%2022March 14 : finished corrections per referee reports
%2022 Jan 24: Added defn of $S^{row*}_\alpha$ and fixed cyclic gen for $X_\alpha$
%
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\usepackage{tabularx}
\usepackage{tikz}
\usepackage{ytableau}
%\usepackage{parskip}

\usepackage{comment}  %Sheila
\newcommand\T{\rule{0pt}{2.7ex}}       % Top strut Sheila
\newcommand\B{\rule[-1.2ex]{0pt}{0pt}} % Bottom strut Sheila

%\newtheorem{theorem}{Theorem}[section]
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\newcommand{\poRI}{\preccurlyeq_{\mathcal{R}{\mathfrak{S}}^\ast _\alpha}}  %Steph's macro

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\newcommand{\poI}{\preccurlyeq_{{\mathfrak{S}}^\ast _\alpha}} 
\newcommand{\poIcover}{\prec_{{\mathfrak{S}}^\ast _\alpha}} 
%Sheila's macros for cover relations

%%%%%%%%%%%%%%%MACROS FROM ACRC   Elizabeth
%%%%%%blackboard letters
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% \tableau{ ... & ... & ... \\ ... }
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\DeclareMathOperator{\sgn}{sgn}
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\DeclareMathOperator{\SSYCT}{SSYCT}
\DeclareMathOperator{\DIRT}{DIRT}
\DeclareMathOperator{\cont}{cont}
\DeclareMathOperator{\comp}{comp}
\DeclareMathOperator{\rev}{rev}
\DeclareMathOperator{\sh}{sh}
\DeclareMathOperator{\set}{set}
\newcommand{\rI}{\mathcal{R}\mathfrak{S}}
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\DeclareMathOperator{\DesRS}{Des_{\RS}}
\DeclareMathOperator{\rw}{rw}
\DeclareMathOperator{\stdz}{stdz}
\DeclareMathOperator{\inv}{inv}
\DeclareMathOperator{\F}{\mathcal{F}}  % sheila for characteristic map
\DeclareMathOperator{\chr}{\mathrm{ch}}  % sheila for quasisymmetric characteristic map
%Steph's macros

%:Sym definitions
\newcommand{\Sym}{\ensuremath{\operatorname{Sym}}}

%:Qsym definitions
\newcommand{\Qsym}{\ensuremath{\operatorname{QSym}}}
\newcommand{\qm}{M}				% monomials
\newcommand{\qf}{F}				% fundamentals
\newcommand{\qc}{C}				% canonicals
\newcommand{\qs}{{\mathcal{S}}}		% quasisymmetric  schurs
\newcommand{\qsy}{\widehat{\mathcal{S}}}	% Young quasisymmetric  schurs
\newcommand{\DI}{{\mathfrak{S}}^\ast}	         % dual immaculate schurs
\newcommand{\RI}{\mathcal{R}{\mathfrak{S}}^\ast} % row-strict dual immaculate schurs
\newcommand{\rqsy}{\mathcal{R}\widehat{\mathcal{S}}}	% row-strict Young quasisymmetric  schurs
\newcommand{\SIT}{\ensuremath{\operatorname{SIT}}} %standard immaculate tableaux
\newcommand{\SET}{\ensuremath{\operatorname{SET}}} %Sheila:  standard imm tab. where ALL cols increase
\newcommand{\NSET}{\ensuremath{\operatorname{NSET}}}%Sheila:  standard tab. where all rows increase left to right, but at least ONE col does NOT increase
\newcommand{\SRCT}{\ensuremath{\operatorname{SRCT}}}%Sheila:  standard reverse composition tab. used to expand quasisym Schurs in fundamentals --see p. 57 of Steph's book


%\newcommand{\Fsym}{\textsl{FSym}}

%:Nsym definitions
\newcommand{\Nsym}{\ensuremath{\operatorname{NSym}}}
\newcommand{\nce}{\mathbf{e}}         	% noncommutative elementary fcns
\newcommand{\nch}{\mathbf{h}}         	% noncommutative complete fcns
\newcommand{\ncr}{\mathbf{r}}           	% noncommutative  ribbon schurs
\newcommand{\ncs}{{\mathbf{s}}}          	% noncommutative  schurs
\newcommand{\ncsy}{\widehat{\mathbf{s}}}      % Young noncommutative  schurs
\newcommand{\nci}{{\mathfrak{S}}}	         %  immaculate schurs
\newcommand{\ncri}{\mathcal{R}{\mathfrak{S}}} % row-strict immaculate schurs

%:NCSym definitions
\newcommand{\NCSym}{\ensuremath{\operatorname{NCSym}}}
\newcommand{\slashp}{\mid}
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\newcommand{\mupi}{\mu_\Pi}
\newcommand{\mul}{\mu_L}
\newcommand{\bx}{\text{\bf x}}
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%\newcommand{\sh}{\mathrm{sh}} %already defined as an operator?
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\newcommand{\jota}{\mathcal{J}^t}
                   

%:coplactic algebra definitions
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\newcommand{\wsum}{\widehat{\sigma}}
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\newcommand{\hn}{H_n(0)}
\newcommand{\smodule}{\mathbf{S}}



%: compositions and tableaux
\newcommand\ol[1]{\overline{#1}}
\newcommand\ul[1]{\underline{#1}}
\newcommand\ch[1]{\check{#1}}
\newcommand\strongof[1]{{#1}^+}
\newcommand\partitionof[1]{\widetilde{#1}}
\newcommand\reverse[1]{{#1}^*}
\newcommand\ccomplement[1]{{#1}^c}
\newcommand\ctranspose[1]{{#1}^t}

%\newcommand{\set}{\mathrm{Set}} % for sets corresponding to compositions
\newcommand{\des}{\mathrm{Des}} % for descent sets 
\newcommand{\desri}{\mathrm{Des}_{\mathcal{R}\mathfrak{S}^\ast}} % for row-strict dual immaculate descent sets 
%\newcommand{\inv}{\mathrm{Inv}} % for inversion sets 
\newcommand{\ninv}{\mathrm{inv}} % for size of inversion sets
%\newcommand{\comp}{\mathrm{comp}} % for compositions corresponding to sets
%\newcommand{\cont}{\mathrm{cont}} % for content
\newcommand{\col}{\mathrm{col}} % for column sequence
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\newcommand{\rem}{\check{\mathfrak{d}}} % for Young removal
\newcommand{\rowremy}{\widehat{\mathfrak{h}}} % for Young row removal
\newcommand{\colremy}{\widehat{\mathfrak{v}}} % for Young column removal
\newcommand{\remy}{\widehat{\mathfrak{d}}} % for removal
\newcommand{\wcol}{w_{col}} % for column reading word
\newcommand{\rT}{\check{T}} % for reverse tableaux
\newcommand{\rtau}{{T}} % for reverse composition tableaux

\newcommand{\refines}{\preccurlyeq}   % for refinement of compositions
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\newcommand{\fatzero}{\mathbf{0}}
\newcommand{\suchthat}{\;|\;}
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\newcommand{\Subsetcy}{\Subset _{\widehat{c}}}    % for skew composition shapes
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\newcommand{\cshape}{C\text{-}shape}
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\newcommand{\rect}{\mathrm{rect}}



