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\title{Parabolic Tamari Lattices in Linear Type $B$}

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\author[W.~Fang, H.~M\"uhle, and J.-C.~Novelli]{Wenjie Fang\thanks{\href{mailto:wenjie.fang@u-pem.fr}{wenjie.fang@u-pem.fr}}\addressmark{1}, Henri M{\"u}hle\thanks{\href{mailto:henri.muehle@tu-dresden.de}{henri.muehle@tu-dresden.de}. Received funding from the European Research Council (Grant Agreement no. 681988, CSP-Infinity.)}\addressmark{2}, \and Jean-Christophe Novelli\thanks{\href{mailto:novelli@univ-mlv.fr}{novelli@univ-mlv.fr}. Partially supported by theCARPLO project of the Agence Nationale de la recherche (ANR-20-CE40-0007).}\addressmark{1}}

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\address{\addressmark{1}LIGM, Universit{\'e} Gustave Eiffel, CNRS, ESIEE Paris, F-77454 Marne-la-Vall{\'e}e, France\\ \addressmark{2}Institut f{\"u}r Algebra, Technische Universit{\"a}t Dresden, 01062 Dresden, Germany}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{We study parabolic aligned elements associated with the type-$B$ Coxeter group and the so-called linear Coxeter element.  These elements were introduced algebraically in (M\"{u}hle and Williams, 2019) for parabolic quotients of finite Coxeter groups and were characterized by a certain forcing condition on inversions.  We focus on the type-$B$ case and give a combinatorial model for these elements in terms of pattern avoidance.  Moreover, we describe an equivalence relation on parabolic quotients of the type-$B$ Coxeter group whose equivalence classes are indexed by the aligned elements.  We prove that this equivalence relation extends to a congruence relation for the weak order.  The resulting quotient lattice is the type-$B$ analogue of the parabolic Tamari lattice introduced for type $A$ in (M\"{u}hle and Williams, 2019).  These lattices have not appeared in the literature before.}

%% put your French abstract here, or comment this out if you don't have one
\resumetitle{R{\'e}sum{\'e}}
%\resume{\textbf{This is a temporary placeholder.} We study parabolic aligned elements associated with the type-$B$ Coxeter group and the so-called linear Coxeter element.  These elements were introduced algebraically in (M{\"u}hle and Williams, 2019) for parabolic quotients of finite Coxeter groups and are characterized by a certain forcing condition on inversions.  We focus on the type-$B$ case and give a combinatorial model for these elements in terms of pattern avoidance.  Moreover, we describe an equivalence relation on parabolic quotients of the type-$B$ Coxeter group whose equivalence classes are indexed by the aligned elements.  We prove that this equivalence relation extends to a congruence relation for the weak order.  The resulting quotient lattice is the type-$B$ analogue of the parabolic Tamari lattice introduced for type $A$ in (M{\"u}hle and Williams, 2019).  These lattices have not appeared in the literature before. \textbf{End of placeholder, and a few more words to reflect the fact that the French translation of an English text is often one or two lines longer.}}
\resume{Nous \'{e}tudions les \'{e}l\'{e}ments paraboliques align\'{e}s associ\'{e}s aux groupes de 
Coxeter de type B et \`{a} l'\'{e}l\'{e}ment de Coxeter lin\'{e}aire. Ces \'{e}l\'{e}ments ont 
\'{e}t\'{e} introduits de fa\c{c}on alg\'{e}brique en (M\"{u}hle et Williams, 2019) pour 
les quotients paraboliques des groupes de Coxeter finis, et ils ont \'{e}t\'{e} 
caract\'{e}ris\'{e}s par une certaine condition de forçage sur les inversions. 
Le cas du type B est consid\'{e}r\'{e}, et nous proposons un mod\`ele combinatoire 
de ces \'{e}l\'{e}ments en termes de motifs exclus. De plus, nous d\'{e}crivons 
une relation d'\'{e}quivalence dans un quotient parabolique du groupe de 
Coxeter de type B, dont les classes d'\'{e}quivalence sont indic\'{e}es par les 
\'{e}l\'{e}ments align\'{e}s. Nous montrons que cette relation d'\'{e}quivalence s'\'{e}tend 
\`{a} une relation de congruence sur l'ordre faible. Le quotient ainsi 
obtenu est un analogue en type B du treillis de Tamari parabolique 
introduit par (M\"{u}hle et Williams, 2019) pour le type A. Ces treillis ne 
sont jamais apparus dans la litt\'{e}rature.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{sign-symmetric permutation, hyperoctahedral group, parabolic quotient, Tamari lattice, Coxeter--Catalan combinatorics}

%% 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
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\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}

Knuth introduced in \cite[Section~2.2.1]{knuth97aop_vol1} the family of \emph{stack-sortable permutations}.  These are permutations that can be converted into the identity permutation by passing through a stack.  Combinatorially, stack-sortable permutations are characterized by avoiding the pattern $312$.  Bj{\"o}rner and Wachs have interpreted this pattern-avoidance by a certain \emph{forcing} of inversions~\cite[Section~9]{bjorner97shellable}.  More precisely, a permutation $\pi$ of $\{1,2,\ldots,n\}$ avoids the pattern $312$ if and only if for every $i<j<k$, when $\pi(i)>\pi(k)$, we have $\pi(j)>\pi(k)$.  

Using a root-theoretic approach, Reading generalized this condition to all finite irreducible Coxeter groups $W$ and all Coxeter elements $c$.  The Coxeter group $W$ generalizes the symmetric group, and the Coxeter element $c$ generalizes the linear order $1<2<\cdots<n$ and the reverse lexicographic order on the transpositions used to describe the above ``forcing'' condition.  These \emph{$c$-aligned elements} of $W$ play an important role in the very active stream of Coxeter--Catalan combinatorics.  First, the number of $c$-aligned elements of $W$ is the $W$-Catalan number (for any choice of Coxeter element $c$).  Second, the $c$-aligned elements provide a bijective bridge between $c$-noncrossing partitions and $c$-clusters associated with $W$~\cite{reading07clusters}.  

On top of that, the $c$-aligned elements behave nicely from a lattice-theoretic point of view.  Inside the weak order, they form the \emph{$c$-Cambrian lattice}, which is semidistributive and trim.  This family of lattices generalizes the famous \emph{Tamari lattice}~\cite{reading06cambrian}.  Another remarkable feature is that these lattices arise from a certain orientation of the $1$-skeleton of the \emph{$W$-associahedron}, a polytope associated with the Coxeter group $W$.  In fact, the $W$-associahedron is the dual polytope of the $c$-cluster complex associated with $W$.  

In \cite{muehle19tamari}, $c$-aligned elements, $c$-noncrossing partitions and $c$-clusters were generalized to \emph{parabolic quotients} of finite irreducible Coxeter groups.  In this generalization, some properties of these objects were conjectured to remain, such as the lattice property of parabolic $c$-aligned elements under weak order (\cite[Conjecture~35]{muehle19tamari}), and when $W$ is of \emph{coincidental type}, the equinumerosity of parabolic $c$-aligned elements, parabolic $c$-noncrossing partitions and parabolic $c$-clusters (\cite[Conjecture~41]{muehle19tamari}).  The case of the symmetric group with $c$ the increasing long cycle was settled in the same article.  Further research has exhibited remarkable connections between parabolic Coxeter--Catalan objects associated with the symmetric group, certain Hopf algebras, and the theory of diagonal harmonics~\cite{ceballos20the}.  For more properties of these objects, readers are referred to \cite{fang21consecutive,krattenthaler19the,muehle21noncrossing}.  

In this extended abstract, we present a first study of parabolic Coxeter--Catalan objects associated with the hyperoctahedral group, \ie, the Coxeter group of type $B$.  We start by setting up a combinatorial model of the elements of the parabolic quotient $\Hyper_{\alpha}$ of the hyperoctahedral group with respect to some type-$B$ composition $\alpha$
% underlying elements in Section~\ref{sec:preliminaries}, 
using colored sign-symmetric permutations, and then describe in Section~\ref{sec:parabolic_aligned_elements} a particular order on the relevant inversions we use to describe the forcing conditions that determine the parabolic aligned elements, and further characterize them using pattern-avoidance. We then prove in Section~\ref{sec:parabolic_tamari_quotient} that these parabolic aligned elements in type $B$ form a quotient lattice of the weak order on the whole parabolic quotient.  This quotient lattice is called the \emph{type-$B$ parabolic Tamari lattice}, denoted by  $\Tamari_{B}(\alpha)$, and has the same lattice-theoretic properties as the $c$-Cambrian lattice.  We state this as the main result of this extended abstract.

