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\title[Modules of the 0-Hecke algebras for genomic Schur functions]{Modules of the 0-Hecke algebras for genomic Schur functions}


\author{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 Korea government (NRF-2020R1A5A1016126 and NRF-2022R1A2C1004045).}\addressmark{1} \and 
Semin Yoo\thanks{\href{mailto:syoo19@kias.re.kr}{syoo19@kias.re.kr}. Semin Yoo was supported by the KIAS Individual Grant (CG082701) at Korea Institute for Advanced Study.}\addressmark{2}}


\address{\addressmark{1}Center for quantum structures in modules and spaces, Seoul National University, Seoul 08826, Republic of Korea \\ \addressmark{2}School of Computational Sciences, Korea Institute for Advanced Study, 85 Hoegiro Dongdae\-mun-gu, Seoul 02455, Republic of Korea}


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\abstract{
We construct an $H_m(0)$-module $\mathbf{G}_{\lambda;m}$ whose image under the quasisymmetric characteristic is the $m$th degree homogeneous component of the genomic Schur function $U_\lambda$ by defining an $H_m(0)$-action on increasing gapless tableaux. We then provide a direct sum decomposition of $\mathbf{G}_{\lambda;m}$ and show that each summand of this decomposition is isomorphic to a weak Bruhat interval module.
}


\keywords{0-Hecke algebra, weak Bruhat order, genomic Schur function, quasisymmetric characteristic}


\usepackage[backend=bibtex]{biblatex}
\addbibresource{references.bib}


\begin{document}

\maketitle



\section{Introduction}

Let $X = \mathrm{Gr}_k(\mathbb{C}^n)$ be the Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$.
Since the early 2000s, several combinatorial interpretations for the \emph{$K$-theoretic Littlewood-Richardson rule} have been introduced.
For instance, see \cite{02Buch, 17PY2, 09TY, 06Vakil}.
In particular, Pechenik and Yong \cite{17PY2} gave a combinatorial interpretation by using \emph{genomic tableaux}.
Therein, they defined a symmetric function $U_\lambda$, called the \emph{genomic Schur function}, as a generating function for genomic tableaux of shape $\lambda$ for all partition $\lambda$.
Further, they proved that $\{U_\lambda \mid \text{$\lambda$ is a partition}\}$ is a basis for the ring of symmetric functions and pointed out that genomic Schur functions are not Schur-positive in general.
As an alternative positivity, Pechenik~\cite{20Pechenik} showed that genomic Schur functions are fundamental positive.
Specifically, for any partition $\lambda$,
\begin{align*}
U_\lambda = \sum_{T \in \mathrm{IGLT}(\lambda)} F_{\mathrm{comp}(T)},
\end{align*}
where $\mathrm{IGLT}(\lambda)$ is the set of \emph{increasing gapless tableaux} of shape $\lambda$, $\mathrm{comp}(T)$ is the composition associated to $T$, and $F_{\mathrm{comp}(T)}$ is the \emph{fundamental quasisymmetric function} associated to $\mathrm{comp}(T)$.
For the precise definitions, see Subsection~\ref{subsec: Genomic Schur}.


The \emph{$0$-Hecke algebra $H_n(0)$} is the $\mathbb{C}$-algebra obtained from the Hecke algebra $H_n(q)$ by specializing $q$ to $0$.
Duchamp, Krob, Leclerc, and Thibon \cite{96DKLT} introduced a ring isomorphism, called \emph{quasisymmetric characteristic},
$$
\mathrm{ch}: \mathcal{G}_{0}(H_{\bullet}(0)) \rightarrow \mathrm{QSym}, \quad [\mathbf{F}_\alpha] \mapsto F_\alpha.
$$
Here, $\mathcal{G}_{0}(H_{\bullet}(0))$ is the Grothendieck ring associated to $0$-Hecke algebras and $\mathrm{QSym}$ is the ring of quasisymmetric functions.
In view of this correspondence, there have been considerable attempts to provide a representation theoretic interpretation of noteworthy quasisymmetric functions by constructing appropriate $0$-Hecke modules.
For instance, see \cite{22BS, 15BBSSZ, 22CKNO, 20Searles, 19TW, 15TW}.
Recently, Jung, Kim, Lee, and Oh \cite{22JKLO} introduced the \emph{weak Bruhat interval module $\mathsf{B}(\sigma, \rho)$} to provide a unified method to study the $H_n(0)$-modules in the papers mentioned above.
Here, $\sigma$ and $\rho$ are permutations in the symmetric group $\mathfrak{S}_n$.

The purpose of this paper is to provide a nice representation theoretic interpretation of genomic Schur functions.
First, for each $1 \le m \le n$, we construct an $H_m(0)$-module $\mathbf{G}_{\lambda;m}$ by defining an $H_m(0)$-action on the $\mathbb{C}$-span of the set $\mathrm{IGLT}(\lambda)_{m}$ of increasing gapless tableaux of shape $\lambda$ with maximum entry $m$.
And, we see that the image of $\mathbf{G}_{\lambda;m}$ under the quasisymmetric characteristic is the $m$th homogeneous component of $U_{\lambda}$.
Next, we define an equivalence relation $\sim_{\lambda;m}$ on $\mathrm{IGLT}(\lambda)_{m}$ and show that the $\mathbb{C}$-span of each equivalence class is closed under the $H_m(0)$-action.
Thus, we obtain a direct sum decomposition 
\begin{equation*}
\mathbf{G}_{\lambda;m} = \bigoplus_{E \in \mathcal{E}_{\lambda;m}} \mathbf{G}_E,
\end{equation*}
where $\mathcal{E}_{\lambda;m}$ is the set of all equivalence classes with respect to $\sim_{\lambda;m}$ and $\mathbf{G}_E$ is the submodule of $\mathbf{G}_{\lambda;m}$ whose underlying space is the $\mathbb{C}$-span of $E$.
Finally, we show that $\mathbf{G}_E$ is isomorphic to a weak Bruhat interval module for $E \in \mathcal{E}_{\lambda;m}$.
To do this, we prove that there exist unique source tableau $T_E$ and sink tableau $T_E'$ in $E$.
In addition, we assign a permutation $\mathsf{read}(T)$, called \emph{standardized reading word}, to each $T \in E$.
With these preparations, we show that
\[
\mathbf{G}_E \cong \mathsf{B}(\mathsf{read}(T_E), \mathsf{read}(T_E')).
\]

We end with providing an avenue for future research.
This paper is an extended abstract of our paper \cite{22KY}.





\section{Preliminaries}\label{Sec: prem}

Given any integers $m$ and $n$, define $[m,n]$ to be the set $\{k \in \mathbb{Z} \mid m \le k \le n\}$ if $m \le n$ or the empty set otherwise.
Throughout this section, $n$ denotes a nonnegative integer.


\subsection{Compositions and Diagrams}\label{subsec: comp and diag}

A \emph{composition} $\alpha$ of a nonnegative integer $n$, 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 $\alpha = (\alpha_1, \alpha_2, \ldots,\alpha_{\ell(\alpha)}) \models n$, we define $\mathrm{set}(\alpha) := \{\alpha_1,\alpha_1+\alpha_2,\ldots, \sum_{i = 1}^{\ell(\alpha)-1} \alpha_i\} \subseteq [1,n-1]$.



If $\lambda = (\lambda_1,\lambda_2,\ldots, \lambda_{\ell(\lambda)}) \models n$ satisfies $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_{\ell(\lambda)}$, then we say that $\lambda$ is a \emph{partition} of $n$ and denote it by $\lambda \vdash n$.
For $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_{\ell(\lambda)}) \vdash n$, 
we define the \emph{Young diagram} $\mathtt{yd}(\lambda)$ of $\lambda$ by a left-justified array of $n$ boxes where the $i$th row from the top has $\lambda_i$ boxes for $1 \le i \le \ell(\lambda)$.
We say that a box in $\mathtt{yd}(\lambda)$ is \emph{in the $i$th row} if it is in the $i$th row from the top and \emph{in the $j$th column} if it is in the $j$th column from the left.
We denote by $(i,j)$ the box in the $i$th row and $j$th column.
Denoting $(i,j) \in \mathtt{yd}(\lambda)$ means that $1 \le i \le \ell(\lambda)$ and $1 \le j \le \lambda_i$.
We also say that a lattice point on $\mathtt{yd}(\lambda)$ is \emph{in the $i$th row} if it is in the $(i+1)$st horizontal line from the top and \emph{in the $j$th column} if it is in the $(j+1)$st vertical line from the left.
We denote by $(\underline{i,j})$ the lattice point in the $i$th row and $j$th column.
For example, if $\lambda = (3,2,2)$, then \\[-1.5ex]
\begin{displaymath}
\def \hp {0.65}
\def \vp {0.65}
\scalebox{0.85}{$
\begin{tikzpicture}
\node[] at (-1, 0.95) {\large $\mathtt{yd}(\lambda) =$};
\draw (0*\hp,0*\vp) -- (2*\hp,0*\vp) -- (2*\hp,2*\vp) -- (3*\hp,2*\vp) -- (3*\hp,3*\vp) -- (0*\hp,3*\vp) -- (0*\hp,0*\vp);
\draw (0*\hp,2*\vp) -- (2*\hp,2*\vp);
\draw (0*\hp,1*\vp) -- (2*\hp,1*\vp);
\draw (1*\hp,0*\vp) -- (1*\hp,3*\vp);
\draw (2*\hp,2*\vp) -- (2*\hp,3*\vp);

\draw[fill = red!50] (2*\hp,2*\vp) -- (2*\hp,3*\vp) -- (3*\hp,3*\vp) -- (3*\hp,2*\vp) -- (2*\hp,2*\vp);
\node at (2.5*\hp,2.5*\vp) {\scriptsize $(1,3)$};

