let $l,r\in [1,\eta].$ 
We begin by first describing our main result for weakly separated $I,J.$ Let $l,r\in [\eta],$ then we have a family of inequalities extracted from Pl\"ucker relations \eqref{Plucker-rel-2}:
\begin{equation}\label{Main-ex:1}
\pm \Big{(}\sum^{l}_{k=1} \Delta_{I_{k,r}}\Delta_{J_{k,r}} - \Delta_{I}\Delta_{J}\Big{)} \geq 0 \quad \forall~ l \geq \eta-r+1,\quad \mbox{over}\quad \Gr^{\geq 0}(m,m+n).
\end{equation}
This provides a novel class of ``Pl\"ucker-type'' determinantal inequalities that hold over the entire TNN Grassmannian for the class of weakly separated $I,J.$
% Then for all $l \geq \eta-r+1$:
% \begin{equation}\label{Main-ex:1}
% (-1)^{l}\Big{(}\sum^{l}_{k=1}(-1)^{k} \Delta_{I_{k,r}^{\uparrow}}\Delta_{J_{k,r}^{\uparrow}} + (-1)^{\eta-r}\Delta_{I^{\uparrow}}\Delta_{J^{\uparrow}}\Big{)} \geq 0 \quad \mbox{over}\quad\Gr^{\geq 0}(m,m+n).
% \end{equation}
% This provides a novel class of ``Pl\"ucker-type'' determinantal inequalities that hold over the entire TNN Grassmannian for the class of weakly separated $I,J.$ 
It also suggests two natural follow-up questions: \textit{(1)} Are there analogous Pl\"ucker-type inequalities that hold over $\Gr^{\geq 0}(m,m+n)$ for the remaining cases of $l,r \in [1,\eta]?$ (Here we continue to work with weakly separated $I,J.$) \textit{(2)} If yes, then does this also have a converse, in that certain Pl\"ucker-type determinantal inequalities do not hold if $I,J$ are not weakly separated? Our main theorem provides an affirmative answer to both of these questions, and in the process also characterizes weak separability. We also provide the precise meaning of ``$\pm$'' in \eqref{Main-ex:1}, and show that it refers to the \textit{sign} of $\Delta_{I_{l,r}}\Delta_{J_{l,r}}$ in the inequalities, defined via:
\begin{align*}
\sgn(I) = (-1)^{\sgn(w)}\quad\mbox{where}\quad w\in \mathrm{S}_{m}\quad \mbox{such that}\quad i_{w(1)} < \dots < i_{w(m)}
\end{align*}
for ordered $m$ element subsets $I:=(i_1,\dots,i_m)$ of $[m+n].$



%We introduce a notation before stating this result: for ordered $m$-element subsets $I:=(i_1,\dots,i_m)$ of $[m+n],$ $\sgn(I) = (-1)^{\sgn(w)}$ where $w\in \mathrm{S}_{m}$ such that $i_{w(1)} < \dots < i_{w(m)}.$
\begin{theorem}[Main result]\label{th:ws1}
Let $I,J$ be ordered $m$-element subsets of $[m+n];$ notation as in \Cref{DIJlr}. Consider the following system of oscillating inequalities for $l,r\in [1,\eta]$ over $\Gr^{\geq 0}(m,m+n)$:
\begin{align}\label{ws11:eq3}
\begin{aligned}
\sgn(I_{l,r})\,\sgn(J_{l,r})\sum^{l}_{k=1} \Delta_{I_{k,r}}\Delta_{J_{k,r}}  & \geq 0 \quad \forall~ l < \eta-r+1, \mbox{ and} \\
\sgn(I_{l,r})\,\sgn(J_{l,r})\Big{(}\sum^{l}_{k=1}\Delta_{I_{k,r}}\Delta_{J_{k,r}} - \Delta_{I}\Delta_{J} \Big{)}  & \geq 0 \quad \forall~ l \geq \eta-r+1.  
\end{aligned}
\end{align}
This system holds for all $l,r\in [1,\eta]$ if and only if $I$ and $J$ are weakly separated.
\end{theorem}
% \begin{remark}[``Uniform'' reformulation]
% \Cref{th:ws1} can be stated using more ``homogeneous'' terminology. Define for $k,r\in [1,\eta],$
% \begin{equation}\label{ws11:eq1}
% \Pi_{k,r} :=
% \begin{cases}
% \Delta_{I_{k,r}}\Delta_{J_{k,r}} - \Delta_{I}\Delta_{J}, & \mbox{if }k=\eta-r+1, \mbox{ and} \\
% \Delta_{I_{k,r}}\Delta_{J_{k,r}}, &\mbox{otherwise.}
% \end{cases}
% \end{equation}
% {\it Then the following system of oscillating inequalities holds:
% \begin{equation}\label{ws11:eq2}
%\sgn(I_{l,r})\,\sgn(J_{l,r})\sum_{k=1}^{l}\Pi_{k,r}\geq 0 \qquad \forall l,r\in [1,\eta],~\mbox{ over }\Gr^{\geq 0}(m,m+n),  
% \end{equation}
% if and only if $I$ and $J$ are weakly separated.}
% \end{remark}
% \begin{example}
% Recent refinements of the classical Karlin's identity for TNN matrices (see Theorems~B in \cite{fallat2024inequalities}) can be seen as Pl\"ucker-type inequalities in \Cref{th:ws1} above. To state the inequality, let $m\geq 1$ be an integer. Suppose $T$ is a subset of $[m]$ and $V:=\{v_1<\cdots<v_{m'} \}:=[m]\setminus T.$ Then for all $p\in [m]$ and set $S,$ such that $S\subseteq [m] \setminus \{p\}$ and $|S|=|T|+1,$
% \begin{equation}\label{Kar-GK-FV}
% (-1)^{1+l}\sum_{k=1}^{l} (-1)^{1+k} \det A\big{(}S\big{|}T\cup \{v_k\}\big{)} \det A\big{(}[n]\setminus\{p\}\big{|}[n]\setminus\{v_k\}\big{)}\geq 0,
% \end{equation}
% for all $l\in [m'],$ and all $m\times m$ totally nonnegative matrices $A.$
% Now, suppose $s:= \max S$ and let $\Gr^{\geq 0}(m,2m+1)_s \subset \Gr^{\geq 0}(m,2m+1)$ corresponds to those representative matrices in which rows $s$ and $s+1$ are identical. It is easy to see that $\Gr^{\geq 0}(m,2m+1)_s$ can be constructed from $\Gr^{\geq 0}(m,2m).$
% Suppose $S=\{s_1<\dots < s_{m-m'} < s \}$ and $p<s.$ ($p>s$ can be dealt with similarly.) Now choose 
% \[
% I=(1,\dots,p-1,p+1,\dots,s,s+1,\dots,m+1),\quad  J = (s_1,\dots,s_{m-m'},2m+2-v_{m'},\dots,2m+2-v_1).
% \]
% Note that $I,J$ are weakly separated, and choosing $i_r=s$ provides us with the required system of inequalities over $\Gr^{\geq 0}(m,2m+1)_s$ which is equivalent to \eqref{Kar-GK-FV}.
% %be constructed by inserting a copy of row $s$ just below it of $A$ et $A\in \Gr^{\geq 0}(n,2n).$ L
% %Taking ordered tuples $I:=(1,\dots, i, \dots,n),$ where the $k^{\mbox{th}}$ coordinate is $k$ for all $k \neq j$, and the $j^{\mbox{th}}$ coordinate is $i$, and $J:=(n+1,\dots, 2n),$ and further setting $r=j,$ i.e., $i_r=i$ would show the oscillating system \eqref{Kar-GK-FV} as a special case of \Cref{th:ws1}.
% \end{example}
% Another special case of \Cref{th:ws1} consists of the Gantmacher--Krein inequalities \cite{GantKreinOsc}, as well as their generalizations in \cite{fallat2024inequalities} (see \Cref{FVthm} above). We discuss these inequalities in \Cref{Sec:GenLapIneq}. Before this, we recall in \Cref{Sec:Pre} the tools needed to prove \Cref{th:ws1}, including Temperley--Lieb immanants and Kauffman diagrams. In \Cref{forward-imp}, we prove one implication of \Cref{th:ws1}: the system \eqref{ws11:eq3} holds for weakly separated Pl\"ucker coordinates. We conclude by showing the reverse implication in \Cref{rev-imp}.