%in Section~\ref{sec:parabolic_231}

\begin{theorem}\label{thm:parabolic_tamari}
	For every type-$B$ composition $\alpha$, the type-$B$ parabolic Tamari lattice
% 	partially ordered set 
	$\Tamari_{B}(\alpha)$ is a lattice.  Moreover, it is a trim and semidistributive quotient lattice of the weak order on the parabolic quotient $\Hyper_{\alpha}$.
% 	$\bigl(\Hyper_{\alpha},\weakorder\bigr)$.
\end{theorem}

Most of the proofs are omitted here due to space limitations.  These proofs, along with more details and background, can be found in the full version~\cite{fang21parabolic}.

\section{Basics}
    \label{sec:preliminaries}
\subsection{The hyperoctahedral group}
	\label{sec:hyperoctahedral_group}
For $n>0$, we define $[n]\defs\{1,2,\ldots,n\}$ and $\pm[n] \defs \{{-}n,-{n}{+}1,\ldots,{-}2,{-}1,1,2,\ldots,n\}$.
%\begin{displaymath}
%	\pm[n] \defs \{{-}n,-{n}{+}1,\ldots,{-}2,{-}1,1,2,\ldots,n{-}1,n\}.
%\end{displaymath}
A permutation $\pi$ of $\pm[n]$ is \defn{sign-symmetric} if $\pi(-i)=-\pi(i)$ for all $i\in[n]$. The group of all sign-symmetric permutations of $\pm[n]$ is the \defn{hyperoctahedral group} of degree $n$, denoted by $\Hyper_{n}$.   It is easily checked that $\bigl\lvert\Hyper_{n}\bigr\rvert=2^{n}n!$.  We usually represent sign-symmetric permutations via their \defn{long one-line notation}, \ie, for $\pi\in\Hyper_{n}$ we write down
\begin{displaymath}
	\pi({-}n),\pi({-}n{+}1),\ldots,\pi({-}1),\pi(1),\ldots,\pi(n{-}1),\pi(n).
\end{displaymath}
By definition, the left half of this word is determined by the right half.  We still want to keep it because it will be easier to spot the relevant type-$B$ $231$-patterns that we introduce in Section~\ref{sec:aligned_elements}.
For stylistic reasons, we represent negative values by an overbar instead of a minus sign, and we add a vertical bar between $\pi(-1)$ and $\pi(1)$ to emphasize the symmetry.  In case of potential ambiguities, we use commas to separate the entries.

Let $\sleft i\sright$ denote the sign-symmetric permutation that exchanges $i$ and $-i$, and let $\lleft i\;j\rright$ denote the one that exchanges $i$ and $j$ (and simultaneously $-i$ and $-j$).  We take
\begin{displaymath}
	T = \Bigl\{\sleft i\sright\mid 1\leq i\leq n\Bigr\} \cup \Bigl\{\lleft i\;j\rright\mid 1\leq i<j\leq n\Bigr\} \cup \Bigl\{\lleft {-}j\;i\rright\mid 1\leq i<j\leq n\Bigr\}.
\end{displaymath}
It is well known that $\Hyper_{n}$ admits the following presentation, where $S=\{s_{0},s_{1},\ldots,s_{n-1}\}$ is the set of generators, and $\id$ is the identity:
\begin{displaymath}
	\Hyper_{n} \defs \bigl\langle S\mid s_{i}^{2}=\id, (s_{0}s_{1})^{4}=\id,(s_{i},s_{i+1})^{3}=\id\;\text{for}\;i>0,(s_{i}s_{j})^{2}=\id\;\text{for}\;j>i+1\bigr\rangle.
\end{displaymath}
This can be realized by taking $s_{0}\mapsto\sleft 1\sright$ and $s_{i}\mapsto\lleft i\;i{+}1\rright$ for $i\in[n-1]$.  Thus, $\Hyper_{n}$ has the structure of a Coxeter group.  Moreover, every $\pi\in\Hyper_{n}$ can be written as a product of elements of $S$. An \defn{$S$-reduced word} for $\pi$ is such a product with minimum length, and we denote its length by $\ell_{S}(\pi)$.

A \defn{(right) inversion} of $\pi$ is some $t\in T$ such that $\ell_{S}(\pi)>\ell_{S}(\pi t)$.  A \defn{cover inversion} of $\pi$ is an inversion $t$ of $\pi$ for which there exists $s\in S$ such that $s \pi=\pi t$.  We write $\invset(\pi)$ (resp. $\covset(\pi)$) for the sets of inversions (resp. cover inversions) of $\pi$.  The \defn{(left) weak order} on $\Hyper_{n}$, denoted by $\weakorder$, is the containment order of inversion sets.

\subsection{Aligned elements}
	\label{sec:aligned_elements}
We fix the long cycle $\linc \defs s_{0}s_{1}\cdots s_{n-1}$, whose long one-line notation is 
\begin{displaymath}
	1,\overline{n},\ldots,\overline{3},\overline{2}\mid 2,3,\ldots,n,\overline{1}.
\end{displaymath}
In \cite{reading07clusters}, N.~Reading defined a family of sign-symmetric permutations associated with~$\linc$, called the \defn{$\linc$-aligned elements}, characterized in two equivalent ways.  Algebraically, a sign-symmetric permutation $\pi\in\Hyper_{n}$ is $\linc$-aligned if, for every $1\leq i<k\leq n$, we have:
\begin{itemize}
	\item if $\sleft i\sright\in\covset(\pi)$, then for all $1\leq j<i$, we have $\sleft j\sright\in\invset(\pi)$ ;
	\item if $\lleft i\;k\rright\in\covset(\pi)$, then for all $i<j<k$, we have $\lleft i\;j\rright\in\invset(\pi)$;
	\item if $\lleft {-}k\;i\rright\in\covset(\pi)$, then 
	\begin{itemize}
		\item $\sleft i\sright\in\invset(\pi)$,
		\item $\lleft {-}j\;i\rright\in\invset(\pi)$ for all $1\leq j<k$, $j\neq i$,
		\item $\lleft {-}k\;j\rright\in\invset(\pi)$ for all $1\leq j<i$.
	\end{itemize}
\end{itemize}
We refer to these implications among inversions and cover inversions as \defn{forcing relations}.  

Combinatorially, $\pi\in\Hyper_{n}$ has a \defn{type-$B$ $231$-pattern} if there exist indices $-n\leq i<j<k\leq n$ such that $j,k>0$, $\pi(j)>\pi(i)$ and either $\pi(i)=\pi(k)+1$ or $\pi(i)=1=-\pi(k)$.
%\henri{$i$ doesn't have to be negative.}  
The following result is the starting point of our work.

\begin{proposition}[{\cite[Lemma~4.9]{reading07clusters}}]
	A sign-symmetric permutation $\pi\in\Hyper_{n}$ is $\linc$-aligned if and only if it avoids type-$B$ $231$-patterns.
\end{proposition}

The main purpose of this extended abstract is to generalize this result to parabolic quotients of $\Hyper_{n}$ together with the appropriate definitions.

\subsection{Parabolic Quotients}
	\label{sec:parabolic_quotients}
% Any subset $J\subseteq S$ generates a subgroup of $\Hyper_{n}$.  
We are interested in minimal-length representatives of the left cosets of $\Hyper_{n}$ by the subgroup generated by some $J \subseteq S$.  This \defn{parabolic quotient} is equivalently defined by
\begin{displaymath}
	\Hyper_{n}^{J} \defs \bigl\{\pi\in\Hyper_{n}\mid \ell_{S}(\pi s)>\ell_{S}(\pi)\;\text{for all}\;s\in J\bigr\}.
\end{displaymath}

Combinatorially, we can describe the parabolic quotient neatly as follows. Let $n>0$. Consider a composition $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{r})$ of $n$, \ie, $\alpha_{1}+\alpha_{2}+\cdots+\alpha_{r}=n$.  We define $p_{0}\defs 0$ and $p_{i}\defs\alpha_{1}+\alpha_{2}+\cdots+\alpha_{i}$.  There are $2^{n-1}$ compositions of $n$, as we may map $\alpha$ to $\{p_{1},p_{2},\ldots,p_{r-1}\} \subseteq [n-1]$.  Conversely, the set $\{k_{1},k_{2},\ldots,k_{r-1}\}\subseteq [n-1]$ is associated with the composition $(k_{1},k_{2}-k_{1},k_{3}-k_{2},\ldots,k_{r-1}-k_{r-2},n-k_{r-1})$.