\node[color = blue] at (0,0) {$\bullet$}; 
\node at (0.5*\hp,-0.4*\vp) {\scriptsize $(\underline{3,0})$};
\node[] at (2.2, 0.75) {,};
\end{tikzpicture}
$}
\end{displaymath}
~\\[-4.5ex]
the box $(1,3)$ is the box filled with red, and 
the lattice point $(\underline{3,0})$ is the point marked by the blue dot.
A \emph{filling of $\mathtt{yd}(\lambda)$} is a function $T: \mathtt{yd}(\lambda) \rightarrow \mathbb{Z}_{>0}$. 
Throughout this paper, we assume that
\[
\begin{array}{ll}
T((i,j)) = \infty & \text{if $(i,j) \in (\mathbb{Z}_{>0} \times \mathbb{Z}_{>0}) \setminus \mathtt{yd}(\lambda)$}
\quad \text{and} \\[0.5ex]
T((i,j)) = -\infty & \text{if $(i,j) \in (\mathbb{Z} \times \mathbb{Z}) \setminus (\mathbb{Z}_{>0} \times \mathbb{Z}_{>0})$.}
\end{array}
\]
For any filling $T$ of $\mathtt{yd}(\lambda)$,
let 
$\max (T) := \max \{T((i,j)) \mid (i,j) \in \mathtt{yd}(\lambda)\}$.




\subsection{The $0$-Hecke algebra and the quasisymmetric characteristic}\label{subsec: 0-Hecke alg}
To begin with, we recall that the symmetric group $\mathfrak{S}_n$ is generated by simple transpositions $s_i := (i,i+1)$ with $1 \le i \le n-1$.
An expression for $\sigma \in \mathfrak{S}_n$ of the form $s_{i_1} s_{i_2} \cdots s_{i_p}$ that uses the minimal number of simple transpositions is called a \emph{reduced expression} for $\sigma$. 
The number of simple transpositions in any reduced expression for $\sigma$, denoted by $\ell(\sigma)$, is called the \emph{length} of $\sigma$.

The \emph{$0$-Hecke algebra $H_n(0)$} is the $\mathbb{C}$-algebra generated by $\pi_1, \pi_2, \ldots,\pi_{n-1}$ subject to the following three 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 $|i-j| \ge 2$.
Pick up any reduced expression $s_{i_1} s_{i_2} \cdots s_{i_p}$ for a permutation $\sigma \in \mathfrak{S}_n$. 
We define the element $\pi_{\sigma}$ of $H_n(0)$ by $\pi_{\sigma} := \pi_{i_1} \pi_{i_2} \cdots \pi_{i_p}$
It is well known that $\pi_{\sigma}$ is independent of the choice of reduced expressions, and $\{\pi_\sigma \mid \sigma \in \mathfrak{S}_n\}$ is a basis for $H_n(0)$.

In \cite{79Norton}, Norton classified all irreducible modules of the $0$-Hecke algebras. 
It was shown that there are $2^{n-1}$ distinct 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 $v_\alpha$ whose $H_n(0)$-action is given by
\[
\pi_i \cdot v_\alpha = \begin{cases}
0 & i \in \mathrm{set}(\alpha),\\
v_\alpha & i \notin \mathrm{set}(\alpha),
\end{cases}
\qquad (1\le i \le n-1).
\]


Let $\mathcal{R}(H_n(0))$ denote the $\mathbb{Z}$-span of the isomorphism classes of finite dimensional $H_n(0)$-modules. 
The isomorphism class corresponding to an $H_n(0)$-module $M$ will be denoted by $[M]$. 
The \emph{Grothendieck group} $\mathcal{G}_0(H_n(0))$ is the quotient of $\mathcal{R}(H_n(0))$ modulo the relations $[M] = [M'] + [M'']$ whenever there exists a short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$. 
The irreducible $H_n(0)$-modules form a free $\mathbb{Z}$-basis for $\mathcal{G}_0(H_n(0))$. 
Let $\mathcal{G}_{0}(H_{\bullet}(0)) := \bigoplus_{n \ge 0} \mathcal{G}_0(H_n(0))$ be the ring equipped with the induction product.
In~\cite{96DKLT}, Duchamp, Krob, Leclerc, and Thibon revealed a deep connection between $\mathcal{G}_{0}(H_{\bullet}(0))$ and the ring $\mathrm{QSym}$ of quasisymmetric functions by providing a ring isomorphism
\begin{align*}
\mathrm{ch} : \mathcal{G}_{0}(H_{\bullet}(0)) \rightarrow \mathrm{QSym}, \quad [\mathbf{F}_{\alpha}] \mapsto F_{\alpha},
\end{align*}
called \emph{quasisymmetric characteristic}.
Here, $F_\alpha$ is the \emph{fundamental quasisymmetric function}.




\subsection{Weak Bruhat interval modules of the $0$-Hecke algebra}\label{subsec: WBIM}




Given $\sigma \in \mathfrak{S}_n$, let
$\mathrm{Des}_L(\sigma) := \{i \in [1,n-1] \mid \ell(s_i \sigma) < \ell(\sigma) \}$.
The \emph{left weak Bruhat order $\preceq_L$} on $\mathfrak{S}_n$ is the partial order on $\mathfrak{S}_n$ whose covering relation $\preceq_L^c$ is defined as follows: $\sigma \preceq_L^c s_i \sigma$ if and only if $i \notin \mathrm{Des}_L(\sigma)$.
Given $\sigma, \rho \in \mathfrak{S}_n$, the \emph{left weak Bruhat interval from $\sigma$ to $\rho$}, denoted by $[\sigma,\rho]_L$, is the closed interval $\{\gamma \in \mathfrak{S}_n \mid \sigma \preceq_L \gamma \preceq_L \rho\}$.



\begin{definition}{\rm (\cite{22JKLO})}
Let $\sigma, \rho \in \mathfrak{S}_n$.
The \emph{weak Bruhat interval module associated to $[\sigma,\rho]_L$}, denoted by $\mathsf{B}(\sigma,\rho)$, is the $H_n(0)$-module with the underlying space $\mathbb{C}[\sigma,\rho]_L$ and with the $H_n(0)$-action defined by \vspace{-1ex}
\begin{align*}
\pi_i \cdot \gamma := \begin{cases}
\gamma & \text{if $i \in \mathrm{Des}_L(\gamma)$}, \\
0 & \text{if $i \notin \mathrm{Des}_L(\gamma)$ and $s_i\gamma \notin [\sigma,\rho]_L$,} \\
s_i \gamma & \text{if $i \notin \mathrm{Des}_L(\gamma)$ and $s_i\gamma \in [\sigma,\rho]_L$}.
\end{cases} 
\end{align*}
\end{definition}



\subsection{Genomic Schur functions}
\label{subsec: Genomic Schur}


Given $\lambda \vdash n$, an \emph{increasing gapless tableau} of shape $\lambda$ is a filling of $\mathtt{yd}(\lambda)$ such that
the entries in each row strictly increase from left to right,
the entries in each column strictly increase from top to bottom, and
the set $T^{-1}(i)$ is nonempty for all $1 \le i \le \max (T)$.
Let $\mathrm{IGLT}(\lambda)$ be the set of all increasing gapless tableaux of shape $\lambda$.
Given $T \in \mathrm{IGLT}(\lambda)$ and $1 \le i \le \max (T)$, let $\mathsf{Top}_i(T)$ (resp. $\mathsf{Bot}_i(T)$) be the highest (resp. lowest) box in $T$ having the entry $i$.
Let 
\begin{align*}
(r_\mathsf{b}^{(i)}(T), c_\mathsf{b}^{(i)}(T)) := \mathsf{Bot}_i(T)
\quad \text{and} \quad 
(r_\mathsf{t}^{(i)}(T), c_\mathsf{t}^{(i)}(T)) := \mathsf{Top}_i(T).
\end{align*}
If $T$ is clear in the context, we simply write $r_\mathsf{b}^{(i)}, c_\mathsf{b}^{(i)}, r_\mathsf{t}^{(i)}$, and $c_\mathsf{t}^{(i)}$ instead of $r_\mathsf{b}^{(i)}(T)$, $c_\mathsf{b}^{(i)}(T)$, $r_\mathsf{t}^{(i)}(T)$, and $c_\mathsf{t}^{(i)}(T)$, respectively.
We call an index $i \in [1, \max (T)-1]$ a \emph{descent of $T$} if there is some instance of $i$ strictly above some instance of $i+1$ in $T$.
Let $\mathrm{Des}(T)$ be the set of all descents of $T$ and let $\mathrm{comp}(T) := \mathrm{comp}(\mathrm{Des}(T))$.




\begin{definition}{\rm (\cite{20Pechenik, 17PY2})}
For $\lambda \vdash n$, the \emph{genomic Schur function} $U_\lambda$ is defined by
\[
U_\lambda := \sum_{T \in \mathrm{IGLT}(\lambda)} F_{\mathrm{comp}(T)}.
\]
\end{definition}

Given $1 \le m \le n$, we define  
$$
\mathrm{IGLT}(\lambda)_{m} := \{T \in \mathrm{IGLT}(\lambda) \mid \max (T) = m\}
\quad \text{and} \quad
U_{\lambda;m} := \sum_{T \in \mathrm{IGLT}(\lambda)_{m}} F_{\mathrm{comp}(T)}.
$$
From the definition, it immediately follows that $U_{\lambda;m}$ is the $m$th degree homogeneous component of $U_{\lambda}$.

