\documentclass[ALCO,Unicode]{cedram} \usepackage[matrix,arrow,tips,curve]{xy} \usepackage{bbm} \newcommand{\rk}{\operatorname{rk}} \newcommand{\col}{\operatorname{col}} \newcommand{\pis}{{\pi,\sigma}} \newcommand{\cw}{\operatorname{cw}} \newcommand{\ccw}{\operatorname{ccw}} \title[Positroid quotients]{Characterizing positroid quotients of uniform matroids} % ALCO: metadata \author{\firstname{Zhixing} \lastname{Chen}} \address{Humboldt-Universit\"at zu Berlin\\ Institut f\"ur Mathematik\\ Unter den Linden 6\\ Berlin\\ 10099 (Germany)} \email{zhixing.chen@student.hu-berlin.de} \author{\firstname{Yumou} \lastname{Fei}} \address{Massachusetts Institute of Technology\\ Dept. of electrical engineering and computer science\\ 32 Vassar st.\\ Cambridge\\ MA 02139 (USA)} \email{yumou415@mit.edu} \author{\firstname{Jiyang} \lastname{Gao}} \address{ Harvard University\\ Dept. of mathematics\\ Cambridge\\ MA 02138 (USA)} \email{jgao@math.harvard.edu} \author{\firstname{Yuxuan} \lastname{Sun}} \address{ University of Minnesota Twin Cities\\ Dept. of mathematics\\ Minneapolis\\ MN 55415 (USA)} \email{Sun00816@umn.edu} \author{\firstname{Yuchong} \lastname{Zhang}} \address{ Harvard University\\ Dept. of mathematics\\ Cambridge\\ MA 02138 (USA)} \email{yzhang@math.harvard.edu} \keywords{positroids, flag positroids, Grassmann necklaces} \subjclass{05B35} % ALCO: end-metadata \begin{document} % ALCO:theorems-block % ── Theorem environment aliases (\equalenv) ───────────────────────────────────── \equalenv{quest}{ques} % ALCO: converted from \newtheorem{quest}[cdrthm]{Question} % ── New theorem environments ───────────────────────────────────────────────────── \newtheorem{axio}[cdrthm]{Axiom} % ALCO:end-theorems-block \begin{abstract} We study two-step flag positroids $(P_1, P_2)$, where $P_1$ is a quotient of $P_{2}$. We provide a complete characterization of all two-step flag positroids that contain a uniform matroid, extending and completing a partial result by Benedetti, Ch\'avez, and Jim\'enez. Our characterization reveals that ``non-local information'' is necessary for characterizing flag positroids. To contrast general positroids with the special case of lattice path matroids, we show that the containment relations of Grassmann necklaces and conecklaces fully characterize flag lattice path matroids, but are insufficient for general flag positroids. Additionally, we prove that the decorated permutations of any elementary quotient pair are related by a cyclic shift, resolving a conjecture of Benedetti, Ch\'avez and Jim\'enez. \end{abstract} \maketitle \theoremstyle{plain} \theoremstyle{definition} \theoremstyle{remark} \section{Introduction} Positroids are an important class of matroids that can be represented by full-rank matrices with nonnegative maximal minors. They were first introduced by Postnikov in his study of the totally nonnegative Grassmannian \cite{Pos06}. The applications of positroids span several domains, including cluster algebra \cite{PS14} and physics \cite{KW11, Arkani21}. Positroids also have many nice properties. They are closed under matroid duality