%\usepackage{lipsum}

%% define your title in the usual way
\title[Row-strict $0$-Hecke algebra modules]{Row-strict dual immaculate functions 
 and $0$-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[E.~Niese, S.~Sundaram, S.~van Willigenburg, J.~Vega, S.~Wang]{Elizabeth Niese\addressmark{1},  Sheila Sundaram\addressmark{2}\thanks{\href{mailto:shsund@comcast.net}{shsund@comcast.net}}, \\Stephanie van Willigenburg\addressmark{3}, Julianne Vega\addressmark{4}, and Shiyun Wang\addressmark{5}.}
% Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}
%\addressmark{1}, \addressmark{2}, \addressmark{3}, \addressmark{4}, \addressmark{5}}
%\addressmark{1}, \and you\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Marshall University, Huntington, WV 25755, USA\\ \addressmark{2}Pierrepont School, Westport, CT 06880, USA\\ \addressmark{3}University of British Columbia, Vancouver BC V6T 1Z2, Canada\\ \addressmark{4}Kennesaw State University, Kennesaw, GA 30144, USA\\ \addressmark{5}University of Southern California, Los Angeles, CA 90089-2532, USA\\}

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

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
%\abstract{\lipsum[1]}
\abstract{We introduce a new basis of quasisymmetric functions, the row-strict dual immaculate functions.  We construct a cyclic, indecomposable $0$-Hecke algebra module for these functions.   Our row-strict dual immaculate functions are related to the dual immaculate functions of Berg--Bergeron--Saliola--Serrano--Zabrocki (2014--15) by the involution $\psi$ on the ring $\Qsym$ of quasisymmetric functions.  We give an explicit description of the effect of $\psi$ on the associated $0$-Hecke modules, via the poset induced by the $0$-Hecke action on standard immaculate tableaux.  This remarkable poset reveals other $0$-Hecke submodules and quotient modules, often cyclic and indecomposable, notably for a row-strict analogue of the extended Schur functions studied in Assaf--Searles (2019). }

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{$0$-Hecke algebra, dual immaculate functions,  indecomposable module, quasisymmetric functions}

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

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
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\usepackage[backend=bibtex]{biblatex}
\addbibresource{ICERMrefsFPSAC.bib}

%\addbibresource{sample.bib}
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\begin{document}

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

\section{Introduction}

In this paper we introduce a new family of quasisymmetric functions, \emph{the row-strict dual immaculate functions}.  Our focus here is the study of the associated $0$-Hecke algebra modules, which we define and analyse.

The row-strict dual immaculate functions $\rdI_\alpha$ are initially defined as generating functions for certain types of tableaux of composition shape $\alpha$. By identifying the correct descent set, we show that the functions 
$\rdI_\alpha$ expand positively in the basis of fundamental quasisymmetric functions, and also that they are the image of the well-studied dual immaculate functions of \cite{BBSSZ2015}, under the involutive algebra automorphism $\psi$ of the Hopf algebra $\Qsym$ of quasisymmetric functions.
% which sends the fundamental quasisymmetric function $F_\alpha$ to $F_{\alpha}^c$, where $\alpha^c$ is the complement of the composition $\alpha$.% \cite{BBSSZ2014, BBSSZ2015}.

The descent set  determines a $0$-Hecke algebra action on the set $\SIT(\alpha)$ of standard immaculate tableaux of composition shape $\alpha$, yielding a cyclic indecomposable module whose quasisymmetric characteristic is $\rdI_\alpha$.  The action defines a partial order on   
$\SIT(\alpha)$ which turns out to be dual to the partial order in \cite{BBSSZ2015}, see Lemma~\ref{lem:sameposet}.  The resulting poset $P\rdI(\alpha)$ has remarkable properties, leading to the discovery of several other $0$-Hecke modules with interesting quasisymmetric characteristics.  Among these is an analogue of the extended Schur functions defined by Assaf and Searles \cite{AS2019}. An examination of the poset (see Figure~\ref{fig:Poset}) reveals various subposets that are closed under either our action or the dual immaculate action of \cite{BBSSZ2015}.  We  investigate the resulting submodules.

The duality in the poset reflects the action of the involution $\psi$ on $\Qsym$: we show that the cyclic generators for the dual immaculate module and for the row-strict dual immaculate module  are respectively the top and bottom elements of the poset; see Definition~\ref{def:bot-elt-top-elt-gen-rs-ext}.  Similarly, the cyclic generators for the extended Schur Hecke-module of \cite{S2020} and our row-strict extended Schur Hecke-module (see Theorem~\ref{thm:Z-indecomp}) are the top and bottom elements of the interval $[S^{col}_\alpha, S^{row}_\alpha]$, cf. Definition~\ref{def:bot-elt-top-elt-gen-rs-ext}.  %More generally we can show that, under the conditions of Lemma~\ref{lem:sameposet}, one $0$-Hecke action in the poset completely determines the other. 
These ideas can be used to explain, for example, the passage between the modules in \cite{TvW2015} and \cite{BS2021}, and are developed further in the full paper \cite{NSvWVW2022b}.

Our proofs are technical,  relying heavily on straightening algorithms  which produce saturated chains in the poset $P\rdI(\alpha).$  See Theorem~\ref{thm:top-bot-elt}.  We prove  indecomposability by following the pioneering work in \cite{TvW2015} and  \cite{BBSSZ2015},  with considerable technical modifications.

Table 1 provides a summary of our results and a comparison with prior work.

\section{Background}

We refer the reader to \cite{LMvW2013} for basic definitions.

A composition of $n$ is a positive sequence of integers $\alpha = (\alpha_1, \alpha_2, \ldots,\alpha_k)$ summing to $n$, which we depict as    a collection of left-justified boxes with $\alpha_i$ boxes in row $i$, where row $1$ is the bottom row, in the ``French'' convention.  
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
\begin{ex}\label{ex:comps}
For $\alpha = (3,1,4)$, the diagram is 
\[\tableau{{}&{}&{}&{}\\{}\\{}&{}&{}}\]
\end{ex}
\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

It is well known that compositions of $n$ are in bijection with subsets of $\{1,2,\ldots,n-1\}$.  Write $\alpha\vDash n$ for a composition $\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_k)$ of $n$; the corresponding set is $\set(\alpha) = \{\alpha_1,\alpha_1+\alpha_2,\ldots,\alpha_1+\cdots+\alpha_{k-1}\}$. % For $\alpha=(3,1,4)$, $\set(\alpha) = \{3,4\} \subseteq \{1,2,\ldots,7\}$.
Given a subset $S=\{s_1,s_2,\ldots,s_j\}$ of $\{1,\ldots,n-1\}$, the corresponding composition of $n$ is $\comp(S)=(s_1,s_2-s_1,\ldots,s_j-s_{j-1},n-s_j)$. 

A function $f\in \bQ[[x_1,x_2,\ldots]]$ is {\em quasisymmetric} if the coefficient of $x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_k^{\alpha_k}$ is the same as the coefficient of $x_{i_1}^{\alpha_1}x_{i_2}^{\alpha_2}\cdots x_{i_k}^{\alpha_k}$ for every $(\alpha_1, \alpha_2,\ldots,\alpha_k)$ and $i_1<i_2<\cdots <i_k$.  The set of all quasisymmetric functions forms a ring graded by degree, $\QSym = \bigoplus_{n} \QSym_n$, where each $\QSym_n$ is a vector space over $\bQ$ with bases indexed by compositions of $n$. 

Given a composition $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_k)$ of $n$, the {\em fundamental quasisymmetric function} indexed by $\alpha$ is 
\[F_\alpha(x_1,x_2,\ldots) = \sum_{\substack{i_1\leq i_2\leq \cdots \leq i_n\\i_j<i_{j+1} \text{ if } j\in\set(\alpha)}} x_{i_1}x_{i_2}\cdots x_{i_n}.\]
%
Note that %$\{M_\alpha:\alpha\vDash n\}$ and 
$\{F_\alpha:\alpha\vDash n\}$ is a basis for $\QSym_n$, the \emph{fundamental} basis.