Hereafter, we assume that $n$ is a positive integer, $m$ is a positive integer less than or equal to $n$, and $\lambda$ is a partition of $n$, unless otherwise stated.





\section{0-Hecke modules from increasing gapless tableaux}
\label{Sec: 0-Hecke action}

\subsection{An $H_{m}(0)$-module for $U_{\lambda;m}$}\label{Subsec: 0-Hecke modules for genomic Shcur functions}

We start by introducing the necessary definitions.

\begin{definition}\label{def: attacking descent}
Given $T \in \mathrm{IGLT}(\lambda)$ and $1 \le i \le \max (T) -1$, we say that $i$ is an \emph{attacking descent} if $i \in \mathrm{Des}(T)$, and either \vspace{-1ex} \begin{enumerate}[label = \textrm{(\alph*)}, itemsep = -0.5ex]
\item
there exists $(j,k) \in \mathtt{yd}(\lambda)$ such that $T( (j,k) ) = i$ and $T( (j +1, k) ) = i + 1$, or

\item 
there exists a box $B \in T^{-1}(i+1)$ placed weakly above $\mathsf{Bot}_i(T)$.
\end{enumerate}
\end{definition}





Take any $1 \le m \le n$.
For each $1 \le i \le m-1$, we define a linear operator $\boldsymbol{\uppi}_i: \mathbb{C} \, \mathrm{IGLT}(\lambda)_{m} \rightarrow \mathbb{C} \, \mathrm{IGLT}(\lambda)_{m}$ by \vspace{-1.5ex}
\begin{align}\label{eq: def of bfpi}
\boldsymbol{\uppi}_i (T) := 
\begin{cases}
T & \text{if $i$ is not a descent of $T$,}\\
0 & \text{if $i$ is an attacking descent of $T$,}\\ 
s_i \cdot T & \text{if $i$ is a non-attacking descent of $T$}
\end{cases}
\end{align}~\\[-2ex]
for $T \in \mathrm{IGLT}(\lambda)_{m}$ and extending it by linearity.
Here, $s_i \cdot T$ is the tableau obtained from $T$ by replacing $i$ and $i+1$ with $i+1$ and $i$, respectively. 
By proving $\boldsymbol{\uppi}_i^2 = \boldsymbol{\uppi}_i~(1\le i \le m-1)$,
$\boldsymbol{\uppi}_i \boldsymbol{\uppi}_{i+1} \boldsymbol{\uppi}_i = \boldsymbol{\uppi}_{i+1} \boldsymbol{\uppi}_i
\boldsymbol{\uppi}_{i+1}~(1\le i \le m-2)$, and 
$\boldsymbol{\uppi}_i \boldsymbol{\uppi}_{j} =  \boldsymbol{\uppi}_j \boldsymbol{\uppi}_i~(1\le i,j \le m-1 \text{ with } |i-j|>1)$, we obtain the following theorem.
For details, see \cite[Subsection 3.1]{22KY}.





\begin{theorem}\label{thm: 0-Hecke action on IGLT}
For any $1 \le m \le n$,
the operators $\boldsymbol{\uppi}_1, \boldsymbol{\uppi}_2,\ldots, \boldsymbol{\uppi}_{m-1}$ satisfy the same relations as the generators $\pi_1, \pi_2, \ldots \pi_{m-1}$ for $H_m(0)$.
In other words, $\boldsymbol{\uppi}_1, \boldsymbol{\uppi}_2,\ldots, \boldsymbol{\uppi}_{m-1}$ define an $H_m(0)$-action on $\mathbb{C} \, \mathrm{IGLT}(\lambda)_{m}$.
\end{theorem}

\ytableausetup{boxsize=2ex}


\begin{example}\label{eg: G 211}
(1) When $T = \begin{array}{l}
\scalebox{0.8}{$
\begin{ytableau}
\color{red} 1 & \large \color{red} 2 & \large \color{red} 3 & \large \color{red} 6 \\
\large \color{red} 2 & \large \color{red} 3 & \large \color{red} 5 & \large 7 \\
\large 4 & \large \color{red} 6
\end{ytableau}$}
\end{array}$, we have
$\boldsymbol{\uppi}_3 (T) = s_3 \cdot T$, $\boldsymbol{\uppi}_4 (T) = T$, and $\boldsymbol{\uppi}_i (T) = 0$ for $i = 1,2,5,6$.
Here, the indices in red are used to indicate the descents of the tableau.
\medskip

\noindent 
(2) Note that
$\mathrm{IGLT}( (2,1,1) ) = \left\{
\begin{array}{l}
\scalebox{0.8}{$
\begin{ytableau}
1 & \color{red} 2\\
\color{red} 3 \\
4
\end{ytableau}
$}
\end{array},\ 
\begin{array}{l}
\scalebox{0.8}{$
\begin{ytableau}
\color{red} 1 & \color{red} 3\\
2 \\
4
\end{ytableau}
$}
\end{array},\ 
\begin{array}{l}
\scalebox{0.8}{$
\begin{ytableau}
\color{red} 1 & 4\\
\color{red} 2 \\
3
\end{ytableau}
$}
\end{array},\ 
\begin{array}{l}
\scalebox{0.8}{$
\begin{ytableau}
\color{red} 1 & \color{red} 2\\
\color{red} 2 \\
3
\end{ytableau}
$}
\end{array},\ 
\begin{array}{l}
\scalebox{0.78}{$
\begin{ytableau}
\color{red} 1 & 3\\
\color{red} 2 \\
3
\end{ytableau}
$}
\end{array}
\right\}$.
One can see that $U_{(2,1,1)} = (F_{(2,1,1)} + F_{(1,2,1)} + F_{(1,1,2)}) + 2F_{(1,1,1)}$.
The following figures illustrate the $H_m(0)$-action on $\mathbf{G}_{(2,1,1);m}$ for $m = 3,4$: 
\vspace{-1ex}
\[
\scalebox{0.78}{$
\begin{tikzpicture}
\def \hp {4}
\def \vp {0.2}
% U_{(2,1,1)}

% 1st column

% 1st row from top
\node[] at (0 -0.5*\hp,2) {
\begin{ytableau}
1 & \color{red} 2\\
\color{red} 3 \\
4
\end{ytableau}
};

% loop action
\node at (0.5-0.5*\hp, 2.1) {} edge [out=40,in=320, loop] ();
\node[] at (1.4-0.5*\hp,2.1) {$\pi_1$};

% outgoing action
% to another tableau
\draw [->] (0-0.5*\hp, 1 + \vp) -- (0-0.5*\hp,0.5 + \vp); 
\node at (0.3-0.5*\hp, 0.75 + \vp) {$\pi_2$};

% to zero
\draw [->] (0.6-0.5*\hp, 1.15 + \vp) -- (1.2-0.5*\hp, 0.75 + \vp);
\node at (1.1-0.5*\hp, 1.1 + \vp) {$\pi_3$};
\node at (1.5-0.5*\hp,0.6 + \vp) {$0$};


% 2nd row from top
\node[] at (0-0.5*\hp,-0.5 + 2*\vp) {
\begin{ytableau}
\color{red} 1 & \color{red} 3 \\
2 \\
4
\end{ytableau}
};

% loop action
\node at (0.5-0.5*\hp, -0.4 + 2*\vp) {} edge [out=40,in=320, loop] ();
\node[] at (1.4-0.5*\hp,-0.4 + 2*\vp) {$\pi_2$};

% outgoing action
% to another tableau
\draw [->] (0-0.5*\hp,-2.5 + 1.05 + 2.5*\vp) -- (0-0.5*\hp,-2.5 + 0.55 + 2.5*\vp); 
\node at (0.3-0.5*\hp, -2.5 + 0.9 + 2*\vp) {$\pi_3$};

% to zero
\draw [->] (0.6-0.5*\hp, -2.5 + 1.15 + 2.5*\vp) -- (1.2-0.5*\hp, -2.5 + 0.75 + 2.5*\vp);
\node at (1.1-0.5*\hp, -2.5 + 1.1 + 2.5*\vp) {$\pi_1$};
\node at (1.5-0.5*\hp, -2.5 + 0.6 + 2.5*\vp) {$0$};


% 3nd row from top
\node[] at (0-0.5*\hp, -2.5 -0.5 + 3.75*\vp) {
\begin{ytableau}
\color{red} 1 & 4 \\
\color{red} 2 \\
3
\end{ytableau}
};

% loop action
\node at (0.5-0.5*\hp, -2.5 -0.4 + 3.75*\vp) {} edge [out=40,in=320, loop] ();
\node[] at (1.4-0.5*\hp, -2.5 -0.4 + 3.75*\vp) {$\pi_3$};

% outgoing action
% to another tableau
\draw [->] (0-0.5*\hp,-5 + 1.05 + 4.5*\vp) -- (0-0.5*\hp,-5 + 0.55 + 4.5*\vp); 
\node at (0.6-0.5*\hp, -5 + 0.9 + 4*\vp) {$\pi_1, \pi_2$};


% 4th row from top
\node[] at (0-0.5*\hp,-4.7 + 4.5*\vp) {$0$};



%2nd column

% 1st row from top
\node[] at (\hp, 2) {
\begin{ytableau}
\color{red} 1 & \color{red} 2\\
\color{red} 2 \\
3
\end{ytableau}
};

% outgoing action
% to another tableau
\draw [->] (0 + \hp, 1 + \vp) -- (0 + \hp,0.5 + \vp); 
\node at (0.7 + \hp, 0.75 + \vp) {$\pi_1, \pi_2$};



% 2nd row from top
\node[] at (\hp, 0.25 + \vp) {$0$};