and cyclic shifts of the ground set. Moreover, they are in bijection with many combinatorial objects, including Grassmann necklaces and decorated permutations \cite{Pos06,Oh11}. An ordered pair of matroids $(M_1,M_2)$ is called a \emph{(two-step) flag matroid} if $M_1$ is a quotient of $M_2$, which means that every circuit of $M_2$ is a union of circuits of $M_1$. If both $M_1$ and $M_2$ are positroids, then the pair is called a \emph{(two-step) flag positroid}. Characterizing flag positroids via circuits is computationally challenging, as enumerating all unions of circuits requires exponential time. Since positroids can be represented concisely by many combinatorial objects such as Grassmann necklaces, it is natural to seek a more practical combinatorial criterion for flag positroids. \begin{quest}[\cite{Oh11}] Find a concise combinatorial criterion of flag positroids. \end{quest} This question was first posed in \cite[Section 7]{Oh11}, prompting numerous attempts to address it \cite{Oh17, Ben19, Ben22, Jon22}. In particular, several necessary conditions are identified for two-step flag positroids, such as \cite[Proposition 6.2]{Oh17} and \cite[Remark 36]{Ben19}. In contrast, finding sufficient conditions appears to be more challenging. To the authors' knowledge, existing sufficient conditions mainly focus on special cases: \cite{Ben22} provides a necessary and sufficient condition for flag lattice path matroids (a subclass of positroids), and \cite{Ben19} gives a sufficient but not necessary condition for elementary quotients (that is, quotients that decrease the rank by at most 1) of uniform matroids. Our main result gives a necessary and sufficient characterization of all two-step flag positroids that contain a uniform matroid, based on their CW-arrows (a close relative of Grassmann necklaces, first defined by \cite{Oh17}). \begin{theo}\label{thm:uniformCW} Given integers $0\leq r\leq k \pi(i)$, and for all such $C_i$, we have $1 \in C_i$. Thus, $A \setminus \{ 1 \}$ contains all $n$ CW-arrows except for those $|W_1(\pi^:)|$ CW-arrows and one additional CW-arrow starting at $1$. Therefore, we have \[\cw_{M}(A\setminus\{x\})=n-|W_1(\pi^:)|-1=\cw_{M}(A)-1.\] If $A\neq [n]$, pick an element $x\in A$ such that $(x-1 \pmod n)\not\in A$. This element exists; otherwise, $A = [n]$. There is at most one CW-arrow that is contained in $A$ and contains $x$, namely, the one starting at $x$. Therefore, $\cw_{M}(A\setminus\{x\})\geq \cw_{M}(A)-1.$ \end{proof} \begin{coro}\label{cor:trickle_up} Let $M$ be a positroid on $[n]$ without coloops. If $\cw_{M}(A)\leq r$ for all $k$-element subsets $A\subseteq [n]$, then $\cw_{M}(A)\leq |A|-k+r$ for all $A\subseteq [n]$ such that $|A|\geq k$. \end{coro} \begin{proof} Take an arbitrary $A$ where $|A| \ge k$. By Lemma~\ref{lem:cwA-1}, we know that there exist elements $x_1, \dots, x_{|A|-k}$ such that $\cw_{M}(A\setminus \{x_1,\dots,x_{|A|-k}\}) \ge \cw_{M}(A) - (|A| - k)$. Combining with the assumption yields the desired inequality. \end{proof} Before heading into the proof of Theorem~\ref{thm:uniformCW}, we record another basic