One may refer to \cite{SV} for several (worked out) novel examples of these oscillating inequalities, including \Cref{FVthm} and the refinement of Karlin's identity. Next we recall the tools needed to prove \Cref{th:ws1}, including Temperley--Lieb immanants and Kauffman diagrams. We demonstrate the proof ideas assuming that $m=n=\eta,$ and refer the reader to \cite{SV} for the the general case.
%In \Cref{forward-imp}, we prove one implication of \Cref{th:ws1}: the system \eqref{ws11:eq3} holds for weakly separated Pl\"ucker coordinates.
\section{Temperley--Lieb immanants}\label{Sec:Pre} Some determinantal inequalities can be stated in terms of polynomials $p({\bf x})$ in the matrix entries ${\bf x}=(x_{ij})_{i,j=1}^{n},$ for integer $n\geq 1$. Such a polynomial is called \emph{totally nonnegative (TNN)} if it attains nonnegative values on TNN matrices.  Next, following Littlewood \cite{LittlewoodTGC} and Stanley \cite{StanPos}, given a function $f: \sn \rightarrow \mathbb C$ define the \emph{$f$-immanant} to be the polynomial
\begin{equation}\label{eq:immdef}
\imm f({\bf x}) \defeq \sum_{w \in \sn} f(w) \permmon xw \in \mathbb C[{\bf x}].
\end{equation}
A particular family of immanants are defined for the \emph{Temperley--Lieb algebra}, defined next. Fix $\xi\in \C$; a \emph{Temperley--Lieb algebra} $T_n(\xi)$ is generated by $t_1,\dots,t_{n-1}$ over $\C$ subject to: 
\begin{align}\label{TLalg:defn}
t_i^2 = \xi t_i ~~\mbox{for}~~ i\in [n-1];~t_i t_j t_i = t_i~~\mbox{if}~~|i-j|=1; \mbox{ and}~~t_i t_j = t_j t_i~~\mbox{if}~~|i-j|\geq 2.    
\end{align}
When $\xi = 2$ we have the isomorphism $T_n(2) \cong \csn/(1 + s_1 + s_2 + s_1s_2 + s_2s_1 + s_1s_2s_1)$ \cite{FanMon}, \cite[Sections 2.1 and 2.11]{GHJ}, and \cite[Section 7]{WestburyTL}. Specifically, the isomorphism is via
\begin{equation}\label{eq:sntotn}
\begin{aligned}
\sigma : \csn \rightarrow \tn ~\mbox{ with }~ s_i \mapsto t_i - 1.
\end{aligned}
\end{equation}
Let $\K_n$ be the multiplicative monoid generated by $t_1,\dotsc,t_{n-1}$ subject to \eqref{TLalg:defn}, taking $\xi = 1,$ i.e. modulo the powers of $\xi$. This is also known as a standard basis of $T_n(\xi)$.
%{(2)\color{blue}The standard basis of $T_n(\xi)$ is given by $\K_n$ which is defined as the multiplicative monoid generated by $t_1,\dotsc,t_{n-1}$ subject to the relations, taking $\xi = 1$.}
It is known that $|\K_n|$ is the $n^{\mbox{th}}$ Catalan number $C_n = \tfrac{1}{n+1}\tbinom{2n}{n}.$ Diagrams of the basis elements of $T_n(\xi),$ made popular by Kauffman~\cite[Section 4]{KauffState} are (undirected) graphs with $2n$ vertices and $n$ noncrossing edges, such that each edge lies in the convex hull of these $2n$ vertices (similar to one of the definitions of Catalan numbers involving $n$ noncrossing chords on a circle with $2n$ vertices). Now define the \textit{Temperley--Lieb immanant} 
$\imm{\tau}({\bf x})$ for each $\tau \in \K_n$ in terms of the function
\begin{equation}\label{eq:ftau}
\begin{aligned}
f_\tau: \sn \rightarrow \mathbb{C} ~\mbox{ with }~ w \mapsto \text{ coefficient of $\tau$ in } \sigma(w),
\end{aligned}
\end{equation}
and extend it linearly over $\csn$. Finally, define the Temperley--Lieb immanants as 
%To economize notation, we will write $\imm{\tau}$  instead of $\imm{f_\tau}$,
\begin{equation*}
\imm{\tau}({\bf x}):=\imm{f_\tau}({\bf x}) 
= \sum_{w \in \sn} f_{\tau}(w)x_{1,w_1} \cdots x_{n,w_n}.
\end{equation*}
Rhoades--Skandera \cite{RSkanTLImmp} showed that Temperley--Lieb immanants are a basis of the space 
\begin{equation}\label{eq:tlspace}
\spn_{\mathbb R} \{ \det {\bf x}_{P,Q}  \det {\bf x}_{P^c,Q^c}  \,|\, P,Q \subseteq [n] \mbox{ with }|P|=|Q| \},
\end{equation} 
and that they are TNN. Infact, these are the extreme rays of the cone of TNN immanants:
%in this space~\cite[Theorem\,10.3]{RSkanTLImmp}.
\begin{thm}[\cite{RSkanTLImmp}]\label{SB}
Given a function $f:\sn \to \R,$ the immanant
%Each immanant of the form
\begin{equation}\label{eq:sumofprodsof2minors}
\imm f({\bf x}) = \sumsb{P,Q \subseteq [n]\\ |P|=|Q|} c_{P,Q}
\det {\bf x}_{P,Q} \det {\bf x}_{P^c, Q^c} 
\end{equation}
is TNN if and only if it is a nonnegative linear combination of Temperley--Lieb immanants.
\end{thm}