% 3rd column

% 1st row from top
\node[] at (2*\hp, 2) {
\begin{ytableau}
\color{red} 1 & 3\\
\color{red} 2 \\
3
\end{ytableau}
};

% outgoing action
% to another tableau
\draw [->] (0 + 2*\hp, 1 + \vp) -- (0 + 2*\hp,0.5 + \vp); 
\node at (0.7 + 2*\hp, 0.75 + \vp) {$\pi_1, \pi_2$};

% 2nd row from top
\node[] at (2*\hp, 0.25 + \vp) {$0$};


\node[] at (1.5*\hp, 2) {\large $\oplus$};

\node[] at (0-0.5*\hp, -4.5 ) {\fontsize{15}{15} $\mathbf{G}_{(2,1,1);4}$};
\node[] at (0 + 1.5*\hp, -0.5) {\fontsize{15}{15} $\mathbf{G}_{(2,1,1);3}$};
\end{tikzpicture}
$}
\]
\end{example}
\vspace{-1ex}

The following proposition follows immediately from \eqref{eq: def of bfpi} and \cref{thm: 0-Hecke action on IGLT}.

\begin{proposition}\label{Prop: characteristic image of G}
For any $\lambda \vdash n$ and $1 \le m \le n$, $\mathrm{ch}([\mathbf{G}_{\lambda;m}]) = U_{\lambda;m}$. 
Consequently, 
$\sum_{1 \le m \le n} \mathrm{ch}([\mathbf{G}_{\lambda;m}]) = U_\lambda$.
\end{proposition}



\begin{remark}
\label{rem: counterexample for indecomposable module}
In \cite[Theorem 4.7]{02DHT}, Duchamp, Hivert, and Thibon described the \emph{Ext-quiver} of $H_m(0)$. 
According to their result, for any $\alpha \models m$, we have $\mathrm{Ext}^1_{H_m(0)}(\mathbf{F}_\alpha, \mathbf{F}_\alpha) = 0$, equivalently, there is no indecomposable $H_m(0)$-module $M$ such that $\mathrm{ch}([M]) = 2 F_\alpha$.
On the other hand, in \cref{eg: G 211}, we see that $U_{(2,1,1);3} = 2 F_{(1,1,1)}$.
Thus, we conclude that there is no indecomposable $H_3(0)$-module $M$ satisfying $\mathrm{ch}([M]) = U_{(2,1,1);3}$.
\end{remark}



\subsection{A direct sum decomposition of $\mathbf{G}_{\lambda;m}$ into $H_m(0)$-submodules}
\label{Subsec: decomposition}

Let us start with necessary definitions and notation.
Given $T \in \mathrm{IGLT}(\lambda)_{m}$, let 
$\mathcal{I}(T) := \{ i \in [1,m] \mid |T^{-1}(i)| > 1 \}$.
Given $i \in \mathcal{I}(T)$, let $\Gamma_i(T)$ be the lattice path from $(\underline{r_\mathsf{b}^{(i)},c_\mathsf{b}^{(i)} - 1})$ to $(\underline{r_\mathsf{t}^{(i)} - 1, c_\mathsf{t}^{(i)}})$ satisfying the following two conditions:\vspace{-1ex}
\begin{enumerate}[label = {\rm (\roman*)}, itemsep = -0.5ex]
\item if the path passes through two boxes horizontally, then the entry at the above box is smaller than $i$ and the entry at the below box is greater than $i$, and 
\item if the path passes through two boxes vertically, then the entry at the left box is smaller than $i$ and the entry at the right box is greater than $i$.
\end{enumerate}

\ytableausetup{boxsize=2.5ex}


\begin{example}\label{Eg: lattice path}
Let 
\[
T = \begin{array}{l}\scalebox{0.75}{$
\begin{ytableau}
1  & 6  & 10 & 14 & 22 & 24 & 26 & \color{blue} 27 \\
2  & 7  & 11 & 15 & 23 & 25 & \color{brown} 29    \\
3  & 8  & 12 & 16 & 28 & \color{brown} 29         \\
4  & 9  & 13 & \color{red} 17                      \\
5  &\color{red} 17 & \color{blue} 27              \\
18 & 20                                \\
19 & \color{violet} 21                             \\
\color{violet} 21                                     
\end{ytableau}$}
\end{array}.
\]
Note that $\mathcal{I}(T) = \{17,21,27,29\}$.
By following the way of defining lattice paths, we obtain the lattice paths $\Gamma_{17}(T),\Gamma_{21}(T),\Gamma_{27}(T)$, and $\Gamma_{29}(T)$ as follows:
\[
\def\pp {0.51}
\def\hp {5.5}
\scalebox{0.75}{$
\begin{tikzpicture}


\node at (0, 0) {$
\begin{ytableau}
1  & 6  & 10 & 14 & 22 & 24 & 26 & \color{blue} 27 \\
2  & 7  & 11 & 15 & 23 & 25 & \color{brown} 29    \\
3  & 8  & 12 & 16 & 28 & \color{brown} 29         \\
4  & 9  & 13 & \color{red} 17                      \\
5  & \color{red} 17 & \color{blue} 27                          \\
18 & 20                                \\
19 & \color{violet} 21                             \\
\color{violet} 21                                     
\end{ytableau}
$};

\draw[line width = 0.5mm, color = red] 
(-3*\pp, -1*\pp) -- (-3*\pp, 0*\pp) -- (-1*\pp, 0*\pp) -- (-1*\pp, 1*\pp) -- (0*\pp, 1*\pp);

\node[] at (0, -2.5) {\fontsize{16}{16} $\Gamma_{17}(T)$};

\node at (0 + \hp,0) {$
\begin{ytableau}
1  & 6  & 10 & 14 & 22 & 24 & 26 & \color{blue} 27 \\
2  & 7  & 11 & 15 & 23 & 25 & \color{brown} 29    \\
3  & 8  & 12 & 16 & 28 & \color{brown} 29         \\
4  & 9  & 13 & \color{red} 17                      \\
5  & \color{red} 17 & \color{blue} 27                          \\
18 & 20                                \\
19 & \color{violet} 21                             \\
\color{violet} 21                                     
\end{ytableau}
$};

\draw[line width = 0.5mm, color = red] (-4*\pp +\hp, -4*\pp) -- (-4*\pp + \hp, -3*\pp) -- (-3*\pp + \hp, -3*\pp) -- (-3*\pp + \hp, -2*\pp) -- (-2*\pp + \hp, -2*\pp);
\node[] at (0 + \hp, -2.5) {\fontsize{16}{16} $\Gamma_{21}(T)$};


\node at (0 + 2*\hp,0) {$
\begin{ytableau}
1  & 6  & 10 & 14 & 22 & 24 & 26 & \color{blue} 27 \\
2  & 7  & 11 & 15 & 23 & 25 & \color{brown} 29    \\
3  & 8  & 12 & 16 & 28 & \color{brown} 29         \\
4  & 9  & 13 & \color{red} 17                      \\
5  & \color{red} 17 & \color{blue} 27                          \\
18 & 20                                \\
19 & \color{violet} 21                             \\
\color{violet} 21                                     
\end{ytableau}
$};

\draw[line width = 0.5mm, color = red] 
(-2*\pp + 2*\hp, -1*\pp) -- (-2*\pp + 2*\hp, 0*\pp) -- (0*\pp + 2*\hp, 0*\pp) -- (0*\pp + 2*\hp, 2*\pp) -- (2*\pp + 2*\hp, 2*\pp) -- (2*\pp + 2*\hp, 3*\pp) -- (3*\pp + 2*\hp, 3*\pp) -- (3*\pp + 2*\hp, 4*\pp) -- (4*\pp + 2*\hp, 4*\pp);
\node[] at (2*\hp, -2.5) {\fontsize{16}{16} $\Gamma_{27}(T)$};


\node at (0 + 3*\hp, 0) {$
\begin{ytableau}
1  & 6  & 10 & 14 & 22 & 24 & 26 & \color{blue} 27 \\
2  & 7  & 11 & 15 & 23 & 25 & \color{brown} 29    \\
3  & 8  & 12 & 16 & 28 & \color{brown} 29         \\
4  & 9  & 13 & \color{red} 17                      \\
5  & \color{red} 17 & \color{blue} 27                          \\
18 & 20                                \\
19 & \color{violet} 21                             \\
\color{violet} 21                                     
\end{ytableau}
$};

\draw[line width = 0.5mm, color = red] 
(1*\pp + 3*\hp, 1*\pp) -- (1*\pp + 3*\hp, 2*\pp) -- (2*\pp + 3*\hp, 2*\pp) -- (2*\pp + 3*\hp, 3*\pp) -- (3*\pp + 3*\hp, 3*\pp);
\node[] at (3*\hp, -2.5) {\fontsize{16}{16} $\Gamma_{29}(T)$};
\end{tikzpicture}
$}
\]
\end{example}


Given a lattice path $\Gamma$, let $V(\Gamma)$ be the set of lattice points through which $\Gamma$ passes.
For two lattice paths $\Gamma$ and $\Gamma'$, we write $\Gamma = \Gamma'$ if $V(\Gamma) = V(\Gamma')$.



