%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2025}
\articlenumber{128}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

\usepackage{lipsum}

\usepackage{graphicx} % Required for inserting images
\usepackage{adjustbox}
\usepackage{amsxtra,tikz,stackengine,bm,mathrsfs,bbm,tikz-cd,comment,hyperref,combelow,amsopn}
\usepackage{enumitem}
\usepackage{ytableau}

%\pagestyle{plain}
\numberwithin{equation}{section}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{question}[theorem]{Question}
\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{note}[theorem]{Note}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{assumption}[theorem]{Assumption}

\newcommand{\PP}{\mathbb{P}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\skewschubert}{\lambda/\mu}
\newcommand{\skewbraid}{\beta_{\lambda/\mu}}

\newcommand{\subs}[1]{\langle #1 \rangle}

\newcommand{\seeds}{\mathsf{mut}}


\newcommand{\Cat}{\mathcal{C}}

\newcommand{\CF}{\mathcal{F}}
\newcommand{\CG}{\mathcal{G}}
\newcommand{\CL}{\mathcal{L}}
\newcommand{\CT}{\mathcal{T}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\CO}{\mathcal{O}}
\newcommand{\I}{\mathcal{I}}
\newcommand{\diag}{\mathrm{diag}}
\newcommand{\Hilb}{\mathrm{Hilb}}
\newcommand{\Aqt}{\mathbb{A}_{q,t}}
\newcommand{\Bqt}{\mathbb{B}_{q,t}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\Aq}{\mathbb{A}_{q}}
\newcommand{\sq}{\square}
\newcommand{\aut}{\theta}
\newcommand{\adj}{\Theta}
\newcommand{\e}{\mathbbm{1}}
\newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\pos}[1]{\Pi^{\circ}_{#1}}
\newcommand{\posone}[1]{\Pi^{\circ,1}_{#1}}
\newcommand{\cpos}[1]{\widehat{\Pi^{\circ}}_{#1}}
\newcommand{\relpos}[1]{\buildrel #1 \over \longrightarrow}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\ch}{ch}
\DeclareMathOperator{\spann}{span}
\DeclareMathOperator{\filt}{F}
\DeclareMathOperator{\rad}{rad}
\DeclareMathOperator{\soc}{soc}
\DeclareMathOperator{\ssn}{ss}
\DeclareMathOperator{\gr}{gr}
\newcommand{\resc}{\mathrm{resc}}
\newcommand{\rev}{\mathrm{rev}}
\newcommand{\rel}[1]{\buildrel #1 \over \rightarrow}

\def\br{\beta}
\def\BS{\mathrm{BS}}
\def\minor{\Delta}
\def\gl{\mathfrak{gl}}
\def\GL{\mathrm{GL}}
\def\last{\mathrm{last}}
\def\mut{\mathrm{M}}
\def\seed{\Sigma}
\def\clvar{\mathbf{x}}
\def\colour{\mathrm{color}}
\def\xun{x^{(1)}}
\def\x2{x^{(2)}}
\def\phipul{\Phi_{\br^1, \br^2}^{\ast}}
\def\u1{u^{(1)}}
\def\w0{\Delta}
\def\Br{\mathrm{Br}}

\newcommand{\skewschvar}{S_{\skewschubert}^\circ}
\newcommand{\Gr}{\mathrm{Gr}}
\newcommand{\cox}{\mathbf{c}}
\newcommand{\id}{\mathbf{1}}
\def\GN{\mathsf{GN}}
\def\BA{\mathsf{BA}}

\newcommand{\iso}{\iota}

\newcommand{\std}{\mathrm{std}}
\newcommand{\ant}{\mathrm{ant}}


\def\QO{Q^{\circ}}
\def\QP{Q^{\times}}

\newcommand{\mubar}{\overline{\mu}}
\newcommand{\lambdabar}{\overline{\lambda}}

\newcommand{\todo}[1]{{\color{red}\textbf{to do: #1}}}
\newcommand{\newword}[1]{\emph{\textbf{#1}}}

\newcommand{\TS}[1]{{\color{purple}{{\bf Tonie}: #1}}}

\newcommand{\EG}[1]{{\color{red}{{\bf Eugene}: #1}}}

\newcommand{\JS}[1]{{\color{cyan}{{\bf Jos\'e}: #1}}}

\newcommand{\SK}[1]{{\color{orange}{{\bf Soyeon}: #1}}}
\definecolor{aquamarine}{rgb}{0.5, 1.0, 0.83}
\definecolor{aqua}{rgb}{0.0, 1.0, 1.0}

%% define your title in the usual way
\title{Skew shaped positroids and double Bott--Samelson varieties}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
%\author[Optional shorter names]{\texorpdfstring{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1} \and you\addressmark{2}}{Longer names me and you}}


\author[E. Gorsky,  S. Kim, T. Scroggin, and J. Simental]{\texorpdfstring{Eugene Gorsky\thanks{\href{mailto:egorskiy@math.ucdavis.edu}{egorskiy@math.ucdavis.edu} E. G. was partially supported by the NSF grant DMS-2302305.}\addressmark{1}, Soyeon Kim\thanks{\href{mailto:syxkim@ucdavis.edu}{syxkim@ucdavis.edu}}\addressmark{1}, Tonie Scroggin\thanks{\href{mailto:tmscroggin@ucdavis.edu}{tmscroggin@ucdavis.edu}}\addressmark{1}, \and Jos\'e Simental\thanks{\href{mailto:simental@im.unam.mx}{simental@im.unam.mx} J.S. was partially supported by CONAHCyT project CF-2023-G-106 and UNAM’s PAPIIT Grant IA102124. }\addressmark{2}}{Eugene Gorsky, Soyeon Kim, Tonie Scroggin, and Jos\'e Simental}}


%\received{November 12th, 2024} 


%\email{simental@im.unam.mx}

\address{\addressmark{1}Department of Mathematics, University of California at Davis, One Shields Avenue, Davis CA 95616 \\
\addressmark{2} Instituto de Matem\'aticas, Universidad Nacional Aut\'onoma de M\'exico. Ciudad Universitaria, CDMX,  M\'exico}

%% then use \addressmark to match authors to institutions here
%\address{\addressmark{1}Department of This, University of That, There \\ \addressmark{2}School of Something, University of Something Else, Somewhere}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{Skew shaped positroid varieties are subvarieties of the Grassmannian defined by determinantal equations, and are special cases of open positroid varieties. Double Bott--Samelson varieties are algebraic varieties defined in terms of configurations of flags depending on a positive braid word. We explicitly realize every skew shaped positroid in $\Gr(k,n)$ as a double Bott--Samelson variety, and use this to construct splicing maps for skew shaped positroid varieties, generalizing those constructed by the first and third authors in the case of maximal positroid cells.}% \SK{Some abstract. Any ideas?}}

%% put your French abstract here, or comment this out if you don't have one


%% put your keywords here, or comment this out if you don't have them yet
\keywords{skew shaped positroids, double Bott--Samelson varieties, cluster algebras, quasi-cluster isomorphism}


%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{sample.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands




\begin{document}



\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above


\section{Introduction}


In this paper, we study the relation between two classes of varieties admitting a cluster structure: \emph{skew shaped positroids}, 
which are a special subclass of positroid varieties in the Grassmannian \cite{KLS} (see also \cite{RW24}); and \emph{double Bott--Samelson varieties} introduced by Elek--Lu \cite{GY06, EL21}. The cluster structures we consider were constructed by Galashin--Lam \cite{GL19} for the case of skew shaped positroids,
and by Shen--Weng \cite{SW} for double Bott--Samelson varieties.

Let us fix positive integers $0 < k < n$, and consider two partitions $\mu \subseteq \lambda$ both fitting inside a $k \times (n-k)$-rectangle. Associated to the pair $(\mu \subseteq \lambda)$, the skew shaped positroid variety $\skewschvar$ is an affine subvariety of the Grassmannian $\Gr(k,n)$ defined by the vanishing of certain minors, and the non-vanishing of other minors. These minors are determined by the skew diagram $\skewschubert$, and the skew shaped positroid $\skewschvar$ is in fact a positroid variety in the sense of \cite{GL19}. By the results of that %JS: I think it should be "that" (as opposed to "this") paper, because "this paper" sounds like it's the paper the reader is currently reading
paper, $\C[\skewschvar]$ admits a cluster algebra structure. In a nutshell, this means that the structure of the coordinate algebra $\C[\skewschvar]$ can be codified into the combinatorics of a quiver and its mutations, see \Cref{sec: cluster alg} for details.