A \defn{type-$B$ composition} of $n>0$ is a composition of $n$ with a possible \defn{zero-component} $\alpha_{0}\defs 0$ at the beginning. A type-$B$ composition is \defn{split} if it has a zero-component, and \defn{join} otherwise. There are $2^n$ type-$B$ compositions of $n$, because each type-$B$ composition $\alpha$ is associated with a unique subset $J_{\alpha}\subseteq S$, with $S$ the generating set of $\Hyper_{n}$. The non-zero components of $\alpha$ determine a subset of $\{s_{1},s_{2},\ldots,s_{n-1}\}$ as before, and we add $s_{0}$ to this set if and only if $\alpha$ is split. Through this bijection, we associate with each type-$B$ composition $\alpha$ the parabolic quotient $\Hyper_{n}^{S\setminus J_{\alpha}}$ of $\Hyper_{n}$ determined by the \textit{complement} $S\setminus J_{\alpha}$, also denoted by $\Hyper_{\alpha}$ hereinafter for simplicity.  Clearly, if $\alpha=(0,1,1,\ldots,1)$, then $\Hyper_{\alpha}=\Hyper_{n}$.

%Therefore, type-$B$ compositions are in natural bijection with the subsets of the generating set $S$ of $\Hyper_{n}$, so that if $\alpha$ is a type-$B$ composition, then there exists a unique parabolic quotient of $\Hyper_{n}$ determined by the \textit{complement} $S\setminus J_{\alpha}$. 

Let $\alpha$ be a type-$B$ composition with $r$ non-zero components $\alpha_{1},\alpha_{2},\ldots,\alpha_{r}$ in that order. As before, we define $p_{i}=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{i}$, and we set $\overline{p}_{i}\defs -p_{i}$. For $i<j$, we use the notation $[i,j]\defs\{i,i{+}1,\ldots,j\}$. We define $\Part(\alpha)$ as the following partition of $\pm[n]$:
\begin{align*}
	%\begin{cases}
		&\bigl\{[\overline{n},\overline{p}_{r-1}{-}1],\ldots,[\overline{p}_{2},\overline{p}_{1}{-}1],
	       [\overline{p}_{1},\overline{1}],[1,p_{1}],[p_{1}{+}1,p_{2}],\ldots,
    	   [p_{r-1}{+}1,n]\bigr\}
	    & (\alpha\;\text{split}),\\
		&\bigl\{[\overline{n},\overline{p}_{r-1}{-}1],\ldots,[\overline{p}_{2},\overline{p}_{1}{-}1],
	       [\overline{p}_{1},p_{1}],[p_{1}{+}1,p_{2}],\ldots,[p_{r-1}{+}1,n]\bigr\}
	    & (\alpha\;\text{join}).
	%\end{cases}
\end{align*}
This means, if $\alpha$ is split, that we partition $\pm[n]$ into $2r$ symmetrically placed blocks, and if $\alpha$ is join, we partition it into $2r-1$ blocks, where the middle block contains positive and negative integers. This partition also explains the terminology ``join'' and ``split'' for the type-$B$ compositions. The partition given by a join composition has a block joining positive and negative integers, while the one given by a split composition does not.

For $a\in[n]$, we say that $a$ is \defn{in the $i\th$ $\alpha$-region} if $p_{i-1}\leq a<p_{i}$, where $p_{0}=0$. We sometimes write $\indicator_{\alpha}(a)=i$ in this situation.

\begin{lemma}\label{lem:increasing_blocks}
	For $\alpha$ a type-$B$ composition, a sign-symmetric permutation $\pi\in\Hyper_{n}$ is in $\Hyper_{\alpha}$ if and only if its long one-line notation, with positions partitioned by $\Part(\alpha)$, is increasing in each part.
\end{lemma}

We use the compositions $\ajoin=(1,2)$ (join) and $\asplit=(0,1,2)$ (split) as our running examples hereinafter.  The partition $\Part(\ajoin)$ has three parts $\{-3,-2\},\{-1,1\},\{2,3\}$, while $\Part(\asplit)$ has four parts $\{-3,-2\},\{-1\},\{1\},\{2,3\}$.  We highlight the parts in the long one-line notation, for instance $\moveme{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{1},3,2,\overline{2},\overline{3},1}{.5}{.25}{white}}$ is a member of $\Hyper_{\asplit}$, while $\moveme{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{1},3,2,\overline{2},\overline{3},1}{.5}{.25}{white}}$ is not in $\Hyper_{\ajoin}$, because the elements in the central block are decreasing.  

The \defn{weak order} on $\Hyper_{\alpha}$ is simply the restriction of the weak order on $\Hyper_{n}$ restricted to the subset $\Hyper_{\alpha}$.  Figure~\ref{fig:weak_join} shows the weak order on $\Hyper_{\ajoin}$ and Figure~\ref{fig:weak_split} shows the weak order on $\Hyper_{\asplit}$.