\begin{definition}\label{def: equiv relation}
Let $\lambda \vdash n$ and $T_{1},T_{2} \in \mathrm{IGLT}(\lambda)_{m}$. 
The equivalence relation $\sim_{\lambda;m}$ on $\mathrm{IGLT}(\lambda)_{m}$ is defined by $T_{1} \sim_{\lambda;m} T_{2}$ if and only if 
\[
\left\{ \left(\Gamma_i(T_1), T_1^{-1}(i) \right) \; \middle| \; i \in \mathcal{I}(T_1) \right\} 
= \left\{ \left(\Gamma_i(T_2), T_2^{-1}(i) \right) \; \middle| \; i \in \mathcal{I}(T_2) \right\}.
\]
\end{definition}

\begin{example}
Let 
$\hspace{-1ex}
\def\pp {0.38}
\def\hp {5.5}
\begin{array}{l}
\begin{tikzpicture}
\node at (-1.5,-0.09) {$T_1 =\!$};
\node at  (0,0) {\scalebox{0.75}{$
\begin{ytableau}
1 & \color{red} 2 & \color{blue} 3 & \color{brown} 5 \\
\color{red} 2 & 4 & \color{brown} 5 & 6 \\
\color{blue} 3 & \color{brown} 5 & 7
\end{ytableau}$}};
\draw[line width = 0.5mm, color = red]  (-2*\pp, -0.5*\pp) -- (-2*\pp, 0.5*\pp) -- (-1*\pp, 0.5*\pp) -- (-1*\pp, 1.5*\pp) -- (0*\pp, 1.5*\pp);
\draw[line width = 0.5mm, color = blue]  (-2*\pp, -1.5*\pp) -- (-2*\pp, -0.5*\pp) -- (-1*\pp, -0.5*\pp)  -- (-1*\pp, 0.5*\pp) -- (0*\pp, 0.5*\pp) -- (0*\pp, 1.5*\pp) -- (1*\pp, 1.5*\pp);
\draw[line width = 0.5mm, color = brown]  (-1*\pp, -1.5*\pp) -- (-1*\pp, -0.5*\pp) -- (0*\pp, -0.5*\pp)  -- (0*\pp, 0.5*\pp) -- (1*\pp, 0.5*\pp) -- (1*\pp, 1.5*\pp) -- (2*\pp, 1.5*\pp);
\end{tikzpicture}
\end{array}\hspace{-1ex}$,\!
%%
$\hspace{-1ex}
\def\pp {0.38}
\def\hp {5.5}
\begin{array}{l}
\begin{tikzpicture}
\node at (-1.5,-0.09) {$T_2 =\!$};
\node at  (0,0) {\scalebox{0.75}{$
\begin{ytableau}
1 & \color{red} 2 & \color{blue} 3 & \color{brown} 5 \\
\color{red} 2 & 4 & \color{brown} 5 & 7 \\
\color{blue} 3 & \color{brown} 5 & 6
\end{ytableau}$}};
\draw[line width = 0.5mm, color = red]  (-2*\pp, -0.5*\pp) -- (-2*\pp, 0.5*\pp) -- (-1*\pp, 0.5*\pp) -- (-1*\pp, 1.5*\pp) -- (0*\pp, 1.5*\pp);
\draw[line width = 0.5mm, color = blue]  (-2*\pp, -1.5*\pp) -- (-2*\pp, -0.5*\pp) -- (-1*\pp, -0.5*\pp)  -- (-1*\pp, 0.5*\pp) -- (0*\pp, 0.5*\pp) -- (0*\pp, 1.5*\pp) -- (1*\pp, 1.5*\pp);
\draw[line width = 0.5mm, color = brown]  (-1*\pp, -1.5*\pp) -- (-1*\pp, -0.5*\pp) -- (0*\pp, -0.5*\pp)  -- (0*\pp, 0.5*\pp) -- (1*\pp, 0.5*\pp) -- (1*\pp, 1.5*\pp) -- (2*\pp, 1.5*\pp);
\end{tikzpicture}
\end{array}\hspace{-1ex}$,\!
%%%
$\hspace{-1ex}
\def\pp {0.38}
\def\hp {5.5}
\begin{array}{l}
\begin{tikzpicture}
\node at (-1.5,-0.09) {$T_3 =\!$};
\node at  (0,0) {\scalebox{0.75}{$
\begin{ytableau}
1 & \color{red} 2 & \color{blue} 3 & \color{brown} 4 \\
\color{red} 2 & \color{blue} 3 & 5 & 6 \\
\color{blue} 3 & \color{brown} 4 & 7
\end{ytableau}$}};
\draw[line width = 0.5mm, color = red]  (-2*\pp, -0.5*\pp) -- (-2*\pp, 0.5*\pp) -- (-1*\pp, 0.5*\pp) -- (-1*\pp, 1.5*\pp) -- (0*\pp, 1.5*\pp);
\draw[line width = 0.5mm, color = blue]  (-2*\pp, -1.5*\pp) -- (-2*\pp, -0.5*\pp) -- (-1*\pp, -0.5*\pp)  -- (-1*\pp, 0.5*\pp) -- (0*\pp, 0.5*\pp) -- (0*\pp, 1.5*\pp) -- (1*\pp, 1.5*\pp);
\draw[line width = 0.5mm, color = brown]  (-1*\pp, -1.5*\pp) -- (-1*\pp, -0.5*\pp) -- (0*\pp, -0.5*\pp)  -- (0*\pp, 0.5*\pp) -- (1*\pp, 0.5*\pp) -- (1*\pp, 1.5*\pp) -- (2*\pp, 1.5*\pp);
\end{tikzpicture}
\end{array}\hspace{-1ex}$,
%%%
and\!
%%%
$\hspace{-1ex}
\def\pp {0.38}
\def\hp {5.5}
\begin{array}{l}
\begin{tikzpicture}
\node at (-1.5,-0.09) {$T_4 =\!$};
\node at  (0,0) {\scalebox{0.75}{$
\begin{ytableau}
1 & \color{red} 2 & \color{blue} 4 & \color{brown} 5 \\
\color{red} 2 & 3 & \color{brown} 5 & 6 \\
\color{blue} 4 & \color{brown} 5 & 7
\end{ytableau}$}};
\draw[line width = 0.5mm, color = red]  (-2*\pp, -0.5*\pp) -- (-2*\pp, 0.5*\pp) -- (-1*\pp, 0.5*\pp) -- (-1*\pp, 1.5*\pp) -- (0*\pp, 1.5*\pp);
\draw[line width = 0.5mm, color = blue]  (-2*\pp, -1.5*\pp) 
-- (-2*\pp, -0.5*\pp + 0.05*\pp) 
-- (-1*\pp - 0.05*\pp, -0.5*\pp + 0.05*\pp)
-- (0*\pp - 0.05*\pp, -0.5*\pp + 0.05*\pp)
-- (0*\pp - 0.05*\pp, 0.5*\pp + 0.05*\pp)
-- (0*\pp - 0.05*\pp, 1.5*\pp)
-- (1*\pp, 1.5*\pp);
\draw[line width = 0.5mm, color = brown]  (-1*\pp, -1.5*\pp)
-- (-1*\pp, -0.5*\pp - 0.05*\pp)
-- (0*\pp + 0.05*\pp, -0.5*\pp - 0.05*\pp)
-- (0*\pp + 0.05*\pp, 0.5*\pp - 0.05*\pp)
-- (1*\pp + 0.05*\pp, 0.5*\pp - 0.05*\pp)
-- (1*\pp + 0.05*\pp, 1.5*\pp)
-- (2*\pp, 1.5*\pp);
\end{tikzpicture}
\end{array}\hspace{-1ex}$.
%%%
Then, $T_1 \sim_{(4,3,2);5} T_2$, but
$T_1 \not \sim_{(4,3,2);5} T_k$ for $k = 3,4$. 
\end{example}

Let $\mathcal{E}_{\lambda;m}$ be the set of equivalence classes of $\mathrm{IGLT}(\lambda)_{m}$ with respect to $\sim_{\lambda;m}$.

\begin{theorem}\label{thm: preserving} 
Let $m$ and $n$ be positive integers with $m \le n$ and let $\lambda \vdash n$.
For any $1 \le i \le m-1$ and $E \in \mathcal{E}_{\lambda;m}$, $\pi_i \cdot \mathbb{C} E \subseteq \mathbb{C} E$.
\end{theorem}



For each $E \in \mathcal{E}_{\lambda;m}$, let $\mathbf{G}_E$ be the $H_m(0)$-submodule of $\mathbf{G}_{\lambda;m}$ whose underlying space is the $\mathbb{C}$-span of $E$.
Then, we have the following direct sum decomposition
\begin{equation*}
\mathbf{G}_{\lambda;m} = \bigoplus_{E \in \mathcal{E}_{\lambda;m}} \mathbf{G}_E.
\end{equation*}

Hereafter, $E$ denotes an equivalence class of $\mathrm{IGLT}(\lambda)_{m}$ with respect to $\sim_{\lambda;m}$ and $T$ denotes a tableau contained in $\mathrm{IGLT}(\lambda)_{m}$ unless otherwise stated.





\section{Source and sink tableaux}
\label{Sec: source and sink}
The goal of this section is to show that there are two distinguished tableaux, called \emph{source} and \emph{sink tableaux}, in each equivalence class $E \in \mathcal{E}_{\lambda; m}$.
To achieve our goal, we first construct two tableaux $\mathsf{source}(T)$ and $\mathsf{sink}(T)$ for each $T \in E$.
Then, we prove that $\mathsf{source}(T)$ (resp. $\mathsf{sink}(T)$) is the unique source tableau (resp. sink tableau) in $E$, where $T$ is an arbitrary chosen element in $E$.


