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%% define your title in the usual way
\title[Regular Schur labeled skew shape posets and their 0-Hecke modules]{
Regular Schur labeled skew shape posets and their 0-Hecke modules}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details




\author[Young-Hun Kim, So-Yeon Lee, and Young-Tak Oh]{Young-Hun Kim\thanks{\href{mailto:ykim.math@gmail.com}{ykim.math@gmail.com}. Young-Hun Kim was supported by the National Research Foundation of Korea(NRF) grant funded by the Korean Government (No. NRF-2020R1A5A1016126) and Basic Science Research Program through NRF funded by the Ministry of Education (No. RS-2023-00240377).}\addressmark{1},  So-Yeon Lee \addressmark{2}\thanks{\href{mailto:sylee0814@sogang.ac.kr}{sylee0814@sogang.ac.kr}. So-Yeon Lee was supported by NRF grant funded by the Korean Government (No. NRF-2020R1F1A1A01071055) and Basic Science Research Program through NRF funded by the Ministry of Education (No. RS-2023-00271282).}, \and Young-Tak Oh\addressmark{2}\thanks{\href{mailto:ytoh@sogang.ac.kr}{ytoh@sogang.ac.kr}. Young-Tak Oh was supported by NRF grant funded by the Korean Government (No. NRF-2020R1F1A1A01071055).
}
}

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\address{\addressmark{1}Center for Quantum structures in Modules and Spaces, Seoul National University, Seoul 08826, Republic of Korea \\ \addressmark{2}Department of Mathematics, Sogang University, Seoul 04107, Republic of Korea}

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\abstract{
Assuming Stanley's $P$-partitions conjecture holds, the regular Schur labeled skew shape posets are precisely the finite posets $P$ with underlying set $\{1, 2, \ldots, |P|\}$ such that the $P$-partition generating function is symmetric and the set of linear extensions of $P$, denoted  $\Sigma_L(P)$, is a left weak Bruhat interval in the symmetric group $\mathfrak{S}_{|P|}$. We describe the permutations in $\Sigma_L(P)$ in terms of reading words of standard Young tableaux when $P$ is a regular Schur labeled skew shape poset, and classify $\Sigma_L(P)$'s up to descent-preserving isomorphism as $P$ ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the $0$-Hecke modules $\mathsf{M}_P$ associated with regular Schur labeled skew shape posets $P$ up to isomorphism. Then we characterize  regular Schur labeled skew shape posets as the finite posets $P$ whose linear extensions form a dual plactic-closed subset of  $\mathfrak{S}_{|P|}$. Using this characterization, we construct  distinguished filtrations of $\mathsf{M}_P$ with respect to the Schur basis when $P$ is a regular Schur labeled skew shape poset.
}




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%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{labeled poset, $P$-partition, weak Bruhat order, $0$-Hecke algebra, representation, skew Schur function}

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\begin{document}

\maketitle

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\section{Introduction}

\emph{Schur labeled skew shape posets} naturally appear in the context of the celebrated  Stanley's $P$-partition conjecture. 
Let $\sfP_n$ be the set of posets on $[n] := \{1,2,\ldots, n\}$. 
To each poset $P\in \sfP_n$, one can associate a quasisymmetric function $K_P$, called the \emph{$P$-partition generating function}.
In 1972, Stanley \cite[p. 81]{72Stanley} proposed a conjecture stating that for $P \in \sfP_n$, $K_P$ is a symmetric function if and only if $P$ is a Schur labeled skew shape poset. 
While this conjecture has been verified to be true for all posets $P$ with $|P| \le 8$, it remains open in the general case (see \cite{06McNamara}).
For the definition of 
%$P$-partition generating functions and 
Schur labeled skew shape posets, see \cref{Regular posets and Schur labeled skew shape posets}.
We denote by $\sfSP_n$ the set of all Schur labeled skew shape posets in $\sfP_n$.




On the other hand, \emph{regular posets} were introduced by Bj\"{o}rner--Wachs \cite{91BW} during their investigation of the convex subsets of the symmetric group $\SG_n$ on $[n]$ under the right weak Bruhat order.
For $P \in \sfP_n$ with the partial order $\preceq$, let $\Sigma_R(P)$ be the set  of permutations $\pi \in \SG_n$ satisfying that if $x \preceq y$, then $\pi^{-1}(x) \le \pi^{-1}(y)$.
They 
observed  that every convex subset  of $\SG_n$ under the right weak Bruhat order appears as $\Sigma_R(P)$ for some $P \in \sfP_n$, and 
every right weak Bruhat interval in $\SG_n$ is convex.
This observation led them to characterize the posets $P \in \sfP_n$ satisfying that $\Sigma_R(P)$ is a right weak Bruhat interval. 
They introduced the notion of regular posets, and  proved that $P \in \sfP_n$ is a regular poset if and only if $\Sigma_R(P)$ is a right weak Bruhat interval in $\SG_n$.
For the definition of regular posets, refer to \cref{def: regular posets}.
We denote by $\sfRP_n$ the set of all regular posets in $\sfP_n$. 
% The results discussed in this paragraph still hold if we consider $\Sigma_L(P) := \{\gamma^{-1} \mid \gamma \in \Sigma_R(P)\}$ and the left weak Bruhat order, instead of $\Sigma_R(P)$ and the right weak Bruhat order.
Let $\Sigma_L(P) := \{\gamma^{-1} \mid \gamma \in \Sigma_R(P)\}$. 
By considering the left Bruhat order and $\Sigma_L(P)$ instead of the right Bruhat order and $\Sigma_R(P)$, we can establish a similar characterization. However, we prefer the former over the latter as it is better suited for handling left $H_n(0)$-modules.


% We note that the results mentioned in this paragraph still hold if we consider $\Sigma_L(P) := \{\gamma^{-1} \mid \gamma \in \Sigma_R(P)\}$ and the left weak Bruhat order on $\SG_n$ instead of
% $\Sigma_R(P)$ and right weak Bruhat order.
% because the map sending is order preserving isomorphism.


Let $\sfRSP_n := \sfRP_n \cap \sfSP_n$.
In the following, we explain the reason why we study regular Schur labeled skew shape posets from the perspective of the representation theory of the $0$-Hecke algebra.


 
In 1996, Duchamp--Krob--Leclerc--Thibon~\cite{96DKLT} 
introduced a ring isomorphism, called the \emph{quasisymmetric characteristic}, from the Grothendieck ring $\mathcal{G}_{0}(H_{\bullet}(0))$ of the tower of $0$-Hecke algebras to the ring $\Qsym$ of quasisymmetric functions.
For the definition of the quasisymmetric characteristic, see \cref{subsec: 0-Hecke alg and QSym}.
In 2002, Duchamp--Hivert--Thibon \cite{02DHT} associated a right $H_n(0)$-module $M_P$ with each poset $P \in \sfP_n$, such that the image of $M_P$ under the quasisymmetric characteristic is $K_P$. 
This was achieved by defining a suitable right $H_n(0)$-action on  $\Sigma_R(P)$.
For technical reasons, we use a slightly different $0$-Hecke module, denoted as $\sfM_P$, instead of Duchamp--Hivert--Thibon's module $M_P$. 
Our $\sfM_P$ is a left $H_n(0)$-module with the basis $\Sigma_L(P)$.
For the precise definition of $\sfM_P$, refer to \cref{def: poset module}.