The {\em complement} of a composition $\alpha$, denoted $\alpha^c$ is the composition obtained from $\alpha$ by taking the complement of the set corresponding to  $\alpha$.  That is, 
$\alpha^c= \comp(\set(\alpha)^c)$.
 In $\QSym$ we have the involutive automorphism   $\psi,$ defined on the fundamental basis by
\begin{equation}
\psi(F_\alpha) = F_{\alpha^c} \label{eqn:psiF}.
\end{equation}
%
The algebra $\NSym=\bQ\langle\nce_1,\nce_2,\ldots\rangle$ of {\em noncommutative symmetric functions},  a Hopf algebra dual to $\Qsym$, see \cite{GKLLRT1995}, %\cite{LMvW2013}, %\cite{GKLLRT1995}
is generated by noncommuting indeterminates $\nce_n$ of degree $n$.  
%
We briefly review concepts we will need from the work of Berg--Bergeron--Saliola--Serrano--Zabrocki, who introduced the 
\emph{immaculate functions} $\mathfrak{S}_\alpha$ as a basis of $\NSym$. % formed by iterated creation operators~\cite{BBSSZ2014}.  
Their dual in $\QSym$ are the {\em dual immaculate functions}, $\dI_\alpha$.  These functions can be defined  as the generating function for {\em immaculate tableaux}.  
\begin{defn}[{\cite[Definition 2.1]{BBSSZ2015}}]
Given $\alpha\vDash n$, an immaculate tableau of shape $\alpha$ is a filling, $D$, of the cells of the diagram of $\alpha$ with positive integers such that 
\begin{enumerate}%[itemsep=1pt]
\item The leftmost column entries strictly increase from bottom to top;
\item The row entries weakly increase from left to right. 
\end{enumerate}
A \emph{standard}  immaculate tableau  of shape $\alpha\vDash n$ is one that is filled with distinct entries taken from $\{1,2,\ldots,n\}.$ 
\end{defn}

Given an immaculate tableau $D$, define $x^D=x_1^{d_1}x_2^{d_2}\cdots x_{k}^{d_k}$, where  $d_i$ is the number of $i$'s in the tableau $D$.  
\begin{defn} \label{def:dIfunction}
The {\em dual immaculate function} indexed by $\alpha\vDash n$ is $\dI_\alpha = \sum_D x^D$, summed over all immaculate tableaux of shape $\alpha$.
\end{defn}
%
\begin{thm}[{\cite[Definition~2.3, Proposition~3.1]{BBSSZ2015}}] %\cite[Proposition~3.37]{BBSSZ2014} 
The set $\{\dI_\alpha\}_{\alpha\vDash n}$ is a basis for $\Qsym_n.$
Given a standard immaculate tableau $S$, its $\dI$-{\em descent set}  $\DesI(S)$ of $S$ is 
\begin{center}$\DesI(S)=\{i: i+1 \text{ appears strictly above }i \text{ in } S \}.$\end{center}  %For the standard immaculate tableau in Example~\ref{ex:imtab}, $\DesI(S) = \{2,5,8,10\}$.
%
Then 
$\dI_\alpha = \sum_S F_{\comp(\DesI(S))},$ summed  over all standard immaculate tableaux of shape $\alpha$.  
\end{thm}

For $\alpha=(1,2),$ the unique standard immaculate tableau $\tableau{2&3\\1}$ 
 has $\dI$-descent set $\{1\}$; \quad thus $\dI_{(1,2)}=F_{\comp \{1\}}=F_{(1,2)}.$

\section{Row-strict dual immaculate functions}
In this section we define a new quasisymmetric function, which we call the {\em row-strict dual immaculate function.}  We will show that these functions form a basis of $\Qsym$. 

\begin{defn} Let $\alpha\vDash n$.  A row-strict immaculate tableau of shape $\alpha$ is a filling $U$  with positive integers such that 
\begin{enumerate}%[itemsep=1pt]
\item The leftmost column entries weakly increase from bottom to top;
\item The row entries strictly increase from left to right.
\end{enumerate}
The {\em row-strict dual immaculate function} indexed by $\alpha$ is $ \rdI_\alpha = \sum_U x^U$ where the sum is over all row-strict immaculate tableaux of shape $\alpha$, and $x^U=x_1^{d_1}x_2^{d_2}\cdots x_{k}^{d_k}$, where  $d_i$ is the number of $i$'s in the tableau $U$.  
\end{defn}

Note that standard row-strict  immaculate tableaux coincide with  standard immaculate tableaux. We denote the set of standard immaculate tableaux of shape $\alpha$ by $\SIT(\alpha)$.

\begin{thm}[{\cite{NSvWVW2022}}] \label{thm:rsfunddecomp}  
%Let $\alpha \vDash n$.  
Define the $\rdI$-descent set of a standard  immaculate tableau $S$ by 
\begin{center}{$\Des_{\rdI}(S)=\{i:i+1\text{ is weakly below } i \text{ in } S\}.$}\end{center} 
%
Then $\rdI_\alpha = \sum_{S} F_{\comp(\Des_{\rdI}(S))}$, 
summed over all standard  immaculate tableaux of shape $\alpha$.
\end{thm}
For $\alpha=(1,2),$ the unique standard immaculate tableau $\tableau{2&3\\1}$ 
has $\rdI$-descent set $\{2\}$; thus $\rdI_{(1,2)}=F_{\comp \{2\}}=F_{(2,1)}.$

Clearly for any standard immaculate tableau $S$, $\des_{\DI}(S) = \des_{\rdI}(S)^c$, and hence applying the involution $\psi$  immediately gives $\psi(\dI_\alpha) =\rdI_\alpha $. Consequently, we have: 
%
\begin{thm}\label{the:RIasDI}
 $\{ \RI _\alpha \suchthat \alpha \vDash n\}$ is a basis for $\Qsym _n$.
\end{thm} 

\section{A $0$-Hecke algebra action for $\mathcal{R}\mathfrak{S^*}$}\label{sec:Hecke-action-RSImm}
%2021 Sept 18

Recall that the symmetric group $S_n$ can be defined via generators $s_i, 1\le i\le n-1,$ the adjacent transpositions, subject to the relations 
\begin{center}${s_i}^2 =1;  \quad  s_i s_{i+1}s_i =s_{i+1}s_is_{i+1};\quad    s_is_j =s_js_i, \ |i-j|\ge 2.$\end{center}
%
\begin{defn}[{\cite{MathasHecke1999}}] Let $\bK$ be any field.  The $0$-Hecke algebra $H_n(0)$ is the $\bK$-algebra with 
generators $\pi_i, 1\le i\le n-1$ and relations 
\begin{center}$ {\pi_i}^2 =\pi_i; \quad 
          \pi_i\pi_{i+1}\pi_i =\pi_{i+1}\pi_i\pi_{i+1};\quad
              \pi_i\pi_j =\pi_j\pi_i, \ |i-j|\ge 2.$\end{center}
The algebra $H_n(0)$ has dimension $n!$ over  $\bK$, with basis elements $\{\pi_\sigma: \sigma\in S_n\},$ where $\pi_\sigma=\pi_{i_1}\cdots \pi_{i_m}$ if $\sigma=s_{i_1}\cdots s_{i_m}$  is a reduced word.  This is well-defined by standard Coxeter group theory, see \cite{MathasHecke1999}.
\end{defn}