To begin with, we give definitions for source tableaux and sink tableaux in $\mathrm{IGLT}(\lambda)_{m}$.

\begin{definition}\label{def: source and sink tableaux}
Let $T \in \mathrm{IGLT}(\lambda)_{m}$. \vspace{-1ex}
\begin{enumerate}[label = {\rm (\arabic*)}, itemsep = -0.5ex]
\item $T$ is said to be a \emph{source tableau} if there do not exist $T' \in \mathrm{IGLT}(\lambda)_{m}$ and $1 \le i \le m-1$ such that $\pi_i \cdot T' = T$ and $T' \neq T$.
\item $T$ is said to be a \emph{sink tableau} if there do not exist $T' \in \mathrm{IGLT}(\lambda)_{m}$ and $1 \le i \le m-1$ such that $\pi_i \cdot T = T'$ and $T' \neq T$.
\end{enumerate}
\end{definition}



To construct the desired tableau $\mathsf{source}(T)$, we need the following preparation.
Given two lattice points $P$ and $P'$ in the same row, we denote the horizontal line from $P$ to $P'$ by $\mathtt{HL}(P,P')$.
For each $i \in \mathcal{I}(T)$, we define a new lattice path
$\widetilde{\Gamma}_i(T)$ by extending $\Gamma_i(T)$ with the following algorithm.

\begin{algorithm}\label{alg: tilde Gamma}
Fix $i \in \mathcal{I}(T)$ and let $\lambda_0 := \lambda_1$. \vspace{-1ex}
\begin{enumerate}[label = {\it Step \arabic*.}, itemsep = -0.5ex, leftmargin=*]
\item For each $j \in \mathcal{I}(T)$, let $\Gamma'_j$ be the lattice path obtained by connecting three lattice paths 
$\mathtt{HL}\left((\underline{r_\mathsf{b}^{(j)},0}), (\underline{r_\mathsf{b}^{(j)},c_\mathsf{b}^{(j)} - 1})\right)$,  
$\Gamma_j(T)$, and $\mathtt{HL}\left((\underline{r_\mathsf{t}^{(j)} -1, c_\mathsf{t}^{(j)}}), (\underline{r_\mathsf{t}^{(j)} - 1, \lambda_{r_\mathsf{t}^{(j)} - 1}})\right)$.
\item Let $r_{\mathsf{t}} = \min\{ r \mid (\underline{r, c}) \in V(\Gamma'_i) \}$ and $c_{\mathsf{t}} = \min\{ c \mid (\underline{r_{\mathsf{t}},c}) \in V(\Gamma'_i) \}$.
\item If there exists $j \in \mathcal{I}(T)$ such that
\begin{align}\label{Eq: top cross condition}
r' < r_\mathsf{t} < r'' \quad \text{and} \quad c', c'' > c_\mathsf{t}
\quad \text{for some $(\underline{r', c'}), (\underline{r'', c''}) \in V(\Gamma'_j)$,}
\end{align}
then go to {\it Step 4}. Otherwise, go to {\it Step 5}.
\item Let $j_0 = \min\{j \mid \text{$\Gamma'_j$ satisfies \eqref{Eq: top cross condition}}\}$ and $c_{0} = \min \left\{ 
c \mid  (\underline{r_{\mathsf{t}}, c}) \in V(\Gamma'_{j_0})
\right\}$.
Then, let $\Gamma$ be the lattice path satisfying that
\[
V(\Gamma) = V(\Gamma'_i) \setminus \{(\underline{r_\mathsf{t}, c}) \mid c \ge c_0\} \cup \{ (\underline{r, c}) \in V(\Gamma'_{j_0}) \mid r \le r_\mathsf{t} \ \text{and} \  c \ge c_0 \}.
\]
Set $\Gamma'_i := \Gamma$. 
Go to {\it Step 2}.
\item Return $\widetilde{\Gamma}_i(T) := \Gamma'_i$ and terminate the algorithm.
\end{enumerate}
\end{algorithm}

If $T$ is clear in the context, we simply write the lattice path $\widetilde{\Gamma}_i(T)$ by $\widetilde{\Gamma}_i$ for $i \in \mathcal{I}(T)$. 

\begin{example}\label{Eg: modified lattice path}
Let us revisit \cref{Eg: lattice path}.
By applying \cref{alg: tilde Gamma} to each $i \in \mathcal{I}(T)$, we obtain $\widetilde{\Gamma}_{17}$, $\widetilde{\Gamma}_{21}$, $\widetilde{\Gamma}_{27}$, and $\widetilde{\Gamma}_{29}$ as follows:
\begin{displaymath}
\def\pp {0.51}
\def\hp {5.5}
\scalebox{0.75}{$
\begin{tikzpicture}
\node at (0 - \hp, 0) {$
\begin{ytableau}
1  & 6  & 10 & 14 & 22 & 24 & 26 & \color{blue} 27 \\
2  & 7  & 11 & 15 & 23 & 25 & \color{brown} 29    \\
3  & 8  & 12 & 16 & 28 & \color{brown} 29         \\
4  & 9  & 13 & \color{red} 17                      \\
5  & \color{red} 17 & \color{blue} 27                          \\
18 & 20                                \\
19 & \color{violet} 21                             \\
\color{violet} 21                                     
\end{ytableau}
$};

\draw[line width = 0.5mm, color = red] 
(-4*\pp - \hp, -1*\pp) -- (-3*\pp - \hp, -1*\pp) -- (-3*\pp - \hp, 0*\pp) -- (-1*\pp - \hp, 0*\pp) -- (-1*\pp - \hp, 1*\pp) -- (0*\pp - \hp, 1*\pp);

\draw[line width = 0.5mm, color = red] 
(0*\pp - \hp, 1*\pp) -- (0*\pp - \hp, 2*\pp) -- (2*\pp - \hp, 2*\pp) -- (2*\pp - \hp, 3*\pp) -- (3*\pp - \hp, 3*\pp) -- (3*\pp - \hp, 4*\pp) -- (4*\pp - \hp, 4*\pp);

\node[] at (-\hp, -2.5) {\fontsize{16}{16} $\widetilde{\Gamma}_{17}(T)$};


\node at (0,0) {$
\begin{ytableau}
1  & 6  & 10 & 14 & 22 & 24 & 26 & \color{blue} 27 \\
2  & 7  & 11 & 15 & 23 & 25 & \color{brown} 29    \\
3  & 8  & 12 & 16 & 28 & \color{brown} 29         \\
4  & 9  & 13 & \color{red} 17                      \\
5  & \color{red} 17 & \color{blue} 27                          \\
18 & 20                                \\
19 & \color{violet} 21                             \\
\color{violet} 21                                     
\end{ytableau}
$};

\draw[line width = 0.5mm, color = red] (-4*\pp, -4*\pp) -- (-4*\pp, -3*\pp) -- (-3*\pp, -3*\pp) -- (-3*\pp, -2*\pp) -- (-2*\pp, -2*\pp);
\node[] at (0, -2.5) {\fontsize{16}{16} $\widetilde{\Gamma}_{21}(T)$};


\node at (0 + \hp,0) {$
\begin{ytableau}
1  & 6  & 10 & 14 & 22 & 24 & 26 & \color{blue} 27 \\
2  & 7  & 11 & 15 & 23 & 25 & \color{brown} 29    \\
3  & 8  & 12 & 16 & 28 & \color{brown} 29         \\
4  & 9  & 13 & \color{red} 17                      \\
5  & \color{red} 17 & \color{blue} 27                          \\
18 & 20                                \\
19 & \color{violet} 21                             \\
\color{violet} 21                                     
\end{ytableau}
$};

\draw[line width = 0.5mm, color = red] 
(-4*\pp + \hp, -1*\pp) -- (-2*\pp + \hp, -1*\pp) -- (-2*\pp + \hp, 0*\pp) -- (0*\pp + \hp, 0*\pp) -- (0*\pp + \hp, 2*\pp) -- (2*\pp + \hp, 2*\pp) -- (2*\pp + \hp, 3*\pp) -- (3*\pp + \hp, 3*\pp) -- (3*\pp + \hp, 4*\pp) -- (4*\pp + \hp, 4*\pp);
\node[] at (\hp, -2.5) {\fontsize{16}{16} $\widetilde{\Gamma}_{27}(T)$};



\node at (0 + 2*\hp, 0) {$
\begin{ytableau}
1  & 6  & 10 & 14 & 22 & 24 & 26 & \color{blue} 27 \\
2  & 7  & 11 & 15 & 23 & 25 & \color{brown} 29    \\
3  & 8  & 12 & 16 & 28 & \color{brown} 29         \\
4  & 9  & 13 & \color{red} 17                      \\
5  & \color{red} 17 & \color{blue} 27                          \\
18 & 20                                \\
19 & \color{violet} 21                             \\
\color{violet} 21                                     
\end{ytableau}
$};

\draw[line width = 0.5mm, color = red] 
(-4*\pp + 2*\hp, 1*\pp) -- (1*\pp + 2*\hp, 1*\pp) -- (1*\pp + 2*\hp, 2*\pp) -- (2*\pp + 2*\hp, 2*\pp) -- (2*\pp + 2*\hp, 3*\pp) -- (4*\pp + 2*\hp, 3*\pp);
\node[] at (2*\hp, -2.5) {\fontsize{16}{16} $\widetilde{\Gamma}_{29}(T)$};
\end{tikzpicture}
$}
\end{displaymath}
\end{example}





For $i \in \mathcal{I}(T)$, we set $\mathsf{p}'_i \in \{1,2,\ldots, |\mathcal{I}(T)|\}$ satisfying the following:
Let $i,j \in \mathcal{I}(T)$. \vspace{-1ex}
\begin{enumerate}[label = {\bf C\arabic*.}, itemsep = -0.5ex]
\item If $r_\mathsf{b}^{(i)} < r_\mathsf{b}^{(j)}$, then $\mathsf{p}'_i < \mathsf{p}'_j$. 
And, if $r_\mathsf{b}^{(i)} > r_\mathsf{b}^{(j)}$, then $\mathsf{p}'_i > \mathsf{p}'_j$.
\item When $r_\mathsf{b}^{(i)} = r_\mathsf{b}^{(j)}$, consider the lowest lattice point $p \in V(\widetilde{\Gamma}_i) \cap V(\widetilde{\Gamma}_j)$ such that neither $p + (\underline{-1, 0})$ nor $p + (\underline{0, 1})$ are contained in $V(\widetilde{\Gamma}_i) \cap V(\widetilde{\Gamma}_j)$.
If $p + (\underline{-1,0}) \in V(\widetilde{\Gamma}_i)$, then $\mathsf{p}'_i < \mathsf{p}'_j$. Otherwise, $\mathsf{p}'_i > \mathsf{p}'_j$.
\end{enumerate}