Since the middle of 2010, various  left $0$-Hecke modules, each equipped with a tableau basis and yielding an important quasisymmetric characteristic image, have been constructed (\cite{20BS, 15BBSSZ, 19Searles, 15TW, 19TW}).
In order to handle these modules in a uniform manner, Jung--Kim--Lee--Oh \cite{22JKLO} introduced a left $H_n(0)$-module $\sfB(I)$, referred to as \emph{the weak Bruhat interval module associated with $I$}, for each left weak Bruhat interval $I$ in $\SG_n$.
Furthermore, they showed that the Grothendieck ring $\bigoplus_{n \ge 0} \mathcal{G}_0(\mathscr{B}_n)$ is isomorphic to $\Qsym$ as Hopf algebras, where $\mathscr{B}_n$ is the category direct sums of finitely many isomorphic copies of weak Bruhat interval modules of $H_n(0)$.
Recently, Choi--Kim--Oh \cite{23CKO} clarified the exact relationship between the weak Bruhat interval modules and the $0$-Hecke modules $M_P$, using Bj\"{o}rner--Wachs' characterization.


The aim of this paper is to give a comprehensive investigation of regular Schur labeled skew shape posets and their associated $0$-Hecke modules. 
Firstly, we provide an explicit description of  $\Sigma_L(P)$ for $P \in \sfRSP_n$.
Next, we study the classification of left weak Bruhat intervals in $\SG_n$ up to descent-preserving poset isomorphism.
Using the classification, we  classify the $H_n(0)$-modules $\sfM_P$ up to isomorphism as $P$ ranges over $\sfRSP_n$.
Then, we characterize regular Schur labeled skew shape posets as the posets such that $\Sigma_L(P)$ is a dual plactic-closed subset of $\SG_n$.
This characterization is applied to show that for $P \in \sfRSP_n$, $\sfM_P$ admits a distinguished filtration with respect to the Schur basis.
A tableau description of $\sfM_P$ for $P \in \sfRSP_n$ is also provided.
Lastly, we discuss further issues concerned with the classification of the $H_n(0)$-modules $\mathsf{M}_P$.


For details and more results, we refer the reader to \cite{23KLO}. 





\section{Preliminaries}
Throughout this paper, $n$ will denote a nonnegative integer unless otherwise stated. 

\subsection{Compositions, Young diagrams, and bijective tableaux}\label{subsec: comp and diag}
A \emph{composition} $\alpha$ of $n$, denoted by $\alpha \models n$, is a finite ordered list of positive integers $(\alpha_1,\alpha_2,\ldots, \alpha_k)$ satisfying $\sum_{i=1}^k \alpha_i = n$.
We call  $k =: \ell(\alpha)$ the \emph{length} of $\alpha$  and $n =:|\alpha|$ the \emph{size} of $\alpha$. 
Given $\alpha = (\alpha_1, \ldots, \alpha_{\ell(\alpha)}) \models n$, we define $\set(\alpha) := \{\alpha_1, \alpha_1 + \alpha_2, \ldots, \sum_{i = 1}^{\ell(\alpha) - 1} \alpha_i \}$.
The \emph{reverse composition $\alpha^\rmr$ of $\alpha$} is the composition $(\alpha_k, \alpha_{k-1}, \ldots, \alpha_1)$ and
the \emph{complement $\alpha^\rmc$ of $\alpha$} is the unique composition satisfying $\set(\alpha^c) = [n-1] \setminus \set(\alpha)$.
If a composition $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k) \models n$ satisfies $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_k$, then it is called a \emph{partition} of $n$ and denoted as $\lambda \vdash n$.
Given two partitions $\lambda$ and $\mu$ with $\ell(\lambda) \ge \ell(\mu)$, we write $\lambda \supseteq \mu$ if $\lambda_i \geq  \mu_i$ for all $1 \le i \le \ell(\mu)$. 
A \emph{skew partition} $\lambda/\mu$ is a pair  $(\lambda, \mu)$ of partitions with $\lambda \supseteq \mu$.
We call $|\lambda/\mu| := |\lambda| - |\mu|$ the \emph{size} of $\lambda/\mu$.

Given a partition $\lambda$, we define the \emph{Young diagram $\tyd(\lambda)$ of $\lambda$} to be the left-justified array of $n$ boxes, where the $i$th row from the top has $\lambda_i$ boxes for $1 \le i \le k$.
Similarly, given a skew partition $\lambda / \mu$, we define the \emph{Young diagram $\tyd(\lambda / \mu)$ of $\lambda / \mu$} to be the Young diagram $\tyd(\lambda)$ with all boxes belonging to $\tyd(\mu)$ removed.
A skew partition is called \emph{basic} if the corresponding Young diagram contains neither empty rows nor empty columns.
In this paper, every skew partition is assumed to be  basic unless otherwise stated.
For skew partitions $\lambda/\mu$ and $\nu/\kappa$, $\lambda/\mu \oplus \nu/\kappa$ is the skew partition whose Young diagram is obtained  by taking a rectangle of empty squares with the same number of rows as  $\tyd(\lambda/\mu)$ and the same number of columns as $\tyd(\nu/\kappa)$, and putting
$\tyd(\nu/\kappa)$ below and $\tyd(\lambda/\mu)$ to the right of this rectangle.
For instance, $\tyd((2) \oplus (1)) = \scalebox{0.7}{
$\begin{array}{l}
\begin{ytableau}
\none & ~ & ~ \\
~
\end{ytableau}
\end{array}$}$.





Given a skew partition $\lambda/\mu$ of size $n$, a \emph{bijective tableau} of shape $\lambda / \mu$ is a filling of $\tyd(\lambda / \mu)$ with distinct entries in $[n]$.
For later use, we denote by $\tau_0^{\lambda/\mu}$  the bijective tableau of shape $\lambda / \mu$ obtained by filling $1, 2, \ldots, n$ from right to left starting with the top row.
A bijective tableau is referred to as a \emph{standard Young tableau} if the elements in each row are arranged in increasing order from left to right, and the elements in each column are arranged in increasing order from top to bottom.
We denote by $\SYT(\lambda / \mu)$ the set of all standard Young tableaux of shape $\lambda / \mu$.
And, we let $\SYT_n := \bigcup_{\lambda \vdash n} \SYT(\lambda)$.


\subsection{Weak Bruhat orders on the symmetric group}\label{subsec: Symmetric group}

Let $\SG_n$ denote the symmetric group on $[n]$. 
For $1 \le i \le n-1$, let $s_i$ be the simple transposition $(i,i+1)$.
For $\sigma \in \SG_n$, let
$$
\Des_L(\sigma):= \{i \in [n-1] \mid \ell(s_i \sigma) < \ell(\sigma)\} \;\; \text{and} \;\;    
\Des_R(\sigma):= \{i \in [n-1] \mid \ell(\sigma s_i) < \ell(\sigma)\},
$$
where $\ell(\sigma)$ is the length of $\sigma$.
The \emph{left weak Bruhat order} $\preceq_L$ (resp., \emph{right weak Bruhat order}  $\preceq_R$)  on $\SG_n$
is the partial order on $\SG_n$ whose covering relation $\preceq_L^{\rmc}$(resp., $\preceq_R^{\rmc}$)  is given as follows: 
$\sigma \preceq_L^\rmc s_i \sigma$ if and only if $i \notin \Des_L(\sigma)$ (resp, $\sigma \preceq_R^\rmc  \sigma s_i$ if and only if $i \notin \Des_R(\sigma)$).
Given $\sigma, \rho \in \SG_n$, the \emph{left weak Bruhat interval $[\sigma, \rho]_L$} (resp., \emph{right weak Bruhat interval $[\sigma, \rho]_R$}) denotes the closed interval $\{\gamma \in \SG_n \mid \sigma \preceq_L \gamma \preceq_L \rho \}$ (resp., $\{\gamma \in \SG_n \mid \sigma \preceq_R \gamma \preceq_R \rho \}$).
Let $\rmInt(n)$ be the set of nonempty left weak Bruhat intervals in $\SG_n$.