On the other hand, for each positive braid $\br \in \Br^{+}_{k}$, the double Bott--Samelson variety $\BS(\br)$ is defined as a space of flag configurations dictated by the braid $\br$, see \Cref{def: braid and DBS} for a precise definition. By the work of several authors, see \cite{EL21,GY06},
the coordinate algebra $\C[\BS(\br)]$ admits a cluster algebra structure, that was made explicit in \cite{SW} and later generalized to the setting of \emph{braid varieties} in \cite{CGGLSS} and \cite{GLSBS22}.  

\begin{theorem}\label{thm: main-skew-vs-dbs}
Let $n > 0$ and $0 < k < n$. For any two partitions $\mu \subseteq \lambda$ fitting inside a $k \times (n-k)$-rectangle, there exist a $k$-stranded braid $\br_{\skewschubert} \in \Br_k^{+}$, $s \geq 0$ and an isomorphism:
\[
\BS(\br_{\skewschubert}) \times (\C^{\times})^{s} \cong \skewschvar.
\]
Moreover, the isomorphism can be chosen so that the natural cluster structure in $\BS(\br_{\skewschubert}) \times (\C^{\times})^{s}$ is quasi-isomorphic to the cluster structure in $\skewschvar$ defined in \cite{GL19}. 
\end{theorem}

We refer the reader to \Cref{sec: cluster alg} for the notion of a cluster quasi-isomorphism. {\parfillskip=0pt \par} 
\noindent Roughly speaking, a quasi-isomorphism preserves all the geometric information of a cluster algebra, while possibly changing the combinatorics. 

As an application of \Cref{thm: main-skew-vs-dbs}, we provide a splicing map for skew shaped positroids, generalizing that constructed in \cite{GS} for the maximal positroid cell. More precisely, we show that double Bott--Samelson varieties admit splicing maps, which are predicted by results in link homology and Fock--Goncharov amalgamation \cite{FockGoncharov, SW}. 



\begin{theorem}[\cite{GKSS-dbs}]\label{thm: main-splicing-dbs-intro}
Let $\br = \br_1\br_2 \in \Br^{+}_{n}$ be a positive braid. Then, there exists an open set $U(\br_1, \br_2) \subseteq \BS(\br)$ admitting a cluster structure induced from that on $\BS({\br})$ and a cluster quasi-isomorphism
\[
U(\br_1, \br_2) \cong \BS(\br_1) \times \BS(\br_2). 
\]
\end{theorem}



We combine \Cref{thm: main-skew-vs-dbs,thm: main-splicing-dbs-intro} as follows. Let $\mu \subseteq \lambda$ be partitions fitting inside a $k \times (n-k)$ rectangle, and choose $1 \leq j \leq n-k$. Let $\lambda^{1,j}$ be the partition obtained by considering only the first $j$ columns of $\lambda$, and define $\mu^{1,j}$ similarly. Note that $\mu^{1,j} \subseteq \lambda^{1,j}$ are partitions fitting inside a $k \times j$-rectangle, so we can consider the skew shaped positroid $S_{\lambda^{1,j}/\mu^{1,j}} \subseteq \Gr(k,k+j)$. Now let $\lambda^{2,j}$ be the partition obtained by considering the $j+1, \dots, n-k$-columns of $\lambda$, and similarly for $\mu^{2,j}$. Note that $\mu^{2,j} \subseteq \lambda^{2,j}$ fit inside a $k \times (n-k-j)$-rectangle, so we can consider the skew shaped positroid $S_{\lambda^{2,j}/\mu^{2,j}}^\circ \subseteq \Gr(k,n-j)$.


\begin{theorem}\label{thm: main3}
Let $n > 0$ and $0 < k < n$. Let $\mu \subseteq \lambda$ be partitions fitting inside a $k \times (n-k)$-rectangle. Choose $1 \leq j \leq n-k$, and let $U_j(\skewschvar)$ be the principal open set in $S_{\skewschubert}^\circ$ defined by the non-vanishing of the cluster variables in the $j$-th column of $\skewschubert$. Then, 
\[
U_j(\skewschvar) \cong S_{\lambda^{1,j}/\mu^{1,j}}^\circ \times S_{\lambda^{2,j}/\mu^{2,j}}^\circ. 
\]
\end{theorem}
%-Thm number 3: We can splice skew Schubert and we have an isomorphism between product... generalizing Eugene--Tonie's paper (diagramatically it is clear)
%\EG{The case when $\mu=\emptyset$ and $\lambda$ is the $k\times (n-k)$ rectangle corresponds to the open positroid $\Pi^{\circ}_{k,n}\subset \Gr(k,n)$,  the corresponding splicing maps were constructed in \cite{GS}.}

{\bf Notation.} If $n$ is a positive integer, we denote by $[n]$ the set $\{1, \dots, n\}$.


\section{Cluster algebras}\label{sec: cluster alg}
%\SK{aim to be 1-1.5 pages: Say what cluster algebra is and what quasi-isomorphism is}

%\textcolor{red}{In this section we summarize skew-symmetric cluster algebras and define a cluster quasi-morphism. \SK{Do we actually need skew symmetrizable in here?}}

 A \newword{quiver} is a directed graph $Q$ on a finite edge set that permits multiple edges but does not allow directed cycles of length one or two. An \newword{ice quiver} is obtained by specifying each vertex of $Q$ as either \newword{mutable} or \newword{frozen}. 
 A \newword{seed} $\Sigma$ is a quiver where all vertices are labeled with algebraically independent variables called \newword{cluster variables}, which are deemed mutable or frozen according to the vertex they label. 
  A \newword{mutation} is a local operation that can be performed at a mutable vertex of $\Sigma$ that takes $\Sigma$ to another seed $\Sigma'$. After mutating at the $j$th vertex of $\Sigma$, the associated cluster variable $x_{j}$ changes to $x'_{j}$ where $x_{j}$ and $x'_{j}$ are related by $x_{j}x'_{j}=A_j+B_j,$ where $A=\prod_{i\rightarrow j}x_i^{\#\{i\rightarrow j\}}$ and $B=\prod_{j\rightarrow k}x_k^{\#\{j\rightarrow k\}}$. We define the \newword{exchange ratio} $\widehat{y}_j$ as the ratio $\frac{A_j}{B_j}$. We denote $\Sigma\sim \Sigma'$ if $\Sigma'$ can be obtained from $\Sigma$ after performing a finite sequence of mutations.
 
 The \newword{cluster algebra} $\A(\Sigma)$ is defined to be the algebra generated by all cluster variables in all possible seeds obtained from mutations on $\Sigma$. If $\Sigma\sim \Sigma'$ then $\A(\Sigma)$ and $\A(\Sigma')$ are isomorphic.
 For more details on cluster algebras, see \cite{FZcluster}.
 
 We say that an affine algebraic variety $X$ is a \newword{cluster variety} if there exists some seed $\Sigma$ such that $X$ is isomorphic to $Spec(\A(\Sigma))$.
 In our setting, skew shaped positroids and double Bott--Samelson varieties are all cluster varieties, \cite{GL19, SW}.

A commutative algebra $\A$ may admit more than one cluster structure, that is, there may exist two seeds $\Sigma, \Sigma'$ which are not mutation equivalent but such that $\A(\Sigma) \cong \A \cong \A(\Sigma')$. Following Fraser \cite{Fraser}, see also \cite[Section 5.2]{LamSpeyerI}, we define a class of morphisms between cluster algebras of perhaps non-mutation-equivalent seeds. We say that two cluster structures $\A(\Sigma) \cong \A \cong \A(\Sigma')$ on the commutative algebra $\mathcal{A}$ are \newword{quasi-cluster equivalent} if:
\begin{enumerate}
    \item Both $\Sigma$ and $\Sigma'$ have the same number of mutable as well as of frozen variables.
    \item Each frozen variable in $\Sigma'$ is a Laurent monomial in the frozen variables of $\Sigma$, and vice versa.
    \item Mutable variables coincide up to multiplication by frozen variables.
    \item The exchange ratios coincide, i.e., $\widehat{y}_i=\widehat{y}_i'$ for $i\in[r]$.
\end{enumerate}
Additionally, we say that two cluster varieties $Spec(\A(\Sigma))$ and $Spec(\A(\Sigma'))$ are \newword{quasi-isomorphic} 
if there is an isomorphism between $Spec(\A(\Sigma))$ and $Spec(\A(\Sigma'))$ and also $\Sigma$ is quasi-cluster equivalent to $\Sigma'$.


\section{Skew shaped positroid varieties}\label{sec: skew-schubert}

 We fix $0 < k < n$ and consider a partition $\lambda$ whose Young diagram fits inside an $(n-k)\times k$-rectangle, that is, $\lambda = (\lambda_1, \dots, \lambda_k)$ with $\lambda_1 \leq n-k$. We draw partitions using the French convention, i.e., the boxes are lower left justified. Given partitions $\lambda$ and $\mu$ with $\mu_i \leq \lambda_i$ for all $1 \leq i \leq k$, the \newword{skew diagram} $\skewschubert$ denotes the set-theoretic difference of the Young diagrams of $\lambda$ and $\mu$.



 Given $1\le a\le n-k,1\le i\le k$ we define the square $\sq_{a,i}$ as the $i$-th box from the bottom 
 in the $a$-th column from the right. 
Let $R(\skewschubert)$ denote the ribbon shaped boundary of $\skewschubert$: a box $\square \in \skewschubert$ belongs to $R(\skewschubert)$ if the box northeast of $\square$ does \emph{not} belong to $\skewschubert$.   
For simplicity, we will assume that $R(\skewschubert)$ is connected. For discussions regarding the general case, see \cite{GKSS-positroid}. 