fact about the CW-function: it provides a simple upper bound for the rank function of a positroid. This helps prove the reverse (and easier) direction that if a positroid $M$ is a quotient of $U_{k,n}$, then $\cw_M(A)\le r$ for any $k$-element subset $A\subseteq [n]$. \begin{lemm}\label{lem:rankupperbound} Let $M$ be a positroid on $[n]$ without coloops. Then for any proper subset $A\subsetneq [n]$, we have $\rk_{M}(A)\leq |A|-\cw_{M}(A)$. \end{lemm} \begin{proof} If $A$ is a cyclic interval, note that $\rk_{M}(A)=|A|-\cw_{M}(A)$ from Lemma~\ref{lem:rankcyclic}. If $A$ is not, it can be written as a disjoint union of cyclic intervals $A =\bigsqcup A_i$. Then by submodularity of the rank function (Proposition~\ref{prop:submodular}), we have \[ \rk_{M}(A) = \rk_{M}\left( \bigsqcup A_i \right)\le \sum \rk_{M}(A_i) = \sum|A_i| - \sum \cw_{M}(A_i) = |A| - \cw_{M}(A).\qedhere \] \end{proof} Now we are ready to prove our theorem. The ``if'' direction is the harder direction, and the main idea is to apply Lemma~\ref{lem:main} to the dual matroid of $M$. \begin{proof}[Proof of Theorem~\ref{thm:uniformCW}] We first prove the ``only if'' direction by contradiction. Given that $M$ is a quotient of $U_{k,n}$, it follows that $M$ has no coloops since coloops are not contained in any circuits. Assume there exist $r+1$ CW-arrows of $M$ whose union has cardinality less than $k+1$, and notice that this is equivalent to saying that there exists a $k$-element subset $A \subseteq [n]$ where $\cw_{M}(A) \ge r+1$. By Lemma~\ref{lem:rankupperbound} we have $\rk_{M}(A) \le k-r-1$, and by Lemma~\ref{lem:rank_char_of_quotient} we know that $M$ is not a quotient of $U_{k,n}$, resulting in a contradiction. We then prove the ``if'' direction. By Remark~\ref{remark:cw le r}, the assumption is equivalent to $\cw_{M}(A)\leq r$ for all $k$-element subset of $[n]$. We firstly argue that $M$ cannot have coloops. Suppose $j$ is a coloop of $M$. Consider the CW-arrows whose end points are in $[j+1,j+k] \setminus W_j(\pi^{:})$, where $\pi^{:}$ is the decorated permutation associated with $M$. By Proposition~\ref{prop:decpermtoneck} we know that $W_{j}(\pi^{:})$ has size $k-r$, and since $\pi^{:}(j)=\underline{j}$ we know that $j\in W_{j}(\pi^{:})$. Thus the number of CW-arrows whose end points are in $[j+1,j+k] \setminus W_j(\pi^{:})$ is at least $k-(k-r-1) = r+1$. Since all these CW-arrows are contained in $[j+1,j+k]$ due to the definition of $W_{j}(\pi^{:})$, this implies that $\cw_{M}([j+1,j+k])\geq r+1$, contradicting the assumption. Now fix a $k$-element subset $A\subseteq [n]$. Let $M^{*}$ be the dual of $M$. By applying the Lemma~\ref{lem:main} to $[n]\setminus A$ and the dual matroid $M^{*}$, we obtain a partition $[n]\setminus A=A_{1}\sqcup\dots\sqcup A_{t}$ such that \begin{equation}\label{eq:apply-main} \rk_{M^{*}}([n]\setminus A)=\sum_{j=1}^{t}\Big(\rk_{M^{*}}([n])-\ccw_{M^{*}}([n]\setminus A_{j})\Big). \end{equation} We thus have \begingroup \allowdisplaybreaks \begin{align*} &\quad\rk_{M}(A)\\ &=|A|-n+\rk_{M}([n])+\rk_{M^{*}}([n]\setminus A) \tag{by Proposition~\ref{prop:rank-dual}}\\ &=|A|-n+\rk_{M}([n])+\sum_{j=1}^{t}\Big(\rk_{M^{*}}([n])-\ccw_{M^{*}}([n]\setminus A_{j})\Big)\tag{by \eqref{eq:apply-main}}\\ &=|A|-n+\rk_{M}([n])+\sum_{j=1}^{t}\Big(n-\rk_{M}([n])-\cw_{M}([n]\setminus A_{j})\Big)\tag{by duality}\\ &\geq |A|-n+\rk_{M}([n])+\sum_{j=1}^{t}\Big(n-\rk(M)-\left|[n]\setminus A_{j}\right|+\rk(M)\Big)\tag{by Corollary~\ref{cor:trickle_up}}\\ &=\rk(M). \end{align*} \endgroup Since $A$ is arbitrary, an application of Lemma~\ref{lem:rank_char_of_quotient} shows that $M$ is a quotient of $U_{k,n}$. \end{proof} \subsection{The case of elementary quotients}\label{subsec:elementary-quotients} In this subsection, we interpret Theorem~\ref{thm:uniformCW} in the context of Section~\ref{sec:3}. Recall from Remark~\ref{rmk:open-question} that, given a decorated permutation $\pi^{:}$ over $[n]$, we want to find a concise characterization of sets $A\subseteq [n]$ such that $\overrightarrow{\rho_{A}}(\pi^{:})\lessdot \pi^{:}$. Since Theorem~\ref{thm:uniformCW} is a necessary and sufficient condition for the quotient, it helps answer this question for the case $\pi^{:}=\pi_{k,n}$. For every proper subset $A\subseteq [n]$, there is a unique way to decompose $A$ into a collection of cyclic intervals that are disjoint and pairwise non-adjacent, written as $A=[a_{1},b_{1}]\sqcup \dots \sqcup [a_{s},b_{s}]$ where no pair $i,j\in [s]$ satisfies $b_{j}+1\equiv a_{i}\pmod n$. We call each of these cyclic intervals a \emph{cyclic component} of $A$. \begin{exam} Let $A\subseteq [9]$ be $\{1,2,4,6,7,9\}$, then the cyclic intervals of $A$ are $\{4\},[6,7]$ and $[9,2]$. \end{exam} Using the concept of cyclic components, we can state the main result of this subsection. \begin{theo}\label{thm:uniform} Let $1\leq k\leq n-1$, and let $\sigma^:$ be a decorated permutation on $[n]$. Then $\sigma^{:}\lessdot \pi_{k,n}$ if and only if $\sigma^: = \overrightarrow{\rho_{A}}(\pi_{k,n})$ for some $A \subseteq [n]$, where the union of any two distinct cyclic components of $A$ has cardinality at most $k-1$. \end{theo} \begin{proof} \textbf{The ``only if'' direction:} the existence of an $A\subseteq[n]$ such that $\sigma^{:}=\overrightarrow{\rho_{A}}(\pi_{k,n})$ follows from Theorem~\ref{thm:shift}. Let $[a_{1},b_{1}]$ and $[a_{2},b_{2}]$ be any two distinct cyclic components of $A$. We show that $|[a_{1},b_{1}]|+|[a_{2},b_{2}]|\leq k-1$. Let $c_{1}=\pi_{k,n}(a_{1}-1)=(a_{1}+k-1\pmod n)$. After the cyclic shift, we would have $\sigma(b_{1}+1)=c_{1}$. By Lemmas~\ref{lemma:Gr contain and shift} and~\ref{lem:vec=shift}, we know that the cyclic interval $S^{\pi}_{c_{1}}=(a_{1}-1,c_1]=[a_{1},c_{1}]$ contains the cyclic interval $S^{\sigma}_{c_{1}}=(b_{1}+1,c_{1}]=[b_{1}+2,c_{1}]$. Therefore, the CW-arrow $[b_{1}+1,c_{1}]$ of $\sigma^{:}$ has cardinality $|[a_{1},c_{1}]|-|[a_{1},b_{1}]|=k-|[a_{1},b_{1}]|$. Similarly, we may define $c_{2}=\pi_{k,n}(a_{2}-1)$ and deduce that the CW-arrow $[b_{2}+1,c_{2}]$ of $\sigma^{:}$ has cardinality $k-|[a_{2},b_{2}]|$. \textbf{Case 1:} the CW-arrows $[b_{1}+1,c_{1}]$ and $[b_{2}+1,c_{2}]$ are disjoint. Then by Theorem~\ref{thm:uniformCW}, the sum