\subsection{Regular posets and Schur labeled skew shape posets}
\label{Regular posets and Schur labeled skew shape posets}
Let $\sfP_n$ be the set of posets on $[n]$.
Given $P \in \sfP_n$, 
we write the partial order of $P$ as $\preceq_P$.





\begin{definition}\label{def: regular posets}
A poset $P \in \sfP_n$ is said to be \emph{regular} if the following holds:
for all $x,y,z \in [n]$ with
$x \preceq_P z$, if $x < y < z$ or $z<y<x$, then $x \preceq_P y$ or $y \preceq_P z$.
\end{definition}
We denote by $\sfRP_n$ the set of all regular posets in $\sfP_n$.
From the result of Bj\"orner--Wachs \cite[Theorem 6.8]{91BW}, it follows that
\begin{enumerate}[label = {\rm (\roman*)}, itemsep = 0ex]
\item for $P \in \sfP_n$, $P$ is regular if and only if $\Sigma_L(P)$ is a left weak Bruhat interval, and 
\item 
the map $\sfRP_n \ra \rmInt(n)$ sending $P$ to  $\Sigma_L(P)$ is a one-to-one correspondence.
\end{enumerate}
Here,
$\Sigma_{L}(P) := \{\sigma \in \SG_n \mid \text{$\sigma(i) \leq \sigma(j)$ for all $i, j\in [n]$ with $i \preceq_P j$}\}$.







Next, let us introduce Schur labeled skew shape posets.
Let $\lambda / \mu$ be a skew partition of size $n$.
Given a bijective tableau $\tau$ of shape $\lambda/\mu$, we define $\sfposet(\tau)$ to be 
the poset $([n], \preceq_\tau)$, where $i \preceq_\tau j$ if and only if  $i$ lies weakly northeast of $j$ in $\tau$.
A \emph{Schur labeling of shape $\lambda/ \mu$} is a bijective tableau of shape $\lambda / \mu$ such that
the entries in each row decrease from left to right and the entries in each column increase from top to bottom.
Let $\sfS(\lambda/\mu)$ be the set of all Schur labelings of shape $\lambda / \mu$. 
Since $\tau_0^{\lambda/\mu}$ is a Schur labeling, $\sfS(\lambda/\mu)$ is nonempty.
Set
$\sfSP(\lambda/\mu) := \{\sfposet(\tau) \mid \tau \in \sfS(\lambda/\mu)
\}$
and
$\sfSP_n := \bigcup_{
|\lambda/\mu| = n
} \sfSP(\lambda/\mu)$.


\begin{definition}\label{def: Schur labeled skew shape posets}
A poset in $\sfP_n$ is  said to be a \emph{Schur labeled skew shape poset} if it is contained in $\sfSP_n$.
\end{definition}



\begin{example}
When $\lambda/\mu = (2,2)$, we have 
$\sfS(\lambda/\mu) = 
\left\{
\tau_1 := \scalebox{0.75}{
$\begin{array}{l}
\begin{ytableau}
     2 & 1\\
     4  & 3 
\end{ytableau}
\end{array}$
},\;
\tau_2 := \scalebox{0.75}{
$\begin{array}{l}
\begin{ytableau}
     3 & 1\\
     4  &  2
\end{ytableau}
\end{array}$}
\right\}$.
Therefore, $\sfSP(\lambda/\mu)$ consists of the following posets:
\[
\def \pp {0.8}
\def \hp {0.6}
\sfposet(\tau_1) = 
\scalebox{0.85}{
$\begin{array}{l}
\begin{tikzpicture}
\node at (0,0) {1};
\node at (1*\hp, -1*\pp) {2};
\node at (1*\hp, 1*\pp) {3};
\node at (2*\hp, 0*\pp) {4};

\draw (0.25*\hp, -0.25*\pp) -- (0.75*\hp, -0.75*\pp);
\draw (1.25*\hp, -0.75*\pp) -- (1.75*\hp, -0.25*\pp);
\draw (0.25*\hp, 0.25*\pp) -- (0.75*\hp, 0.75*\pp);
\draw (1.25*\hp, 0.75*\pp) -- (1.75*\hp, 0.25*\pp);
\end{tikzpicture}
\end{array}$} \quad \text{and} \quad
\sfposet(\tau_2) = 
\scalebox{0.85}{
$\begin{array}{l}
\begin{tikzpicture}
\node at (0,0) {1};
\node at (1*\hp, -1*\pp) {3};
\node at (1*\hp, 1*\pp) {2};
\node at (2*\hp, 0*\pp) {4};

\draw (0.25*\hp, -0.25*\pp) -- (0.75*\hp, -0.75*\pp);
\draw (1.25*\hp, -0.75*\pp) -- (1.75*\hp, -0.25*\pp);
\draw (0.25*\hp, 0.25*\pp) -- (0.75*\hp, 0.75*\pp);
\draw (1.25*\hp, 0.75*\pp) -- (1.75*\hp, 0.25*\pp);
\end{tikzpicture}
\end{array}$}.
\]
\end{example}

 
For simplicity, we set $\sfRSP_n := \sfRP_n \cap \sfSP_n$.



\subsection{The $0$-Hecke algebra and the quasisymmetric characteristic}\label{subsec: 0-Hecke alg and QSym}
The $0$-Hecke algebra $H_n(0)$ is the associative $\C$-algebra with $1$ generated by $\pi_1,\pi_2,\ldots,\pi_{n-1}$ subject to the following relations: (1) $\pi_i^2 = \pi_i$ for $1\le i \le n-1$, (2) $\pi_i \pi_{i+1} \pi_i = \pi_{i+1} \pi_i \pi_{i+1}$ for $1\le i \le n-2$, and (3) $\pi_i \pi_j =\pi_j \pi_i$ if $|i-j| \ge 2$.
According to \cite{79Norton}, there are $2^{n-1}$ pairwise nonisomorphic irreducible $H_n(0)$-modules which are naturally parametrized by compositions of $n$.
For each $\alpha \models n$,
the irreducible module $\mathbf{F}_\alpha$ corresponding to $\alpha$ is the $1$-dimensional $H_n(0)$-module spanned by a vector $v_\alpha$, which is annihilated by $\pi_i$ if $i \in \mathrm{set}(\alpha)$ or fixed by $\pi_i$ otherwise for all $1 \leq i \leq n-1$.


Let $\mathcal{G}_0(H_n(0))$ be the \emph{Grothendieck group} of the category of finitely generated left $H_n(0)$-modules and $\mathcal{G}_{0}(H_{\bullet}(0)) := \bigoplus_{n \ge 0} \mathcal{G}_0(H_n(0))$  the ring equipped with the induction product.
In \cite{96DKLT},
Duchamp--Krob--Leclerc--Thibon 
showed that the linear map
$$
\mathrm{ch} : \mathcal{G}_{0}(H_{\bullet}(0)) \rightarrow \mathrm{QSym}, \quad [\mathbf{F}_{\alpha}] \mapsto F_{\alpha},
$$
called \emph{quasisymmetric characteristic}, is a ring isomorphism.
Here, $\Qsym$ is the ring of quasisymmetric functions and  $F_\alpha$ is the \emph{fundamental quasisymmetric function}.