Given a square $\sq_{a,i} \in (n-k)^{k}$, we define $\mu_{a,i}$ as the minimal Young diagram such that $\mu \subseteq \mu_{a,i}$ and $\sq_{a,i} \in \mu_{a,i}$. Note that $\mu_{a,i} = \mu$ if and only if $\sq_{a,i} \in \mu$,  and if $\sq_{a,i} \in \skewschubert$ then $\mu_{a,i} \subseteq \lambda$. We will identify $\mu_{a,i}$ with its \newword{boundary path} $P_{a,i}$ from the southeast to the northwest corner of the $(n-k)\times k$-rectangle. We label all the steps of $P_{a,i}$ with numbers from $1$ to $n$, starting at the southeast corner and increasing as we go west and north. The \newword{label} of $\sq_{a,i}$ is the $k$-element subset $I'(\sq_{a,i}) \subseteq [n]$ consisting of the labels of the vertical steps of $P_{a,i}$. We denote by $I_{\mu} := I'(\sq_{a,i})$ for any square $\sq_{a,i} \in \mu$. 

\begin{example}\label{ex: running}
    Our running example will be $n = 12, k = 5$, $\lambda = (7,7,5,3,1)$ and $\mu = (3,3,2) \subseteq \lambda$. The next figure shows that $I'(\sq_{4,2}) = \{4,5,8,11,12\}$. 
    \begin{center}
        \begin{tikzpicture}[scale=0.7]
        \filldraw[color=lightgray] (0,0) -- (0,3) -- (2,3) -- (2,2) -- (3,2) -- (3,0) -- cycle;
\draw (0,0) -- (0,5) -- (1,5) -- (1,4) -- (3,4) -- (3,3) -- (5,3) -- (5,2) -- (7,2) -- (7,0)-- cycle;
\draw (0,1) to (7,1);
\draw (0,2) to (7,2);
\draw (0,3) to (5,3);
\draw (0,4) to (3,4);
\draw (1,0) to (1,5);
\draw (2,0) to (2,4);
\draw (3,0) to (3,4);
\draw (4,0) to (4,3);
\draw (5,0) to (5,2);
\draw (6,0) to (6,2);
\draw (3.5, 1.5) node {\tiny $\ast$};
\draw[dashed, ->] (3.5, 1.5) to (3.5, 0.5);
\draw (3.7, 0.5) node {\tiny $i$};
\draw[dashed, ->] (3.5, 1.5) to (6.5, 1.5);
\draw (6.5, 1.3) node {\tiny $a$};
\draw[line width=1] (7,0) -- (4,0) -- (4,2) -- (2,2) -- (2,3) -- (0,3) -- (0,5);
\draw (6.5, 0.2) node {\tiny $1$};
\draw (5.5, 0.2) node {\tiny $2$};
\draw (4.5, 0.2) node {\tiny $3$};
\draw (4.2, 0.5) node {\tiny $4$};
\draw (4.2, 1.5) node {\tiny $5$};
\draw (3.5, 2.2) node {\tiny $6$};
\draw (2.5, 2.2) node {\tiny $7$};
\draw (2.2, 2.5) node {\tiny $8$};
\draw (1.5, 3.2) node {\tiny $9$};
\draw (0.5, 3.2) node {\tiny $10$};
\draw (0.2, 3.5) node {\tiny $11$};
\draw (0.2, 4.5) node {\tiny $12$};
    \end{tikzpicture}
    \end{center}
\end{example}



\begin{definition}
    Given $\mu \subseteq \lambda$ as above, the \emph{skew shaped positroid variety} $\skewschvar \subseteq \Gr(k,n)$ consists of those subspaces $V$ admitting a matrix representative $M_V \in \mathrm{Mat}_{k \times n}(\C)$ satisfying the following conditions:
    \begin{enumerate}
        \item $\minor_{I'(a,i)}(M_V) = 0$ for $\sq_{a,i} \in (n-k)^{k} /\lambda$.
        \item $\minor_{I'(a,i)}(M_V) \neq 0$ for $\sq_{a,i} \in R(\skewschubert)$.
        \item $\minor_{I_\mu}(M_V) \neq 0$. 
    \end{enumerate}
    By Condition (3), {\bf we can and will restrict to the affine chart of $\Gr(k,n)$ where $\minor_{I_{\mu}} = 1$. }
\end{definition}


By definition, $\skewschvar$ is a locally closed subset of $\Gr(k,n)$. In fact, any skew shaped positroid is in fact a \emph{positroid variety} inside the Grassmannian. Introduced by Knutson--Lam--Speyer \cite{KLS}, positroid varieties provide a stratification of the Grassmannian.  By the work of Galashin--Lam \cite{GL19}, the coordinate ring of open positroid variety is a cluster algebra, and hence coordinate ring of a skew shaped posiroid admits a cluster algebra structure. We simplify their initial seed for the case of skew shaped positroids as follows.


\begin{theorem} \label{thm: skew schubert cluster}
Consider a skew diagram $\skewschubert$. An initial seed for the cluster structure on $\skewschvar$ can be described as follows:

{\bf Cluster variables:} For each box $\sq_{a,i} \in \skewschubert$, we have a cluster variable $x_{a,i} := \minor_{I'(a,i)}$.

{\bf Frozen variables:} A variable $x_{a,i}$ is frozen if the box $\sq_{a,i}$ belongs to the ribbon shaped boundary $R(\skewschubert)$.

{\bf Quiver:} The vertices of the quiver $Q_{\skewschubert}$ are in bijection with the boxes $\sq_{a,i} \in \skewschubert$. There are arrows:
\begin{center}
$\sq_{a, i} \to\sq_{a+1, i}$, \qquad \qquad $\sq_{a,i} \to \sq_{a, i-1}$, \qquad \qquad $\sq_{a,i} \to \sq_{a-1, i+1}$
\end{center} 
provided at least one of these boxes corresponds to a mutable vertex.
\end{theorem}
\begin{remark}
In fact, the sets $I'(a,i)$ for $\sq_{a,i} \in R(\skewschubert)$, together with $I_{\mu}$ form a \emph{source Grassmann necklace}. Consequently, we also get a bounded affine permutation. See \cite{KLS} for details. 
\end{remark}

\begin{remark}
In part of the literature, there is an extra frozen variable $x_{\mu} = \minor_{I_{\mu}}$, and consequently the quiver $Q_{\skewschubert}$ has an extra frozen vertex. This yields a cluster structure on the affine cone over $\skewschvar$. Since we work inside $\Gr(k,n)$ by setting $\minor_{I_{\mu}} = 1$, we do not see this frozen variable.
\end{remark}



\begin{example}\label{ex: running quiver}
Continuing with $n, k, \lambda$ and $\mu$ as in our running \Cref{ex: running} we obtain the following quiver, where the frozen variables are drawn in blue, and the mutable variables in red. 
\begin{center}
    \begin{tikzpicture}[scale=0.45]
    \filldraw[color=cyan, scale=2] (7,0) -- (7,2) -- (5,2) -- (5,3) -- (3,3) -- (3,4) -- (1,4) -- (1,5) -- (0,5) -- (0,3) -- (2,3) -- (2,2) -- (4,2) -- (4,1) -- (6, 1) -- (6,0) -- cycle;
        \filldraw[color=lightgray, scale=2] (0,0) -- (0,3) -- (2,3) -- (2,2) -- (3,2) -- (3,0) -- cycle;
\draw[scale=2] (0,0) -- (0,5) -- (1,5) -- (1,4) -- (3,4) -- (3,3) -- (5,3) -- (5,2) -- (7,2) -- (7,0)-- cycle;
\draw[scale=2] (0,1) to (7,1);
\draw[scale=2] (0,2) to (7,2);
\draw[scale=2] (0,3) to (5,3);
\draw[scale=2] (0,4) to (3,4);
\draw[scale=2] (1,0) to (1,5);
\draw[scale=2] (2,0) to (2,4);
\draw[scale=2] (3,0) to (3,4);
\draw[scale=2] (4,0) to (4,3);
\draw[scale=2] (5,0) to (5,2);
\draw[scale=2] (6,0) to (6,2);


%%mu
\node at (3,3) {\tiny $\begin{matrix} 5, 6, 8 \\ 11, 12 \end{matrix}$};

%frozens

\node at (13,1) {\tiny $\textcolor{blue}{\begin{matrix} 1, 6, 8 \\ 11, 12 \end{matrix}}$};
\node at (13,3) {\tiny $\textcolor{blue}{\begin{matrix} 1, 2, 8 \\ 11, 12 \end{matrix}}$};
\node at (11,3) {\tiny $\textcolor{blue}{\begin{matrix} 2, 3, 8 \\ 11, 12 \end{matrix}}$};
\node at (9,3) {\tiny $\textcolor{blue}{\begin{matrix} 3, 4, 8 \\ 11, 12 \end{matrix}}$};
\node at (9,5) {\tiny $\textcolor{blue}{\begin{matrix} 3, 4, 5 \\ 11, 12 \end{matrix}}$};
\node at (7,5) {\tiny $\textcolor{blue}{\begin{matrix} 4, 5, 6 \\ 11, 12 \end{matrix}}$};
\node at (5,5) {\tiny $\textcolor{blue}{\begin{matrix} 5, 6, 7 \\ 11, 12 \end{matrix}}$};
\node at (5,7) {\tiny $\textcolor{blue}{\begin{matrix} 5, 6, 7 \\ 8, 12 \end{matrix}}$};
\node at (3,7) {\tiny $\textcolor{blue}{\begin{matrix} 5, 6, 8 \\ 9, 12 \end{matrix}}$};
\node at (1,7) {\tiny $\textcolor{blue}{\begin{matrix} 5, 6, 8 \\ 10, 12 \end{matrix}}$};
\node at (1,9) {\tiny $\textcolor{blue}{\begin{matrix} 5, 6, 8 \\ 10, 11 \end{matrix}}$};