It is known   \cite{PamelaBromwichNorton1979} that the $0$-Hecke algebra admits  precisely $2^{n-1}$  simple modules $L_\alpha$, 
%
one for each composition $\alpha\vDash n$, and all one-dimensional. 
%
The well-known  Frobenius characteristic defined on the Grothendieck ring of the symmetric groups has the following analogue for finite-dimensional $\hn$-modules.  See also \cite{DKLT1996}.
\begin{defn}[{\cite[Section 5.4]{KrobThibon1997}}] Let $M$ be a finite-dimensional $\hn$-module;  let $M=M_1\supset M_2\supset \cdots \supset M_k\supset M_{k+1}=\mathbb{K}$ be a composition series of submodules for $M$, where  each successive quotient $M_i/M_{i+1}$ is simple, and thus equal to $L_{\alpha^i}$ for some composition $\alpha^i\vDash n$. The \emph{quasisymmetric characteristic} of the module $M$, $\chr(M)$, is then defined to be the quasisymmetric function equal to the sum of fundamentals $\sum_{i=1}^k F_{\alpha^i}$.
%
In particular $\chr(L_\alpha)=F_\alpha$ for each $\alpha\vDash n$.
\end{defn}
%
The following is our restatement of the main result of~\cite{BBSSZ2015}.
\begin{thm}[{\cite[Theorem 3.12]{BBSSZ2015}}] \label{thm:BBSSZ2dualImm} There is an indecomposable cyclic $0$-Hecke algebra module $\mathcal{W}_\alpha$ whose quasisymmetric characteristic is the dual immaculate function $\dI_\alpha, $  
$ \chr(\mathcal{W}_\alpha)= \dI_\alpha.$
%
The module $\mathcal{W}_\alpha$ has dimension equal to the number of standard immaculate tableaux of shape $\alpha.$
%
The $\dI$-action of the  $0$-Hecke algebra generator $\pi_i$  on the set of standard immaculate tableaux of shape $\alpha$,  for $\alpha\vDash n$, may be described as follows:
\begin{equation}\label{eqn:defn-dualImm-pi(T)}\pi_i^{\dI}(T)=\begin{cases} T, &\text{if $i+1$ is in a row weakly below }i,\\
%
0  & \text{if $i,i+1$ are in column 1 of } T,\\
s_i(T) & \hspace{-6pt}\text{\begin{tabular}{l} if $i+1$ is strictly above $i$ in $T$ \\ and $i,i+1$ are \textbf{not} in column 1, \end{tabular}}
\end{cases}\end{equation}
where $s_i(T)$ is the standard immaculate tableau obtained from $T$ by swapping $i$ and $i+1.$ 
%
\end{thm}

The goal of this section is to find an analogous module $\mathcal{V}_\alpha$ for our newly defined row-strict dual immaculate functions, that is, so that $\chr(\mathcal{V}_\alpha)= \rdI_\alpha.$


 Following \cite{BBSSZ2015}, we consider the vector space 
$\mathcal{V}_\alpha$ whose basis vectors are the standard immaculate tableaux of shape $\alpha.$ 
Define, for each $1\le i\le n-1$ and each standard immaculate tableau $T$ of shape $\alpha,$ the $\rdI$-action of the generator $\pi_i$ on $T$ to be 
%
\begin{equation}\label{eqn:defn-RSdualImm-pi(T)}\pi_i(T)=\pi_i^{\rdI}(T)=\begin{cases} T   &\text{if $i+1$ is strictly above $i$ },\\
0  &\text{if $i$ and $i+1$ are in the same row of $T$},\\
s_i(T) &\text{if $i+1$ is strictly below $i$ in $T$},
\end{cases}\end{equation}
where $s_i(T)$ is  as above. 
%
To avoid cumbersome notation, we write simply $\pi_i(T)$ for the row-strict immaculate action, using 
   $\pi_i^{\rdI}$ and $\pi_i^{\dI}$ only when there is explicit need to distinguish between the actions of~\eqref{eqn:defn-RSdualImm-pi(T)} and~\eqref{eqn:defn-dualImm-pi(T)}.  We refer to these as the $\rdI$-action and the $\dI$-action respectively.  Likewise we may refer to the resulting $\hn$-modules as the  $\rdI$-Hecke module and the $\dI$-Hecke module respectively.
%
\begin{ex} For the standard  immaculate tableau
$S=\tableau{6\\4&5&8&10\\3&7\\1&2&9}$, 
%
from \eqref{eqn:defn-RSdualImm-pi(T)} we have $\pi^{\rdI}_i\!(T)=T \text{ for } i\in\{2,3,5,7,9\}, \ \ \pi^{\rdI}_1\!(T)=0=\pi^{\rdI}_4\!(T)$, and %\qquad \qquad
%
%
$\pi^{\rdI}_6\!(T)=s_6(T)$ $=\tableau{{\bf 7}\\4&5& 8&10\\3& {\bf 6}\\1&2&9} , \quad  \pi^{\rdI}_8\!(T)=s_8(T)=\tableau{6\\4&5&  {\bf 9}&10\\3&7\\1&2& {\bf 8}}.$
\end{ex}


\begin{thm}\label{thm:Hecke-action-RSImm} The operators $\pi_i^{\rdI}$ define an action of   
$\hn$ on the vector space $\mathcal{V}_\alpha.$ 
\end{thm}
%
Figure~\ref{fig:Poset} shows the row-strict dual immaculate action on the standard immaculate tab\-leaux of shape $223,$ with labelled arrows indicating which generator maps one tab\-leau to another.  We note that this poset is the dual of the poset appearing in \cite{BBSSZ2015}.

The cover relation for our poset $P\rdI_\alpha$ for  standard immaculate tableaux of shape $\alpha\vDash n$  is 
\begin{equation}\label{eqn:rdIcover} 
S\poRIcover  T
\text{ if and only if there exists $i$ such that }  T=\pi_i^{\rdI}(S),
\end{equation}
with respect to the \noindent\textbf{row-strict $0$-Hecke} action defined by Theorem~\ref{thm:Hecke-action-RSImm}.

On the other hand, the cover relation for the poset $P\dI_\alpha$ for standard immaculate tableaux of shape $\alpha\vDash n,$ as described in \cite{BBSSZ2015}, is
\begin{equation}\label{eqn:dIcover} 
S\poIcover  T
\text{ if and only if there exists $i$ such that }  S=\pi^{\dI}_i(T),
\end{equation}
with respect to the \noindent\textbf{dual immaculate $0$-Hecke} action defined  by Theorem~\ref{thm:BBSSZ2dualImm}.

The following lemma shows that in each case,  a cover relation is determined by a unique generator of the $0$-Hecke algebra.
\begin{lem}\label{lem:uniqueness1} Let $\alpha\vDash n$ and $T\in \SIT(\alpha).$  If $\pi_i(T)=s_i(T), \pi_j(T)=s_j(T)\in \SIT(\alpha)$ and  $\pi_i(T)=\pi_j(T),$ then necessarily $i=j.$
\end{lem}

By means of a filtration defined on the poset $P\rdI_\alpha$, just as in \cite{TvW2015} and  \cite{BBSSZ2015}, we show that 
\begin{thm}\label{the:bigone}
Let $\alpha \vDash n$ and $\rtau _1 \in \SIT (\alpha)$ be the minimal element under any  total order  imposed on the poset $P\rdI_\alpha$. Then $\rtau_1$ determines a unique $\hn$-module $\mathcal{V}_\alpha$ whose quasisymmetric characteristic is the row-strict dual immaculate function $\RI_{\alpha}$.\end{thm}

However, we analyse the poset $P\rdI_\alpha$ in more detail.  We observe first that 
\begin{lem}\label{lem:sameposet} Let $\alpha\vDash n$ and $S,T\in \SIT(\alpha).$ Then 
$S\poIcover  T \text{ if and onlf if } S\poRIcover  T.$
Hence  the two posets $P\dI_\alpha$ and $P\rdI_\alpha$ are isomorphic.
\end{lem}
 
\begin{defn}\label{def:bot-elt-top-elt-gen-rs-ext} Let $\alpha\vDash n$ be of length $\ell=\ell(\alpha).$ We single out three special  standard immaculate tableaux: 

Define $S^0_\alpha$ to be the standard tableau of shape $\alpha$ with entries $1, 2,\ldots, \ell$ in column 1, increasing bottom to top, and then fill the remaining rows, top to bottom, left to right with consecutive integers starting at $\ell+1$ and ending at $n.$ 

 Define   
$ S^{row}_\alpha$ to be the \textit{row superstandard} tableau of shape $\alpha,$ whose  rows are  filled  left to right, bottom to top, beginning with the first row, with the numbers $1,2, \ldots,  n$ in consecutive order. 