\begin{figure}[!tbh]
	\centering
	\begin{subfigure}[t]{.3\textwidth}
		\centering
		\begin{tikzpicture}[scale=.8, every node/.style={scale=0.8}]
			\def\x{1.25};
			\def\y{1};
			%%
			\coordinate(n1) at (2*\x,1*\y);
			\coordinate(n2) at (2*\x,2*\y);
			\coordinate(n3) at (3*\x,3*\y);
			\coordinate(n4) at (1*\x,3*\y);
			\coordinate(n5) at (2*\x,4*\y);
			\coordinate(n6) at (0*\x,4*\y);
			\coordinate(n7) at (2*\x,5*\y);
			\coordinate(n8) at (0*\x,5*\y);
			\coordinate(n9) at (3*\x,6*\y);
			\coordinate(n10) at (1*\x,6*\y);
			\coordinate(n11) at (2*\x,7*\y);
			\coordinate(n12) at (2*\x,8*\y);
			%%
			\draw[thick](n1) -- (n2);
			\draw[thick](n2) -- (n3);
			\draw[thick](n2) -- (n4);
			\draw[thick](n3) -- (n5);
			\draw[thick](n4) -- (n5);
			\draw[thick](n4) -- (n6);
			\draw[thick](n5) -- (n7);
			\draw[thick](n6) -- (n8);
			\draw[thick](n7) -- (n9);
			\draw[thick](n7) -- (n10);
			\draw[thick](n8) -- (n10);
			\draw[thick](n9) -- (n11);
			\draw[thick](n10) -- (n11);
			\draw[thick](n11) -- (n12);
			%%
			\draw(n1) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{3},\overline{2},\overline{1},1,2,3}{.5}{.25}{white!50!gray}};
			\draw(n2) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{3},\overline{1},\overline{2},2,1,3}{.5}{.25}{white!50!gray}};
			\draw(n3) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{2},\overline{1},\overline{3},3,1,2}{.5}{.25}{white!50!gray}};
			\draw(n4) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{3},1,\overline{2},2,\overline{1},3}{.5}{.25}{white!50!gray}};
			\draw(n5) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{2},1,\overline{3},3,\overline{1},2}{.5}{.25}{white!50!gray}};
			\draw(n6) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{3},2,\overline{1},1,\overline{2},3}{.5}{.25}{white!50!gray}};
			\draw(n7) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{1},2,\overline{3},3,\overline{2},1}{.5}{.25}{white!50!gray}};
			\draw(n8) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{2},3,\overline{1},1,\overline{3},2}{.5}{.25}{white!50!gray}};
			\draw(n9) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{1,2,\overline{3},3,\overline{2},\overline{1}}{.5}{.25}{white!50!gray}};
			\draw(n10) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{1,3,\overline{2},2,\overline{3},\overline{1}}{.5}{.25}{white!50!gray}};
			\draw(n11) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{1,3,\overline{2},2,\overline{3},\overline{1}}{.5}{.25}{white!50!gray}};
			\draw(n12) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{2,3,\overline{1},1,\overline{3},\overline{2}}{.5}{.25}{white!50!gray}};
			%%
			\begin{pgfonlayer}{background}
				\fill[white!50!gray,rounded corners](-.88*\x,3.65*\y) -- (-.88*\x,5.35*\y) -- (.95*\x,5.35*\y) -- (.95*\x,3.65*\y) -- cycle;
			\end{pgfonlayer}
		\end{tikzpicture}
		\caption{The weak order on $\Hyper_{(1,2)}$.}
		\label{fig:weak_join}
	\end{subfigure}
	\hspace*{.25cm}
	\begin{subfigure}[t]{.55\textwidth}
		\centering
		\begin{tikzpicture}[scale=.8, every node/.style={scale=0.8}]
			\def\x{1.75};
			\def\y{1};
			%%
			\coordinate (n1) at (3*\x,1*\y);
			\coordinate (n2) at (2*\x,2*\y);
			\coordinate (n3) at (4*\x,2*\y);
			\coordinate (n4) at (2*\x,3*\y);
			\coordinate (n5) at (4*\x,3*\y);
			\coordinate (n6) at (5*\x,3*\y);
			\coordinate (n7) at (1*\x,4*\y);
			\coordinate (n8) at (2*\x,4*\y);
			\coordinate (n9) at (4*\x,4*\y);
			\coordinate (n10) at (5*\x,4*\y);
			\coordinate (n11) at (2*\x,5*\y);
			\coordinate (n12) at (3*\x,5*\y);
			\coordinate (n13) at (4*\x,5*\y);
			\coordinate (n14) at (5*\x,5*\y);
			\coordinate (n15) at (2*\x,6*\y);
			\coordinate (n16) at (3*\x,6*\y);
			\coordinate (n17) at (5*\x,6*\y);
			\coordinate (n18) at (6*\x,6*\y);
			\coordinate (n19) at (2*\x,7*\y);
			\coordinate (n20) at (3*\x,7*\y);
			\coordinate (n21) at (5*\x,7*\y);
			\coordinate (n22) at (3*\x,8*\y);
			\coordinate (n23) at (5*\x,8*\y);
			\coordinate (n24) at (4*\x,9*\y);
			%%
			\draw[thick](n1) -- (n2);
			\draw[thick](n1) -- (n3);
			\draw[thick](n2) -- (n4);
			\draw[thick](n3) -- (n5);
			\draw[thick](n3) -- (n6);
			\draw[thick](n4) -- (n7);
			\draw[thick](n4) -- (n8);
			\draw[thick](n5) -- (n9);
			\draw[thick](n5) -- (n10);
			\draw[thick](n6) -- (n10);
			\draw[thick](n7) -- (n11);
			\draw[thick](n8) -- (n11);
			\draw[thick](n8) -- (n12);
			\draw[thick](n9) -- (n12);
			\draw[thick](n9) -- (n13);
			\draw[thick](n10) -- (n14);
			\draw[thick](n11) -- (n15);
			\draw[thick](n12) -- (n16);
			\draw[thick](n13) -- (n16);
			\draw[thick](n13) -- (n17);
			\draw[thick](n14) -- (n17);
			\draw[thick](n14) -- (n18);
			\draw[thick](n15) -- (n19);
			\draw[thick](n15) -- (n20);
			\draw[thick](n16) -- (n20);
			\draw[thick](n17) -- (n21);
			\draw[thick](n18) -- (n21);
			\draw[thick](n19) -- (n22);
			\draw[thick](n20) -- (n22);
			\draw[thick](n21) -- (n23);
			\draw[thick](n22) -- (n24);
			\draw[thick](n23) -- (n24);
			%%
			\draw(n1) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},\overline{2},\overline{1},1,2,3}{.5}{.25}{white!50!gray}};
			\draw(n2) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},\overline{2},1,\overline{1},2,3}{.5}{.25}{white!50!gray}};
			\draw(n3) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},\overline{1},\overline{2},2,1,3}{.5}{.25}{white!50!gray}};
			\draw(n4) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},\overline{1},2,\overline{2},1,3}{.5}{.25}{white!50!gray}};
			\draw(n5)++(-.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},1,\overline{2},2,\overline{1},3}{.5}{.25}{white!50!gray}};
			\draw(n6)++(.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{2},\overline{1},\overline{3},3,1,2}{.5}{.25}{white!50!gray}};
			\draw(n7)++(.05*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{2},\overline{1},3,\overline{3},1,2}{.5}{.25}{white!50!gray}};
			\draw(n8)++(.3*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},1,2,\overline{2},\overline{1},3}{.5}{.25}{white!50!gray}};
			\draw(n9)++(-.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},2,\overline{1},1,\overline{2},3}{.5}{.25}{white!50!gray}};
			\draw(n10)++(.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{2},1,\overline{3},3,\overline{1},2}{.5}{.25}{white!50!gray}};
			\draw(n11)++(-.4*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{2},1,3,\overline{3},\overline{1},2}{.5}{.25}{white!50!gray}};
			\draw(n12)++(-.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},2,1,\overline{1},\overline{2},3}{.5}{.25}{white!50!gray}};
			\draw(n13)++(.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{2},3,\overline{1},1,\overline{3},2}{.5}{.25}{white!50!gray}};
			\draw(n14)++(.4*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{1},2,\overline{3},3,\overline{2},1}{.5}{.25}{white!50!gray}};
			\draw(n15)++(-.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{1},2,3,\overline{3},\overline{2},1}{.5}{.25}{white!50!gray}};
			\draw(n16)++(.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{2},3,1,\overline{1},\overline{3},2}{.5}{.25}{white!50!gray}};
			\draw(n17)++(-.3*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{1},3,\overline{2},2,\overline{3},1}{.5}{.25}{white!50!gray}};
			\draw(n18)++(-.05*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{1,2,\overline{3},3,\overline{2},\overline{1}}{.5}{.25}{white!50!gray}};
			\draw(n19)++(-.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{1,2,3,\overline{3},\overline{2},\overline{1}}{.5}{.25}{white!50!gray}};
			\draw(n20)++(.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{1,3,2,\overline{2},\overline{3},\overline{1}}{.5}{.25}{white!50!gray}};
			\draw(n21) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{1,3,\overline{2},2,\overline{3},\overline{1}}{.5}{.25}{white!50!gray}};
			\draw(n22) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{1,3,2,\overline{2},\overline{3},\overline{1}}{.5}{.25}{white!50!gray}};
			\draw(n23) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{2,3,\overline{1},1,\overline{3},\overline{2}}{.5}{.25}{white!50!gray}};
			\draw(n24) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{2,3,1,\overline{1},\overline{3},\overline{2}}{.5}{.25}{white!50!gray}};
			%%
			\begin{pgfonlayer}{background}
				\fill[white!50!gray,rounded corners](3.4*\x,1.65*\y) -- (3.4*\x,2.35*\y) -- (3.25*\x,2.65*\y) -- (3.25*\x,4.35*\y) -- (3.55*\x,4.65*\y) -- (3.55*\x,5.35*\y) -- (4.77*\x,5.35*\y) -- (4.77*\x,4.65*\y) -- (4.47*\x,4.35*\y) -- (4.47*\x,2.65*\y) -- (4.62*\x,2.35*\y) -- (4.62*\x,1.65*\y) -- cycle;
				\fill[white!50!gray,rounded corners](2.27*\x,4.65*\y) -- (2.27*\x,5.35*\y) -- (2.56*\x,5.65*\y) -- (2.56*\x,6.35*\y) -- (3.79*\x,6.35*\y) -- (3.79*\x,5.65*\y) -- (3.5*\x,5.35*\y) -- (3.5*\x,4.65*\y) -- cycle;
				\fill[white!50!gray,rounded corners](4.52*\x,2.65*\y) -- (4.52*\x,4.35*\y) -- (4.82*\x,4.65*\y) -- (4.82*\x,5.35*\y) -- (4.08*\x,5.65*\y) -- (4.08*\x,6.35*\y) -- (4.4*\x,6.65*\y) -- (4.4*\x,8.35*\y) -- (5.62*\x,8.35*\y) -- (5.62*\x,6.65*\y) -- (6.66*\x,6.35*\y) -- (6.66*\x,5.65*\y) -- (6.04*\x,5.35*\y) -- (6.04*\x,4.65*\y) -- (5.84*\x,4.35*\y) -- (5.84*\x,2.65*\y) -- cycle;
			\end{pgfonlayer}
		\end{tikzpicture}
		\caption{The weak order on $\Hyper_{(0,1,2)}$.}
		\label{fig:weak_split}
	\end{subfigure}
	\caption{The parabolic quotients with respect to certain type-$B$ compositions of $3$ in the weak order.  The grey regions indicate congruence classes with respect to the congruence relation $\Theta_{\alpha}$ defined in Section~\ref{sec:parabolic_tamari_quotient}.}
	\label{fig:weak_order}
\end{figure}

\section{Aligned Elements in $\Hyper_{\alpha}$}
	\label{sec:parabolic_aligned_elements}
The forcing relations from Section~\ref{sec:aligned_elements} that determine the $\linc$-aligned elements of $\Hyper_{n}$ come from a certain total order on $T$ together with a  decomposition of the \defn{roots} of $\Hyper_{n}$, which are vectors associated with the elements of $T$ that are important in the reflection representation of the group $\Hyper_{n}$.  For lack of space, we do not go into detail here.  Instead, we present a uniform construction that works for all type-$B$ compositions of $n$.