In fact, each complementary product of minors is a $0$-$1$ linear combination of 
{\parfillskip=0pt \par} \noindent
Temperley--Lieb immanants~\cite[Proposition 4.4]{RSkanTLImmp}. Next we briefly demonstrate how \Cref{SB} is used in identifying inequalities that involve products of pairs of Pl\"ucker coordinates. For this, recall that the $n\times n$ TNN matrices embed inside $\Gr^{\geq 0}(n,2n)$:
\begin{align}\label{Plucker-embed}
\{\mbox{$n\times n$ TNN matrices}\}\hookrightarrow \Gr^{\geq 0}(n,2n)\quad\mbox{via}\quad {\bf x} \mapsto \overline{{\bf x}}:= \begin{pmatrix} {\bf x} \\ W_{0} \end{pmatrix},
\end{align}
where $W_{0}:=\big((-1)^{i+1}\cdot\delta_{j,n-i+1}\big)^{n}_{i,j=1}.$ This yields a one-to-one correspondence between the minors of ${\bf x}$ and the maximal minors of $\overline{{\bf x}}$ via
$\det {\bf x}_{P,Q} =\det \overline{{\bf x}}_{I,[m]}=\Delta_{I}(\overline{{\bf x}}),$ for all $P\subseteq [n],Q \subseteq [n]$ with $|P|=|Q|,$ where $I:= P \cup \{ 2n+1-j \,|\, j \in [n] \setminus Q\}
.$ This, along with the density of $\Gr^{>}(n,2n)$ in $\Gr^{\geq 0}(n,2n),$ and the projective geometry of $\Gr(n,2n),$ yield a \textit{correspondence} between inequalities in products of pairs of minors of TNN matrices and inequalities in products of pairs of Pl\"ucker coordinates over the TNN Grassmannian, which is in fact compatible with the Temperley--Lieb immanant idea: 

\begin{thm}[\cite{RSkanTLImmp,SV}]\label{t:oneprod}
Suppose $I$ runs over $n$-element subsets of $[2n],$ and $c_I\in \R.$ Then
\begin{align}\label{eq:pl0}
\sum_{I} c_{I}\Delta_{I} \Delta_{I^{\mathrm{c}}} \geq 0 \quad\mbox{over}\quad\Gr^{\geq 0}(n,2n),
\end{align}
if and only if $\sum_{I} c_{I}\Delta_{I}(\overline{{\bf x}}) \Delta_{I^{\mathrm{c}}}(\overline{{\bf x}})$ is a nonnegative linear combination of Temperley--Lieb immanants. Moreover, we have for each $I$ that 
\begin{equation}\label{eq:pl1}
\Delta_{I}(\overline{{\bf x}})\Delta_{I^c}(\overline{{\bf x}}) = \sum_{\tau \in \K_n}b_{\tau}\imm{\tau}({\bf x}),    
\end{equation}
where $b_{\tau}=1$ if each edge in the Kauffman diagram of $\tau$ connects elements from $I$ and $I^c$, and $0$ otherwise.
\end{thm}
%\begin{remark}
In light of \Cref{t:oneprod}, it is necessary and sufficient for the validity of \eqref{eq:pl0} that 
\begin{align*}
\mbox{the replacements}\quad\Delta_{I} \Delta_{I^{\mathrm{c}}} \leftrightarrow \sum_{\tau \in \K_n}b_{\tau}\imm{\tau}({\bf x}) \quad\mbox{in}\quad\eqref{eq:pl0}
\end{align*}
yield a nonnegative linear combination of immanants $\imm{\tau}({\bf x}),$ where $b_{\tau}=1$ if each edge in the Kauffman diagram of $\tau$ connects elements from $I$ and $I^c$, and $0$ otherwise. Now, one of the possible methods to keep track of these immanants is to count 2-colorings of the Kauffman diagrams, where each edge connects vertices of opposite colors. We show this via the following example (see \cite{SV} for more details). 
%\end{remark}

%One of the possible methods to show the nonnegativity of an element from \eqref{eq:tlspace}, and equivalently the validity of the corresponding \eqref{eq:pl0}, is to count 2-colorings of the Kauffman diagrams, where each edge connects vertices of opposite colors. We show this via the following example (see \cite{skan2022sosk} for more details). 

\begin{example}\label{ex:1}
Suppose we aim to verify if
\begin{align}\label{eq:ex:1}
\Delta_{(1,2,4)}\Delta_{(3,5,6)} - \Delta_{(1,2,3)}\Delta_{(4,5,6)} \geq 0  \quad \mbox{over}\quad \Gr^{\geq 0}(3,6).
\end{align}
For this, we make the following replacement/identification in \eqref{eq:ex:1}:
\begin{align*}
\Delta_{(1,2,3)}\Delta_{(4,5,6)} \leftrightarrow \imm{\tau_1}({\bf x}), \mbox{ and } \Delta_{(1,2,4)}\Delta_{(3,5,6)}\leftrightarrow\imm{\tau_1}({\bf x})+\imm{\tau_2}({\bf x}),   
\end{align*}
where the diagrams $\tau_1$ and $\tau_2$ are shown in \Cref{eq:ex1}, and the vertices corresponding to the sets $I$ and $I^c$ are colored in black and white, respectively. The first coloring admits only one bi-colored matching $\tau_1,$ while the second coloring admits exactly two bi-colored matchings $\tau_1$ and $\tau_2$. We substitute these replacements in \eqref{eq:ex:1} to get, 
\begin{align*}
\Delta_{(1,2,4)}\Delta_{(3,5,6)} - \Delta_{(1,2,3)}\Delta_{(4,5,6)} \leftrightarrow \big{(}\imm{\tau_1}({\bf x})+\imm{\tau_2}({\bf x})\big{)} - \imm{\tau_1}({\bf x}) = \imm{\tau_2}({\bf x}) \geq 0.
\end{align*}
\end{example}

\begin{figure}
\begin{center}
\begin{tikzpicture}
\Vertex[x=-1,y=0,size=0.1,color=white,position=right,label=$6$,fontscale=1]{0-6}
\Vertex[x=-1,y=0.6,size=0.1,color=white,position=right,label=$5$,fontscale=1]{0-5}
\Vertex[x=-1,y=1.2,size=0.1,color=white,position=right,label=$4$,fontscale=1]{0-4}

\Vertex[x=-2,y=1.2,size=0.1,color=black,position=left,label=$3$,fontscale=1]{0-3}
\Vertex[x=-2,y=0.6,size=0.1,color=black,position=left,label=$2$,fontscale=1]{0-2}
\Vertex[x=-2,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{0-1}

\Edge(0-1)(0-6) \Edge(0-2)(0-5) \Edge(0-3)(0-4)