\subsection{Modules arising from posets and weak Bruhat interval modules}



\begin{definition}\label{def: poset module}{\rm (cf. \cite[Definition 3.18]{02DHT})}
Let $P \in \sfP_n$. 
Define $\sfM_P$ to be the left $H_n(0)$-module with $\C \Sigma_{L}(P)$ as the underlying space and with the $H_n(0)$-action defined by
\begin{align*}
\pi_{i} \cdot \gamma :=
\begin{cases}
\gamma & \text{if $i \in \Des_L(\gamma)$}, \\
0 & \text{if $i \notin \Des_L(\gamma)$ and $s_i\gamma \notin \Sigma_{L}(P)$,} \\
s_i \gamma & \text{if $i \notin \Des_L(\gamma)$ and $s_i\gamma \in \Sigma_{L}(P)$}.
\end{cases} 
\end{align*}
\end{definition}


For $P \in \sfP_n$, a map $f: [n] \ra \Z_{\ge 0}$ is called a  \emph{$P$-partition} if (i) $f(i) \le f(j)$ for all $i \preceq_P j$ and (ii) $f(i) < f(j)$ for all $i \preceq_P j$ with $i > j$.
The \emph{$P$-partition generating function} is defined by
$
K_{P} := \sum_{f:  \text{$P$-partition}} x_1^{|f^{-1}(1)|} x_2^{|f^{-1}(2)|} \cdots.
$



\begin{theorem}\label{thm: char of Mp}{\rm (\cite[Theorem 3.21(i)]{02DHT})}
For $P \in \sfP_n$, we have $\ch([\sfM_P]) = \uppsi(K_{P})$, where  $\uppsi$ is the involution  of $\Qsym$ defined by
$\uppsi(F_\alpha) = F_{\alpha^\rmc}$.
\end{theorem}


In order to provide a unified method for dealing with $H_n(0)$-modules constructed using tableaux in \cite{20BS, 15BBSSZ, 19Searles, 15TW,19TW}, Jung--Kim--Lee--Oh \cite{22JKLO} introduced the \emph{weak Bruhat interval module} $\sfB(I)$  associated with a left weak Bruhat interval $I$ in $\SG_n$.
For $I \in \rmInt(n)$,
$\sfB(I)$ can be defined as $\sfM_{P}$, where $P$ is the  unique poset in $\sfRP_n$ such that $\Sigma_L(P) = I$.



\section{The weak Bruhat interval structure of $\Sigma_L(P)$ for $P \in \sfRSP_n$}



First, we  introduce a specific Schur labeling associated with a Schur labeled skew shape poset.
For  $P \in \sfSP_n$,
we define $\tau_P$ to be the unique Schur labeling such that
\begin{align}\label{eq: cond for tauP}
\text{$\sh(\tau_P)$ is basic, \quad  $\sfposet(\tau_P) = P$,\quad  and \quad $\mathrm{min}_i(\tau_P) < \mathrm{min}_j(\tau_P)$ \  for $1 \leq i < j \leq k$.} 
\end{align}
Here, $\mathrm{min}_i(\tau_P)$ is the smallest entry in the $i$th connected component of $\tau_P$ from the top for all $1 \leq i \leq k$ and $k$ is the number of connected components of $P$.


\begin{example}\label{eg: dist Schur labeling}
Let
$\def \pp {0.8}
\def \hp {0.6}
P = \scalebox{0.85}{$
\begin{array}{l}
\begin{tikzpicture}
\node at (0,0) {1};
\node at (1*\hp, -1*\pp) {2};
\node at (1*\hp, 1*\pp) {3};
\node at (2*\hp, 0*\pp) {4};
\node at (3* \hp, 0) {5};

\draw (0.25*\hp, -0.25*\pp) -- (0.75*\hp, -0.75*\pp);
\draw (1.25*\hp, -0.75*\pp) -- (1.75*\hp, -0.25*\pp);
\draw (0.25*\hp, 0.25*\pp) -- (0.75*\hp, 0.75*\pp);
\draw (1.25*\hp, 0.75*\pp) -- (1.75*\hp, 0.25*\pp);
\end{tikzpicture}
\end{array}$}$\hspace{-0.7ex}.
There are two Schur labelings $\tau$ such that $\sfposet(\tau) = P$, more precisely, 
\[
\tau_1 := \scalebox{0.75}{$\begin{array}{c} 
\begin{ytableau}
\none &  2 & 1 \\
\none & 4 & 3 \\
5
\end{ytableau}
\end{array}$
} \quad \text{and} \quad
\tau_2 := \scalebox{0.75}{$\begin{array}{c} 
\begin{ytableau}
\none & \none & 5 \\
2 & 1 \\
 4 & 3
\end{ytableau}
\end{array}$
}.
\]
Since $\tau_1$ is a Schur labeling and satisfies \cref{eq: cond for tauP},  $\tau_P = \tau_1$.
\end{example}






\begin{definition}\label{def: read tau}
Let $P \in \sfSP_n$ and $\lambda/\mu = \sh(\tau_P)$. 
The \emph{$\tau_P$-reading} is the map 
$$
\rmread_{\tau_P}: \SYT(\lambda/\mu) \ra \SG_n, \quad T \mapsto \rmread_{\tau_P}(T),
$$ 
where $\rmread_{\tau_P}(T)$ is the permutation in $\SG_n$ given by
$\rmread_{\tau_P}(T)(k) = T_{{\tau_P}^{-1}(k)}$, the entry of $T$ in the box $\tau^{-1}(k)$,   for $1 \le k \le n$.

\end{definition}

With these notions, we state the following theorem.
\begin{theorem}
Let $P \in \sfSP_n$ and $\lambda/\mu = \sh(\tau_P)$.
Then, $\Sigma_L(P) = \rmread_{\tau_P}(\SYT(\lambda/\mu))$.
In particular, if $P \in \sfRSP_n$, then  
\[
\Sigma_L(P) =  [\rmread_{\tau_P}(T_{\lambda/\mu}), \rmread_{\tau_P}(T'_{\lambda/\mu})]_L.
\]
Here,  $T_{\lambda/\mu}$(resp. $T'_{\lambda/\mu}$) is the standard Young tableau obtained by filling $\tyd(\lambda/\mu)$ by $1, 2, \ldots, n$ from left to right starting with the top row (resp. from top to bottom starting with leftmost column).

\end{theorem}



\begin{example}
Let $P$ be the poset given in \cref{eg: dist Schur labeling} and $\lambda/\mu =  (3,3,1)/(1,1)$.
Note that 
\[
T_{\lambda/\mu} = \scalebox{0.75}{$\begin{array}{c} 
\begin{ytableau}
\none & 1 & 2\\
\none & 3 & 4\\
5
\end{ytableau}
\end{array}$}, \quad  T'_{\lambda/\mu} = \scalebox{0.75}{$\begin{array}{c} 
\begin{ytableau}
\none & 2 & 4\\
\none & 3 & 5\\
1
\end{ytableau}
\end{array}$}, \quad \text{and} \quad 
\tau_P = 
\scalebox{0.75}{$\begin{array}{c} 
\begin{ytableau}
\none &  2 & 1 \\
\none & 4 & 3 \\
5
\end{ytableau}
\end{array}$
}.
\]
Since
$\rmread_{\tau_P}(T_{\lambda/\mu}) = 21435$ and $\rmread_{\tau_P}(T'_{\lambda/\mu}) = 42531$, we have  $\Sigma_L(P) = [21435, 42531]_L$.
\end{example}



\section{An equivalence relation on $\rmInt(n)$}

For $I_1, I_2 \in \rmInt(n)$, a poset isomorphism $f: (I_1, \preceq_L) \ra (I_2, \preceq_L)$ is called  \emph{descent-preserving} if $\Des_L(\gamma) = \Des_L(f(\gamma))$ for all $\gamma \in I_1$.
We define an equivalence relation $\Deq$ on $\rmInt(n)$ by $I_1 \Deq I_2$ if there is a descent-preserving poset isomorphism between $I_1$ and $I_2$.
In \cite[Section 4]{23KLO}, we show that  $\sfB(I_1) \cong \sfB(I_2)$ for all $I_1, I_2 \in \rmInt(n)$ with $I_1 \Deq I_2$. 
This leads us to study the equivalence classes under $\Deq$.
% Understanding the equivalence classes under $\Deq$ helps us to classify the weak Bruhat interval modules because 
%Let  $\Deq$ be an equivalence relation on $\rmInt(n)$ defined as follows: $I_1 \Deq I_2$ if there is a descent-preserving poset isomorphism between $I_1$ and $I_2$.
The following theorem provides significant information regarding equivalence classes under $\Deq$.
\begin{theorem}\label{thm: desc pres equiv}
Let $C$ be an equivalence class under $\Deq$.
Then, 
$\{\sigma \mid [\sigma, \rho]_L \in C\}$ is a right weak Bruhat interval in $\SG_n$.
\end{theorem}