Define $S^{col}_\alpha$ to be the \textit{column superstandard} tableau of shape $\alpha,$ whose columns are  filled   bottom to top and left to right, beginning with the first column, with the numbers $1,2,\ldots, n$   in consecutive order.% In Figure~\ref{fig:Poset}, $S^{col}_\alpha$ is the lowest-ranked  tableau in red. 
\end{defn}

\begin{ex}\label{ex:bot-elt1} We have 
\[S^0_{43423}=\tableau{\bf{5} &6 &7\\
                      \bf{4} &8\\
                      \bf{3}&9 & 10 & 11\\
                      \bf{2} & 12 & 13\\
                      \bf{1} & 14 & 15 & 16}\ , \qquad  
S^{row}_{43423}=\tableau{\bf{14} &15 &16\\
                      \bf{12} &13\\
                      \bf{8}&9 & 10 & 11\\
                      \bf{5} & 6 & 7\\
                      \bf{1} & 2 & 3 & 4}\ ,\qquad
S^{col}_{43423}=\tableau{\bf{5} &10 &14\\
                      \bf{4} &9\\
                      \bf{3}&8& 13 & 16\\
                      \bf{2} & 7 & 12\\
                      \bf{1} & 6 & 11 & 15}\ .\]
\end{ex}

By means of two important straightening algorithms, we show that 
\begin{thm}\label{thm:top-bot-elt} The poset $P\rdI_\alpha$ has a unique bottom element $S^0_\alpha$ and a unique top element $S^{row}_\alpha.$ Also, for any $T\in \SIT(\alpha)$, there are saturated chains from $S^0_\alpha$ to $T$, and from $T$ to $S^{row}_\alpha$. 
\end{thm}

\begin{ex}\label{ex:rdI-straightening1}  Let $\alpha=223,$ so that 
 $S^0_\alpha=\tableau{3 &4 &5\\2 &6\\1 &7},$ and let $T=\tableau{5 &6 &7\\2 &4\\1&3}$.
 First we straighten column 1 of $T$ to match column 1 of $S^0_\alpha$: Start with the lowest entry
% fixed typo top-most -->lowest 2022 Mar 7
 $a_j$  in column 1 of $T$ such that $a_j\ne j,$ and swap it with $a_j-1$; continue in this manner until column 1 has the entries $3,2,1$ from top to bottom.
% replace-->swap 2022 Mar 7
 \[\small{T=T_0\stackrel{\pi_4}{\longleftarrow} T_1=\tableau{\bf{4} &6 &7\\2 &5\\1&3}
 \stackrel{\pi_3}{\longleftarrow} T_2=\tableau{\bf{3} &6 &7\\2 &5\\1&4}}\ .\]
 Next we work on the top row of $T_2,$ starting with the smallest entry which differs from the corresponding entry in the same cell of $S^0_\alpha,$ and continuing until the top rows match:
 \[T_2 \!  \stackrel{\pi_5}{\longleftarrow}\! T_3=\tableau{3 &\bf{5} &7\\2 &6\\1&4}
  \stackrel{\pi_4}{\longleftarrow} T_4=\tableau{3 &\bf{4} &7\\2 &6\\1&5}
  \stackrel{\pi_6}{\longleftarrow} T_5=\tableau{3 &4 &\bf{6}\\2 &7\\1&5}
 \stackrel{\pi_5}{\longleftarrow} T_6=\tableau{3 &4 &\bf{5}\\2 &7\\1&6}\ .
  \]
  Now move down to the next row from the top, and proceed in the same manner, finding the smallest entry that differs from the corresponding entry in $S^0_\alpha$:
\[T_6   \stackrel{\pi_6}{\longleftarrow} T_7=\tableau{3 &4 &5\\2 &\bf{6}\\1&7}=S^0_\alpha. \qquad \text{ We end up with  } T=\pi_4\pi_3\pi_5\pi_4\pi_6\pi_5\pi_6(S^0_\alpha).\]
\end{ex}
%
From this we obtain:
%
\begin{thm}\label{thm:cyclic-rdI}  The $\rdI$-Hecke module $\mathcal{V}_\alpha$  of Theorem~\ref{the:bigone}, with $\chr(\mathcal{V}_\alpha)=\rdI_\alpha$, is cyclic, and generated by the unique minimal element $S^0_\alpha$  of the poset $P\rdI_\alpha$.
\end{thm}
%
We also recover the result of \cite{BBSSZ2015}:
\begin{thm}\label{thm:cyclic-dI}  The $\dI$-Hecke  module $\mathcal{W}_\alpha$ of Theorem~\ref{thm:BBSSZ2dualImm}, with $\chr(\mathcal{W}_\alpha)=\dI_\alpha$, is cyclic, and generated by the unique maximal element $S^{row}_\alpha$  of the poset $P\rdI_\alpha$.
\end{thm}


\begin{figure}[t]% \centering
\[
%\begin{center}
		%\scalebox{0.8}{%			
			\begin{tikzpicture}[scale=0.61, every node/.style={scale=0.75}]
			\newcommand*{\xdist}{*3}
			\newcommand*{\ydist}{*2.2}
			
			\node (n0) at (0.00\xdist,0\ydist) 
			{ \textcolor{blue}{
				$S^0_{223} =
				\tableau{ 
				 3 & 4 & 5 \\ 
				 2 & 6 \\
                      1 & 7}  $ }
			}; 
			\node (n11) at (-1\xdist,1\ydist) 
			{\textcolor{blue}{
				$\tableau{ 
				 3 & 4 & 6 \\ 
				 2 & 5 \\
                      1 & 7}  $  }
			}; 
\node (n12) at (1\xdist,1\ydist) 
			{\textcolor{blue}{
				$\tableau{ 
				 3 & 4 & 5 \\ 
				 2 & 7 \\
                      1 & 6}  $ }
			}; 
			\node (n21) at (-1.5\xdist,2\ydist) 
			{\textcolor{blue}{
				$\tableau{ 
				 3 & 5 & 6 \\ 
				 2 & 4 \\
                      1 & 7}  $ }
			}; 
\node (n22) at (0\xdist,2\ydist) 
			{\textcolor{blue}{
				$\tableau{ 
				 3 & 4 & 7 \\ 
				 2 & 5 \\
                      1 & 6}  $ }
			}; 
\node (n23) at (1.5\xdist,2\ydist) 
			{ \textcolor{blue}{
				$\tableau{ 
				 3 & 4 & 6 \\ 
				 2 & 7 \\
                      1 & 5}  $ }
			}; 
						\node (n31) at (-2\xdist,3\ydist) 
			{
				$\tableau{ 
				 4 & 5 & 6 \\ 
				 2 & 3 \\
                      1 & 7}  $
			}; 
			\node (n32) at (-1\xdist,3\ydist) 
			{\textcolor{blue}{
				$\tableau{ 
				 3 & 5 & 7 \\ 
				 2 & 4 \\
                      1 & 6}  $ }
			}; 
\node (n33) at (1\xdist,3\ydist) 
			{\textcolor{blue}{
				$\tableau{ 
				 3 & 4 & 7 \\ 
				 2 & 6 \\
                      1 & 5}  $}
			}; 
\node (n34) at (2\xdist,3\ydist) 
			{\textcolor{blue}{
				$\tableau{ 
				 3 & 5 & 6\\ 
				 2 & 7 \\
                      1 & 4}  $ }
			}; 

						\node (n41) at (-2\xdist,4\ydist) 
			{
				$\tableau{ 
				 4 & 5 & 7 \\ 
				 2 & 3 \\
                      1 & 6}  $
			}; 
			\node (n42) at (-.5\xdist,4\ydist) 
			{\textcolor{blue}{
				$\tableau{ 
				 3 & 6 & 7 \\ 
				 2 & 4 \\
                      1 & 5}  $ }
			}; 
\node (n43) at (.5\xdist,4\ydist) 
			{\textcolor{blue}{
				$\tableau{ 
				 3 & 5 & 7 \\ 
				 2 & 6 \\
                      1 & 4}  $ }
			}; 
\node (n44) at (2\xdist,4\ydist) 
			{
				$\tableau{ 
				 4 & 5 & 6\\ 
				 2 & 7 \\
                      1 & 3}  $
			}; 
% about to do line 5...