\subsection{The $\linc$-sorting word of the longest element}
	\label{sec:parabolic_longest}
By \cite[Theorem~4.1]{bjorner88generalized}, the set $\Hyper_{\alpha}$ has a unique element of maximum length, which is denoted by $\poa$.  For our constructions, we are interested in a particular $S$-reduced word of $\poa$, namely the \defn{$\linc$-sorting word} of $\poa$.  This is the $S$-reduced word of $\poa$ that appears as far right as possible as a subword in the half-infinite word 
\begin{displaymath}
	~^{\infty}\linc \defs \ldots \mid s_{n-1}\cdots s_{1}s_{0} \mid s_{n-1}\cdots s_{1}s_{0}.
\end{displaymath}

Given integers $\lambda_{1}\geq \lambda_{2}\geq\cdots$, a \defn{Ferrers diagram} of shape $(\lambda_{1},\lambda_{2},\ldots)$ is a left-aligned arrangement of unit boxes, where the $i\th$ row consists of $i$ boxes.  For two Ferrers diagrams $\lambda$, $\mu$, the associated \defn{skew diagram} $\lambda/\mu$ consists of all boxes of $\lambda$ that are not boxes of $\mu$.  For a type-$B$ composition $\alpha$ of $n$ with non-zero components $\alpha_{1},\alpha_{2},\ldots,\alpha_{r}$, we define the \defn{$\alpha$-shape} to be the skew shape $\Skew(\alpha)\defs\lambda_{(\alpha)}/\mu_{(\alpha)}$, where the $k\th$ entry of $\mu_{(\alpha)}$ is $\mu_{k}\defs\alpha_{\indicator_{\alpha}(k)}+\alpha_{\indicator_{\alpha}(k)+1}+\cdots+\alpha_{r}$ and the $k\th$ entry of $\lambda_{(\alpha)}$ is
\begin{displaymath}
	\lambda_{k} \defs \begin{cases}2n-\alpha_{1}, & \text{if}\;\alpha\;\text{is join and}\;k\leq\alpha_{1},\\2n+1-k, & \text{otherwise}.\end{cases}
\end{displaymath}
We now fill $\Skew(\alpha)$ with elements of $S$ as follows.  Let $k\in[n]$.  If $\alpha$ is split or $k>\alpha_{1}$, then we fill the $k\th$ row of $\Skew(\alpha)$ with the elements $s_{0},s_{1},\ldots$ \emph{from right to left} until all boxes in that row are filled.  If $\alpha$ is join and $k\leq\alpha_{1}$, then we fill the $k\th$ row of $\Skew(\alpha)$ in the same way with $s_{\alpha_{1}+1-k},s_{\alpha_{1}+2-k},\ldots$ instead.  We denote by $\wb_{\alpha}$ the word obtained from reading the filling of $\Skew(\alpha)$ from bottom to top, left to right.

\begin{proposition}
	For every type-$B$ composition $\alpha$, the word $\wb_{\alpha}$ is the $\linc$-sorting word of $\poa$.
\end{proposition}

\begin{corollary}
	If $\alpha$ is a type-$B$ composition of $n$ with non-zero components $\alpha_{1},\alpha_{2},\ldots,\alpha_{r}$, then 
	\begin{displaymath}
		\ell_{S}(\poa) = n^{2}-\sum_{i=1}^{r}\binom{\alpha_{i}}{2}-\begin{cases}\binom{\alpha_{1}+1}{2}, & \text{if}\;\alpha\;\text{is join},\\0, & \text{if}\;\alpha\;\text{is split}.\end{cases}
	\end{displaymath}
\end{corollary}

\begin{figure}
	\centering
	\begin{subfigure}[t]{.45\textwidth}
		\centering
		\begin{tikzpicture}\small	
			\def\x{.75};
			%%
			\draw(0*\x,0*\x) -- (4*\x,0*\x) -- (4*\x,1*\x) -- (5*\x,1*\x) -- (5*\x,3*\x) -- (3*\x,3*\x) -- (3*\x,2*\x) -- (2*\x,2*\x) -- (2*\x,0*\x);
			\draw[white!50!gray,thin](3*\x,0*\x) -- (3*\x,2*\x) -- (5*\x,2*\x);
			\draw[white!50!gray,thin](2*\x,1*\x) -- (4*\x,1*\x) -- (4*\x,3*\x);
			\begin{pgfonlayer}{background}
				\fill[c1](2*\x,2*\x) -- (3*\x,2*\x) -- (3*\x,3*\x) -- cycle;
				\fill[c2](0*\x,0*\x) -- (2*\x,0*\x) -- (2*\x,2*\x) -- cycle;
			\end{pgfonlayer}
			%%
			\draw(2.5*\x,.5*\x) node{$s_{1}$};
			\draw(3.5*\x,.5*\x) node{$s_{0}$};
			\draw(2.5*\x,1.5*\x) node{$s_{2}$};
			\draw(3.5*\x,1.5*\x) node{$s_{1}$};
			\draw(4.5*\x,1.5*\x) node{$s_{0}$};
			\draw(3.5*\x,2.5*\x) node{$s_{2}$};
			\draw(4.5*\x,2.5*\x) node{$s_{1}$};
		\end{tikzpicture}
		\caption{The skew shape $\Skew\bigl((1,2)\bigr)$.}
		\label{fig:skew_join}
	\end{subfigure}
	\hspace*{1cm}
	\begin{subfigure}[t]{.45\textwidth}
		\centering
		\begin{tikzpicture}\small	
			\def\x{.75};
			%%
			\draw(0*\x,0*\x) -- (4*\x,0*\x) -- (4*\x,1*\x) -- (5*\x,1*\x) -- (5*\x,2*\x) -- (6*\x,2*\x) -- (6*\x,3*\x) -- (3*\x,3*\x) -- (3*\x,2*\x) -- (2*\x,2*\x) -- (2*\x,0*\x);
			\draw[white!50!gray,thin](3*\x,0*\x) -- (3*\x,2*\x) -- (5*\x,2*\x) -- (5*\x,3*\x);
			\draw[white!50!gray,thin](2*\x,1*\x) -- (4*\x,1*\x) -- (4*\x,3*\x);
			\begin{pgfonlayer}{background}
				\fill[c1](2*\x,2*\x) -- (3*\x,2*\x) -- (3*\x,3*\x) -- cycle;
				\fill[c2](0*\x,0*\x) -- (2*\x,0*\x) -- (2*\x,2*\x) -- cycle;
			\end{pgfonlayer}
			%%
			\draw(2.5*\x,.5*\x) node{$s_{1}$};
			\draw(3.5*\x,.5*\x) node{$s_{0}$};
			\draw(2.5*\x,1.5*\x) node{$s_{2}$};
			\draw(3.5*\x,1.5*\x) node{$s_{1}$};
			\draw(4.5*\x,1.5*\x) node{$s_{0}$};
			\draw(3.5*\x,2.5*\x) node{$s_{2}$};
			\draw(4.5*\x,2.5*\x) node{$s_{1}$};
			\draw(5.5*\x,2.5*\x) node{$s_{0}$};
		\end{tikzpicture}
		\caption{The skew shape $\Skew\bigl((0,1,2)\bigr)$.}
		\label{fig:skew_split}
	\end{subfigure}
	\caption{Skew shapes associated with certain type-$B$ compositions of $3$.  The colorful triangles on the left indicate the composition.}
	\label{fig:skew_shapes}
\end{figure}	
	
Figure~\ref{fig:skew_join} shows the skew shape $\Skew(\ajoin)$ and Figure~\ref{fig:skew_split} shows the skew shape $\Skew(\asplit)$, both together with their fillings.  It is easy to check that
\begin{align*}
	\pi_{\mathsf{o};\ajoin} & = \moveme{\AlphaPerm{2,2,2}{c2,c1,c2}{2,3,\overline{1},1,\overline{3},\overline{2}}{.5}{.25}{white}} = s_{1}s_{0}s_{2}s_{1}s_{0}s_{2}s_{1} = \mathbf{w}_{\ajoin},\\
	\pi_{\mathsf{o};\asplit} & = \moveme{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{2,3,1,\overline{1},\overline{3},\overline{2}}{.5}{.25}{white}} = s_{1}s_{0}s_{2}s_{1}s_{0}s_{2}s_{1}s_{0} = \mathbf{w}_{\asplit}.
\end{align*}

\subsection{The inversion order of $\poa$ with respect to $\linc$}
	\label{sec:parabolic_inversion_order}
Let $w = a_{1}a_{2}\cdots a_{k}$ be an $S$-reduced word of some sign-symmetric permutation $\pi\in\Hyper_{n}$.  For $i\in[k]$, we define $t_{i}=a_{k}a_{k-1}\cdots a_{k-i+1}\cdots a_{k-1}a_{k}$.  By \cite[Section~1.3]{bjorner05combinatorics}, the inversion set of $\pi$ is then $\invset(\pi)=\{t_{1},t_{2},\ldots,t_{k}\}$.  The total order of $\invset(\pi)$ induced by $w$ is the \defn{inversion order} of $\pi$. The inversion order associated with the $\linc$-sorting word $\woa$ of $\poa$ can be obtained from $\Skew(\alpha)$ as follows.

\begin{enumerate}[label={(\roman*)}]
\item The first $n$ columns are labeled by $\poa(n)$, $\poa(n-1)$, $\ldots$, $\poa(1)$ from left to right.  The next $n-\alpha_{1}$ columns are labeled from $n$ to $\alpha_1 + 1$.  If $\alpha$ is split, we label the remaining $\alpha_{1}$ columns by $\alpha_{1},\alpha_{1}-1,\ldots,1$.  If $\alpha$ is join, there are no columns left.
\item If $\alpha$ is split, then we label the rows by $-1$, $-2$, $\ldots$, $-n$ from top to bottom.  If $\alpha$ is join, then we label the first $\alpha_{1}$ rows by $\alpha_{1},\alpha_{1}-1,\ldots,1$, and the remaining $n-\alpha_{1}$ rows by $-\alpha_{1}-1,-\alpha_{1}-2,\ldots,-n$.
\end{enumerate}