Given $i,j \in \mathcal{I}(T)$, if there exist $(\underline{r', c'}), (\underline{r'', c''}) \in V(\widetilde{\Gamma}_{j})$ such that $r' < r^{(i)}_\mathsf{b} < r''$ and $c',c'' < c_\mathsf{b}^{(i)}$, then we say that \emph{$\widetilde{\Gamma}_{j}$ crosses the bottom path of $\widetilde{\Gamma}_{i}$}.
By rearranging $\mathsf{p}_i'$'s with the following algorithm,
we define a bijection $\mathsf{p}_T: \mathcal{I}(T) \rightarrow \{1,2,\ldots, |\mathcal{I}(T)|\}$.


\begin{algorithm}\label{alg: tsfp} 
For each $i \in \mathcal{I}(T)$, let $p_i := \mathsf{p}'_i$, where $\mathsf{p}'_i$ is the index defined above. \vspace{-1ex}
\begin{enumerate}[label = {\it Step \arabic*.}, itemsep = -0.5ex, leftmargin=*]
\item Let $k = 1$.
\item Take $i_k$ and $i_{k+1}$ in $\mathcal{I}(T)$ such that $p_{i_k} = k$ and $p_{i_{k+1}} = k+1$.
\item If $\widetilde{\Gamma}_{i_{k+1}}$ crosses the bottom path of $\widetilde{\Gamma}_{i_{k}}$, then set $p_{i_k} := k + 1$ and $p_{i_{k+1}} := k$ and go to {\it Step 1}. Otherwise, go to {\it Step 4}.
\item If $k < |\mathcal{I}(T)| - 1$, then set $k = k+1$ and go to {\it Step 2}. 
Otherwise, set $\mathsf{p}_T(i) := p_i$ for each $i\in \mathcal{I}(T)$ and go to {\it Step 5}. 
\item Return $(\mathsf{p}_T(i))_{i \in \mathcal{I}(T)}$ and terminate the algorithm.
\end{enumerate}
\end{algorithm}


Given $u \in [1,|\mathcal{I}(T)|\,]$, let $\mathsf{A}_{u}$ be the subdiagram of $\mathtt{yd}(\lambda)$ consisting of the boxes located above $\widetilde{\Gamma}^{(u)}$. 
Let
$$
\mathsf{D}_{u}^{(1)}(T) :=
\mathsf{A}_{u} \setminus \Big(\bigcup_{1 \le v < u} \left(\mathsf{A}_{v} \cup T^{-1}(\mathsf{p}_T^{-1}(v)) \right)\Big)
\quad \text{and} \quad 
\mathsf{D}_{u}^{(2)}(T):= T^{-1}(\mathsf{p}_T^{-1}(u)).
$$



Now, we construct the desired tableau $\mathsf{source}(T)$ with the following algorithm.

\begin{algorithm}\label{Alg: construct source of T}
Let $T \in \mathrm{IGLT}(\lambda)_{m}$.
Set $e_0 = 0$ and $M_0 = 0$.
For $1 \le u \le |\mathcal{I}(T)|$, let $e_u := |\mathsf{D}_u^{(1)}(T)| + 1$ and $M_u = \sum_{v = 0}^{u} e_v$.
\vspace{-1ex}
\begin{enumerate}[label = {\it Step \arabic*.}, itemsep = -0.5ex, leftmargin=*]
\item Set $v := 1$.
\item Fill the boxes in $\mathsf{D}_v^{(1)}(T)$ by $M_{v-1} + 1, M_{v-1} + 2, \ldots, M_{v-1} + e_v - 1$ from left to right starting from the top.
\item Fill the boxes in $\mathsf{D}_v^{(2)}(T)$ by $M_v$.
\item If $v < |\mathcal{I}(T)|$, then set $v := v + 1$ and go to {\it Step 2}. 
Otherwise, fill the remaining boxes by $M_{|\mathcal{I}(T)|} + 1, M_{|\mathcal{I}(T)|} + 2,\ldots, m$ from left to right starting from the top.
Set $\mathsf{source}(T)$ to be the resulting filling.
Return $\mathsf{source}(T)$ and terminate the algorithm.
\end{enumerate}
\end{algorithm}


\begin{example}\label{Eg: sfp and sfD compute}
Let us revisit \cref{Eg: modified lattice path}.
One can easily see that $\mathsf{p}'_{17} = 2,
\mathsf{p}'_{21} = 4,  
\mathsf{p}'_{27} = 3$, and $\mathsf{p}'_{29} = 1$.
By applying \cref{alg: tsfp}, we have
\[
\mathsf{p}_T(17) = 1, \quad 
\mathsf{p}_T(21) = 4, \quad 
\mathsf{p}_T(27) = 2, \quad \text{and} \quad
\mathsf{p}_T(29) = 3.
\]
In addition, by applying \cref{Alg: construct source of T}, we have
\[
\def\pp {0.51}
\def\hp {12}
\def\sp {0.035}
\scalebox{0.75}{$
\begin{tikzpicture}

\node at (-3.5 - \hp,0) {\fontsize{16.5}{16.5} $T \ \  = \quad $};
\node at (-\hp, 0) {$
\begin{ytableau}
1  & 6  & 10 & 14 & 22 & 24 & 26 & \color{blue} 27 \\
2  & 7  & 11 & 15 & 23 & 25 & \color{brown} 29    \\
3  & 8  & 12 & 16 & 28 & \color{brown} 29         \\
4  & 9  & 13 & \color{red} 17                      \\
5  &\color{red} 17 & \color{blue} 27              \\
18 & 20                                \\
19 & \color{violet} 21                             \\
\color{violet} 21                                     
\end{ytableau}
$};

% tGam_{17}

\draw[line width = 0.5mm, color = red] 
(-4*\pp - \hp , -1*\pp + \sp) -- 
(-3*\pp - \hp  - \sp, -1*\pp + \sp) -- 
(-3*\pp - \hp  - \sp, 0*\pp + \sp) -- 
(-1*\pp - \hp  - \sp, 0*\pp + \sp) -- 
(-1*\pp - \hp  - \sp, 1*\pp + \sp) -- 
(0*\pp  - \hp - \sp, 1*\pp + \sp) --
(0*\pp  - \hp - \sp, 2*\pp + \sp) -- 
(2*\pp  - \hp - \sp, 2*\pp + \sp) -- 
(2*\pp  - \hp - \sp, 3*\pp + \sp) -- 
(3*\pp  - \hp - \sp, 3*\pp + \sp) -- 
(3*\pp  - \hp - \sp, 4*\pp + \sp) -- 
(4*\pp  - \hp, 4*\pp + \sp);


% tGam_{21}

\draw[line width = 0.5mm, color = violet] 
(-4*\pp - \hp, -4*\pp) -- 
(-4*\pp - \hp, -3*\pp) -- 
(-3*\pp - \hp, -3*\pp) -- 
(-3*\pp - \hp, -2*\pp) -- 
(-2*\pp - \hp, -2*\pp);


% tGam_{27}

\draw[line width = 0.5mm, color = blue] 
(-4*\pp - \hp, -1*\pp - \sp) -- 
(-2*\pp - \hp + \sp, -1*\pp - \sp) -- 
(-2*\pp - \hp + \sp, 0*\pp - \sp) -- 
(0*\pp  - \hp+ \sp, 0*\pp - \sp) -- 
(0*\pp  - \hp+ \sp, 2*\pp - \sp) -- 
(2*\pp  - \hp+ \sp, 2*\pp - \sp) -- 
(2*\pp  - \hp+ \sp, 3*\pp - \sp) -- 
(3*\pp  - \hp+ \sp, 3*\pp - \sp) -- 
(3*\pp  - \hp+ \sp, 4*\pp - \sp) -- 
(4*\pp  - \hp, 4*\pp - \sp);


% tGam_{29}

\draw[line width = 0.5mm, color = brown] 
(-4*\pp- \hp , 1*\pp) -- 
(1*\pp - \hp, 1*\pp) -- 
(1*\pp - \hp, 2*\pp) -- 
(2*\pp - \hp, 2*\pp) -- 
(2*\pp - \hp, 3*\pp) -- 
(4*\pp - \hp, 3*\pp);


\node at (-7.5,0.5) {\small \cref{Alg: construct source of T}};
\draw[->] (-8.5,0) -- (-6.5,0);



\node at (-4.3,0) {\fontsize{16.5}{16.5} $\mathsf{source}(T) \  = \quad $};
\node at (2.5,0) {.};
\node at (0, 0) {$
\begin{ytableau}
1  & 2 & 3 & 4 & 5 & 6 & 7 & \color{blue} 23 \\
8  & 9 & 10& 11& 12& 13& \color{brown} 25      \\
14 & 15& 16& 17& 24  & \color{brown} 25           \\
18 & 19& 20& \color{red} 22                     \\
21 & \color{red} 22 & \color{blue} 23    \\
26   & 27                               \\
28   & \color{violet} 29                               \\
\color{violet} 29
\end{ytableau}
$};