						\node (n51) at (-2\xdist,5\ydist) 
			{
				$\tableau{ 
				 4 & 6 & 7 \\ 
				 2 & 3 \\
                      1 & 5}  $
			}; 
			\node (n52) at (-1\xdist,5\ydist) 
			{\textcolor{magenta}{
				$S^{col}_{223}=\tableau{ 
				 3 & 6 & 7 \\ 
				 2 & 5 \\
                      1 & 4}  $}
			}; 
\node (n53) at (1\xdist,5\ydist) 
			{
				$\tableau{ 
				 4 & 5 & 7 \\ 
				 2 & 6 \\
                      1 & 3}  $
			}; 
\node (n54) at (2\xdist,5\ydist) 
			{
				$\tableau{ 
				 4 & 5 & 6\\ 
				 3 & 7 \\
                      1 & 2}  $
			}; 
% about to do line 6

	\node (n61) at (-1.5\xdist,6\ydist) 
			{
				$\tableau{ 
				 5 & 6 & 7 \\ 
				 2 & 3 \\
                      1 & 4}  $
			}; 
			\node (n62) at (0\xdist,6\ydist) 
			{\textcolor{red}{
				$\tableau{ 
				 4 & 6 & 7 \\ 
				 2 & 5 \\
                      1 & 3}  $}
			}; 
\node (n63) at (1.5\xdist,6\ydist) 
			{
				$\tableau{ 
				 4 & 5 & 7 \\ 
				 3 & 6 \\
                      1 & 2}  $
			}; 

%about to do line 7
\node (n71) at (-1\xdist,7\ydist) 
			{\textcolor{red}{
				$\tableau{ 
				 5 & 6 & 7 \\ 
				 2 & 4 \\
                      1 & 3}  $ }
			}; 
\node (n72) at (1\xdist,7\ydist) 
			{\textcolor{red}{
				$\tableau{ 
				  4 & 6 & 7 \\ 
				 3& 5 \\
                      1 & 2}  $ }
			}; 

			\node (n8) at (0\xdist,8\ydist) 
			{\textcolor{red}{
				$S^{row}_{223} =
				\tableau{ 
				 5 & 6 & 7 \\ 
				 3 & 4 \\
                      1 & 2}  $ }
			}; 
			
			\draw [thick, ->] (n0) -- (n11) node [near start, left] {$\pi_{5}$};
                 \draw [thick, ->] (n0) -- (n12) node [near start, right] {$\pi_{6}$};
			\draw [thick, ->] (n11) -- (n21) node [near start, left] {$\pi_{4}$}; 
                \draw [thick, ->] (n11) -- (n22) node [near start, right] {$\pi_{6}$};
			\draw [thick, ->] (n12) -- (n23) node [near start, right] {$\pi_{5}$}; 
			
                \draw [thick, ->] (n21) -- (n31) node [near start, left] {$\pi_{3}$}; 
                \draw [thick, ->] (n21) -- (n32) node [near start, right] {$\pi_{6}$}; 

                \draw [thick, ->] (n22) -- (n32) node [near start, left] {$\pi_{4}$}; 
                \draw [thick, ->] (n22) -- (n33) node [near start, right] {$\pi_{5}$}; 

                \draw [thick, ->] (n23) -- (n33) node [near start, left] {$\pi_{6}$}; 
                \draw [thick, ->] (n23) -- (n34) node [near start, right] {$\pi_{4}$}; 

\draw [thick, ->] (n31) -- (n41) node [near start, left] {$\pi_{6}$}; 
\draw [thick, ->] (n32) -- (n41) node [near start, left] {$\pi_{3}$}; 
\draw [thick, ->] (n32) -- (n42) node [near start, right] {$\pi_{5}$}; 

\draw [thick, ->] (n33) -- (n43) node [near start, left] {$\pi_{4}$}; 
\draw [thick, ->] (n34) -- (n43) node [near start, left] {$\pi_{6}$}; 
\draw [thick, ->] (n34) -- (n44) node [near start, right] {$\pi_{3}$}; 

\draw [thick, ->] (n41) -- (n51) node [near start, left] {$\pi_{5}$}; 
\draw [thick, ->] (n42) -- (n51) node [near start, left] {$\pi_{3}$}; 
\draw [thick, ->] (n42) -- (n52) node [near start, right] {$\pi_{4}$}; 

\draw [thick, ->] (n43) -- (n52) node [left, near start] {$\pi_{5}$};

\draw [thick, ->] (n43) -- (n53) node [near start, right] {$\pi_{3}$}; 
\draw [thick, ->] (n44) -- (n53) node [near start, left] {$\pi_{6}$}; 
\draw [thick, ->] (n44) -- (n54) node [near start, right] {$\pi_{2}$}; 

\draw [thick, ->] (n51) -- (n61) node [near start, left] {$\pi_{4}$}; 
\draw [thick, ->] (n52) -- (n62) node [near start, left] {$\pi_{3}$}; 
\draw [thick, ->] (n53) -- (n62) node [near start, right] {$\pi_{5}$};

\draw [thick, ->] (n53) -- (n63) node [near start, right] {$\pi_{2}$}; 
\draw [thick, ->] (n54) -- (n63) node [near start, left] {$\pi_{6}$}; 

\draw [thick, ->] (n61) -- (n71) node [near start, right] {$\pi_{3}$}; 
\draw [thick, ->] (n62) -- (n71) node [near start, left] {$\pi_{4}$}; 
\draw [thick, ->] (n62) -- (n72) node [near start, right] {$\pi_{2}$};
\draw [thick, ->] (n63) -- (n72) node [near start, left] {$\pi_{5}$}; 

  \draw [thick, ->] (n71) -- (n8) node [near start, right] {$\pi_{2}$}; 
\draw [thick, ->] (n72) -- (n8) node [near start, left] {$\pi_{4}$};               	
			\end{tikzpicture}	
%}			
%\end{center}
\]
% SHEILA'S NOTE TO HERSELF: \label{fig:} ALWAYS GOES AFTER %\caption, otherwise numbering is messed up	
\caption{The row-strict dual immaculate poset $P\rdI_{223}$; the red tableaux and $S^{col}_{223}$ are increasing along rows and up columns, so are in $\textcolor{red}{\SET(223)=[S^{col}_{223}, S^{row}_{223}]}$; the blue tableaux and $S^{col}_{223}$ are the elements in $\textcolor{blue}{\SIT^*(223)=[S^0_{223}, S^{col}_{223}]}$, with the smallest elements in the first column.}\label{fig:Poset}
\end{figure}