of their cardinalities, $(k-|[a_{1},b_{1}]|)+(k-|[a_{2},b_{2}]|)$, is at least $k+1$, and thus $|[a_{1},b_{1}]|+|[a_{2},b_{2}]|\leq k-1$. \textbf{Case 2:} the CW-arrows $[b_{1}+1,c_{1}]$ and $[b_{2}+1,c_{2}]$ are not disjoint. Without loss of generality, assume that $b_{2}+1\in [b_{1}+1,c_{1}]$. Then, since $[a_1, b_1]$ and $[a_2, b_2]$ are distinct cyclic components, we have $b_{1}+1\not\in [a_{2},b_{2}]$, and moreover, $[a_{2},b_{2}]\subseteq [b_{1}+2,c_{1}]$. Therefore, we have \[|[a_{1},b_{1}]|+|[a_{2},b_{2}]|\leq |[a_{1},b_{1}]\cup[b_{1}+2,c_{1}]|\le |[a_1, c_1]|-1=k-1.\] \textbf{The ``if'' direction:} assume $\sigma^{:}=\overrightarrow{\rho_{A}}(\pi_{k,n})$ for some $A\subseteq [n]$, and the union of any two distinct cyclic components of $A$ has cardinality at most $k-1$. By \cite[Theorem 26]{Ben19}, we know that $\rk(\sigma^{:})=k-1$. We then show that the union of any two CW-arrows of $\sigma^{:}$ has cardinality at least $k+1$, which implies $\sigma^{:}\lessdot \pi_{k,n}$ due to Theorem~\ref{thm:uniformCW}. Let $[b_{1}+1,c_{1}]$ and $[b_{2}+1,c_{2}]$ be the two CW arrows and assume, on the contrary, that the cardinality of their union is at most $k$. Then we must have $b_{1}+1\notin A$, since otherwise $c_{1}=\sigma(b_{1}+1)=\pi_{k,n}(b_1+1)=(b_{1}+1+k\pmod n)$ and $|[b_{1}+1,c_{1}]|=k+1$. Similarly, we have $b_{2}+1\notin A$. We again divide our proof into two cases. \textbf{Case 1:} the CW-arrows $[b_{1}+1,c_{1}]$ and $[b_{2}+1,c_{2}]$ are disjoint. Observe that $(\pi^{-1}_{k,n}(c_{1}),b_{1}]$ and $(\pi_{k,n}^{-1}(c_{2}),b_{2}]$ are both cyclic components of $A$. Therefore, \[|[b_{1}+1,c_{1}]|+|[b_{2}+1,c_{2}]|= k-\left|(\pi_{k,n}^{-1}(c_{1}),b_{1}]\right|+k-\left|(\pi_{k,n}^{-1}(c_{2}),b_{2}]\right|\geq 2k-(k-1)=k+1. \] \textbf{Case 2:} the CW-arrows $[b_{1}+1,c_{1}]$ and $[b_{2}+1,c_{2}]$ are not disjoint. Without loss of generality, assume $b_{2}+1\in [b_{1}+1,c_{1}]$. Since $b_{1}+1,b_{2}+1\notin A$ and $\sigma^{:}=\overrightarrow{\rho_{A}}(\pi_{k,n})$, there exists some $x\in [b_{1}+1,b_{2}]$ such that $\sigma(b_{2}+1)=\pi_{k,n}(x)$, that is, $c_{2}=\pi_{k,n}(x)$. Thus, \[|[b_{1}+1,c_{1}]\cup[b_{2}+1,c_{2}]|\geq \left|[b_{1}+1,x]\cup[x+1,\pi_{k,n}(x)]\right|\geq k+1. \qedhere\] \end{proof} \begin{exam} Consider $\pi_{4,8} = 56781234$. If $A=\left\{1,3,5,8\right\}$, then the size of the union of any two components in $\{3\}, \{5\},[8,1]$ does not exceed $3 = 4 - 1$. By Theorem~\ref{thm:uniform}, we have $\overrightarrow{\rho_{1358}}\left(\pi_{4,8}\right)\lessdot \pi_{4,8}$. If $A = \left\{1,2,5,8\right\}$, then the union of two components $\left\{5\right\}$ and $[8,2]$ has size $4$, violating the condition. As a result, $\overrightarrow{\rho_{1258}}(\pi_{4,8})$ is not a quotient of $\pi_{4,8}$. \end{exam} \begin{rema} The paper \cite{Ben19} also focuses on characterizing elementary positroid quotients of uniform matroids. They proved that the condition $|A|\leq k-1$ is sufficient for $\overrightarrow{\rho_{A}}(\pi_{k,n})\lessdot \pi_{k,n}$ (\cite[Theorem 28]{Ben19}), while a necessary condition is that each \emph{individual} cyclic component