\Vertex[x=4,y=0,size=0.1,color=white,position=right,label=$6$,fontscale=1]{1-6}
\Vertex[x=4,y=0.6,size=0.1,color=white,position=right,label=$5$,fontscale=1]{1-5}
\Vertex[x=4,y=1.2,size=0.1,color=black,position=right,label=$4$,fontscale=1]{1-4}

\Vertex[x=3,y=1.2,size=0.1,color=white,position=left,label=$3$,fontscale=1]{1-3}
\Vertex[x=3,y=0.6,size=0.1,color=black,position=left,label=$2$,fontscale=1]{1-2}
\Vertex[x=3,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{1-1}

\Edge(1-1)(1-6) \Edge(1-2)(1-5) \Edge(1-3)(1-4)

\Vertex[x=6.5,y=0,size=0.1,color=white,position=right,label=$6$,fontscale=1]{2-6}
\Vertex[x=6.5,y=0.6,size=0.1,color=white,position=right,label=$5$,fontscale=1]{2-5}
\Vertex[x=6.5,y=1.2,size=0.1,color=black,position=right,label=$4$,fontscale=1]{2-4}

\Vertex[x=5.5,y=1.2,size=0.1,color=white,position=left,label=$3$,fontscale=1]{2-3}
\Vertex[x=5.5,y=0.6,size=0.1,color=black,position=left,label=$2$,fontscale=1]{2-2}
\Vertex[x=5.5,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{2-1}

\Edge(2-1)(2-6) \Edge[bend=-45](2-2)(2-3) \Edge[bend=-45](2-4)(2-5)

\end{tikzpicture}
\end{center}
\caption{Kauffman diagrams $\tau_1;$ $\tau_1$ and $\tau_2,$ respectively, as refered to in \Cref{ex:1}.}
\label{eq:ex1}
\end{figure}

\section{Weak separability and Pl\"ucker inequalities}

In the theory of cluster algebras, one of the central examples is the cluster algebra of the Grassmannian $\Gr(m,m+n)$ \cite{JScott2006}. Clusters of minors are in correspondence with subsets of Pl\"ucker coordinates which can be described by set-theoretic properties. To recall, two $m$-element sets $I,J\subset [m+n]$ are weakly separated if $I\setminus J$ and $J \setminus I$ can be separated by a chord on the circle labeled with $1,2,\dots,m+n$ enumerated clockwise. A family of Pl\"ucker coordinates is said to be weakly separated if all pairs of Pl\"ucker coordinates in it are weakly separated \cite{LW}. It is known that maximal (by inclusion) weakly separated sets of Pl\"ucker coordinates are in bijection with the clusters consisting of Pl\"ucker coordinates in the cluster algebra of $\Gr{(m,m+n)}$ \cite{FarPost}. This makes weak separability important in the theory of cluster algebras (for instance also see \cite{OPostSpey2015}). Now recall our main result: weak separability provides the exact classification of indices $I,J$ for which the Pl\"ucker-type inequalities hold. We demonstrate one part of this via the next proof idea.

\begin{proof}[Proof idea for \Cref{th:ws1}: weak separability $\implies$ the system of inequalities \eqref{ws11:eq3}] ~ First we 
    {\parfillskip=0pt \par} \noindent
    consider the case when $m=n=\eta=4,$ the weakly separated (ordered) sets are $I=(1,2,3,4)$ and $J=(5,6,7,8),$ and $r=4,$ i.e. $i_r=i_4=4.$ Thus we have:
\begin{align*}
I_{1,r}=(1,2,3,5),~J_{1,r} = (4,6,7,8);\qquad &~I_{2,r}=(1,2,3,6),~J_{2,r} = (5,4,7,8);\\
I_{3,r}=(1,2,3,7),~J_{3,r} = (5,6,4,8);\qquad &~I_{4,r}=(1,2,3,8),~J_{4,r} = (5,6,7,4).
\end{align*}
The set of inequalities that we need to verify over $\Gr^{\geq 0}(4,8)$ are 
\begin{align}\label{proofidea:eq1}
\begin{aligned}
\Delta_{I_{1,r}^{\uparrow}}\Delta_{J_{1,r}^{\uparrow}} - \Delta_{I^{\uparrow}}\Delta_{J^{\uparrow}}&\geq 0 ,\\
-\big{(}\Delta_{I_{1,r}^{\uparrow}}\Delta_{J_{1,r}^{\uparrow}} - \Delta_{I^{\uparrow}}\Delta_{J^{\uparrow}}\big{)} + \Delta_{I_{2,r}^{\uparrow}}\Delta_{J_{2,r}^{\uparrow}} &\geq 0,\\
\big{(}\Delta_{I_{1,r}^{\uparrow}}\Delta_{J_{1,r}^{\uparrow}} - \Delta_{I^{\uparrow}}\Delta_{J^{\uparrow}}\big{)} -\Delta_{I_{2,r}^{\uparrow}}\Delta_{J_{2,r}^{\uparrow}} + \Delta_{I_{3,r}^{\uparrow}}\Delta_{J_{3,r}^{\uparrow}} &\geq 0,
\end{aligned}
\end{align}
and the last inequality is the Pl\"ucker relation itself. For this verification, we identify
\begin{align}\label{proofidea:eq2}
\begin{aligned}
\Delta_{I^\uparrow} \Delta_{J^\uparrow}\leftrightarrow \imm{\tau_0}({\bf x}),&\qquad \Delta_{I_{1,r}^\uparrow} \Delta_{J_{1,r}^\uparrow} \leftrightarrow \imm{\tau_0}({\bf x}) + \imm{\tau_1}({\bf x}),\\
\Delta_{I_{2,r}^\uparrow} \Delta_{J_{2,r}^\uparrow} \leftrightarrow \imm{\tau_1}({\bf x}) + \imm{\tau_2}({\bf x}),&\qquad \Delta_{I_{3,r}^\uparrow} \Delta_{J_{3,r}^\uparrow} \leftrightarrow \imm{\tau_2}({\bf x}) + \imm{\tau_3}({\bf x}),
\end{aligned}
\end{align}
where the corresponding Kauffman diagrams are shown in \Cref{proofidea:diagram1}. Indeed these inequalities are true since upon making immanant replacements in \eqref{proofidea:eq1} we get nonnegative linear combinations of Temperley--Lieb immanants.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\Vertex[x=1,y=0,size=0.1,color=white,position=right,label=$8$,fontscale=1]{0-8}
\Vertex[x=1,y=0.6,size=0.1,color=white,position=right,label=$7$,fontscale=1]{0-7}
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\Vertex[x=1,y=1.8,size=0.1,color=white,position=right,label=$5$,fontscale=1]{0-5}