%
A long series of technical lemmas culminates in our main result for this section, namely:
%
\begin{thm}\label{thm:Indecomp} Let $\mathcal{V}_\alpha$ be the cyclic $\rdI$-Hecke module generated by $S^0_\alpha.$ Then $\mathcal{V}_\alpha$ is indecomposable, and $\chr(\mathcal{V}_\alpha)=\rdI_\alpha$.
\end{thm}

\section{A row-strict analogue for extended Schur functions}\label{sec:row-strict-ext}

This section is motivated by  a closer examination of the poset $P\rdI_\alpha.$  

\begin{defn}[{\cite{S2020}}]\label{def:SET} Let $\mathrm{SET}(\alpha)$ be the set of all standard immaculate tableaux of shape $\alpha$, called 
\textit{s}tandard \textit{e}xtended \textit{t}ableaux, in which \textbf{all} columns increase from bottom to top. %Equivalently, $\mathrm{SET}(\alpha)$ is simply the set of all standard tableaux of shape $\alpha$ with strictly increasing rows, left to right, and strictly increasing columns, bottom to top. 
\end{defn}
%
Searles \cite{S2020} constructs a quotient $\hn$-module with basis $\SET(\alpha)$, from a larger parent module, whose quasisymmetric characteristic is the  extended Schur function $\mathcal{E}_\alpha$ defined in \cite[Section 7]{CFLSX2014} and \cite{AS2019}, and whose cyclic generator is $S^{row}_\alpha.$  First we restate Searles' formulation of the Assaf--Searles theorem as follows:
%
\begin{thm}[{\cite{AS2019},\cite[Theorem 2.7]{S2020}}]  Let $\alpha\vDash n.$ Then the extended Schur function 
$\mathcal{E}_\alpha$ expands positively in the fundamental basis of $\QSym$ as follows:
%
$\mathcal{E}_\alpha= \sum_{T\in \SET(\alpha)} F_{\comp(\Des_{\dI}(T))}.$
\end{thm}
%
%
%Also, 
\begin{thm}[{\cite{CFLSX2014}}]\label{thm:AS19}  The extended Schur functions $\{\mathcal{E}_\alpha\}_{\alpha\vDash n}$ form a basis of $\QSym_n,$ such that when the composition $\alpha$  is a partition $\lambda$ of $n,$  $\mathcal{E}_\lambda$ coincides with the Schur function $s_\lambda.$
\end{thm}

Let $\mathcal{Z}_\alpha=\mathrm{span}\langle\SET(\alpha)\rangle.$ Figure~\ref{fig:Poset} shows the elements of $\SET(223)$ in red; notice that they form a closed interval $[S^{col}_\alpha, S^{row}_\alpha]$ of $P\rdI_{223}$  with respect to our $\rdI$-Hecke action defined 
in~\eqref{eqn:defn-RSdualImm-pi(T)}.  This is no accident. 

\begin{thm}\label{thm:Z-indecomp} The tableau $S^{col}_\alpha$ is the unique minimal element of $\SET(\alpha)$. The  $\hn$-submodule $\mathcal{Z}_\alpha$ of $\mathcal{V}_\alpha$ is cyclically generated by $S^{col}_\alpha$ and is indecomposable.  It has characteristic 
\[
\mathcal{R}\mathcal{E}_\alpha=\sum_{T\in\SET(\alpha)} F_{\comp(\Des_{\rdI}(T))} = \psi(\mathcal{E}_\alpha).
\]
 Hence the functions $\{\mathcal{R}\mathcal{E}_\alpha\}_{\alpha\vDash n}$ also form a basis for 
$\QSym_n$. Furthermore, 
when the composition $\alpha$  is a partition $\lambda$ of $n,$  $\mathcal{R}\mathcal{E}_\lambda$ coincides with the Schur function $s_{\lambda^t},$ where $\lambda^t$ is the transpose of $\lambda$.
%
\end{thm}

We call the  $\mathcal{R}\mathcal{E}_\alpha$  \textit{row-strict extended Schur} functions.

The extended Schur functions are dual to the \textit{shin} basis of noncommutative symmetric functions of \cite{CFLSX2014}. %
Hence our 
row-strict extended Schur functions $\mathcal{R}\mathcal{E}_\alpha$ also give rise to a dual basis of $\Nsym,$ which we call the \textit{Rshin} basis.

Next we examine   the quotient module ${\mathcal{V}_\alpha/\mathcal{Z}_\alpha}$ arising from the submodule $\mathcal{Z}_\alpha$  of $\mathcal{V}_\alpha.$ 

\begin{defn}\label{def:NSET} Let $\mathrm{NSET}(\alpha)$ be the set of standard  tableaux of shape $\alpha$ where all  rows increase left to right, but at least one \textbf{column} does \emph{not} increase from bottom to top. 
\end{defn}

\begin{thm}\label{thm:another-bigone}
The quotient module $\mathcal{\overline V}_{\alpha}={\mathcal{V}_\alpha/\mathcal{Z}_\alpha}$ is also an 
 $\hn$-module  for the $\rdI$-action, with basis indexed by $\NSET(\alpha)\cap \SIT(\alpha).$  The module $\mathcal{\overline{V}}_{\alpha}$ is nonzero if and only if $\alpha$ has at least two parts of size greater than or equal to 2. In the latter case it is cyclically generated by $S^0_\alpha+\mathcal{Z}_\alpha$
and indecomposable, with characteristic equal to the quasisymmetric function
\[
\overline{\mathcal{R}\mathcal{E}}_\alpha= \sum_{T\in \NSET(\alpha)\cap\SIT(\alpha)} F_{\mathrm{comp}(\Des_{\rdI}(T))}.
\]
%
Equivalently, $\ \rdI_\alpha-\overline{\mathcal{R}\mathcal{E}}_\alpha=\sum_{\rtau \in \SET(\alpha)}F_{\comp(\desri(\rtau))}=\mathcal{R}\mathcal{E}_\alpha.$
\end{thm}

\begin{def}\label{def:top-elt-SIT*}  Define $S^{row*}_\alpha$ to be the standard immaculate tableau in $\SIT(\alpha)$ whose first column consists of $1,2,\ldots,\ell(\alpha)$, and whose remaining cells  are filled with the entries $\ell+1, \ldots, n$ in consecutive order along  rows, bottom to top and left to right.  Clearly  $S^{row*}_\alpha$ has strictly increasing columns bottom to top, and hence $S^{row*}_\alpha\in \SIT^*(\alpha)\cap \SET(\alpha)$. 