According to \cref{thm: desc pres equiv}, every equivalence class $C$ can be expressed as follows:
\[
C = \{[\gamma,  \xi_C\gamma]_L \mid \gamma \in [\sigma_0, \sigma_1]_R\},
\]
where $\xi_C := \rho \sigma^{-1}$ for any $[\sigma, \rho]_L \in C$ and $\sigma_0, \sigma_1 \in \SG_n$ with $[\sigma_0, \sigma_1]_R = \{\sigma \mid [\sigma, \rho]_L \in C \}$.
When $P \in \sfRSP_n$, we explicitly describe the equivalence class of $\Sigma_L(P)$ in the following theorem.
% The following theorem explicitly describes the equivalence class of $\Sigma_L(P)$ for $P \in \sfRSP_n$.


\begin{theorem}\label{thm: equiv class X lam mu}
Let $P \in \sfRSP_n$ and $C$ the equivalence class of $\Sigma_L(P)$ under $\Deq$.
Then, 
\[
C = \{\Sigma_L(Q) \mid \text{$Q \in \sfRSP_n$ with $\sh(\tau_Q) = \sh(\tau_P)$} \}.
\]
\end{theorem}

\cref{thm: equiv class X lam mu} tells us that $\{\Sigma_L(P) \mid P \in \sfRSP_n \}$ is closed under $\Deq$ and the equivalence classes in this set are  parametrized by skew partitions of size $n$.
To be precise,  
for any skew partition $\lambda/\mu$ of size $n$, let 
\[
C_{\lambda/\mu} = \{\Sigma_L(P) \mid \text{$P \in \sfRSP_n$ with $\sh(\tau_P)= \lambda/\mu$} \}.
\]
This set is nonempty since $\sfposet(\tau_0^{\lambda/\mu}) \in C_{\lambda/\mu}$, and therefore it is an equivalence class
by \cref{thm: desc pres equiv}.
To summarize, 
$\{\Sigma_L(P) \mid P \in \sfRSP_n \} = \bigsqcup_{|\lambda/\mu| = n} C_{\lambda / \mu}$ $($disjoint union$)$.

%This set is nonempty since  $\sfposet(\tau_0^{\lambda/\mu}) \in \sfRSP_n$ and  $\tau_0^{\lambda/\mu} = \tau_{\sfposet(\tau_0^{\lambda/\mu})}$\,. $C_{\lambda/\mu}$ is an equivalence class.


\section{The classification of $\sfM_P$'s  for $P \in \sfRSP_n$}

In \cite{72Stanley}, Stanley proposed the following conjecture, called \emph{Stanley's $P$-partitions conjecture}.


\begin{conjecture}{\rm (\cite[p. 81]{72Stanley})}\label{conj: Stanley}
For $P \in \sfP_n$, if $K_P$ is symmetric, then $P \in \sfSP_n$.
\end{conjecture}


Assuming Stanley's $P$-partitions conjecture holds, \cref{thm: char of Mp} implies that for any $P \in \sfRSP_n$ and $Q \in \sfRP_n \setminus \sfSP_n$,
$\ch([\sfM_P])$ is symmetric but $\ch([\sfM_Q])$ is not symmetric, thus
$\sfM_P \not\cong \sfM_Q$.
In addition, by the correspondence between $\sfRP_n$ and $\rmInt(n)$ in \cref{Regular posets and Schur labeled skew shape posets},  
\[
\{\sfM_P \mid P \in \sfRSP_n\} = 
\{\sfB(I) \mid I \in \rmInt(n) \text{ and } \ch([\sfB(I)]) \in \Sym \} 
.
\]
This leads us to  consider the classification problem for $\{\sfM_P \mid P \in \sfRSP_n\}$.
We solve this problem by determining the  projective cover and injective hull of $\sfM_P~(P\in \sfRSP_n)$ up to isomorphism.

It is well known that there is a one-to-one correspondence between the set of irreducible $H_n(0)$-modules and that of projective indecomposable $H_n(0)$-modules.
For $\alpha \models n$,
let $\bfP_\alpha$ be the projective indecomposable module corresponding to $\bfF_\alpha$, that is, $\bfP_\alpha / \rad (\bfP_\alpha) \cong \bfF_\alpha$.
In \cite[Propsition 4.1]{02DHT}, it was shown that $H_n(0)$ is a Frobenius algebra. 
Thus, an $H_n(0)$-module $M$ is projective if and only if it is injective (see \cite[Proposition 1.6.2]{91Benson}).




A \emph{generalized composition $\bal$ of $n$} is a formal sum $\alpha^{(1)} \oplus \alpha^{(2)} \oplus \cdots \oplus  \alpha^{(k)}$,
where $\alpha^{(i)} \models n_i$ for positive integers $n_i$'s with $n_1 + n_2 + \cdots + n_k = n$.
For a generalized composition $\bal = \alpha^{(1)} \oplus \alpha^{(2)} \oplus \cdots \oplus \alpha^{(k)}$ of $n$, set $
\bal^\rmc  := (\alpha^{(1)})^\rmc \oplus (\alpha^{(2)})^\rmc \oplus \cdots \oplus (\alpha^{(k)})^\rmc$ and $\bal^\rmr  := (\alpha^{(k)})^\rmr \oplus (\alpha^{(k-1)})^\rmr \oplus \cdots \oplus (\alpha^{(1)})^\rmr$.
And, define
$\bfP_\bal := \bfP_{\alpha^{(1)}} \otimes \bfP_{\alpha^{(2)}} \otimes \cdots \otimes \bfP_{\alpha^{(k)}} \uparrow_{H_{n_1}(0) \otimes H_{n_2}(0) \otimes \cdots \otimes H_{n_k}(0)}^{H_n(0)}$\,,
where $n_i := |\alpha_i|$ for $1 \leq i \leq k$.
This module is projective and its decomposition into projective indecomposable modules was provided in \cite[Theorem 3.3]{16Huang}.


For a connected skew partition $\lambda / \mu$ of size $n$, define
\[
\balproj(\lambda/\mu) := (\lambda_1 - \mu_1, \lambda_2 - \mu_2, \ldots , \lambda_{\ell(\lambda)} - \mu_{\ell(\lambda)}).
\]
And, for a disconnected skew partition $\lambda / \mu$ of size $n$, define
\[
\balproj(\lambda/\mu) := \balproj(\lambda^{(1)} / \mu^{(1)}) 
\oplus \balproj(\lambda^{(2)} / \mu^{(2)})
\oplus \cdots \oplus 
\balproj(\lambda^{(k)} / \mu^{(k)}),
\]
where $\lambda/\mu = \lambda^{(1)} / \mu^{(1)} \oplus \lambda^{(2)} / \mu^{(2)} \oplus \cdots \oplus \lambda^{(k)} / \mu^{(k)}$ with connected $\lambda^{(i)} / \mu^{(i)}$'s $(1\le i \le k)$.
Set
\[
\balinj(\lambda/\mu) := (\balproj(\lambda^\rmt / \mu^\rmt)^\rmc \,)^\rmr,
\]
where $\lambda^\rmt$ and $\mu^\rmt$ denote the transpose of $\lambda$ and $\mu$, respectively.