Now we fill the cells of $\Skew(\alpha)$. Consider a cell with row label $r$ and column label $c$.

\begin{enumerate}[label={(\roman*)}]
\item If $r>0$ and $c>0$, then we fill this cell by $\lleft r\;c\rright$.  This only happens when $\alpha$ is join, in which case we necessarily have $r<c$.
\item If $r<0$ and $c=-r$, then we fill this cell by $\sleft r\sright$.
\item If $r<0$ and $c>0$ with $c > -r$, then we fill this cell by $\lleft {-}c\;{-}r\rright$.
\item If $r<0$ and $c>0$ with $c < -r$, then we fill this cell by $\lleft r\;c\rright$.
\item If $r<0$ and $c<0$, then we fill this cell by $\lleft {-}c\;{-}r\rright$.  By construction, we are in the first $n$ columns, thus $r<c$.
\end{enumerate}

\begin{proposition}\label{prop:alpha_inversion_order}
	The filling of $\Skew(\alpha)$, read from top to bottom and right to left, yields the inversion order of $\poa$ with respect to its $\linc$-sorting word.
\end{proposition}

\begin{figure}
	\centering
	\begin{subfigure}[t]{.43\textwidth}
		\centering
		\begin{tikzpicture}\small	
			\def\x{1.25};
			\def\y{.75};
			%%
			\draw(0*\x,0*\y) -- (4*\x,0*\y) -- (4*\x,1*\y) -- (5*\x,1*\y) -- (5*\x,3*\y) -- (3*\x,3*\y) -- (3*\x,2*\y) -- (2*\x,2*\y) -- (2*\x,0*\y);
			\draw[white!50!gray,thin](3*\x,0*\y) -- (3*\x,2*\y) -- (5*\x,2*\y);
			\draw[white!50!gray,thin](2*\x,1*\y) -- (4*\x,1*\y) -- (4*\x,3*\y);
			\begin{pgfonlayer}{background}
				\fill[c1](2*\x,2*\y) -- (3*\x,2*\y) -- (3*\x,3*\y) -- cycle;
				\fill[c2](0*\x,0*\y) -- (2*\x,0*\y) -- (2*\x,2*\y) -- cycle;
			\end{pgfonlayer}
			%%
			\draw(.5*\x,.15*\y) node[scale=.67,text=purple!50!gray]{$-2$};
			\draw(1.5*\x,.15*\y) node[scale=.67,text=purple!50!gray]{$-3$};
			\draw(2.5*\x,2.15*\y) node[scale=.67,text=purple!50!gray]{$1$};
			\draw(3.5*\x,3.15*\y) node[scale=.67,text=purple!50!gray]{$3$};
			\draw(4.5*\x,3.15*\y) node[scale=.67,text=purple!50!gray]{$2$};
			%%
			\draw(4*\x,.5*\y) node[anchor=west,scale=.67,text=purple!50!gray]{$-3$};
			\draw(5*\x,1.5*\y) node[anchor=west,scale=.67,text=purple!50!gray]{$-2$};
			\draw(5*\x,2.5*\y) node[anchor=west,scale=.67,text=purple!50!gray]{$1$};
			%%
			\draw(2.5*\x,.5*\y) node[scale=.85]{$\lleft {-}3\;1\rright$};
			\draw(3.5*\x,.5*\y) node[scale=.85]{$\sleft 3\sright$};
			\draw(2.5*\x,1.5*\y) node[scale=.85]{$\lleft {-}2\;1\rright$};
			\draw(3.5*\x,1.5*\y) node[scale=.85]{$\lleft {-}3\;2\rright$};
			\draw(4.5*\x,1.5*\y) node[scale=.85]{$\sleft 2\sright$};
			\draw(3.5*\x,2.5*\y) node[scale=.85]{$\lleft 1\;3\rright$};
			\draw(4.5*\x,2.5*\y) node[scale=.85]{$\lleft 1\;2\rright$};
		\end{tikzpicture}
		\caption{The inversion order with respect to the composition $(1,2)$.}
		\label{fig:invorder_join}
	\end{subfigure}
	\hspace*{0.5cm}
	\begin{subfigure}[t]{.5\textwidth}
		\centering
		\begin{tikzpicture}\small	
			\def\x{1.25};
			\def\y{.75};
			%%
			\draw(0*\x,0*\y) -- (4*\x,0*\y) -- (4*\x,1*\y) -- (5*\x,1*\y) -- (5*\x,2*\y) -- (6*\x,2*\y) -- (6*\x,3*\y) -- (3*\x,3*\y) -- (3*\x,2*\y) -- (2*\x,2*\y) -- (2*\x,0*\y);
			\draw[white!50!gray,thin](3*\x,0*\y) -- (3*\x,2*\y) -- (5*\x,2*\y) -- (5*\x,3*\y);
			\draw[white!50!gray,thin](2*\x,1*\y) -- (4*\x,1*\y) -- (4*\x,3*\y);
			\begin{pgfonlayer}{background}
				\fill[c1](2*\x,2*\y) -- (3*\x,2*\y) -- (3*\x,3*\y) -- cycle;
				\fill[c2](0*\x,0*\y) -- (2*\x,0*\y) -- (2*\x,2*\y) -- cycle;
			\end{pgfonlayer}
			%%
			\draw(.5*\x,.15*\y) node[scale=.67,text=purple!50!gray]{$-2$};
			\draw(1.5*\x,.15*\y) node[scale=.67,text=purple!50!gray]{$-3$};
			\draw(2.5*\x,2.15*\y) node[scale=.67,text=purple!50!gray]{$-1$};
			\draw(3.5*\x,3.15*\y) node[scale=.67,text=purple!50!gray]{$3$};
			\draw(4.5*\x,3.15*\y) node[scale=.67,text=purple!50!gray]{$2$};
			\draw(5.5*\x,3.15*\y) node[scale=.67,text=purple!50!gray]{$1$};
			%%
			\draw(4*\x,.5*\y) node[anchor=west,scale=.67,text=purple!50!gray]{$-3$};
			\draw(5*\x,1.5*\y) node[anchor=west,scale=.67,text=purple!50!gray]{$-2$};
			\draw(6*\x,2.5*\y) node[anchor=west,scale=.67,text=purple!50!gray]{$-1$};
			%%
			\draw(2.5*\x,.5*\y) node[scale=.85]{$\lleft 1\;3\rright$};
			\draw(3.5*\x,.5*\y) node[scale=.85]{$\sleft 3\sright$};
			\draw(2.5*\x,1.5*\y) node[scale=.85]{$\lleft 1\;2\rright$};
			\draw(3.5*\x,1.5*\y) node[scale=.85]{$\lleft {-}3\;2\rright$};
			\draw(4.5*\x,1.5*\y) node[scale=.85]{$\sleft 2\sright$};
			\draw(3.5*\x,2.5*\y) node[scale=.85]{$\lleft {-}3\;1\rright$};
			\draw(4.5*\x,2.5*\y) node[scale=.85]{$\lleft {-}2\;1\rright$};
			\draw(5.5*\x,2.5*\y) node[scale=.85]{$\sleft 1\sright$};
		\end{tikzpicture}
		\caption{The inversion order with respect to the composition $(0,1,2)$.}
		\label{fig:invorder_split}
	\end{subfigure}
	\caption{The inversion orders associated with certain type-$B$ compositions of $3$.}
	\label{fig:skew_shapes_inversions}
\end{figure}	

Figure~\ref{fig:invorder_join} shows the filling of the skew shape $\Skew(\ajoin)$ with inversions, and Figure~\ref{fig:invorder_split} shows that of $\Skew(\asplit)$.  The corresponding inversion order of $\ajoin$ is
\begin{displaymath}\begin{aligned}
	& \lleft 1\;2\rright & \prec && \lleft 1\;3\rright && \prec && \sleft 2\sright && \prec && \lleft {-}3\;2\rright && \prec && \lleft {-}2\;1\rright && \prec && \sleft 3\sright && \prec && \lleft{-}3\;1\rright,
\end{aligned}\end{displaymath}
while that of $\asplit$ is
\begin{displaymath}\begin{aligned}
	& \sleft 1\sright & \prec && \lleft {-}2\;1\rright && \prec && \lleft {-}3\;1\rright && \prec && \sleft 2\sright && \prec && \lleft {-}3\;2\rright && \prec && \lleft 1\;2\rright && \prec && \sleft 3\sright && \prec && \lleft 1\;3\rright.
\end{aligned}\end{displaymath}
This can also be checked directly using the $\linc$-sorting words $\mathbf{w}_{\ajoin}$ and $\mathbf{w}_{\asplit}$.