%mutables

\node at (11,1) {\tiny $\textcolor{red}{\begin{matrix} 2, 6, 8 \\ 11, 12 \end{matrix}}$};
\node at (9,1) {\tiny $\textcolor{red}{\begin{matrix} 3, 6, 8 \\ 11, 12 \end{matrix}}$};
\node at (7,1) {\tiny $\textcolor{red}{\begin{matrix} 4, 6, 8 \\ 11, 12 \end{matrix}}$};
\node at (7,3) {\tiny $\textcolor{red}{\begin{matrix} 4, 5, 8 \\ 11, 12 \end{matrix}}$};

\draw[thick, -Stealth] (7, 2.5) to (7, 1.5);
\draw[thick, -Stealth] (7, 4.5) to (7, 3.5);

\draw[thick, -Stealth] (9, 2.5) to (9, 1.5);
\draw[thick, -Stealth] (11, 2.5) to (11, 1.5);

\draw[thick, -Stealth] (12.5, 1) to (11.5, 1);
\draw[thick, -Stealth] (10.5, 1) to (9.5, 1);

\draw[thick, -Stealth] (8.5, 1) to (7.5, 1);

\draw[thick, -Stealth] (8.5, 3) to (7.5, 3);

\draw[thick, -Stealth] (7.6, 1.5) to (8.6, 2.5);

\draw[thick, -Stealth] (7.6, 3.5) to (8.6, 4.5);

\draw[thick, -Stealth] (9.6, 1.5) to (10.6, 2.5);

\draw[thick, -Stealth] (11.6, 1.5) to (12.6, 2.5);



%%mutables




    \end{tikzpicture}
    \end{center}
    \end{example}

    \begin{remark}
            Skew shaped positroids differ from the skew Schubert varieties of \cite{SSBW19}, and are closely related to Grassmannian Richardson varieties, see \cite[Remark 3.2.7]{GKSS-positroid}.  
    \end{remark}

    





\section{Double Bott--Samelson varieties}


We now define double Bott--Samelson varieties. We will work with the positive braid monoid on $k$ strands, i.e. $\Br_k^{+}$ is the monoid with generators $\sigma_1, \dots, \sigma_{k-1}$ satisfying the usual braid relations: $\sigma_i\sigma_j = \sigma_j\sigma_i$ if $|i-j|>1$ and $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$. We will picture an element $\br \in \Br_k^{+}$ via its \newword{wiring diagram}, obtained by taking $k$ horizontal strands, numbered from bottom to top, representing a generator $\sigma_i$ by a crossing between the $i$-th and $(i+1)$-st strands, and concatenating left-to-right as we read $\br$ in the same direction. 

We will also need the flag variety $\F(k)$ of complete flags in $\C^k$. If $F_1^{\bullet} = (0 \subseteq F_1^{1} \subseteq \cdots \subseteq F_1^{k} = \C^k)$ and $F_2^{\bullet}$ are two flags in $\C^k$, and $j \in \{1, \dots, k-1\}$, then we say that $F_1^{\bullet}$ is in position $s_j$ with respect to $F_2^{\bullet}$ if 
\[
F_1^{j} \neq F_2^{j}, \qquad \text{and} \qquad F_1^{i} = F_{2}^{i} \; \text{for} \; i \neq j. 
\]
we write $F_1^{\bullet} \rel{j} F_2^\bullet$ to express that $F_1^{\bullet}$ is in position $s_j$ with respect to $F_2^{\bullet}$.
On the other hand, we say that the flags $F_1^{\bullet}$ and $F_2^{\bullet}$ are transverse if, for each $j = 1, \dots, k-1$
\[
F_1^{j} \cap F_2^{k-j} = \{0\}, \qquad \text{equivalently} \qquad F_1^{j} + F_2^{k-j} = \C^k.  
\]
We write $F_1^{\bullet}  \pitchfork F_2^{\bullet}$ to express the fact that $F_1^{\bullet}$ and $F_2^{\bullet}$ are transverse. 
Finally, we will need two special flags: the standard and antistandard flags, defined by
\[
\begin{array}{l}
F_{\std}^\bullet = \{0\} \subseteq \subs{e_1} \subseteq \subs{e_1, e_2} \subseteq \cdots \subseteq \subs{e_1, e_2, \dots, e_k} = \C^k, \\
F_{\ant}^\bullet = \{0\} \subseteq \subs{e_k} \subseteq \subs{e_k, e_{k-1}} \subseteq \cdots \subseteq \subs{e_k, e_{k-1}, \dots, e_1} = \C^k. 
\end{array}
\]

\begin{definition}\label{def: braid and DBS}
    Let $\br = \sigma_{i_1}\cdots \sigma_{i_{\ell}} \in \Br_{+}^{k}$.  We define the \emph{double Bott--Samelson variety} $\BS(\beta)$ as follows:
        \[
        \BS(\br) = \{(F_0^{\bullet}, \dots, F_{\ell}^{\bullet}) \in \F(k)^{\ell+1} \mid F_0^{\bullet} = F_{\std}^{\bullet} \rel{i_1} F_1^{\bullet} \rel{i_2} \cdots \rel{i_{\ell}} F_{\ell}^{\bullet} \pitchfork F_{\ant}^{\bullet}\}.
        \]
\end{definition}

In fact, for any braid $\br = \sigma_{i_1}\cdots \sigma_{i_{\ell}}$, the double Bott--Samelson variety $\BS(\br)$ is a smooth, affine algebraic variety, that is a principal open set in the affine space $\C^{\ell}$. To see this, let us give coordinates to the double Bott--Samelson variety $\BS(\br)$. We identify the flag variety $\F(k)$ with the quotient $\GL(k)/B(k)$, where $B(k) \subseteq \GL(k)$ is the group of upper triangular matrices, as follows. If $M = (m_1|m_2|\cdots|m_k) \in \GL(k)$ has columns $m_1, \dots, m_k$, we define the flag
\[
F^{\bullet}_{M} = \{0\} \subseteq \subs{m_1} \subseteq \subs{m_1, m_2} \subseteq \cdots \subseteq \subs{m_1, m_2, \dots, m_k} = \C^k.
\]
Note that $F^{\bullet}_{M} = F^{\bullet}_{N}$ if and only if there exists an upper-triangular matrix $U$ such that $M = NU$, and this provides the identification $\F(k) \cong \GL(k)/B(k)$. Now we need the following results. Recall that if $I, J \subseteq [k]$ are sets of the same size and $M \in \GL(n)$, then the minor $\Delta_{I,J}(M)$ is the determinant of the $|I|\times|J|$-submatrix of $M$ obtained by deleting all rows (resp. columns) that do not belong to $I$ (resp. $J$). 

\begin{lemma}\label{lem:principal-minors}
    Let $M \in \GL(k)$. Then $F^{\bullet}_{M} \pitchfork F^{\bullet}_{ant}$ if and only if for each $i = 1, \dots, k$, the minor $\Delta_{[i], [i]}(M)$ is nonzero. 
\end{lemma}

\begin{lemma}\label{lem:matrix-parametrization}
    Let $M \in \GL(k)$ and let $j \in [k-1]$. The space of all flags $F^{\bullet} \in \F(k)$ such that $F^{\bullet}_{M} \rel{j} F^{\bullet}$ forms an affine line $\C$, and consists of all the flags of the form $F^{\bullet}_{MB_j(z)}$, where $z \in \C$ and $B_j(z)$ is the matrix
    \[
    B_j(z) = I_{k} + (z-1)E_{i,i} - E_{i+1,i+1} - E_{i,i+1} + E_{i+1,1}
    \]
     where $E_{r,s}$ is the matrix with a $1$ on the $(r,s)$-position and zeroes everywhere else.  %\EG{Say that is called braid matrix elsewhere?}
\end{lemma}

If $\br = \sigma_{i_1}\cdots \sigma_{i_{\ell}} \in \Br^{+}_{k}$ is a positive braid and $(z_1, \dots, z_{\ell}) \in \C^{\ell}$, define $B_{\br}(z_1, \dots, z_{\ell}) := B_{i_1}(z_1)\cdots B_{i_r}(z_r) \in \GL(n)$. \Cref{lem:principal-minors,lem:matrix-parametrization} have the following consequence.

\begin{theorem}\label{thm:param-dbs}
    Let $\br = \sigma_{i_1}\cdots \sigma_{i_{\ell}} \in \Br_k^{+}$. Then,
    \[
    \BS(\br)\cong \{(z_1, \dots, z_{\ell}) \in \C^{\ell} \mid \Delta_{[i], [i]}(B_{\br}(z_1, \dots, z_{\ell})) \neq 0 \; \text{for every} \; i \in [k]\}.
    \]
\end{theorem}

Let us give another equivalent characterization of the variety $\BS(\br)$ that will be useful later. The wiring diagram of $\br$ separates the horizontal strip where the wiring diagram is drawn in several connected components, which moreover can be arranged by \lq\lq levels,\rq\rq \, i.e., by horizontal sections of the diagram. Then, an element in $\BS(\br)$ is an assignment of a subspace $V_i$ of dimension $i$ to each region of level $i$ satisfying the following \newword{incidence conditions}:
\begin{enumerate}
    \item[(I1)] To the unique left-unbounded region of level $i$, we assign the subspace $\subs{e_1, \dots, e_i}$.
    \item[(I2)] If $V_i$ is the subspace assigned to the unique right-unbounded region of level $i$, then $V_i \cap \subs{e_{i+1}, \dots, e_k} = \{0\}$.
    \item[(I3)] If $V_i$ and $V_{i+1}$ are assigned to regions with
    intersecting boundaries, then $V_i \subseteq V_{i+1}$.
    \item[(I4)] If $V_i, V'_i$ are assigned to regions separated by a single crossing, then $V_i \neq V'_i$. 
\end{enumerate}

To finish this section, we describe the initial seed in the cluster structure on $\BS(\br)$ constructed by Shen and Weng in \cite{SW}. We set $\br = \sigma_{i_1}\cdots\sigma_{i_{\ell}} \in \Br_k^+$ and use the parametrization of $\BS(\br)$ given by \Cref{thm:param-dbs}.