of $A$ has cardinality at most $k-1$ (\cite[Theorem 26]{Ben19}). In comparison, our results show that the precise necessary and sufficient condition lies between these two: specifically, the union of any two cyclic components must have cardinality at most $k-1$. \end{rema} \section{Flag LPMs and necklace containment}\label{sec:5} The main objective of this section is to examine the ``non-locality'' phenomenon in Theorem~\ref{thm:uniformCW}, as highlighted in Remark~\ref{rmk:nonlocality}. To illustrate this concept, we introduce and prove Theorem~\ref{thm:LPM}, which serves as a contrasting example. \subsection{Locality of necklace containment} Recall that in the proof of Theorem~\ref{thm:shift}, we crucially utilized the observation that a flag positroid satisfies the ``necklace containment relation.'' That is, for any flag positroid $(M,N)$, if the Grassmann necklace of $M$ and $N$ are $I^{\sigma}$ and $I^{\pi}$, respectively, we have $I^{\sigma}_{i}\subseteq I^{\pi}_{i}$ for all $i\in[n]$ (due to Propositions~\ref{prop:quotient-implies-containment} and~\ref{prop:necklace-minimizes}). By Definition~\ref{def:conecklace}, we know that the Grassmann conecklaces $J^{\sigma}$ and $J^{\pi}$ of $M$ and $N$ also satisfy $J^{\sigma}_{i}\subseteq J^{\pi}_{i}$ for all $i\in [n]$. We record this observation as follows. \begin{prop}\label{prop:Ne CoNe Cont} Given a flag positroid $(M,N)$, let $I^{\sigma},I^{\pi}$ be the Grassmann necklaces of $M$ and $N$, and let $J^{\sigma},J^{\pi}$ be the Grassmann conecklaces of $M$ and $N$, respectively. Then $I^{\sigma}_{i}\subseteq I^{\pi}_{i}$ and $J^{\sigma}_{i}\subseteq J^{\pi}_{i}$ for all $i\in [n]$. \end{prop} The perspective developed in Section~\ref{sec:3} also reveals that necklace containment relations can be translated into containment relations of Grassmann intervals (which are essentially equivalent to CW-arrows). We formalize this useful observation in the following proposition. The first item is merely a restatement of Corollary~\ref{cor:neckcontain=vecdom}, while the second follows easily from duality (see also the beginning of the proof of Lemma~\ref{A}). \begin{prop}\label{prop:neck-coneck-containment} Given decorated permutations $\sigma^{:}$ and $\pi^{:}$ over $[n]$, let $S^{\sigma}_{1},\dots,S^{\sigma}_{n}$ and $S^{\pi}_{1},\dots,S^{\pi}_{n}$ be their Grassmann intervals, as defined in Definition~\ref{def:grassmann-matrix}. Then \begin{enumerate}[label=(\arabic*)] \item $I^{\sigma}_{i}\subseteq I^{\pi}_{i}$ for all $i\in [n]$ if and only if $S^{\sigma}_{i}\subseteq S^{\pi}_{i}$ for all $i\in [n]$. \item $J^{\sigma}_{i}\subseteq J^{\pi}_{i}$ for all $i\in [n]$ if and only if $S^{\sigma}_{\sigma(i)}\subseteq S^{\pi}_{\pi(i)}$ for all $i\in [n]$. \end{enumerate} \end{prop} We claim that both the necklace containment $I^{\sigma}_{i}\subseteq I^{\pi}_{i}$ and the conecklace containment $J^{\sigma}_{i}\subseteq J^{\pi}_{i}$ can be viewed as ``local criteria.'' As in Remark~\ref{rmk:nonlocality}, consider the case where all CW-arrows of $\sigma^{:}$ and $\pi^{:}$ are of length much smaller than $n$. In other words, each $S^{\sigma}_{i}$ or $S^{\pi}_{i}$ covers only a few consecutive positions on the circle of elements in $[n]$. However, Theorem~\ref{thm:uniformCW} reveals that even when all CW-arrows are short, characterizing quotient relations requires accounting for multiple CW-arrows that may be arbitrarily far apart. Consequently, it is natural that ``local criteria'' such as the containment of necklaces and conecklaces fail to characterize quotient relations among positroids. We illustrate this with a concrete example. \begin{exam}\label{cex:Oh conj} Let $M$ be a positroid given by the decorated permutation $261534$ and take $N$ to be the uniform positroid $U_{4,6}$. The necklace and the conecklace of $M$ are $(134,234,346,461,\allowbreak 561,613)$ and $(356,561,562,623,234,235)$ respectively. For $N$, they are $(1234,2345,3456,4561,\allowbreak 5612,6123)$ and $(3456,4561,5612,6123,1234,2345)$. They satisfy the containment conditions. However, $\rk_{M}([6])-\rk_{M}(1245)=1$ while $\rk_{N}([6])-\rk_{N}(1245)=0$. This means $M$ is \emph{not} a quotient of $N$, since the rank condition of quotient in Proposition~\ref{prop:rank-quotient} is violated. Therefore, the converse of Proposition~\ref{prop:Ne CoNe Cont} is not true. \end{exam} \subsection{A conjecture of Oh and Xiang} Oh and Xiang \cite{Oh17} conjectured that flag positroids can be characterized by certain ``covering'' relations of CCW-arrows (Definition~\ref{def:CCW-arrows}). \begin{conj}[{\cite[Conjecture 6.3]{Oh17}}]\label{conj:Oh-Xiang} Let $M$ and $N$ be positroids on $[n]$ without loops or coloops. If every CCW-arrow of $M$ is the union of some CCW-arrows of $N$, then $M$ is a quotient of $N$. \end{conj} It is not hard to see that Example~\ref{cex:Oh conj} disproves this conjecture. In fact, if $M$ is the positroid given by the decorated permutation $261534$, all CCW-arrows of $M$ have length at least 3. Since every cyclic interval of length 3 is a CCW-arrow of $U_{4,6}$, it follows that every CCW-arrow of $M$ is the union of some CCW-arrows of $U_{4,6}$. However, $M$ is not a quotient of $U_{4,6}$, as argued in Example~\ref{cex:Oh conj}. \subsection{The case of LPMs} Lattice path matroids (LPMs) are a special class of positroids that is defined solely by the first term of its Grassmann necklace and conecklace (see Definition~\ref{def:LPM}). By Proposition~\ref{prop:Ne CoNe Cont}, we know that if the LPM $M[U',L']$ is a quotient of $M[U,L]$, then $U'\subseteq U$ and $L'\subseteq L$. The work of Benedetti and Knauer \cite{Ben22} provides a necessary and sufficient criterion on $U',L',U$ and $L$ for which $M[U',L']$ is a quotient of $M[U,L]$. \begin{theo}[{\cite[Theorem 19]{Ben22}}] \label{thm:LPM-greedy-pairing} Consider an LPM $M[U,L]$ over $[n]$. Order the elements of $U$ and $L$ as $u_{1}<\dotsi_{s}$ or $u_{i_{s}}-\ell_{j_{s}}>i_{s}-j_{s}$. We consider the smallest $s$ such that this is the case. \textbf{Case 1:} $j_{s}>i_{s}$. Let $\sigma^{:}$ and $\pi^:$ be the decorated permutation of $N'$ and $N$, respectively. It is easy to see that $\pi(\ell_{i_{s}})=u_{i_{s}}$. Let $x=\ell_{i_{s}}$. By the minimality