\subsection{Aligned elements of $\Hyper_{\alpha}$}
	\label{sec:parabolic_231}
We now put everything together and define the $\linc$-aligned elements of $\Hyper_{\alpha}$.  A sign-symmetric permutation $\pi\in\Hyper_{\alpha}$ is \defn{$\linc$-aligned} if, for all $1\leq i<k\leq n$, we have%\henri{Merged the alignment-conditions for join and split compositions in one. Can you check?}\wfang{Checked}
\begin{enumerate}[label={(\arabic*)}]
	\item if $\sleft i\sright\in\covset(\pi)$, then $\sleft j\sright\in\invset(\pi)$ for all $1\leq j<i$ with $\indicator_{\alpha}(j)<\indicator_{\alpha}(i)$;
	\item if $\lleft i\;k\rright\in\covset(\pi)$, then $\lleft i\;j\rright\in\invset(\pi)$ for all $i<j<k$ with $\indicator_{\alpha}(i)<\indicator_{\alpha}(j)<\indicator_{\alpha}(k)$;
	\item if $\lleft {-}k\;i\rright\in\covset(\pi)$, then
	\begin{enumerate}[label={(3\alph*)}]
		\item $\sleft i\sright\in\invset(\pi)$ when $i>\alpha_{1}$ or $\alpha$ is split,
		\item $\lleft {-}j\;i\rright\in\invset(\pi)$ for $1\leq j<k$ with $\indicator_{\alpha}(j)<\indicator_{\alpha}(k)$ when $\alpha$ is split or $j>\alpha_{1}$,
		\item $\lleft j\;k\rright\in\invset(\pi)$ when $j\leq \alpha_{1}$, $j\neq i$ and $\alpha$ is join,
		\item $\lleft {-}k\;j\rright\in\invset(\pi)$ for $1\leq j<i$ with $\indicator_{\alpha}(j)<\indicator_{\alpha}(i)$ when $\alpha$ is split or $j>\alpha_{1}$,
		\item $\lleft j\;i\rright\in\invset(\pi)$ when $i>j>\alpha_{1}$ and $\alpha$ is join.
	\end{enumerate}
\end{enumerate}

% \begin{remark}
% 	The forcing conditions above are not really the definition of $\linc$-alignment, but rather a characterization of this property.  The underlying definition is \cite[Definition~33]{muehle19tamari}, which demands that the presence of certain cover inversions entails that of some other inversions.  This \emph{forcing} comes from a decomposition of the normal vectors of the reflection hyperplanes associated with transpositions in the reflection representation of $\Hyper_{n}$.  Due to the space limitations, we cannot go into more detail here.
% \end{remark}

The alignment condition can be expressed combinatorially as pattern avoidance.  For $k\in\pm[n]$, we define
\begin{displaymath}
	k^{+} \defs \begin{cases} k+1, & \text{if}\;k\neq -1,\\1, & \text{if}\;k=-1.\end{cases}
\end{displaymath}
Let $\pi\in\Hyper_{\alpha}$.  A \defn{type-$B$ $(\alpha,231)$-pattern} is a triple $(i,j,k)$ with $-n\leq i<j<k\leq n$ such that $i,j,k$ are in different $\alpha$-regions, $j>0$, $\pi(i)=\pi(k)^{+}$ and
\begin{displaymath}
	\begin{cases}
		\pi(i)<\pi(j), & \text{when}\;\alpha\;\text{is split}\;\text{or}\;j>\alpha_{1},\\
		\pi(j)<\pi(k), & \text{when}\;\alpha\;\text{is join}\;\text{and}\;j\leq\alpha_{1}.
	\end{cases}
\end{displaymath}

\begin{proposition}\label{prop:parabolic_alignment_patterns}
	For any type-$B$ composition $\alpha$ of $n$, it holds that $\pi\in\Hyper_{\alpha}$ is $\linc$-aligned if and only if it does not have a type-$B$ $(\alpha,231)$-pattern.
\end{proposition}

Consider $\pi=\moveme{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{2},3,\overline{1},1,\overline{3},2}{.5}{.25}{white}}\in\Hyper_{(1,2)}$.  We can check easily that $\lleft{-}3\;2\rright\in\covset(\pi)$, but $\lleft 1\;3\rright\notin\invset(\pi)$.  This violates Condition (3c) above, and $\pi$ is therefore not $\linc$-aligned.  This is, equivalently, witnessed by the $(\ajoin,231)$-pattern in positions $(-2,1,3)$.

Now take $\pi'=\moveme{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{1,2,\overline{3},3,\overline{2},\overline{1}}{.5}{.25}{white}}\in\Hyper_{(0,1,2)}$.  There, we see that $\sleft 3\sright\in\covset(\pi')$, but $\sleft 1\sright\notin\invset(\pi')$.  This violates Condition (1) above, and $\pi'$ is thus not $\linc$-aligned, which is again manifested in the $(\asplit,231)$-pattern in positions $(-3,1,3)$.

\medskip

We denote by $\Hyper_{\alpha}(231)$ the set of elements of $\Hyper_{\alpha}$ avoiding type-$B$ $(\alpha,231)$-patterns.  By Proposition~\ref{prop:parabolic_alignment_patterns}, this is also the set of $\linc$-aligned elements of $\Hyper_{\alpha}$.  The \defn{type-$B$ parabolic Tamari lattice} is the partially ordered set $\Tamari_{B}(\alpha)\defs\bigl(\Hyper_{\alpha}(231),\weakorder\bigr)$.  This name is justified in the next section.  Figure~\ref{fig:tamari_join} shows $\Tamari_{B}\bigl((1,2)\bigr)$ and Figure~\ref{fig:tamari_split} shows $\Tamari_{B}\bigl((0,1,2)\bigr)$.