\begin{lemma}\label{lem: proj and inj of MP}
For $P \in \sfRSP_n$ and $\lambda/\mu = \sh(\tau_P)$,
$\bfP_{\balproj(\lambda/\mu)}$ $($resp. $\bfP_{\balinj(\lambda/\mu)}$$)$ is the projective cover $($resp. the injective hull$)$ of $\sfM_P$.
\end{lemma}


Using this lemma, 
we establish the following  classification theorem of $\sfM_P$'s for $P \in \sfRSP_n$ up to $H_n(0)$-module isomorphism.


\begin{theorem}\label{thm: classification}
Let $P, Q \in \sfRSP_n$.
Then
$\sfM_{P} \cong \sfM_{Q}$ if and only if $\sh(\tau_P) = \sh(\tau_Q)$.
\end{theorem}

The ``if'' part can be derived from \cref{thm: equiv class X lam mu}.
Let us briefly explain how we prove the ``only if'' part.  
Considering \cref{lem: proj and inj of MP} together with Huang's decomposition of $\bfP_\bal$ in \cite[Theorem 3.3]{16Huang}, we prove that for $P,Q \in \sfRSP_n$, $\sfM_P$ and $\sfM_Q$ have either nonisomorphic projective covers or nonisomorphic injective hulls if $\tau_P$ and $\tau_Q$ have different shapes.









\section{A characterization of regular Schur labeled skew shape posets $P$  and distinguished filtrations of $\sfM_P$}

We first characterize regular Schur labeled skew shape posets from the viewpoint of dual plactic congruence.
The \emph{Robinson–Schensted correspondence} is a one-to-one correspondence between $\SG_n$ and $\bigcup_{\lambda \vdash n} \SYT(\lambda) \times \SYT(\lambda)$.
For $\sigma \in \SG_n$,
we use the notation $(\Ptab(\sigma), \Qtab(\sigma))$ to represent the image of $\sigma$ under this bijection.
The \emph{dual plactic congruence} is an equivalence relation $\dKeq$ on $\SG_n$ defined by $\sigma \dKeq \rho$ if  $\Qtab(\sigma) = \Qtab(\rho)$.
A subset $S$ of $\SG_n$ is called \emph{dual plactic-closed} if $S$ is a union of equivalence classes under the dual plactic congruence.



In \cite[Theorem 1]{93M}, Malvenuto proved that if $\Sigma_L(P)$ is dual plactic-closed, then $P \in \sfSP_n$.
We improve Malvenuto's result by providing the following characterization of regular Schur labeled skew shape posets.


\begin{theorem}\label{thm: RSP iff DPC}
For $P \in \sfP_n$, $P$ is a regular Schur labeled skew shape poset if and only if $\Sigma_L(P)$ is dual plactic-closed.
\end{theorem}


\begin{example}
Consider the posets
$
\def \pp {0.8}
\def \hp {0.6}
P = 
\scalebox{0.85}{
$\begin{array}{l}
\begin{tikzpicture}
\node at (0,0) {1};
\node at (1*\hp, -1*\pp) {3};
\node at (2*\hp, -0.5*\pp) {2};

\draw (0.25*\hp, -0.25*\pp) -- (0.75*\hp, -0.75*\pp);
\end{tikzpicture}
\end{array}$}$
and
$\def \pp {0.8}
\def \hp {0.6}
Q =
\scalebox{0.85}{
$\begin{array}{l}
\begin{tikzpicture}
\node at (0,0) {1};
\node at (1*\hp, -1*\pp) {2};
\node at (2*\hp, -0.5*\pp) {3};
\draw (0.25*\hp, -0.25*\pp) -- (0.75*\hp, -0.75*\pp);
\end{tikzpicture}
\end{array}$}$ in $\sfSP_3$.
One sees that (i) $P$ is non-regular and $Q$ is regular and that (ii)
$\Sigma_L(P) = \{231, 312, 321\}$ is not dual plactic-closed and $\Sigma_L(Q) = \{213, 312, 321\}$ is dual plactic-closed.
\end{example}



Using the characterization given in \cref{thm: RSP iff DPC}, 
we construct a filtration of $\sfM_P$ ($P \in \sfRSP_n$) which provides a representation theoretic interpretation of 
$s_{\lambda/\mu} = \sum_{\nu \vdash n }c^{\lambda}_{\mu, \nu} s_{\nu}$, the expansion of the skew Schur function $s_{\lambda/\mu}$ in the  Schur basis $\{s_\nu \mid \nu \vdash n\}$.
Here, $\lambda/\mu = \sh(\tau_P)$ and  $c^{\lambda}_{\mu, \nu}$ is the Littlewood--Richardson coefficient.
To handle such filtrations in a uniform manner, we introduce the notion of distinguished filtrations.

\begin{definition}\label{def: dist filt}
Let $\mathcal{B} = \{\mathcal{B}_\alpha \mid \alpha \in I\}$ be a linearly independent subset of $\Qsym_n$ 
with the property that  $\mathcal{B}_\alpha$ is $F$-positive for all $\alpha \in I$,
where $I$ is an index set.
Given a finite dimensional $H_n(0)$-module $M$,
a \emph{distinguished filtration of $M$ with respect to $\mathcal{B}$} is an $H_n(0)$-submodule series  of $M$ 
\[
0 =: M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_l := M
\]
such that for all $1 \leq k \leq l$, $\ch([M_k / M_{k-1}]) = \mathcal{B}_\alpha$ for some $\alpha \in I$.
\end{definition}

It should be remarked that a distinguished filtration of $M$ with respect to $\calB$ may not exist even if $\ch([M])$ expands positively in $\calB$.
For instance, see \cite[Example 6.6]{23KLO}.
If such a filtration exists, we have a representation theoretic interpretation of the expansion of $\ch([M])$ in $\mathcal{B}$.



\begin{theorem}\label{thm: filt of X lam mu}
For every $P \in \sfRSP_n$,
$\sfM_P$ admits a distinguished filtration  with respect to the Schur basis $\{s_{\lambda} \mid \lambda \vdash n\}$.
\end{theorem}

 
\begin{example}\label{eg: distinguished filt}
Let $P = \sfposet(\tau_0^{(4,2,1)/(2,1)})$.
The set $\{\Qtab(\gamma) \mid \gamma \in \Sigma_L(P)\}$ is equal to
\[
\left\{
Q_1 := \scalebox{0.75}{$\begin{array}{c} 
\begin{ytableau}
1 \\
2 \\
3 \\ 
4
\end{ytableau}
\end{array}$}, \;
Q_2 := \scalebox{0.75}{$\begin{array}{c} 
\begin{ytableau}
1 & 3\\
2 \\
4
\end{ytableau}
\end{array}$}, \;
Q_3 := \scalebox{0.75}{$\begin{array}{c} 
\begin{ytableau}
1 & 4\\
2 \\
3
\end{ytableau}
\end{array}$}, \;
Q_4 := \scalebox{0.75}{$\begin{array}{c} 
\begin{ytableau}
1 & 3 \\
2 & 4
\end{ytableau}
\end{array}$}, \;
Q_5 := \scalebox{0.75}{
$\begin{array}{c} 
\begin{ytableau}
1 & 3 & 4 \\
2
\end{ytableau}
\end{array}$
}
\right\}.
\]
For $0 \leq k \leq 5$, let
$B_k := \{\gamma \in  \SG_4 \mid \text{$\Qtab(\gamma) = Q_l$ for some $1 \le l \le k$}\}$.
Then, $0 = \C B_0  \subset \C B_1 \subset \C B_2 \subset \C B_3 \subset \C B_4 \subset \C B_5 = \sfM_P$
is a filtration of $\sfM_P$, as seen in the figure: 
\[
\def \vp {1.5}
\def \hp {2.7}
\scalebox{0.875}{$
\begin{tikzpicture}[scale = 0.8]
%%%%%%%%%%%%%%%dashed lines
\node[right] at (-1.5*\hp, 0*\vp) {\textcolor{violet}{\small $B_5 \setminus B_4$}};
\draw[violet, dashed, thick] (-0.4*\hp, -2.2*\vp) -- (-1.4*\hp, -1.4*\vp) -- (-1.4*\hp, -0.6*\vp) -- (-0.4*\hp, 0.2*\vp) -- (0.4*\hp, 0.2*\vp) -- (0.4*\hp, -2.2*\vp) -- (-0.4*\hp, -2.2*\vp);