\Vertex[x=0,y=1.8,size=0.1,color=black,position=left,label=$4$,fontscale=1]{0-4}
\Vertex[x=0,y=1.2,size=0.1,color=black,position=left,label=$3$,fontscale=1]{0-3}
\Vertex[x=0,y=0.6,size=0.1,color=black,position=left,label=$2$,fontscale=1]{0-2}
\Vertex[x=0,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{0-1}

\Edge(0-1)(0-8) \Edge(0-2)(0-7) \Edge(0-3)(0-6) \Edge(0-4)(0-5)


\Vertex[x=3.5,y=0,size=0.1,color=white,position=right,label=$8$,fontscale=1]{1-8}
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\Vertex[x=2.5,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{1-1}

\Edge(1-1)(1-8) \Edge(1-2)(1-7) \Edge(1-3)(1-6) \Edge(1-4)(1-5)


\Vertex[x=5.5,y=0,size=0.1,color=white,position=right,label=$8$,fontscale=1]{2-8}
\Vertex[x=5.5,y=0.6,size=0.1,color=white,position=right,label=$7$,fontscale=1]{2-7}
\Vertex[x=5.5,y=1.2,size=0.1,color=white,position=right,label=$6$,fontscale=1]{2-6}
\Vertex[x=5.5,y=1.8,size=0.1,color=black,position=right,label=$5$,fontscale=1]{2-5}

\Vertex[x=4.5,y=1.8,size=0.1,color=white,position=left,label=$4$,fontscale=1]{2-4}
\Vertex[x=4.5,y=1.2,size=0.1,color=black,position=left,label=$3$,fontscale=1]{2-3}
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\Vertex[x=4.5,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{2-1}

\Edge(2-1)(2-8) \Edge(2-2)(2-7) \Edge[bend=-45](2-5)(2-6) \Edge[bend=-45](2-3)(2-4)
%==============================================


\Vertex[x=8,y=0,size=0.1,color=white,position=right,label=$8$,fontscale=1]{3-8}
\Vertex[x=8,y=0.6,size=0.1,color=white,position=right,label=$7$,fontscale=1]{3-7}
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\Vertex[x=8,y=1.8,size=0.1,color=white,position=right,label=$5$,fontscale=1]{3-5}

\Vertex[x=7,y=1.8,size=0.1,color=white,position=left,label=$4$,fontscale=1]{3-4}
\Vertex[x=7,y=1.2,size=0.1,color=black,position=left,label=$3$,fontscale=1]{3-3}
\Vertex[x=7,y=0.6,size=0.1,color=black,position=left,label=$2$,fontscale=1]{3-2}
\Vertex[x=7,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{3-1}

\Edge(3-1)(3-8) \Edge(3-2)(3-7) \Edge[bend=-45](3-5)(3-6) \Edge[bend=-45](3-3)(3-4)

\Vertex[x=10,y=0,size=0.1,color=white,position=right,label=$8$,fontscale=1]{4-8}
\Vertex[x=10,y=0.6,size=0.1,color=white,position=right,label=$7$,fontscale=1]{4-7}
\Vertex[x=10,y=1.2,size=0.1,color=black,position=right,label=$6$,fontscale=1]{4-6}
\Vertex[x=10,y=1.8,size=0.1,color=white,position=right,label=$5$,fontscale=1]{4-5}

\Vertex[x=9,y=1.8,size=0.1,color=white,position=left,label=$4$,fontscale=1]{4-4}
\Vertex[x=9,y=1.2,size=0.1,color=black,position=left,label=$3$,fontscale=1]{4-3}
\Vertex[x=9,y=0.6,size=0.1,color=black,position=left,label=$2$,fontscale=1]{4-2}
\Vertex[x=9,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{4-1}

\Edge(4-1)(4-8) 

%\Edge(4-2)(4-5) 
\draw[-,very thick] (4-2) to[out=35, in=215, looseness=1.5] (4-5);

\Edge[bend=-45](4-6)(4-7) \Edge[bend=-45](4-3)(4-4)
%==============================================


\Vertex[x=12.5,y=0,size=0.1,color=white,position=right,label=$8$,fontscale=1]{5-8}
\Vertex[x=12.5,y=0.6,size=0.1,color=black,position=right,label=$7$,fontscale=1]{5-7}
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\Vertex[x=11.5,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{5-1}

\Edge(5-1)(5-8) 

%\Edge(5-2)(5-5) 
\draw[-,very thick] (5-2) to[out=35, in=215, looseness=1.5] (5-5);
\Edge[bend=-45](5-6)(5-7) \Edge[bend=-45](5-3)(5-4)

\Vertex[x=14.5,y=0,size=0.1,color=white,position=right,label=$8$,fontscale=1]{6-8}
\Vertex[x=14.5,y=0.6,size=0.1,color=black,position=right,label=$7$,fontscale=1]{6-7}
\Vertex[x=14.5,y=1.2,size=0.1,color=white,position=right,label=$6$,fontscale=1]{6-6}
\Vertex[x=14.5,y=1.8,size=0.1,color=white,position=right,label=$5$,fontscale=1]{6-5}

\Vertex[x=13.5,y=1.8,size=0.1,color=white,position=left,label=$4$,fontscale=1]{6-4}
\Vertex[x=13.5,y=1.2,size=0.1,color=black,position=left,label=$3$,fontscale=1]{6-3}
\Vertex[x=13.5,y=0.6,size=0.1,color=black,position=left,label=$2$,fontscale=1]{6-2}
\Vertex[x=13.5,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{6-1}

\Edge(6-1)(6-6) \Edge(6-2)(6-5) \Edge[bend=-45](6-7)(6-8) \Edge[bend=-45](6-3)(6-4)
\end{tikzpicture}
\end{center}
\caption{For \eqref{proofidea:eq2}: $\Delta_{I^\uparrow}\Delta_{J^\uparrow} \leftrightarrow \imm{\tau_0}({\bf x}),$ $\Delta_{I_{1,r}^\uparrow}\Delta_{J_{1,r}^\uparrow}\leftrightarrow \imm{\tau_0}({\bf x})+\imm{\tau_1}({\bf x}),$ $\Delta_{I_{2,r}^\uparrow}\Delta_{J_{2,r}^\uparrow}\leftrightarrow \imm{\tau_1}({\bf x})+\imm{\tau_2}({\bf x}),$ and $\Delta_{I_{3,r}^\uparrow}\Delta_{J_{3,r}^\uparrow}\leftrightarrow \imm{\tau_2}({\bf x})+\imm{\tau_3}({\bf x}),$ respectively.}
\label{proofidea:diagram1}
\end{figure}