{\bf Cluster variables.} For each $e \in [\ell]$, set $x_e := \Delta_{[i_e], [i_e]}(B_{\sigma_{i_1}\cdots\sigma_{i_e}}(z_1, \dots, z_e))$. The cluster variables in $\Sigma$ are $x_1, \dots, x_\ell$. We picture these variables in the wiring diagram of $\br$, by putting the variable $x_j$ on the region to the right of the crossing $\sigma_{i_j}$.

{\bf Frozen variables.} A variable $x_e$ is frozen if and only if there is no $d > e$ with $i_e = i_d$. Note that these correspond to the rightmost regions in the wiring diagram of $\br$. 

{\bf Quiver.} The quiver $Q_{\br}$ is read from the wiring diagram of $\br$, see \cite{SW} and \Cref{ex:running k-strand} below. 
  
\section{Skew shaped positroids and double Bott--Samelson varieties} 
Let $0 < k < n$ and take partitions $\mu \subseteq \lambda \subseteq (n-k)^{k}$. We define a $k$-stranded braid $\br_{\skewschubert}$ via the following procedure.
\begin{enumerate}
    \item Cross out the top box of every column of $\lambda$.
    \item For each column of $(n-k)^{k}$, put strands starting on the lower left corner of each box in this column, so that there are precisely $k$ strands. We join the strands to corners in the right border of this column, as follows:
    \begin{enumerate}
    \item Join the strand starting in the lower left corner of the crossed box belonging to $\skewschubert$ (from Step 1) in this column to the lower right corner of the lowest box in the same column belonging to $\skewschubert$. If the crossed box also belongs to $\mu$, then simply join the strand to the lower right corner of the same box.
    \item If a strand starts in the lower left corner of a non-crossed box in $\skewschubert$, join it to the upper right corner of the same box.
    \item If a strand starts on the lower left corner of any box which is not of the above type, join it (in a straight line) to the lower righg corner of the same box.  
    \end{enumerate}
    \item The braid $\br_{\skewschubert}$ is defined to be the concatenation of the $(n-k)$-braids from Step 2. In formulas,
    \begin{equation}\label{eq: k-strand-braid}
\br_{\skewschubert} = C_1C_2\cdots C_{n-k}, \qquad C_{j} = \sigma_{\lambda_{j}^{t}-1}\sigma_{\lambda_j^{t}-2}\cdots \sigma_{\mu_j^{t}+1}
    \end{equation}
    where $C_j$ is the empty braid if $\mu_j^{t}+1 > \lambda_{j}^{t}-1$. 
\end{enumerate}

\begin{example}\label{ex:running k-strand}
    In our running \Cref{ex: running}, we draw the braid $\br_{\skewschubert}$ as well as the quiver $Q_{\br_{\skewschubert}}$. Note that $\br_{\skewschubert} = (\sigma_4)()(\sigma_3)(\sigma_2\sigma_1)(\sigma_2\sigma_1)(\sigma_1)(\sigma_1)$, and that the quiver $Q_{\br}$ is the opposite of the quiver obtained from $Q_{\skewschubert}$ by deleting the vertices corresponding to crossed out boxes.  
\begin{center}
\begin{tikzpicture}[scale=0.6]
\filldraw[color=lightgray] (0,0) -- (0,3) -- (2,3) -- (2,2) -- (3,2) -- (3,0) -- cycle;
\draw (0,0) -- (0,5) -- (1,5) -- (1,4) -- (3,4) -- (3,3) -- (5,3) -- (5,2) -- (7,2) -- (7,0)-- cycle;
\draw (0,1) to (7,1);
\draw (0,2) to (7,2);
\draw (0,3) to (5,3);
\draw (0,4) to (3,4);
\draw (1,0) to (1,5);
\draw (2,0) to (2,4);
\draw (3,0) to (3,4);
\draw (4,0) to (4,3);
\draw (5,0) to (5,2);
\draw (6,0) to (6,2);

\draw (0,5) to (1,4); \draw (0,4) to (1,5);
%\filldraw[color=pink] (0.1,4.1) -- (0.9,4.1) -- (0.9,4.9) -- (0.1,4.9) -- cycle;
\draw (1,3) to (2,4); \draw (1,4) to (2,3);
%\filldraw[color=pink] (1.1, 3.1) -- (1.9, 3.1) -- (1.9, 3.9) -- (1.1, 3.9) -- cycle;
\draw (2,3) to (3,4); \draw (2,4) to (3,3);
%\filldraw[color=pink] (2.1, 3.1) -- (2.9, 3.1) -- (2.9, 3.9) -- (2.1, 3.9) -- cycle;
\draw (3,2) to (4,3); \draw (3,3) to (4,2); \draw(4,2) to (5,3); \draw (4,3) to (5,2);
%\filldraw[color=pink] (3.1, 2.1) -- (3.9, 2.1) -- (3.9, 2.9) -- (3.1, 2.9) -- cycle;
%\filldraw[color=pink] (4.1, 2.1) -- (4.9, 2.1) -- (4.9, 2.9) -- (4.1, 2.9) -- cycle;
\draw(5,1) to (6,2); \draw (5,2) to (6,1); \draw (6,1) to (7,2); \draw (6,2) to (7,1);
%\filldraw[color=pink] (5.1, 1.1) -- (5.9, 1.1) -- (5.9, 1.9) -- (5.1, 1.9) -- cycle;
%\filldraw[color=pink] (6.1, 1.1) -- (6.9, 1.1) -- (6.9, 1.9) -- (6.1, 1.9) -- cycle;

\draw[color=blue, thick] (0,4) to[out=0, in=180] (1,3) to[out=0, in=180] (2,3) to[out=0, in=180] (3,2)  to[out=0, in=180] (4,0) to[out=0, in=180] (5,1) to[out=0, in=180] (6,0) to[out=0, in=180] (7,1);
\draw[color=blue, thick] (0,3) to[out=0, in=180] (1,4) to[out=0, in=180] (7,4);
\draw[color=blue, thick] (0,2) to[out=0, in=180] (2,2) to[out=0, in=180] (3,3) to[out=0, in=180] (7,3);
\draw[color=blue, thick] (0,1) to[out=0, in=180] (3,1) to[out=0, in=180] (4,2) to[out=0, in=180] (5,0) to[out=0, in=180] (6,1) to[out=0, in=180] (7,0);
\draw[color=blue, thick] (0,0) to[out=0, in=180] (3,0) to[out=0, in=180] (4,1) to[out=0, in=180] (5,2) to[out=0, in=180] (7,2);
\end{tikzpicture}\qquad \qquad 
\begin{tikzpicture}[scale=0.6]
\draw (0,4) to[out=0, in=180] (1,3) to[out=0, in=180] (2,3) to[out=0, in=180] (3,2)  to[out=0, in=180] (4,0) to[out=0, in=180] (5,1) to[out=0, in=180] (6,0) to[out=0, in=180] (7,1);
\draw (0,3) to[out=0, in=180] (1,4) to[out=0, in=180] (7,4);
\draw (0,2) to[out=0, in=180] (2,2) to[out=0, in=180] (3,3) to[out=0, in=180] (7,3);
\draw (0,1) to[out=0, in=180] (3,1) to[out=0, in=180] (4,2) to[out=0, in=180] (5,0) to[out=0, in=180] (6,1) to[out=0, in=180] (7,0);
\draw (0,0) to[out=0, in=180] (3,0) to[out=0, in=180] (4,1) to[out=0, in=180] (5,2) to[out=0, in=180] (7,2);

\node at (2.5, 3.5) {\tiny \color{blue} $\bullet$};
\node at (3.5, 2.5) {\tiny \color{blue} $\bullet$};
\node at (5.5, 1.5) {\tiny \color{blue} $\bullet$};
\node at (7, 0.5) {\tiny \color{blue} $\bullet$};

\node at (6, 0.5) {\tiny \color{red} $\bullet$};
\node at (5, 0.5) {\tiny \color{red} $\bullet$};
\node at (4, 0.5) {\tiny \color{red} $\bullet$};
\node at (3.9, 1.5) {\tiny \color{red} $\bullet$};