%%%%

\node[right] at (2.2*\hp, -1.3*\vp) {\textcolor{brown}{\small $B_4  \setminus B_3$}};
\draw[brown, dashed, thick] (1.6*\hp, -2.2*\vp) -- (0.6*\hp, -1.4*\vp) -- (0.6*\hp, -0.8*\vp) -- (1.4*\hp, -0.8*\vp) -- (2.4*\hp, -1.6*\vp) -- (2.4*\hp, -2.2*\vp) -- (1.6*\hp, -2.2*\vp);


%%%%

\node[right] at (-3.1*\hp, -2.5*\vp) {\textcolor{teal}{\small $B_3 \setminus B_2$}};
\draw[teal, dashed, thick] (-1.4*\hp, -4.2*\vp) -- (-2.4*\hp, -3.4*\vp) -- (-2.4*\hp, -1.8*\vp) -- (-1.6*\hp, -1.8*\vp) -- (-1.6*\hp, -3*\vp) -- (-0.6*\hp, -3.8*\vp) -- (-0.6*\hp, -4.2*\vp) -- (-1.4*\hp, -4.2*\vp);


%%%%%%
\node[right] at (2.2*\hp, -3.7*\vp) {\textcolor{blue}{\small $B_2 \setminus B_1$}};
\draw[blue, dashed, thick] (0.6*\hp, -4.2*\vp) -- (-0.4*\hp, -3.4*\vp) -- (-0.4*\hp, -2.8*\vp) -- (2.4*\hp, -2.8*\vp) -- (2.4*\hp, -3.4*\vp) -- (1.4*\hp, -4.2*\vp) -- (0.6*\hp, -4.2*\vp);


%%%%%%
\node[left] at (-0.5*\hp, -5*\vp) {\textcolor{red}{\small $B_1$}};
\draw[red, dashed, thick]
(-0.4*\hp, -5.2*\vp)-- (-0.4*\hp, -4.8*\vp) -- (0.4*\hp, -4.8*\vp) -- (0.4*\hp, -5.2*\vp) -- (-0.4*\hp, -5.2*\vp);



%%%node
\node[] at (0*\hp, 0*\vp) {$2134$};

\node[] at (-1*\hp, -1*\vp) {$3124$};
\node[] at (1*\hp, -1*\vp) {$2143$};

\node[] at (-2*\hp, -2*\vp) {$3214$};
\node[] at (0*\hp, -2*\vp) {$4123$};
\node[] at (2*\hp, -2*\vp) {$3142$};

\node[] at (-2*\hp, -3*\vp) {$4213$};
\node[] at (-0*\hp, -3*\vp) {$4132$};
\node[] at (2*\hp, -3*\vp) {$3241$};

\node[] at (-1*\hp, -4*\vp) {$4312$};
\node[] at (1*\hp, -4*\vp) {$4231$};


\node[] at (0*\hp, -5*\vp) {$4321$};

%%%%%%%%%%%%%%%%%%loop%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (0*\hp + 0.2*\hp, 0*\vp) {} edge [out=40,in=320, loop] ();
\node at (0*\hp + 0.625*\hp, 0*\vp) {\footnotesize $\pi_1$};

\node at (-1*\hp + 0.2*\hp, -1*\vp) {} edge [out=40,in=320, loop] ();
\node at (-1*\hp + 0.625*\hp, -1*\vp) {\footnotesize $\pi_2$};
\node at (1*\hp + 0.2*\hp, -1*\vp) {} edge [out=40,in=320, loop] ();
\node at (1*\hp + 0.75*\hp, -1*\vp) {\footnotesize $\pi_1, \pi_3$};


\node at (-2*\hp + 0.2*\hp, -2*\vp) {} edge [out=40,in=320, loop] ();
\node at (-2*\hp + 0.75*\hp, -2*\vp) {\footnotesize $\pi_1, \pi_2$};
\node at (0*\hp + 0.2*\hp, -2*\vp) {} edge [out=40,in=320, loop] ();
\node at (0*\hp + 0.625*\hp, -2*\vp) {\footnotesize $\pi_3$};
\node at (2*\hp + 0.2*\hp, -2*\vp) {} edge [out=40,in=320, loop] ();
\node at (2*\hp + 0.625*\hp, -2*\vp) {\footnotesize $\pi_2$};


\node at (-2*\hp + 0.2*\hp, -3*\vp) {} edge [out=40,in=320, loop] ();
\node at (-2*\hp + 0.75*\hp, -3*\vp) {\footnotesize $\pi_1, \pi_3$};
\node at (0*\hp + 0.2*\hp, -3*\vp) {} edge [out=40,in=320, loop] ();
\node at (0*\hp + 0.75*\hp, -3*\vp) {\footnotesize $\pi_2, \pi_3$};
\node at (2*\hp + 0.2*\hp, -3*\vp) {} edge [out=40,in=320, loop] ();
\node at (2*\hp + 0.75*\hp, -3*\vp) {\footnotesize $\pi_1, \pi_2$};

\node at (-1*\hp + 0.2*\hp, -4*\vp) {} edge [out=40,in=320, loop] ();
\node at (-1*\hp + 0.75*\hp, -4*\vp) {\footnotesize $\pi_2, \pi_3$};
\node at (1*\hp + 0.2*\hp, -4*\vp) {} edge [out=40,in=320, loop] ();
\node at (1*\hp + 0.75*\hp, -4*\vp) {\footnotesize $\pi_1, \pi_3$};

\node at (0*\hp + 0.2*\hp, -5*\vp) {} edge [out=40,in=320, loop] ();
\node at (0*\hp + 0.875*\hp, -5*\vp) {\footnotesize $\pi_1, \pi_2, \pi_3$};


%%%%%%%%%%%%%%%%%%%%%%EDGE%%%%%%%%%%%%%%%%%%%%%%%%
\draw[->] (-0.2*\hp, -0.25*\vp) -- (-0.8*\hp, -0.7*\vp);
\node at (-0.6*\hp, -0.4*\vp) {\footnotesize $\pi_2$};
\draw[->] (0.2*\hp, -0.25*\vp) -- (0.8*\hp, -0.7*\vp);
\node at (0.6*\hp, -0.4*\vp) {\footnotesize $\pi_3$};

%%%%
\draw[->] (-1.2*\hp, -1.25*\vp) -- (-1.8*\hp, -1.7*\vp);
\node at (-1.6*\hp, -1.4*\vp) {\footnotesize $\pi_1$};
\draw[->] (-0.8*\hp, -1.25*\vp) -- (-0.2*\hp, -1.7*\vp);
\node at (-0.4*\hp, -1.4*\vp) {\footnotesize $\pi_3$};


\draw[->] (1.2*\hp, -1.25*\vp) -- (1.8*\hp, -1.7*\vp);
\node at (1.6*\hp, -1.4*\vp) {\footnotesize $\pi_2$};
%%%%
\draw[->] (-2*\hp, -2.25*\vp) -- (-2*\hp, -2.7*\vp);
\node at (-2.1*\hp, -2.45*\vp) {\footnotesize $\pi_3$};