To show another possibility in the combinatorics of the general proof, we discuss another (slightly more complicated) example. Suppose $m=n=\eta=5,$ the weakly separated sets are $K=(1,2,3,4,10)$ and $L=(5,6,7,8,9).$ Therefore $(k_1,k_2,k_3,k_4,k_5)=(10,1,2,3,4)$ and $(\ell_1,\ell_2,\ell_3,\ell_4,\ell_5)=(5,6,7,8,9).$ Fix $s:=r=3,$ i.e. $k_s=2.$ We have:
\begin{align*}
K_{1,s}=(1,5,3,4,10),~L_{1,s} = (2,6,7,8,9);\quad &~K_{2,s}=(1,6,3,4,10),~L_{2,s} = (5,2,7,8,9);\\
K_{3,s}=(1,7,3,4,10),~L_{3,s} = (5,6,2,8,9);\quad &~K_{4,s}=(1,8,3,4,10),~L_{4,s} = (5,6,7,2,9);\\
K_{5,s}=(1,9,3,4,10),~L_{5,s} = (5,6,7,8,2).\quad &
\end{align*}

%PASTE THE DIAGRAMS HERE

\begin{figure}
\begin{center}
\begin{tikzpicture}
\Vertex[x=1,y=0,size=0.1,color=black,position=right,label=$10$,fontscale=1]{0-10}
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\Edge(0-1)(0-8) \Edge(0-2)(0-7) \Edge(0-3)(0-6) \Edge[bend=-45](0-4)(0-5) \Edge[bend=-45](0-9)(0-10)
%==============================================

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\Vertex[x=4,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{1-1}

%\Edge(1-1)(1-8)
\draw[-,very thick] (1-1) to[out=35, in=215, looseness=1.5] (1-8);

\Edge[bend=-45](1-2)(1-3) \Edge(1-5)(1-6) \Edge(1-4)(1-7) \Edge[bend=-45](1-9)(1-10)

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\Vertex[x=6.5,y=0.6,size=0.1,color=white,position=left,label=$2$,fontscale=1]{2-2}
\Vertex[x=6.5,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{2-1}

\Edge[bend=-45](2-1)(2-2) \Edge(2-3)(2-8) \Edge(2-4)(2-7) \Edge(2-5)(2-6) \Edge[bend=-45](2-9)(2-10)
%==============================================


% \Vertex[x=8,y=0,size=0.1,color=black,position=right,label=$10$,fontscale=1]{3-10}
% \Vertex[x=8,y=0.6,size=0.1,color=white,position=right,label=$9$,fontscale=1]{3-9}
% \Vertex[x=8,y=1.2,size=0.1,color=white,position=right,label=$8$,fontscale=1]{3-8}
% \Vertex[x=8,y=1.8,size=0.1,color=white,position=right,label=$7$,fontscale=1]{3-7}
% \Vertex[x=8,y=2.4,size=0.1,color=black,position=right,label=$6$,fontscale=1]{3-6}

% \Vertex[x=7,y=2.4,size=0.1,color=white,position=left,label=$5$,fontscale=1]{3-5}
% \Vertex[x=7,y=1.8,size=0.1,color=black,position=left,label=$4$,fontscale=1]{3-4}
% \Vertex[x=7,y=1.2,size=0.1,color=black,position=left,label=$3$,fontscale=1]{3-3}
% \Vertex[x=7,y=0.6,size=0.1,color=white,position=left,label=$2$,fontscale=1]{3-2}
% \Vertex[x=7,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{3-1}

% \Edge(3-1)(3-8) \Edge(3-2)(3-3) \Edge(3-4)(3-7) \Edge(3-5)(3-6) \Edge[bend=-45](3-9)(3-10)

% \Vertex[x=10,y=0,size=0.1,color=black,position=right,label=$10$,fontscale=1]{4-10}
% \Vertex[x=10,y=0.6,size=0.1,color=white,position=right,label=$9$,fontscale=1]{4-9}
% \Vertex[x=10,y=1.2,size=0.1,color=white,position=right,label=$8$,fontscale=1]{4-8}
% \Vertex[x=10,y=1.8,size=0.1,color=white,position=right,label=$7$,fontscale=1]{4-7}
% \Vertex[x=10,y=2.4,size=0.1,color=black,position=right,label=$6$,fontscale=1]{4-6}

% \Vertex[x=9,y=2.4,size=0.1,color=white,position=left,label=$5$,fontscale=1]{4-5}
% \Vertex[x=9,y=1.8,size=0.1,color=black,position=left,label=$4$,fontscale=1]{4-4}
% \Vertex[x=9,y=1.2,size=0.1,color=black,position=left,label=$3$,fontscale=1]{4-3}
% \Vertex[x=9,y=0.6,size=0.1,color=white,position=left,label=$2$,fontscale=1]{4-2}
% \Vertex[x=9,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{4-1}

% \Edge[bend=-45](4-1)(4-2) \Edge(4-3)(4-8) \Edge(4-4)(4-7) \Edge(3-5)(3-6) \Edge(4-5)(4-6) \Edge[bend=-45](4-9)(4-10)
% %==============================================

% \Vertex[x=12,y=0,size=0.1,color=black,position=right,label=$10$,fontscale=1]{5-10}
% \Vertex[x=12,y=0.6,size=0.1,color=white,position=right,label=$9$,fontscale=1]{5-9}
% \Vertex[x=12,y=1.2,size=0.1,color=white,position=right,label=$8$,fontscale=1]{5-8}
% \Vertex[x=12,y=1.8,size=0.1,color=white,position=right,label=$7$,fontscale=1]{5-7}
% \Vertex[x=12,y=2.4,size=0.1,color=black,position=right,label=$6$,fontscale=1]{5-6}

% \Vertex[x=11,y=2.4,size=0.1,color=white,position=left,label=$5$,fontscale=1]{5-5}
% \Vertex[x=11,y=1.8,size=0.1,color=black,position=left,label=$4$,fontscale=1]{5-4}
% \Vertex[x=11,y=1.2,size=0.1,color=black,position=left,label=$3$,fontscale=1]{5-3}
% \Vertex[x=11,y=0.6,size=0.1,color=white,position=left,label=$2$,fontscale=1]{5-2}
% \Vertex[x=11,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{5-1}

% \Edge[bend=-45](5-1)(5-2) \Edge(5-3)(5-8) \Edge[bend=-45](5-4)(5-5) \Edge[bend=-45](5-6)(5-7) \Edge[bend=-45](5-9)(5-10)

% \Vertex[x=14,y=0,size=0.1,color=black,position=right,label=$10$,fontscale=1]{6-10}
% \Vertex[x=14,y=0.6,size=0.1,color=white,position=right,label=$9$,fontscale=1]{6-9}
% \Vertex[x=14,y=1.2,size=0.1,color=white,position=right,label=$8$,fontscale=1]{6-8}
% \Vertex[x=14,y=1.8,size=0.1,color=white,position=right,label=$7$,fontscale=1]{6-7}
% \Vertex[x=14,y=2.4,size=0.1,color=black,position=right,label=$6$,fontscale=1]{6-6}