For example, we have
 \[S^{row*}_{223}=S^{col}_{223}=\tableau{3 &6 &7\\2&5 \\1& 4}\!,\qquad
S^{row*}_{332}=\tableau{3 &8 \\2 &6 &7 \\1 & 4 & 5}\ne S^{col}_{332},\qquad S^{row*}_{1323}
=\tableau{4 &8  &9 \\3 &7  \\2 & 5 & 6\\1}\ne S^{col}_{1323}.\]
\end{def}
%
Let $\SIT^*\!(\alpha)$ denote the set of standard immaculate tableaux whose first column consists of the integers $\{1,2,\ldots,\ell(\alpha)\}$. Examining the poset $P\rdI(\alpha)$ again, we observe that the tableaux in blue in Figure~\ref{fig:Poset} form a closed interval $[S^0_\alpha, S^{col}_\alpha=S^{row*}_\alpha]$ for the \textbf{dual immaculate action} of Theorem~\ref{thm:BBSSZ2dualImm} (reverse the arrows, thanks to Lemma~\ref{lem:sameposet}). We obtain 
%
\begin{thm}\label{thm:dualImm-submodule-quotient} Let $\alpha\vDash n,$ and let $\mathcal{W}_\alpha$ be the $\hn$-module with basis $\SIT(\alpha)$ as in Theorem~\ref{thm:BBSSZ2dualImm}.  Then 
\begin{enumerate}
\item
$\SIT^*\!(\alpha)\!=\![S^0_\alpha, S^{row*}_\alpha]$  spans an $\hn$-submodule $\mathcal{X}_\alpha$ of $\mathcal{W}_\alpha$, of dimension $\binom{n-\ell(\alpha)}{\alpha_1-1, \ldots, \alpha_\ell-1}.$
\item $\mathcal{X}_\alpha$ is cyclically generated by $S^{row*}_\alpha$;  it has characteristic 
\[
\chr(\mathcal{X}_\alpha)= \sum_{T\in \SIT^*\!(\alpha)} F_{\comp(\Des_{\dI}(T))}.
\]
\item Assume $\alpha$ has at least two parts of size greater than 1. Then the quotient module $\mathcal{W}_\alpha/\mathcal{X}_\alpha$ is  nonzero, indecomposable, and cyclically generated by $S^{row}_\alpha+\mathcal{X}_\alpha$,  with characteristic 
\[
\chr(\mathcal{W}_\alpha/\mathcal{X}_\alpha)
= \sum_{T\in \SIT(\alpha)\setminus \SIT^*(\alpha)} F_{\comp(\Des_{\dI}(T))}.
\]
\end{enumerate}
If $\alpha$ has at most one part greater than 1, then $\mathcal{W}_\alpha=\mathcal{X}_\alpha$ is the $\hn$-irreducible $L_\alpha$.
\end{thm}
%
We have the following pleasing expression for the characteristic  of the module $\mathcal{X}_\alpha.$
%
\begin{prop}\label{prop:ElizabethSIT*-char} Let $\alpha\vDash n$ have length $\ell$, and let $\overline{\alpha}$ be the composition $(\alpha_1-1, \alpha_2-1, \ldots, \alpha_\ell-1)$ of $n-\ell,$ where we omit any part that is zero.  Then
\[
\chr(\mathcal{X}_\alpha)= 
\sum_{k\ge \ell} e_{\ell-1}(x_1,\ldots,x_{k-1}) x_k h_{\overline{\alpha}}(x_k,x_{k+1},\ldots ),
\]
where $e_r$ is the $r$th elementary symmetric function,  $h_\beta=h_{\beta_1}h_{\beta_2}\cdots $ is the product of the homogeneous symmetric functions indexed by the parts of the composition $\beta.$
\end{prop}
%

Finally, we observe (contrast with \cite{S2020}) that the poset $P\dI(\alpha)$ itself gives a quotient module of  $\mathcal{W}_\alpha$ whose characteristic is the extended Schur function $\mathcal{E}_\alpha$. Reversing arrows in  Figure~\ref{fig:Poset} shows that  the tableaux \textbf{not} in $\SET(\alpha)$ are  
  closed under the $\dI$-action.
%
\begin{prop}\label{prop:shin-module} The set $\NSET(\alpha)\cap\SIT(\alpha)$ indexes a basis for a submodule $\mathcal{Y}_\alpha$ of $\mathcal{W}_\alpha$ for the $\dI$-action.   The resulting quotient module 
$\mathcal{W}_\alpha/\mathcal{Y}_\alpha$ has basis $\SET(\alpha)$ with characteristic $\mathcal{E}_\alpha$, and is cyclically generated by $S^{row}_\alpha+\mathcal{Y}_\alpha$.  It  is indecomposable.
\end{prop}

A summary of our results for row-strict dual immaculates appears in Table~\ref{table:1} below.

\begin{table}[h]
\begin{center}
\scalebox{0.8}{
\begin{tabular}{|c|c|c|l|}
\hline
\tclb{Immaculate Tableaux $\rightarrow$}  &  $\dI_\alpha\ $ \cite{BBSSZ2015} &  \tclb{$\rdI_\alpha$} \\
 &  Dual immaculate &  \tclb{ Row-strict dual imm.}\\
[2pt]\hline\hline
 \bf{1st Col} \bf{bottom to top} &strict $\nearrow$ &\tclb{weak $\nearrow$ }\\[2pt]\hline
 \bf{Rows left to right}
& weak $\nearrow$&\tclb{strict  $\nearrow$} \\[2pt]\hline
%
Descents (for fund. expansion) &$\{i: i+1 \text{ strictly above } i\}$
  & \tclb{$\{i: i+1 \text{ weakly below } i\}$}\\[2pt]\hline\hline
% 
Action of $\psi$  in $\QSym$ 
%
& $\dI_\alpha(x_1,\ldots,x_n)$ & \tclb{$\rdI_\alpha=\psi(\dI_\alpha)$}\\[2pt]\hline
%
 \textcolor{blue}{$0$-Hecke action on $\SIT(\alpha)$}&\textit{Note that  standard}  &\textit{tableaux are the same} \\[2pt]\hline\hline
 %
 $\pi_i(T)=T$ & $ i+1 \text{ weakly below } i$ 
& \tclb{$ i+1 \text{ strictly above } i$}
 \\[2pt]\hline
% 
$\pi_i(T)=0$ & $i, i+1$ in 1st column & \tclb{$i, i+1$ in same row}\\[2pt]\hline
 $T, s_i(T)$ standard, &$ i+1 $\text{ strictly above }$ i$,& \tclb{$ i+1 \text{ strictly below } i$}\\[2pt]
and $\pi_i(T)=s_i(T)$ & $i,i+1$ NOT both in 1st column & \phantom{\tclb{ $i, i+1$ NOT in same row}}\\[2pt]\hline\hline
 \textcolor{blue}{Partial order on $\SIT(\alpha)$} &Poset $P\dI(\alpha)=[S^{0}_\alpha, S^{row}_\alpha]$&\tclb{ Poset $P\rdI(\alpha)\simeq P\dI(\alpha)$}\\[2pt]\hline\hline
Cover relation & $S\!\poIcover\!  T \text{ if and onlf if }  S=\pi_i^{\dI}(T)$ & \tclb{$S\!\poRIcover\!  T \text{ if and onlf if }  T=\pi_i^{\rdI} (S)$}\\[2pt]\hline
% 
Imm. module generated by& top element: $\mathcal{W}_\alpha=\langle S^{row}_\alpha\rangle$  & \tclb{bottom  element: $\mathcal{V}_\alpha=\langle S^0_\alpha\rangle$} \\[2pt]\hline
%
%
 Indecomposable? & Yes \cite{BBSSZ2015} & \textcolor{blue}{Yes}\\[2pt]\hline\hline
Extended Schur fn basis & $\mathcal{E}_\alpha,$ $\mathcal{E}_\lambda=s_\lambda$\cite{AS2019} & \tclb{$\psi(\mathcal{E}_\alpha)=\mathcal{R}\mathcal{E}_\alpha$, $\mathcal{R}\mathcal{E}_\lambda=s_{\lambda^t}$}  \\[2pt]\hline
Module, Extended Schur fn & cyclic $\langle S^{row}_\alpha\rangle$, indecomp.~\cite{S2020} & \tclb{cyclic $\langle S^{col}_\alpha\rangle$, indecomp.}\\[2pt]
Basis \tclb{$\SET(\alpha)$} & (quotient of larger module) & \tclb{submodule of $\mathcal{V}_\alpha$}\\[2pt]\hline
Quotient module, Ext Schur fn& None & \tclb{Yes, cyclic $\langle S^{row}_\alpha\rangle$, indecomp.} \\[2pt]
Basis \tclb{$\NSET(\alpha)\cap\SIT(\alpha)$}&  & \tclb{quotient of $\mathcal{V}_\alpha$}\\[2pt]\hline
\end{tabular}
}%end \scalebox
\end{center}
\caption{$\dI_\alpha$ versus the new basis $\rdI_\alpha$.}
\label{table:1}
\end{table}
%\vskip .2in

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\end{document}