\draw[thick, ->] (4.2, 0.5) to (4.8, 0.5);
\draw[thick, ->] (3.9, 0.6) to (3.9, 1.4);
\draw[thick, ->] (5.2, 0.5) to (5.8, 0.5);
\draw[thick, ->] (6.2, 0.5) to (6.8, 0.5);

\draw[thick, ->] (5.4, 1.4) to (4.1, 0.6);
\draw[thick, ->] (4, 1.5) to (5.3, 1.5);



\end{tikzpicture}
\end{center}
\end{example}


\begin{theorem}\label{thm: skew schubert as dbs}
    There exist $s \geq 0$ and an isomorphism of algebraic varieties
    \[
\BS(\br_{\skewschubert}) \times (\C^{\times})^{s} \cong \skewschvar
    \]
    that is a quasi-isomorphism of cluster varieties.
\end{theorem}

The proof of this theorem makes use of the following key concept. The \newword{short labeling} $J(\sq_{a,i}) \subseteq I'(\sq_{a,i})$ of a box $\sq_{a,i} \in \skewschubert$, consists of the first $i$ (with respect to the usual order of $[n]$) elements of $I'(\sq_{a,i})$. 
The short labels provide a map $\Phi: \skewschvar \to \BS(\br_{\skewschubert})$. More precisely, given an element $V \in \skewschvar$, pick a matrix $M_V = (v_1|\dots|v_n)$ representing it. The matrix $M_V$ is unique if we require that $v_{b_1} = e_1, \dots, v_{b_k} = e_k$, where $I_{\mu} = \{b_1 < \cdots < b_k\}$ and $e_1, \dots, e_k$ is the standard basis of $\C^k$. 

The element $\Phi(V)$ is specified by labeling the region to the right of a crossing of $\br_{k,n}$, which in turn corresponds to a non-crossed box $\sq_{i,a}$ of $\skewschubert$, by the subspace spanned by the elements $v_{j}$ where $j$ belongs to the short labeling $J(\sq_{i,a})$. 

The inverse map $\Psi: \BS(\br_{\skewschubert}) \to S_{\skewschubert}^\circ$ is obtained by reverse-engineering the construction above, using the fact that $\br_{\skewschubert}$ is the product of interval braids, and the following key identity:
\begin{equation}\label{eq: multiplying-partial-coxeter}
(v_1|\dots|v_k)B_{\sigma_{j-1}\cdots\sigma_{i}}(z_1, \dots, z_{j-i}) = (v_1|\cdots|v_{i-1}|v'_{i}|-v_{i}|\cdots|-v_{j-1}|v_{j+1}|\cdots|v_{k})
\end{equation}
where $v'_i = v_j + \sum_{s = 1}^{j-i}z_sv_{i+s-1}$. As we will see below in an example, this will allow us to construct the matrix $\Psi(z_1, \dots, z_\ell) = (v_1|\dots|v_n)$ for $(z_1, \dots, z_\ell) \in \BS(\br_{\skewschubert})$. We will get, however, that for every crossed box $\boxtimes_{a,i} \in \skewschubert$, $\Delta_{I'(\boxtimes_{a,i})}(\Psi(z_1, \dots, z_\ell)) = \pm 1$. This accounts for the $(\C^{\times})^{s}$-factor in \Cref{thm: skew schubert as dbs}. 



\begin{example}
    Let us carefully construct the maps in our running example. First, we construct the map $\Phi: \skewschvar \to \BS(\br_{\skewschubert})$. Let $V \in \skewschvar$ with $M_V = (v_1|\dots|v_{12})$. Since $I_{\mu} = \{5,6,8,11,12\}$, we set $v_{5} = e_1, v_6 = e_6, v_8 = e_3, v_{11} = e_4, v_{12} = e_5$. Reading the short labels of the boxes corresponding to the crossings of $\br_{\skewschubert}$ (see \Cref{ex: running quiver}), we obtain that $\Phi(V)$ is given as in \Cref{fig: from skew to dbs}. That this labeling satisfies the incidence conditions (I1)--(I4) follows from the equations defining $\skewschvar$. 
    For example, let us verify that $\subs{v_1} \subseteq \subs{v_3, v_4}$. From the Grassmann necklace $\mathcal{I}_{\skewschubert}$ and Remark 2.4 in \cite{MS17}, we can find the bounded affine permutation $f_{\skewschubert}$ which is in bijection with $\mathcal{I}_{\skewschubert}$. In this case, $f_{\skewschubert}(1)=3$ and this implies that $\subs{v_1,v_2}=\subs{v_2,v_3}$. Thus, $\subs{v_1} \subseteq \subs{v_2, v_3}$. Similarly, we obtain $f_{\skewschubert}(2)=4$, and thus $\subs{v_2,v_3}=\subs{v_3,v_4}$. Therefore, $\subs{v_1} \subseteq \subs{v_2, v_3} = \subs{v_3, v_4}$.
    Let us now construct the map $\Psi: \BS(\br_{\skewschubert}) \to \skewschvar$. Let $(z_1, \dots, z_8) \in \BS(\br_{\skewschubert})$. We define $V := \Psi(z_1, \dots, z_8)$ by specifying the columns of the matrix $M_V = (v_1|\dots|v_{12})$. First, we set $(v_5|v_6|v_8|v_{11}|v_{12}) = I_{k}$, the $k \times k$-identity matrix. After $C_1$ in \Cref{fig: from skew to dbs} we see the new vector $v_{10}$, so we define $v_{10}$ so that
    \[
    (v_5|v_6|v_8|v_{11}|v_{12})B_{4}(z_1) = (v_5|v_6|v_8|v_{10}|-v_{11})
    \]
    that we can do by \eqref{eq: multiplying-partial-coxeter}. The braid $C_2$ is empty so we skip it. Looking at $C_3$ define $v_7$ so that
    \[
    (v_5|v_6|v_8|v_{10}|-v_{11})B_{3}(z_2) = (v_5|v_6|v_7|-v_8|-v_{11}).
    \]
    Moving on to $C_4$, define $v_4$ so that
    \[
    (v_5|v_6|v_7|-v_8|-v_{11})B_{\sigma_2\sigma_1}(z_3, z_4) = (v_4|-v_5|-v_6|-v_8|-v_{11})
    \]
    where we have once again used \eqref{eq: multiplying-partial-coxeter}. This allows us to recover every vector $v_1, \dots, v_{12}$, \emph{except for} $v_9$, since $9$ does not appear in \Cref{fig: from skew to dbs}. However, the equations of $\skewschvar$ imply that $v_9$ is a scalar multiple of $v_{10}$, and we can recover $v_9$ by requiring that $\Delta_{5,6,8,9,12} = 1$. 
    Let us discuss the compatibility between cluster structures, by looking at the cluster variables in $\BS(\br_{\skewschubert})$ attached to the crossings of $C_4 = \sigma_2\sigma_1$. These are
    \begin{align*}
    \minor_{[2],[2]}B_{C_1C_2C_3\sigma_2}(z_1, z_2, z_3) = &\quad \minor_{[2],[2]}B_{C_1C_2C_3C_4}(z_1, z_2, z_3, z_4) \\ & \quad \text{and} \; \minor_{[1],[1]}B_{C_1C_2C_3C_4}(z_1, z_2, z_3, z_4).
    \end{align*}
    By construction,
    \[
    \begin{array}{rl}
    \minor_{[2],[2]}B_{C_1C_2C_3C_4}(z_1, z_2, z_3, z_4) &= \minor_{[2],[2]}(v_4|-v_5|-v_6|-v_8|-v_{11}) \\ & = -\det(v_4|v_5|e_{3}|e_{4}|e_{5}) = -\det(v_4|v_5|v_8|v_{11}|v_{12}) 
    \end{array}
    \]
    and similarly $\minor_{[1],[1]}B_{C_1C_2C_3C_4}(z_1, z_2, z_3, z_4) = \det(v_4|v_6|v_8|v_{11}|v_{12})$. Looking at \Cref{ex: running quiver} these are, possibly up to signs, the corresponding cluster variables in $\skewschvar$. 
\end{example}


\begin{figure}
    \centering
\begin{tikzpicture}[scale=0.8]
\draw[color=blue, thick] (-1,4) to (0,4) to[out=0, in=180] (1,3) to[out=0, in=180] (2,3) to[out=0, in=180] (3,2)  to[out=0, in=180] (4,0) to[out=0, in=180] (5,1) to[out=0, in=180] (6,0) to[out=0, in=180] (7,1);
\draw[color=blue, thick] (-1,3) to (0,3) to[out=0, in=180] (1,4) to[out=0, in=180] (7,4);
\draw[color=blue, thick] (-1, 2) to (0,2) to[out=0, in=180] (2,2) to[out=0, in=180] (3,3) to[out=0, in=180] (7,3);
\draw[color=blue, thick] (-1,1) to (0,1) to[out=0, in=180] (3,1) to[out=0, in=180] (4,2) to[out=0, in=180] (5,0) to[out=0, in=180] (6,1) to[out=0, in=180] (7,0);
\draw[color=blue, thick] (-1,0) to (0,0) to[out=0, in=180] (3,0) to[out=0, in=180] (4,1) to[out=0, in=180] (5,2) to[out=0, in=180] (7,2);