\draw[->] (-0.2*\hp, -2.25*\vp) -- (-1.8*\hp, -2.7*\vp);
\node at (-1.25*\hp, -2.45*\vp) {\footnotesize $\pi_1$};
\draw[->] (0*\hp, -2.25*\vp) -- (0*\hp, -2.7*\vp);
\node at (-0.1*\hp, -2.45*\vp) {\footnotesize $\pi_2$};

\draw[->] (1.8*\hp, -2.25*\vp) -- (0.2*\hp, -2.7*\vp);
\node at (0.75*\hp, -2.45*\vp) {\footnotesize $\pi_3$};
\draw[->] (2*\hp, -2.25*\vp) -- (2*\hp, -2.7*\vp);
\node at (2.1*\hp, -2.45*\vp) {\footnotesize $\pi_1$};

%%%%
\draw[->] (-1.8*\hp, -3.25*\vp) -- (-1.2*\hp, -3.7*\vp);
\node at (-1.4*\hp, -3.4*\vp) {\footnotesize $\pi_2$};

\draw[->] (0.2*\hp, -3.25*\vp) -- (0.8*\hp, -3.7*\vp);
\node at (0.6*\hp, -3.4*\vp) {\footnotesize $\pi_1$};
\draw[->] (1.8*\hp, -3.25*\vp) -- (1.2*\hp, -3.7*\vp);
\node at (1.4*\hp, -3.4*\vp) {\footnotesize $\pi_3$};

%%%%
\draw[->] (-0.8*\hp, -4.25*\vp) -- (-0.2*\hp, -4.7*\vp);
\node at (-0.4*\hp, -4.4*\vp) {\footnotesize $\pi_1$};
\draw[->] (0.8*\hp, -4.25*\vp) -- (0.2*\hp, -4.7*\vp);
\node at (0.4*\hp, -4.4*\vp) {\footnotesize $\pi_2$};
\end{tikzpicture}$}
\]
Moreover, 
since $\ch([\C B_{k}/\C B_{k-1}]) = s_{\sh(Q_k)^\rmt}$ for all $1 \leq k \leq 5$,
it is a distinguished filtration with respect to $\{s_{\lambda} \mid \lambda \vdash 4\}$.
\end{example}














\section{A tableau description of $\sfM_P$ for $P \in \sfRSP_n$}\label{sec: appendix}



Let $\lambda/\mu$ be a skew partition of size $n$.
Define $X_{\lambda/\mu}$ to be the $H_n(0)$-module 
with the underlying space $\C\SYT(\lambda/\mu)$ and 
with the $H_n(0)$-action given by 
\[
\pi_i \cdot T =
\begin{cases}
T & \text{if $i$ is strictly left of $i+1$ in $T$,}\\
0 & \text{if $i$ and $i+1$ are in the same column of $T$,}\\
s_i \cdot T & \text{if $i$ is strictly right of $i + 1$ in $T$}
\end{cases}
\]
for $1 \leq i \leq n-1$ and $T \in \SYT(\lambda/\mu)$.
Here, $s_i \cdot T$ is the tableau obtained from $T$ by swapping $i$ and $i+1$.
One can see that 
this $H_n(0)$-action is well defined and 
%$X_{\lambda/\mu}$ is a well defined $H_n(0)$-module and
$X_{\lambda/\mu} \cong \sfM_{\sfposet(\tau_0^{\lambda/\mu})}$.
\cref{thm: classification} says that $\sfM_P \cong X_{\sh(\tau_P)}$ for $P \in \sfRSP_n$, and $X_{\lambda/\mu} \not\cong X_{\nu/\kappa}$ for distinct skew partitions $\lambda/\mu$, $\nu/\kappa$ of size $n$.









%The following proposition shows how $X_{\lambda/\mu}$'s behave with respect to induction product and restrictions.

\begin{proposition}\label{prop: property of X}
We have the following isomorphisms.
\begin{enumerate}[label = {\rm (\arabic*)}]
    \item For skew partitions $\lambda/\mu$ of size $n$ and $\nu/\kappa$ of size $m$, 
    \[
    X_{\lambda/\mu} \otimes X_{\nu/\kappa} \uparrow_{H_n(0) \otimes H_m(0)}^{H_{n+m}(0)} \cong X_{\lambda/\mu \oplus \nu/\kappa} \quad \text{as $H_{n+m}(0)$-modules.}
    \]
    \item For a skew partition $\lambda/\mu$ of size $n$ and
     $1 \leq k \leq n-1$, 
    \[
    X_{\lambda/\mu} \downarrow_{H_k(0) \otimes H_{n-k}(0)} \cong \bigoplus_{\substack{|\nu/\mu| = k \\ \mu \subset \nu \subset \lambda}} X_{\overline{\nu/\mu}} \otimes  X_{\overline{\lambda/\nu}} \quad \text{as $H_k(0) \otimes H_{n-k}(0)$-modules}.
    \]
    Here, $\overline{\nu/\mu}$ and $\overline{\lambda/\nu}$ denote the basic skew partitions whose Young diagrams are obtained from $\tyd(\nu/\mu)$ and $\tyd(\lambda/\nu)$, respectively,  by removing empty rows and empty columns.
\end{enumerate}
\end{proposition}









\section{Final remarks}
In \cref{thm: classification}, we show that for $P, Q \in \sfRSP_n$,
\begin{align}\label{eq: final rem class}
\sfM_P \cong \sfM_Q \quad \text{if and only if} \quad \sh(\tau_P) = \sh(\tau_Q).
\end{align}
Since $\sfRSP_n = \sfRP_n \cap \sfSP_n$, 
 it would be natural to consider the classification problem for $\{\sfM_P \mid P \in \sfSP_n\}$ and $\{\sfM_P \mid P \in \sfRP_n\}$.

\begin{enumerate}[label = {\rm (\arabic*)}, leftmargin=4ex]
\item 
Although the notion `the shape of $\tau_P$' is available for $P \in \sfSP_n$, \cref{eq: final rem class} does not hold for $P, Q \in \sfSP_n$ in general (see \cite[Section 7.1.1]{23KLO}).  
%Although the notion `the shape of $\tau_P$' is available for $P \in \sfSP_n$, we see that classification given in ~\cref{eq: final rem class} cannot be extended to $\{\sfM_P  \mid P \in \sfSP_n\}$ (see \cite[Section 7.1.1]{23KLO}).

\item Unlike (i), the notion `the shape of $\tau_P$' is not available for $P \in \sfRP_n$ in general.
For this reason, we modify \cref{eq: final rem class}  in the following form:
for $P, Q \in \sfRSP_n$, 
%This leads us to introduce the following classification: for $P, Q \in \sfRSP_n$,
\begin{align}\label{eq: conj on class}
\sfM_P \cong \sfM_Q \quad \text{if and only if} \quad \Sigma_L(P) \Deq \Sigma_L(Q),
\end{align}
which can be obtained by combining \cref{thm: equiv class X lam mu}
and \cref{thm: classification}.
Since the equivalence relation $\Deq$ is defined on $\rmInt(n) = \{\Sigma_L(P) \mid P \in \sfRP_n\}$,  
we expect that this classification can be extended  to $\sfRP_n$ in its current form.
The validity of this expectation has been checked for values of $n$ up to 6 with the aid of the computer program \textsc{SageMath}.
Also, we show that \cref{eq: conj on class} holds when $P \in \sfRSP_n$, $Q \in \sfRP_n$, and $\ch([\sfM_P])$ is a Schur function.
For more detail, see ~\cite[Section 7.1.2]{23KLO}. 
\end{enumerate}

\printbibliography

\end{document}