% \Vertex[x=13,y=2.4,size=0.1,color=white,position=left,label=$5$,fontscale=1]{6-5}
% \Vertex[x=13,y=1.8,size=0.1,color=black,position=left,label=$4$,fontscale=1]{6-4}
% \Vertex[x=13,y=1.2,size=0.1,color=black,position=left,label=$3$,fontscale=1]{6-3}
% \Vertex[x=13,y=0.6,size=0.1,color=white,position=left,label=$2$,fontscale=1]{6-2}
% \Vertex[x=13,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{6-1}

% \Edge(6-1)(6-8) \Edge[bend=-45](6-2)(6-3) \Edge[bend=-45](6-4)(6-5) \Edge[bend=-45](6-6)(6-7) \Edge[bend=-45](6-9)(6-10)
\end{tikzpicture}
\end{center}
\caption{For \eqref{proofidea:eq4}: $\Delta_{K^\uparrow}\Delta_{L^\uparrow} \leftrightarrow \imm{\tau_0'}({\bf x}),$ and $\Delta_{K_{1,s}^\uparrow}\Delta_{L_{1,s}^\uparrow}\leftrightarrow \imm{\tau_1'}({\bf x})+\imm{\tau_2'}({\bf x}).$}
\label{proofidea:diagram2}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}
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\Vertex[x=1,y=0.6,size=0.1,color=white,position=right,label=$9$,fontscale=1]{3-9}
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%\Edge(3-1)(3-8) 
\draw[-,very thick] (3-1) to[out=35, in=215, looseness=1.5] (3-8);

\Edge[bend=-45](3-2)(3-3) \Edge(3-4)(3-7) \Edge(3-5)(3-6) \Edge[bend=-45](3-9)(3-10)

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\Edge[bend=-45](4-1)(4-2) \Edge(4-3)(4-8) \Edge(4-4)(4-7) \Edge(3-5)(3-6) \Edge(4-5)(4-6) \Edge[bend=-45](4-9)(4-10)
%==============================================

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\Edge[bend=-45](5-1)(5-2) \Edge(5-3)(5-8) \Edge[bend=-45](5-4)(5-5) \Edge[bend=-45](5-6)(5-7) \Edge[bend=-45](5-9)(5-10)

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\Vertex[x=8.5,y=2.4,size=0.1,color=black,position=right,label=$6$,fontscale=1]{6-6}

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\Vertex[x=7.5,y=1.8,size=0.1,color=black,position=left,label=$4$,fontscale=1]{6-4}
\Vertex[x=7.5,y=1.2,size=0.1,color=black,position=left,label=$3$,fontscale=1]{6-3}
\Vertex[x=7.5,y=0.6,size=0.1,color=white,position=left,label=$2$,fontscale=1]{6-2}
\Vertex[x=7.5,y=0,size=0.1,color=black,position=left,label=$1$,fontscale=1]{6-1}

%\Edge(6-1)(6-8) 
\draw[-,very thick] (6-1) to[out=35, in=215, looseness=1.5] (6-8);

\Edge[bend=-45](6-2)(6-3) \Edge[bend=-45](6-4)(6-5) \Edge[bend=-45](6-6)(6-7) \Edge[bend=-45](6-9)(6-10)
\end{tikzpicture}
\end{center}
\caption{For \eqref{proofidea:eq4}: $\Delta_{K_{2,s}^\uparrow}\Delta_{L_{2,s}^\uparrow}\leftrightarrow \imm{\tau_1'}({\bf x})+\imm{\tau_2'}({\bf x})+\imm{\tau_3'}({\bf x})+\imm{\tau_4'}({\bf x}).$}
\label{proofidea:diagram3}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}
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%==============================================

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%\Edge(1-1)(1-8)

\draw[-,very thick] (1-1) to[out=35, in=215, looseness=1.5] (1-8);

\Edge[bend=-45](1-2)(1-3) \Edge[bend=-45](1-4)(1-5) \Edge[bend=-45](1-6)(1-7) \Edge[bend=-45](1-9)(1-10)

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%\Edge(2-1)(2-6) 

\Edge[bend=-45](2-2)(2-3) \Edge[bend=-45](2-4)(2-5) \Edge[bend=-45](2-7)(2-8) \Edge[bend=-45](2-9)(2-10)

\draw[-,very thick] (2-1) to[out=45, in=225, looseness=1.5] (2-6);
%==============================================


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\Edge[bend=-45](3-1)(3-2)

%\Edge(3-3)(3-6) 
\draw[-,very thick] (3-3) to[out=35, in=215, looseness=1.5] (3-6);

\Edge[bend=-45](3-4)(3-5) \Edge[bend=-45](3-7)(3-8) \Edge[bend=-45](3-9)(3-10)

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%=============================================
\end{tikzpicture}
\end{center}
\caption{For \eqref{proofidea:eq4}: $\Delta_{K_{3,s}^\uparrow}\Delta_{L_{3,s}^\uparrow}\leftrightarrow \imm{\tau_3'}({\bf x})+\imm{\tau_4'}({\bf x})+\imm{\tau_5'}({\bf x})+\imm{\tau_6'}({\bf x}) + \imm{\tau_0'}({\bf x}).$}
\label{proofidea:diagram4}
\end{figure}
\begin{figure}
\begin{center}
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%==============================================


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% %==============================================