\node at (-0.5, 0.5) {\scriptsize $\subs{v_5}$};
\node at (-0.5, 1.5) {\scriptsize $\subs{v_5,v_6}$};
\node at (-0.5, 2.5) {\scriptsize $\subs{v_5,v_6,v_8}$};
\node at (-0.7, 3.5) {\scriptsize $\subs{v_5,v_6,v_8,v_{11}}$};

\node at (1.8, 3.5) {\scriptsize $\subs{v_5,v_6,v_8,v_{10}}$};

\node at (3.5, 2.5) {\scriptsize $\subs{v_5,v_6,v_7}$};

\node at (4, 1.5) {\tiny $\subs{v_4,v_5}$};
\node at (5.3, 1.5) {\scriptsize $\subs{v_3,v_4}$};

\node at (4, 0.5) {\scriptsize $\subs{v_4}$};

\node at (5, 0.5) {\scriptsize $\subs{v_3}$};
\node at (6, 0.5) {\scriptsize $\subs{v_2}$};
\node at (7, 0.5) {\scriptsize $\subs{v_1}$};


\end{tikzpicture}
 \caption{The braid $\br_{\skewschubert} = (\sigma_4)()(\sigma_3)(\sigma_2\sigma_1)(\sigma_2\sigma_1)(\sigma_1)(\sigma_1)$ associated to $\skewschubert$ where $\lambda=(7,7,5,3,1)$ and $\mu=(3,3,2)$.}
 \label{fig: from skew to dbs}
\end{figure}


\section{Splicing}
%\SK{Aim for 1 pages}
Let $\br = \br_1\br_2$ be a decomposition of the braid $\br \in \Br_k^+$. We let $U(\br_1,\br_2) \subseteq \BS(\br)$ be the open set obtained by the non-vanishing of the cluster variables corresponding to the last appearance of each letter in $\br_1$. Note that $U(\br_1,\br_2)$ has a cluster structure, obtained from the cluster structure in $\BS(\br)$ by freezing the cluster variables just described. 

\begin{theorem}\label{thm: main-splicing-dbs}
Let $\br = \br_1\br_2 \in \Br^{+}_{k}$ and $U(\br_1,\br_2)$ be as above. Then, we have a quasi-cluster isomorphism
\[
U(\br_1,\br_2) \cong \BS(\br_1) \times \BS(\br_2). 
\]
\end{theorem}

The idea behind the previous result is as follows. We let $\beta_1 = \sigma_{i_1}\cdots \sigma_{i_{\ell_1}}$, $\beta_2 = \sigma_{j_1}\cdots \sigma_{j_{\ell_2}}$. Then, $\BS(\br)$ consists of sequences of flags:
\begin{equation}\label{eq: b1 + b2}
F_{\std}^{\bullet} = F_0^{\bullet} \rel{i_1} F_1^{\bullet} \rel{i_2} \cdots \rel{i_{\ell_1}} {\color{red} F_{\ell_1}^{\bullet}} \rel{j_1} F_{\ell_1+1}^{\bullet} \rel{j_2} \cdots \rel{j_{\ell_2}} F_{\ell_1+\ell_2}^{\bullet} \pitchfork F_{\ant}^{\bullet}. 
\end{equation}
The conditions on $U(\br_1,\br_2)$ imply that $F_{\ell}^{\bullet} \pitchfork F_{\ant}^{\bullet}$, so we can try to separate \eqref{eq: b1 + b2} as:
\begin{equation}\label{eq: b1 b2 separated}
 F_0^{\bullet} \rel{i_1} F_1^{\bullet} \rel{i_2} \cdots \rel{i_{\ell_1}} {\color{red} F_{\ell_1}^{\bullet}} \qquad \text{and} \qquad {\color{red} F_{\ell_1}^{\bullet}} \rel{j_1} F_{\ell_1+1}^{\bullet} \rel{j_2} \cdots \rel{j_{\ell_2}} F_{\ell_1+\ell_2}^{\bullet}.
\end{equation}
The first chain of flags belongs to $\BS(\br_1)$. The second chain of flags, however, does not belong to $\BS(\br_2)$, because it would have to start with the standard flag for this to be the case. So the isomorphism in \Cref{thm: main-splicing-dbs} is obtained upon a simultaneous translation of all the flags in the second chain of \eqref{eq: b1 b2 separated}. 

Now let $\mu \subseteq \lambda$ be partitions fitting inside a $k \times (n-k)$-rectangle. By definition, we have a decomposition $\br_{\skewschubert} = C_1\cdots C_{n-k}$. Pick $1 \leq j \leq n-k$, and let $\br_1 = C_1\cdots C_j$, and $\br_2 = C_{j+1}\cdots C_{n-k}$, so that we have a decomposition $\br_{\skewschubert} = \br_1\br_2$ and we can apply \Cref{thm: main-splicing-dbs}. Note that the set $U = U_j$ can be obtained as the non-vanishing locus all the cluster variables corresponding to the $j$-th column of the $k \times (n-k)$-box which, up to variables associated to crossed boxes, correspond to the letters of $C_j$. Note also that $\br_1 = \br_{\lambda^{1,j}/\mu^{1,j}}$ and $\br_2 = \br_{\lambda^{2,j}/\mu^{2,j}}$, where $\lambda^{1,j}, \mu^{1,j}$ and $\lambda^{2,j}, \mu^{2,j}$ are as defined in the introduction. Thus, we obtain the following result:

\begin{theorem}
With the notation as above, we have a quasi-isomorphism of cluster varieties.
\[
U_j(\skewschvar) \cong S_{\lambda^{1,j}/\mu^{1,j}}^\circ \times S_{\lambda^{2,j}/\mu^{2,j}}^\circ
\]
\end{theorem}

In \cite{GKSS-positroid} we also give a way of understanding this result in a linear algebraic way.

\begin{example}
In our running \Cref{ex: running}, take $j = 5$. Then, we need to freeze the variable $x_{7-5,1} = x_{2,1} = \Delta_{3,6,8,11,12}$, which is the only non-frozen variable in the fifth column. We have $\lambda^{1,5} = (5,5,5,3,1)$, $\mu^{1,5} = \mu = (3,3,2)$, while $\lambda^{2,5} = (2,2)$ and $\mu^{2,5} = \emptyset$.
\begin{center}
    \begin{tikzpicture}[scale=0.3]
    \filldraw[color=cyan, scale=2] (7,0) -- (7,2) -- (5,2) -- (5,3) -- (3,3) -- (3,4) -- (1,4) -- (1,5) -- (0,5) -- (0,3) -- (2,3) -- (2,2) -- (4,2) -- (4,1) -- (6, 1) -- (6,0) -- cycle;
        \filldraw[color=lightgray, scale=2] (0,0) -- (0,3) -- (2,3) -- (2,2) -- (3,2) -- (3,0) -- cycle;
\draw[scale=2] (0,0) -- (0,5) -- (1,5) -- (1,4) -- (3,4) -- (3,3) -- (5,3) -- (5,2) -- (7,2) -- (7,0)-- cycle;
\draw[scale=2] (0,1) to (7,1);
\draw[scale=2] (0,2) to (7,2);
\draw[scale=2] (0,3) to (5,3);
\draw[scale=2] (0,4) to (3,4);
\draw[scale=2] (1,0) to (1,5);
\draw[scale=2] (2,0) to (2,4);
\draw[scale=2] (3,0) to (3,4);
\draw[scale=2] (4,0) to (4,3);
\draw[scale=2] (5,0) to (5,2);
\draw[scale=2] (6,0) to (6,2);


%%mu
%\node at (3,3) {\tiny $\begin{matrix} 5, 6, 8 \\ 11, 12 \end{matrix}$};

%frozens

\node at (13,1) {\tiny $\textcolor{blue}{\bullet}$};
\node at (13,3) {\tiny $\textcolor{blue}{\bullet}$};
\node at (11,3) {\tiny $\textcolor{blue}{\bullet}$};
\node at (9,3) {\tiny $\textcolor{blue}{\bullet}$};
\node at (9,5) {\tiny $\textcolor{blue}{\bullet}$};
\node at (7,5) {\tiny $\textcolor{blue}{\bullet}$};
\node at (5,5) {\tiny $\textcolor{blue}{\bullet}$};
\node at (5,7) {\tiny $\textcolor{blue}{\bullet}$};
\node at (3,7) {\tiny $\textcolor{blue}{\bullet}$};
\node at (1,7) {\tiny $\textcolor{blue}{\bullet}$};
\node at (1,9) {\tiny $\textcolor{blue}{\bullet}$};

%mutables

\node at (11,1) {\tiny $\textcolor{red}{\bullet}$};
\node at (9,1) {\tiny $\textcolor{blue}{\bullet}$};
\node at (7,1) {\tiny $\textcolor{red}{\bullet}$};
\node at (7,3) {\tiny $\textcolor{red}{\bullet}$};

\draw[thick, -Stealth] (7, 2.5) to (7, 1.5);
\draw[thick, -Stealth] (7, 4.5) to (7, 3.5);

%\draw[thick, -Stealth] (9, 2.5) to (9, 1.5);
\draw[thick, -Stealth] (11, 2.5) to (11, 1.5);

\draw[thick, -Stealth] (12.5, 1) to (11.5, 1);
\draw[thick, -Stealth] (10.5, 1) to (9.5, 1);

%\draw[thick, blue, -Stealth] (12.5, 3) to (11.5, 3);
%\draw[thick, blue, -Stealth] (10.5, 3) to (9.5, 3);