of the choice of $s$, we must have $x\in L'$. Since $j_{s}>i_{s}$, the number of elements in $L'$ that are at most $x$ is at least $i_{s}-s+1$, and thus the number of elements in $U'$ that are at most $\sigma(x)$ is at least $i_{s}-s+1$. Therefore $\sigma(x)>u_{i_{s}}$, and thus we have $\pi(x)=u_{i_s}<\sigma(x)\leq x$. But by Proposition~\ref{prop:neck-coneck-containment}, we have $S^{\sigma}_{\sigma(x)}\subseteq S^{\pi}_{\pi(x)}$, leading to a contradiction (note that since $x\in L'$, if $\sigma(x)=x$ then $\sigma^{:}(x)=\underline{x}$ and $S^{\sigma}_{\sigma(x)}=[n]$). \textbf{Case 2:} $u_{i_{s}}-\ell_{j_{s}}>i_{s}-j_{s}$. Let $\sigma^{:}$ and $\pi^:$ be the decorated permutation of $N'$ and $N$, respectively. Let $y$ be the smallest element of $[n]\setminus L$ greater than $\ell_{j_{s}}$, or equivalently, the $(\ell_{j_{s}}-j_{s}+1)$-th smallest element of $[n]\setminus L$. Such a $y$ exists because $$\ell_{j_{s}}-j_{s}\leq u_{i_{s}}-i_{s}-1\leq n-|U|-1=n-|L|-1.$$ Now $\pi(y)$ is the $(\ell_{j_{s}}-j_{s}+1)$-th smallest element of $[n]\setminus U$. Since $u_{i_{s}}-i_{s}>\ell_{j_{s}}-j_{s}$, it follows that $u_{i_{s}}>\pi(y)$. This means the number of elements in $[n]\setminus U'$ that are at most $\pi(y)$ is at most $(\ell_{j_{s}}-j_{s}+1)+(s-1)$, while the number of elements in $[n]\setminus L'$ that are at most $y$ is at least $(\ell_{j_{s}}-j_{s}+1)+s$. Therefore, we have $\sigma(y)>\pi(y)\geq y$. But by Proposition~\ref{prop:neck-coneck-containment}, we have $S^{\sigma}_{\sigma(y)}\subseteq S^{\pi}_{\pi(y)}$, leading to a contradiction (note that since $y\not\in L$, if $\pi(y)=y$ then $\pi^{:}(y)=\overline{y}$ and $S^{\pi}_{\pi(y)}=\emptyset$). \end{proof} Theorem~\ref{thm:LPM-formal} (and \cite[Proof of Theorem 41]{Ben22}) stands in contrast to Example~\ref{cex:Oh conj}, demonstrating that in some sense, it is much easier to ensure quotient relations between LPMs than between general positroids. \begin{rema} Note that in the proof of Theorem~\ref{thm:LPM-formal}, while the conecklace containment $J'_{i}\subseteq J_{i}$ is used for all $i\in [n]$, we only used the necklace containment $I'_{i}\subseteq I_{i}$ for $i=1$. By symmetry, we could also use the necklace containment for each component and only use the conecklace containment for $i=1$. It is certainly impossible not to use the conecklace containment at all. For example, consider $M[14, 57]$ and $M[145, 467]$. In this case, we have \[ I' =(14,24,34,45,56,16,17) \quad \text{and} \quad I=(145,245,345,456,156,167,147). \] So we have $I'_i\subseteq I_i$ for all $i=1,2, \dots, 7$, but $M[14, 57]$ is not a quotient of $M[145,467]$ because $J'_{1}\not\subseteq J_{1}$. \end{rema} % ALCO: longthanks \longthanks{This research was carried out as part of the PACE program in the summer of 2023 at Peking University, Beijing, partially supported by NSFC grant 12426507. We thank Shiliang Gao, Yibo Gao, and Lauren Williams for helpful discussions. We thank anonymous referees for helpful suggestions.} % ALCO: end-longthanks \bibliographystyle{mersenne-plain-nobysame} \bibliography{ALCO_Chen_1415} \end{document}