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\end{tikzpicture}
\end{center}
\caption{For \eqref{proofidea:eq4}: $\Delta_{K_{4,s}^\uparrow}\Delta_{L_{4,s}^\uparrow}\leftrightarrow \imm{\tau_5'}({\bf x})+\imm{\tau_6'}({\bf x})+\imm{\tau_7'}({\bf x})+\imm{\tau_8'}({\bf x}).$}
\label{proofidea:diagram5}
\end{figure}
\noindent The set of inequalities that we need to verify over $\Gr^{\geq 0}(5,10)$ are:
\begin{align}\label{proofidea:eq3}
\begin{aligned}
\Delta_{K_{1,s}^{\uparrow}}\Delta_{L_{1,s}^{\uparrow}} &\geq 0 ,\\
-\Delta_{K_{1,s}^{\uparrow}}\Delta_{L_{1,s}^{\uparrow}} + \Delta_{K_{2,s}^{\uparrow}}\Delta_{L_{2,s}^{\uparrow}} &\geq 0,\\
\Delta_{K_{1,s}^{\uparrow}}\Delta_{L_{1,s}^{\uparrow}} -\Delta_{K_{2,s}^{\uparrow}}\Delta_{L_{2,s}^{\uparrow}} + \big{(} \Delta_{K_{3,s}^{\uparrow}}\Delta_{L_{3,s}^{\uparrow}} - \Delta_{K^{\uparrow}}\Delta_{L^{\uparrow}} \big{)} &\geq 0,\\
-\Delta_{K_{1,s}^{\uparrow}}\Delta_{L_{1,s}^{\uparrow}} + \Delta_{K_{2,s}^{\uparrow}}\Delta_{L_{2,s}^{\uparrow}} - \big{(} \Delta_{K_{3,s}^{\uparrow}}\Delta_{L_{3,s}^{\uparrow}} - \Delta_{K^{\uparrow}}\Delta_{L^{\uparrow}} \big{)} + \Delta_{K_{4,s}^{\uparrow}}\Delta_{L_{4,s}^{\uparrow}} &\geq 0,
\end{aligned}
\end{align}
and the last inequality is the Pl\"ucker relation itself. We perform the replacements:
\begin{align}\label{proofidea:eq4}
\begin{aligned}
\Delta_{K^\uparrow} \Delta_{L^\uparrow} &\leftrightarrow \imm{\tau_0'}({\bf x}),\\
\Delta_{K_{1,s}^\uparrow} \Delta_{L_{1,s}^\uparrow} &\leftrightarrow \imm{\tau_1'}({\bf x}) + \imm{\tau_2'}({\bf x}),\\
\Delta_{K_{2,s}^\uparrow} \Delta_{L_{2,s}^\uparrow} &\leftrightarrow \imm{\tau_1'}({\bf x}) + \imm{\tau_2'}({\bf x})+\imm{\tau_3'}({\bf x}) + \imm{\tau_4'}({\bf x}),\\ \Delta_{K_{3,s}^\uparrow} \Delta_{L_{3,s}^\uparrow} &\leftrightarrow \imm{\tau_3'}({\bf x}) + \imm{\tau_4'}({\bf x})+\imm{\tau_5'}({\bf x}) + \imm{\tau_6'}({\bf x}) + \imm{\tau_0'}({\bf x}),\\
\Delta_{K_{4,s}^\uparrow} \Delta_{L_{4,s}^\uparrow} &\leftrightarrow \imm{\tau_5'}({\bf x}) + \imm{\tau_6'}({\bf x})+\imm{\tau_7'}({\bf x}) + \imm{\tau_8'}({\bf x}),\\
\Delta_{K_{5,s}^\uparrow} \Delta_{L_{5,s}^\uparrow} &\leftrightarrow \imm{\tau_7'}({\bf x}) + \imm{\tau_8'}({\bf x}),
\end{aligned}
\end{align}
where the corresponding Kauffman diagrams are shown in \Cref{proofidea:diagram2,proofidea:diagram3,proofidea:diagram4,proofidea:diagram5}. (We skipped the Kauffman diagrams for $\Delta_{K_{5,s}^\uparrow} \Delta_{L_{5,s}^\uparrow}$ as the last inequality is the Pl\"ucker relation.) One can see that these replacements yield nonnegative linear combinations of Temperley--Lieb immanants for each inequality in \eqref{proofidea:eq3}, as desired. 
\end{proof}

\begin{proof}[Towards the general proof]
Suppose we call $i_r=i_4=4$ and $k_s=k_3=2$ as the \textit{moving indices} in the two examples in the proof idea above. Note that $i_r$ is an end-point of $(i_1,\dots,i_4),$ while $k_s$ is not so in $(k_1,\dots,k_5).$ There are a few things that one can observe:
\begin{enumerate}
    \item The moving indices $i_r,k_s$ provide the ``mid-points'' ($1$ for $i_r$ and $3$ for $k_s$) at which the term $\Delta_{I^{\uparrow}}\Delta_{J^{\uparrow}}$ and $\Delta_{K^{\uparrow}}\Delta_{L^{\uparrow}}$ appear in the systems of inequalities. In particular, the moving index is an end-point if and only if the mid-point is so. And in general it is given by $\eta - r + 1$ for the moving index given by $i_r$ (see the main \Cref{th:ws1}). 
    \item Suppose we disregard $\imm{\tau_0}({\bf x})$ ($\leftrightarrow \Delta_{I^\uparrow}\Delta_{J^\uparrow}$) and $\imm{\tau_0'}({\bf x})$ ($\leftrightarrow\Delta_{K^\uparrow}\Delta_{L^\uparrow}$) that refer to ``no switch'' by the moving indices, in this and (3) below. Then $i_r$ being the end-point and $k_s$ being not so -- along with weak separability -- causes each $\Delta_{I_{-,r}^{\uparrow}}\Delta_{J_{-,r}^{\uparrow}}$ to identify with the sum of either $1$ or $2$ immanants, and each $
\Delta_{K_{-,s}^{\uparrow}}\Delta_{L_{-,s}^{\uparrow}}$ with that of $2$ or $4.$ In general, it is $1,2$ if the moving index is an end-point, and $2,4$ otherwise.


\item Moreover, one can observe the telescoping phenomena as we successively verified the inequalities in \eqref{proofidea:eq1} and \eqref{proofidea:eq3}, respectively via \eqref{proofidea:eq2} and \eqref{proofidea:eq4}: the immanant(s) for $\Delta_{*_{1,-}^{\uparrow}}\Delta_{*_{1,-}^{\uparrow}}$ constitute a half of $\Delta_{*_{2,-}^{\uparrow}}\Delta_{*_{2,-}^{\uparrow}};$ the other half of $\Delta_{*_{2,-}^{\uparrow}}\Delta_{*_{2,-}^{\uparrow}}$ match half of those in  $\Delta_{*_{3,-}^{\uparrow}}\Delta_{*_{3,-}^{\uparrow}};$ the other half of $\Delta_{*_{3,-}^{\uparrow}}\Delta_{*_{3,-}^{\uparrow}}$ match half of those in $\Delta_{*_{4,-}^{\uparrow}}\Delta_{*_{4,-}^{\uparrow}};~\dots$ 
\end{enumerate}
The proof in the general case demands a bit more work. However, because of the weak separability, the story about the mid-point, the number of immanants ($1,2$ or $2,4$), and the telescoping phenomenon -- all remain the same. In fact, if one goes back to Gantmacher--Krein \cite{GantKreinOsc} and its various refinements in \cite{fallat2024inequalities}, it is this telescoping which is at the center of the theory of oscillating inequalities, and which we completely capture in the main \Cref{th:ws1} as Pl\"ucker inequalities.    
\end{proof}

To conclude here, we refer the reader to \cite{SV} for the detailed general proof, and the proof of the converse, i.e., \textit{the system of inequalities \eqref{ws11:eq3} $\implies$ weak separability of $I,J$.}

\acknowledgements{We thank the American Institute of Mathematics (AIM, CalTech Campus) for their stimulating environment during a workshop on Theory and Applications of Total Positivity, held in July 2023. This allowed us to interact fruitfully with several other researchers which led to the beginning of this work. In particular, we thank Projesh Nath Choudhury, Dominique Guillot, Himanshu Gupta, Chi-Kwong Li, and Shahla Nasserasr. We also thank Michael Gekhtman for helpful discussions, and Apoorva Khare for suggesting expository improvements. We sincerely appreciate Steven Karp for writing to us and providing feedback that helped in further improving the work. We also thank Pavlo Pylyavskyy and Shaun Fallat for useful conversations. Finally, we thank the two anonymous referees for their helpful feedback on the paper.


\printbibliography
\end{document}