\begin{figure}
	\centering
	\begin{subfigure}[t]{.3\textwidth}
		\centering
		\begin{tikzpicture}
			\def\x{1.25};
			\def\y{1};
			%%
			\coordinate(n1) at (2*\x,1*\y);
			\coordinate(n2) at (2*\x,2*\y);
			\coordinate(n3) at (3*\x,3*\y);
			\coordinate(n4) at (1*\x,3*\y);
			\coordinate(n5) at (2*\x,4*\y);
			\coordinate(n6) at (0*\x,4*\y);
			\coordinate(n7) at (2*\x,5*\y);
			\coordinate(n9) at (3*\x,6*\y);
			\coordinate(n10) at (1*\x,6*\y);
			\coordinate(n11) at (2*\x,7*\y);
			\coordinate(n12) at (2*\x,8*\y);
			%%
			\draw[thick](n1) -- (n2);
			\draw[thick](n2) -- (n3);
			\draw[thick](n2) -- (n4);
			\draw[thick](n3) -- (n5);
			\draw[thick](n4) -- (n5);
			\draw[thick](n4) -- (n6);
			\draw[thick](n5) -- (n7);
			\draw[thick](n6) -- (n10);
			\draw[thick](n7) -- (n9);
			\draw[thick](n7) -- (n10);
			\draw[thick](n9) -- (n11);
			\draw[thick](n10) -- (n11);
			\draw[thick](n11) -- (n12);
			%%
			\draw(n1) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{3},\overline{2},\overline{1},1,2,3}{.5}{.25}{white}};
			\draw(n2) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{3},\overline{1},\overline{2},2,1,3}{.5}{.25}{white}};
			\draw(n3) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{2},\overline{1},\overline{3},3,1,2}{.5}{.25}{white}};
			\draw(n4) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{3},1,\overline{2},2,\overline{1},3}{.5}{.25}{white}};
			\draw(n5) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{2},1,\overline{3},3,\overline{1},2}{.5}{.25}{white}};
			\draw(n6) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{3},2,\overline{1},1,\overline{2},3}{.5}{.25}{white}};
			\draw(n7) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{\overline{1},2,\overline{3},3,\overline{2},1}{.5}{.25}{white}};
			\draw(n9) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{1,2,\overline{3},3,\overline{2},\overline{1}}{.5}{.25}{white}};
			\draw(n10) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{1,3,\overline{2},2,\overline{3},\overline{1}}{.5}{.25}{white}};
			\draw(n11) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{1,3,\overline{2},2,\overline{3},\overline{1}}{.5}{.25}{white}};
			\draw(n12) node[scale=.67]{\AlphaPerm{2,2,2}{c2,c1,c2}{2,3,\overline{1},1,\overline{3},\overline{2}}{.5}{.25}{white}};
		\end{tikzpicture}
		\caption{The lattice $\Tamari_{B}\bigl((1,2)\bigr)$.}
		\label{fig:tamari_join}
	\end{subfigure}
	\hspace*{.25cm}
	\begin{subfigure}[t]{.55\textwidth}
		\centering
		\begin{tikzpicture}
			\def\x{1.75};
			\def\y{1};
			%%
			\coordinate (n1) at (3*\x,1*\y);
			\coordinate (n2) at (2*\x,2*\y);
			\coordinate (n3) at (4*\x,2*\y);
			\coordinate (n4) at (2*\x,3*\y);
			\coordinate (n5) at (4*\x,3*\y);
			\coordinate (n6) at (4.8*\x,3*\y);
			\coordinate (n7) at (1*\x,4*\y);
			\coordinate (n8) at (2*\x,4*\y);
			\coordinate (n9) at (4*\x,4*\y);
			\coordinate (n10) at (5*\x,4*\y);
			\coordinate (n11) at (2*\x,5*\y);
			\coordinate (n12) at (3*\x,5*\y);
			\coordinate (n13) at (4*\x,5*\y);
			\coordinate (n14) at (5*\x,5*\y);
			\coordinate (n15) at (2*\x,6*\y);
			\coordinate (n16) at (3*\x,6*\y);
			\coordinate (n17) at (5*\x,6*\y);
			\coordinate (n18) at (6*\x,6*\y);
			\coordinate (n19) at (2*\x,7*\y);
			\coordinate (n20) at (3*\x,7*\y);
			\coordinate (n21) at (5*\x,7*\y);
			\coordinate (n22) at (3*\x,8*\y);
			\coordinate (n23) at (5*\x,8*\y);
			\coordinate (n24) at (4*\x,9*\y);
			%%
			\draw[thick](n1) -- (n2);
			\draw[thick](n1) -- (n3);
			\draw[thick](n2) -- (n4);
			\draw[thick](n3) -- (n12);
			\draw[thick](n3) -- (n6);
			\draw[thick](n4) -- (n7);
			\draw[thick](n4) -- (n8);
			\draw[thick](n6) -- (n24);
			\draw[thick](n7) -- (n11);
			\draw[thick](n8) -- (n11);
			\draw[thick](n8) -- (n12);
			\draw[thick](n11) -- (n15);
			\draw[thick](n12) -- (n20);
			\draw[thick](n15) -- (n19);
			\draw[thick](n15) -- (n20);
			\draw[thick](n19) -- (n22);
			\draw[thick](n20) -- (n22);
			\draw[thick](n22) -- (n24);
			%%
			\draw(n1) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},\overline{2},\overline{1},1,2,3}{.5}{.25}{white}};
			\draw(n2) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},\overline{2},1,\overline{1},2,3}{.5}{.25}{white}};
			\draw(n3) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},\overline{1},\overline{2},2,1,3}{.5}{.25}{white}};
			\draw(n4) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},\overline{1},2,\overline{2},1,3}{.5}{.25}{white}};
			\draw(n6)++(.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{2},\overline{1},\overline{3},3,1,2}{.5}{.25}{white}};
			\draw(n7)++(-.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{2},\overline{1},3,\overline{3},1,2}{.5}{.25}{white}};
			\draw(n8)++(.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},1,2,\overline{2},\overline{1},3}{.5}{.25}{white}};
			\draw(n11)++(-.4*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{2},1,3,\overline{3},\overline{1},2}{.5}{.25}{white}};
			\draw(n12)++(-.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{3},2,1,\overline{1},\overline{2},3}{.5}{.25}{white}};
			\draw(n15)++(-.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{\overline{1},2,3,\overline{3},\overline{2},1}{.5}{.25}{white}};
			\draw(n19)++(-.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{1,2,3,\overline{3},\overline{2},\overline{1}}{.5}{.25}{white}};
			\draw(n20)++(.15*\x,0) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{1,3,2,\overline{2},\overline{3},\overline{1}}{.5}{.25}{white}};
			\draw(n22) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{1,3,2,\overline{2},\overline{3},\overline{1}}{.5}{.25}{white}};
			\draw(n24) node[scale=.67]{\AlphaPerm{2,1,1,2}{c2,c1,c1,c2}{2,3,1,\overline{1},\overline{3},\overline{2}}{.5}{.25}{white}};
		\end{tikzpicture}
		\caption{The lattice $\Tamari_{B}\bigl((0,1,2)\bigr)$.}
		\label{fig:tamari_split}
	\end{subfigure}
	\caption{The type-$B$ parabolic Tamari lattices associated with certain type-$B$ compositions of $3$.}
	\label{fig:tamari}
\end{figure}

\section{The parabolic Tamari lattice as a quotient lattice}
	\label{sec:parabolic_tamari_quotient}
Let $n>0$.  By \cite[Theorem~4.1]{bjorner88generalized}, the partially ordered set $\bigl(\Hyper_{\alpha},\weakorder\bigr)$ is a lattice for every type-$B$ composition $\alpha$ of $n$.  We now construct a congruence relation on $\bigl(\Hyper_{\alpha},\weakorder\bigr)$ that exhibits $\Tamari_{B}(\alpha)$ as a quotient lattice.  Key to this construction is the following lemma. %\henri{Maybe add the definitions?}\wfang{Probably not, it breaks the flow.}

\begin{lemma}\label{lem:alpha_projection}
	For every $\pi\in\Hyper_{\alpha}$ there exists a unique element $\pi_{\downarrow}\in\Hyper_{\alpha}(231)$ with $\invset(\pi_{\downarrow})\subseteq\invset(\pi)$ such that $\invset(\pi_{\downarrow})$ is maximal by inclusion for any such element.
\end{lemma}

We may therefore consider the assignment $\pi\mapsto\pi_{\downarrow}$ as a map from $\Hyper_{\alpha}$ to $\Hyper_{\alpha}(231)$.  By construction, the elements of $\Hyper_{\alpha}(231)$ are the fixed points of this map.  We also define an equivalence relation $\Theta_{\alpha}$ on $\Hyper_{\alpha}$ by taking $(\sigma,\pi)\in\Theta_{\alpha}$ if and only if $\sigma_{\downarrow}=\pi_{\downarrow}$.  

\begin{proof}[Sketch of proof of Theorem~\ref{thm:parabolic_tamari}]
	The lattice property of $\Tamari_{B}(\alpha)$ is a direct consequence of Lemma~\ref{lem:alpha_projection}.  Indeed, one consequence of Lemma~\ref{lem:alpha_projection} is that the map $\pi\mapsto\pi_{\downarrow}$ is order-preserving.  As a consequence, $\Hyper_{\alpha}(231)$ is closed under taking joins.  Since the identity belongs to $\Hyper_{\alpha}(231)$, the poset $\Tamari_{B}(\alpha)$ has a unique smallest element.  This is enough to guarantee that it is a lattice.
	
	Moreover, it follows that the equivalence classes of $\Theta_{\alpha}$ form intervals in $\bigl(\Hyper_{\alpha},\weakorder\bigr)$.
% 	Next, we prove that the assignment $\pi\mapsto\pi_{\downarrow}$ is order-preserving, and the equivalence classes of $\Theta_{\alpha}$ form intervals in $\bigl(\Hyper_{\alpha},\weakorder\bigr)$.  
% 	
	The bottom elements of these intervals are precisely the members of $\Hyper_{\alpha}(231)$.  Finally, we prove that the map that sends $\pi\in\Hyper_{\alpha}$ to the top element of the interval induced by elements equivalent to $\pi$ in $\Theta_{\alpha}$ is also order-preserving.  In view of \cite[Section~3]{reading06cambrian}, this is enough to conclude that $\Theta_{\alpha}$ is a congruence relation on $\bigl(\Hyper_{\alpha},\weakorder\bigr)$.  The corresponding quotient lattice is isomorphic to $\Tamari_{B}(\alpha)$.
	
	Thanks to the quotient property, the semidistributivity of $\Tamari_{B}(\alpha)$ is inherited from that of the weak order on $\Hyper_{\alpha}$.  For the trimness of $\Tamari_{B}(\alpha)$, given the semidistributivity, by \cite[Theorem~1.4]{thomas19rowmotion}, we only need to show that $\Tamari_{B}(\alpha)$ is extremal in the sense that the number of join-irreducible elements equals the maximum length of a maximal chain.  Extremality is established by showing that each suffix of the $\linc$-sorting word of $\poa$ is in $\Hyper_{\alpha}(231)$ and that for each inversion $t\in\invset(\poa)$ there exists a unique $\pi\in\Hyper_{\alpha}(231)$ whose only cover inversion is $t$.    
\end{proof}

In Figures~\ref{fig:weak_join} and \ref{fig:weak_split} we highlight the equivalence classes with grey boxes.  The reader is invited to check that the weak order on representatives of these equivalence classes is isomorphic to the lattices shown in Figures~\ref{fig:tamari_join} and \ref{fig:tamari_split} respectively.



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