% tGam_{17}

\draw[line width = 0.5mm, color = red] 
(-4*\pp , -1*\pp + \sp) -- 
(-3*\pp  - \sp, -1*\pp + \sp) -- 
(-3*\pp  - \sp, 0*\pp + \sp) -- 
(-1*\pp  - \sp, 0*\pp + \sp) -- 
(-1*\pp  - \sp, 1*\pp + \sp) -- 
(0*\pp  - \sp, 1*\pp + \sp) --
(0*\pp  - \sp, 2*\pp + \sp) -- 
(2*\pp  - \sp, 2*\pp + \sp) -- 
(2*\pp  - \sp, 3*\pp + \sp) -- 
(3*\pp  - \sp, 3*\pp + \sp) -- 
(3*\pp  - \sp, 4*\pp + \sp) -- 
(4*\pp , 4*\pp + \sp);


% tGam_{21}

\draw[line width = 0.5mm, color = violet] 
(-4*\pp , -4*\pp) -- 
(-4*\pp , -3*\pp) -- 
(-3*\pp , -3*\pp) -- 
(-3*\pp , -2*\pp) -- 
(-2*\pp , -2*\pp);


% tGam_{27}

\draw[line width = 0.5mm, color = blue] 
(-4*\pp , -1*\pp - \sp) -- 
(-2*\pp  + \sp, -1*\pp - \sp) -- 
(-2*\pp  + \sp, 0*\pp - \sp) -- 
(0*\pp  + \sp, 0*\pp - \sp) -- 
(0*\pp  + \sp, 2*\pp - \sp) -- 
(2*\pp  + \sp, 2*\pp - \sp) -- 
(2*\pp  + \sp, 3*\pp - \sp) -- 
(3*\pp  + \sp, 3*\pp - \sp) -- 
(3*\pp  + \sp, 4*\pp - \sp) -- 
(4*\pp , 4*\pp - \sp);


% tGam_{29}

\draw[line width = 0.5mm, color = brown] 
(-4*\pp , 1*\pp) -- 
(1*\pp , 1*\pp) -- 
(1*\pp , 2*\pp) -- 
(2*\pp , 2*\pp) -- 
(2*\pp , 3*\pp) -- 
(4*\pp , 3*\pp);


\end{tikzpicture}$}
\]
\end{example}

\begin{theorem}\label{thm: uniqueness of source}
For any $T \in E$, $\mathsf{source}(T)$ is the unique source tableau in $E$.
\end{theorem}

Similarly, we construct $\mathsf{sink}(T)$ for each $T \in E$ and prove the following theorem.

\begin{theorem}\label{thm: uniqueness of sink}
For any $T \in E$, $\mathsf{sink}(T)$ is the unique sink tableau in $E$.
\end{theorem}

We denote by $T_E$ and $T_E'$ the unique source tableau and sink tableau in $E$, respectively.

\section{A weak Bruhat interval module description of $\mathbf{G}_{E}$}
\label{Sec: weak bruhat interval module}

Throughout this section, we let $\mathrm{Des}(T_E) = \{d_1< d_2 < \cdots < d_k\}$, $d_0 := 0$, and $d_{k+1} := m$.
For each $1 \le j \le k+1$, let 
$\mathtt{H}_j := T_E^{-1}([d_{j-1}+1, d_{j}])$.


\begin{example}\label{Eg: tH_j}
When
\[
T_{E} = \begin{array}{l}
\scalebox{0.75}{$
\begin{ytableau}
1 & 2 & 3 & 4 & 5 & 11 & 12 & 14 & 15 \\
6 & 7 & 8 & 10 & 11 & 14 \\
9 & 10 \\
13 & 14
\end{ytableau}
$}
\end{array},
\]
we have that $\mathrm{Des}(T_{E}) = \{5,8,12\}$. In this case, $\mathtt{H}_j~(1 \le j \le 4)$ are given as follows: 
\[
\begin{array}{c}
\scalebox{0.75}{$
\begin{ytableau}
*(red!20) \mathtt{H}_1 & *(red!20) \mathtt{H}_1 & *(red!20) \mathtt{H}_1 & *(red!20) \mathtt{H}_1 & *(red!20) \mathtt{H}_1 & *(gray!20) \mathtt{H}_3 & *(gray!20) \mathtt{H}_3  & *(magenta!20) \mathtt{H}_4 & *(magenta!20) \mathtt{H}_4\\
*(blue!20) \mathtt{H}_2 & *(blue!20) \mathtt{H}_2 & *(blue!20) \mathtt{H}_2 & *(gray!20) \mathtt{H}_3 & *(gray!20) \mathtt{H}_3 & *(magenta!20) \mathtt{H}_4 \\
*(gray!20) \mathtt{H}_3 & *(gray!20) \mathtt{H}_3 \\
*(magenta!20) \mathtt{H}_4 & *(magenta!20) \mathtt{H}_4
\end{ytableau}
$}
\end{array}
\]
\end{example}

For each $1\le j \le k$, let $\mathrm{w}^{(j)}(T)$ be the word obtained by reading the entries of $T$ contained in $\mathtt{H}_j$ from right to left. 
Note that if an integer $i$ appears multiple times in $\mathrm{w}^{(j)}(T)$, then the integer $i$'s are placed consecutively.
We define $\overline{\mathrm{w}}^{(j)}(T)$ as the word obtained from $\mathrm{w}^{(j)}(T)$ by erasing all $i$'s except one $i$ for each $i$ that appears in $\mathrm{w}^{(j)}(T)$.

\begin{definition}\label{def: standard reading}
For $T \in E$, \emph{the standardized reading word $\mathsf{read}(T)$ of $T$} is defined to be the word $\overline{\mathrm{w}}^{(1)}(T) \overline{\mathrm{w}}^{(2)}(T) \cdots \overline{\mathrm{w}}^{(k+1)}(T)$ obtained by concatenating $\overline{\mathrm{w}}^{(j)}(T)$ for $1 \le j \le k+1$.
\end{definition}



\begin{example}\label{eg: reading}
We revisit \cref{Eg: tH_j}. 
One can see that
\[
\mathsf{read}(T_{E}) = 5 \ 4 \ 3 \ 2 \ 1 \ 8 \ 7 \ 6 \ 12 \ 11 \ 10 \ 9 \ 15 \ 14 \ 13 \in \mathfrak{S}_{15}. 
\]
\end{example}





\begin{theorem}
For each $E \in \mathcal{E}_{\lambda;m}$, $\mathbf{G}_E \cong 
\mathsf{B}(\mathsf{read}(T_E), \mathsf{read}(T'_E))$ as $H_m(0)$-modules.
\end{theorem}




\section{Further avenue}

In \cite{20Pechenik}, Pechenik proved that for all $\lambda = (\lambda_1, \lambda_2) \vdash n$,
\begin{align}\label{eq: schur expansion for genomic}
U_\lambda = \sum_{l_\lambda \le m \le n} \  \sum_{\mu \in \mathsf{Par}(\lambda; m)} s_{\mu},
\end{align}
where $l_\lambda := \max \{\lambda_1, \lambda_2 +1\}$, $\mathsf{Par}(\lambda; n) := \{(\lambda_1, \lambda_2)\}$, and
\begin{align*}
\mathsf{Par}(\lambda; m) & := 
\begin{cases}
\left\{(\lambda_1 - k_m, \lambda_1 - k_m, 1^{k_m})\right\} & \text{if $\lambda_1 = \lambda_2$,}\\
\left\{(\lambda_1 - k_m, \lambda_2 - k_m, 1^{k_m}), (\lambda_1 - k_m, \lambda_2 - k_m + 1, 1^{k_m-1}) \right\} & \text{if $\lambda_1 > \lambda_2$}
\end{cases}
\end{align*}
for all $l \le m < n$.
Here, $k_m := n-m$ and $s_\mu := 0$ if $\mu$ is not a partition.
And, for each $\lambda \vdash m$, 
Searles \cite{20Searles} introduced the $H_m(0)$-module $X_\lambda$ such that $\mathrm{ch}([X_\lambda]) = s_\lambda$.
The study of representation theoretic interpretation for \eqref{eq: schur expansion for genomic} will be pursued in the near future by using $\mathbf{G}_{\lambda;m}$ and $X_\mu$.
In this direction, we leave the following conjecture.

\begin{conjecture}\label{conj: repn interpretation}
Let $\lambda = (\lambda_1, \lambda_2) \vdash n$.
For each $l_\lambda \le m \le n$, there exists a partition $\{\mathcal{E}_\mu \mid \mu \in \mathsf{Par}(\lambda; m)\}$ of $\mathcal{E}_{\lambda;m}$ satisfying the following:
For each $\mu \in \mathsf{Par}(\lambda; m)$,
\vspace{-1ex}
\begin{enumerate}[label = {\rm (\arabic*)}, itemsep = -0.5ex]
\item $\sum_{E \in \mathcal{E}_\mu} \mathrm{ch}([\mathbf{G}_E]) = s_{\mu}$,
\item there exist a total order $\prec_\mu$ on $\mathcal{E}_\mu = \{E_1 \prec_\mu \cdots \prec_\mu E_{|\mathcal{E}_\mu|}\}$ and a filtration 
$M_0 = \{0\} \subseteq M_1 \subseteq \cdots \subseteq M_{|\mathcal{E}_\mu|} = X_\mu$
such that $\mathbf{G}_{E_i} \cong M_{i}/M_{i-1}$ for all $1 \le i \le |\mathcal{E}_\mu|$.
\end{enumerate}
\end{conjecture}

\acknowledgements{
The authors are grateful to the anonymous referees for their careful readings of the manuscript and valuable advice.
}




\printbibliography



\end{document}