%\draw[thick, blue, -Stealth] (8.5, 5) to (7.5, 5);
%\draw[thick, blue, -Stealth] (6.5, 5) to (5.5, 5);

%\draw[thick, blue, -Stealth] (4.5, 7) to (3.5, 7);
%\draw[thick, blue, -Stealth] (2.5, 7) to (1.5, 7);

%\draw[thick, blue, -Stealth] (9, 4.5) to (9, 3.5);
%\draw[thick, blue, -Stealth] (5, 6.5) to (5, 5.5);
%\draw[thick, blue, -Stealth] (1, 8.5) to (1, 7.5);

%\draw[thick, blue, -Stealth] (13, 2.5) to (13, 1.5);


\draw[thick, -Stealth] (8.5, 1) to (7.5, 1);

\draw[thick, -Stealth] (8.5, 3) to (7.5, 3);

\draw[thick, -Stealth] (7.6, 1.5) to (8.6, 2.5);

\draw[thick, -Stealth] (7.6, 3.5) to (8.6, 4.5);

%\draw[thick, -Stealth] (9.6, 1.5) to (10.6, 2.5);

\draw[thick, -Stealth] (11.6, 1.5) to (12.6, 2.5);

%%%%%%%%%
%%%%%%%%%
%%%%%%%%%

\node at (15, 5) {$\cong$}; 

%%%%%%%%%
%%%%%%%
%%%%%%%

\filldraw[color=lightgray, scale=2] (8,0) -- (11,0) -- (11,2) -- (10,2) -- (10, 3) -- (8, 3) -- cycle;
\filldraw[color=cyan, scale=2] (13,0) -- (13,3) -- (11,3) -- (11,4) -- (9,4) -- (9,5) -- (8,5) -- (8,3) -- (10,3) -- (10,2) -- (12,2) -- (12,0) -- (13,0);
\draw[scale=2] (8,0) -- (13,0) -- (13,3) -- (11,3) -- (11,4) -- (9, 4) -- (9,5) -- (8,5) -- cycle;

\draw[scale=2] (8,1) to (13,1);
\draw[scale=2] (8, 2) to (13,2);
\draw[scale=2] (8,3) to (11,3);
\draw[scale=2] (8,4) to (9,4);

\draw[scale=2] (9,0) to (9,4);
\draw[scale=2] (10,0) to (10,4);
\draw[scale=2] (11,0) to (11,3);
\draw[scale=2] (12,0) to (12,3);

\node at (25,1) {\tiny $\textcolor{blue}{\bullet}$};
\node at (25,3) {\tiny $\textcolor{blue}{\bullet}$};
\node at (25,5) {\tiny $\textcolor{blue}{\bullet}$};
\node at (23,5) {\tiny $\textcolor{blue}{\bullet}$};
\node at (21,5) {\tiny $\textcolor{blue}{\bullet}$};
\node at (21,7) {\tiny $\textcolor{blue}{\bullet}$};
\node at (19,7) {\tiny $\textcolor{blue}{\bullet}$};
\node at (17,7) {\tiny $\textcolor{blue}{\bullet}$};
\node at (17,9) {\tiny $\textcolor{blue}{\bullet}$};

\node at (23,1) {\tiny $\textcolor{red}{\bullet}$};
\node at (23,3) {\tiny $\textcolor{red}{\bullet}$};

\draw[thick, -Stealth] (24.5, 1) to (23.5, 1);
\draw[thick, -Stealth] (24.5, 3) to (23.5, 3);
\draw[thick, -Stealth] (23, 2.5) to (23, 1.5);
\draw[thick, -Stealth] (23, 4.5) to (23, 3.5);

\draw[thick, -Stealth] (23.6, 1.5) to (24.6, 2.5);
\draw[thick, -Stealth] (23.6, 3.5) to (24.6, 4.5);


%%%%%%%
%%%%%
%%%%%%

\node at (27, 5) {$\times$};


%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%



\filldraw[color=cyan, scale=2] (16,0) -- (16,2) -- (14,2) -- (14,1) -- (15,1) -- (15,0) -- (16,0);

\draw[scale=2] (14,0) -- (16,0) -- (16,2) -- (14,2) -- cycle;

\draw[scale=2] (14,1) -- (16,1);
\draw[scale=2] (15,0) -- (15,2);

\node at (31,1) {\tiny $\textcolor{blue}{\bullet}$};
\node at (31,3) {\tiny $\textcolor{blue}{\bullet}$};
\node at (29,3) {\tiny $\textcolor{blue}{\bullet}$};

\node at (29,1) {\tiny $\textcolor{red}{\bullet}$};

\draw[thick, -Stealth] (30.5, 1) to (29.5, 1);

\draw[thick, -Stealth] (29, 2.5) to (29, 1.5);

\draw[thick, -Stealth] (29.6, 1.5) to (30.6, 2.5);








    \end{tikzpicture}
    \end{center}
\end{example}


%We can also understand this result in a purely linear-algebraic way, as follows. Let $V=(v_1|\cdots|v_n)\in S_{\skewschubert}\subseteq \Gr(k,n)$ be a given matrix. Assume that $1\le a\le \lambdabar_a$. We define ordered subset \begin{equation}
   % I'(a,s,i)=\begin{cases}
    %    I_{1}(a,i) & \text{if $1\le s\le i-1$}\\
     %   I_{2}(a,i) & \text{if $s>i-1$.}
   % \end{cases}
%\end{equation}
%where $I_{1}(a,i)$ is obtained from $I'(a,i)$ by replacing $a+s-1$ by $a+i-1$ while preserving the order, and $I_{2}(a,i)$ is obtained by replacing $s$th element in $I'(n-k-\lambda_{i}+1,i)$ with $b_{s}$. Similarly, we define an ordered subset $I'(a,t,i)$ by replacing $b_{n-t+1}$ by $a+i-1$.  
%From this, we first define a vector \begin{equation}\label{eq: for $u_{i}$}
%    u_{i}=v_{a+i}-\sum_{\textcolor{red}{s=1}}^{\textcolor{red}{i-1}}\frac{\Delta_{I'(a,s,i)}}{\Delta_{I'(a,i)}}v_{a+s}=\sum_{\textcolor{red}{t=1}}^{\textcolor{red}{k-i-1}}\frac{\Delta_{I''(a,t,i)}}{\Delta_{I'(a,i)}}v_{b_{k-t}}
%\end{equation}
%for $1\le a\le k-1$. 
%Let $\lambda_{a,1}, \lambda_{a,2}$, $\mu_{a,1}, \mu_{a,2}$ be the partitions obtained by splitting $\lambda$ and $\mu$ along their $a$-th column, respectively.
%Using this $u_{i}$, we consider two submatrices of $V$:
%\begin{equation}\label{eq: V1 and V2}
%V_{1}=\langle v_{j}: j_1\in J(a',i'), v_{a}, u_{i} \rangle, \quad 
%V_{2}=\langle v_{j}: j_2\in J(a'',i''), v_{a}, v_{n}\rangle
%\end{equation}
%for some $\sq_{a',i'}\in \lambda_{a,1}/\mu_{a,1}$ and $\sq_{a'',i''}\in \lambda_{a,2}/\mu_{a,2}$.
%\begin{theorem}
%    The map $\Phi_{a}:V\to (V_{1},V_{2})$ defined by \ref{eq: V1 and V2} is an isomorphism between $U_{a}\subset S_{\skewschubert}^{\circ}$ and the product $S_{\lambda_{a,1}/\mu_{a,1}}^{\circ}\times S_{\lambda_{a,2}/\mu_{a,2}}$. 
%\end{theorem}

%\begin{example}
 %   In our running \Cref{ex: running}, we let $j = 5$. 
%\end{example}




%\subsection{Application: Splicing skew Schubert varieties}

%\textcolor{red}{Assume that we already defined $\beta_{\skewschubert}$ in the previous section.}

%As an application of \Cref{thm: main-splicing-dbs}, we obtain the cluster compatible splicing map for $S_{\skewschubert}$ as follows:

%\begin{theorem}\label{thm: main3}

%Let $n > 0$ and $0 < k \leq n$. Let $\mu \subseteq \lambda$ be partitions fitting inside a $k \times (n-k)$-rectangle. Choose $1 \leq j \leq n-k$, and let $S^{\circ}_{\skewschubert}$ be the principal open set in $S_{\skewschubert}$ defined by the non-vanishing of the cluster variables in the $j$-th column of $\skewschubert$. Let $\lambda_1, \lambda_2$, $\mu_1, \mu_2$ be the partitions obtained by splitting $\lambda$ and $\mu$ along their $j$-th column, respectively. Then, there is a quasi-cluster isomorphism
%\[
%S^{\circ}_{\skewschubert} \cong S_{\lambda_1/\mu_1} \times S_{\lambda_2/\mu_2}. 
%\]
%\end{theorem}













\acknowledgements{
The authors would like to thank Mikhail Gorsky, Melissa Sherman-Bennett, Thomas Lam, David Speyer, and Daping Weng for the useful discussions. %\JS{also, should be alphabetically ordered} for the useful discussions. 
%J.S. was partially supported by CONAHCyT project CF-2023-G-106 and UNAM’s PAPIIT Grant IA102124. 
}





%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
\printbibliography



\end{document}