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%\newcommand{\defn}[1]{{\color{darkred}\emph{#1}}} % emphasis of a definition
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\usepackage[colorinlistoftodos]{todonotes}
\newcommand{\anne}[1]{\todo[size=\tiny,color=red!30]{#1 \\ \hfill --- Anne}}
\newcommand{\Anne}[1]{\todo[size=\tiny,inline,color=red!30]{#1
      \\ \hfill --- Anne}}
      
\newcommand{\Wencin}[1]{\todo[size=\tiny,inline,color=green!30]{#1
      \\ \hfill --- Wencin}}
 \newcommand{\Jianping}[1]{\todo[size=\tiny,inline,color=blue!30]{#1
 		\\ \hfill --- Jianping}}
 
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		\\ \hfill --- Reviewer}}
	

\newcommand{\wt}{\operatorname{wt}}
\newcommand{\et}{\overline{e}}
\newcommand{\ft}{\overline{f}}
\newcommand{\st}{\overline{s}}
\newcommand{\epst}{\overline{\varepsilon}}
\newcommand{\phit}{\overline{\varphi}}
\newcommand{\len}{\mathsf{len}}

\newcommand{\HH}{\mathcal{H}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
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\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
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%% define your title in the usual way
%\title[Optional shorter title]{Something to do with math}
\title[Crystal for stable Grothendieck polynomials]{Crystal for stable Grothendieck polynomials}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}

\author[Jennifer Morse, Jianping Pan, Wencin Poh, and Anne Schilling]{Jennifer Morse\addressmark{1}\thanks{\href{mailto:morsej@virginia.edu}{morsej@virginia.edu}. 
Supported by NSF grant DMS--1833333.}, 
Jianping Pan\thanks{\href{mailto:jpgpan@ucdavis.edu}{jpgpan@ucdavis.edu}.}\addressmark{2}, 
Wencin Poh\thanks{\href{mailto:wpoh@ucdavis.edu}{wpoh@ucdavis.edu}.}\addressmark{2}, \and 
Anne Schilling\addressmark{2}\thanks{\href{mailto:anne@math.ucdavis.edu}{anne@math.ucdavis.edu}. Supported by NSF grants DMS--1760329, DMS--1764153.}}

\begin{comment}
\author[J.~Morse]{Jennifer Morse}
\address[J. Morse]{Department of Mathematics, University of Virginia, Charlottesville, VA, USA}
\email{morsej@virginia.edu}

\author[J.~Pan]{Jianping Pan}
\address[J. Pan]{Department of Mathematics, UC Davis, One Shields Ave., Davis, CA 95616-8633, U.S.A.}
\email{jpgpan@ucdavis.edu}

\author[W.~Poh]{Wencin Poh}
\address[W. Poh]{Department of Mathematics, UC Davis, One Shields Ave., Davis, CA 95616-8633, U.S.A.}
\email{wpoh@ucdavis.edu}

\author[A.~Schilling]{Anne Schilling}
\address[A. Schilling]{Department of Mathematics, UC Davis, One Shields Ave., Davis, CA 95616-8633, U.S.A.}
\email{anne@math.ucdavis.edu}
\end{comment}

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\address{\addressmark{1} Department of Mathematics, University of Virginia, Charlottesville, VA, USA \\ \addressmark{2} Department of Mathematics, UC Davis, CA, USA}

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\abstract{We introduce a type $A$ crystal structure on decreasing factorizations on $321$-avoiding elements in the 0-Hecke 
monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing factorizations 
in the symmetric group of the first and last author. We prove that under the residue map the $\star$-crystal intertwines with the 
crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from 
decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke 
insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.}

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

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\keywords{crystal bases, 0-Hecke monoid, stable Grothendieck polynomials}
%\subjclass[2010]{Primary 05E05, 05E10; Secondary 14N15, 20G42}

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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The Grassmannian Grothendieck polynomials~\cite{LascouxSchutzenberger.1982,LascouxSchutzenberger.1983,
FominKirillov.1994} represent Schubert classes in the $K$-theory of  the Grassmannian. They can be 
expressed~\cite{Buch.2002} as generating functions of semi\-standard set-valued tableaux
\begin{equation}
\label{equation.straight G}
	\mathfrak{G}_\lambda(x_1,\ldots,x_m; \beta) = \sum_{T \in \SVT^m(\lambda)} \beta^{\ext(T)} x^{\wt(T)},
\end{equation}
where $\SVT^m(\lambda)$ is the set of semistandard set-valued tableaux of shape $\lambda$ in the 
alphabet $[m]:=\{1,2,\ldots,m\}$ and $\ext(T)$ is the excess of $T$. Recently, Monical, Pechenik
and Scrimshaw~\cite{MPS.2018} provided a type $A_{m-1}$-crystal structure on $\SVT^m(\lambda)$.
In particular, this implies that the Grassmannian Grothendieck polynomial $\mathfrak{G}_\lambda(x_1,\ldots,x_m;\beta)$ 
is a positive sum of Schur polynomials $\sum_\mu \beta^{|\mu|-|\lambda|}
M_\lambda^\mu \; s_\mu(x_1,\ldots,x_m)$,
where $M_\lambda^\mu$ is the number of highest weight set-valued tableaux of weight $\mu$
in the crystal $\SVT^m(\lambda)$. A different combinatorial formula for the multiplicities $M_\lambda^\mu$
was given by Lenart~\cite[Theorem~2.2]{Lenart.2000} in terms of flagged increasing tableaux.

Stable Grothendieck polynomials~\cite{FominKirillov.1994}, which are labeled by permutations $w\in \mathbb{S}_n$,
generalize the Grassmannian Grothendieck polynomials. They are defined as
\[
	\mathfrak{G}_w(x_1,\ldots,x_m;\beta) = \sum_{(\mathbf{k},\mathbf{h})} \beta^{\ell(\mathbf{h})-\ell(w)} x^{\mathbf{k}},
\]
where the sum is over decreasing factorizations $[\mathbf{k},\mathbf{h}]^t$ of $w$ in the 0-Hecke algebra.
When $\beta=0$, $\mathfrak{G}_w$ specializes to the Stanley symmetric function $F_w$ \cite{Stanley.1984}.

In this paper, we define a type $A$ crystal structure on decreasing factorizations of $w$ in the 0-Hecke algebra, when $w$ 
is $321$-avoiding. A permutation $w$ is $321$-avoiding if its reduced expressions do not contain any braids or equivalently $w$ is
fully commutative. The residue map (see \cref{section.residue map}) shows that $321$-avoiding permutations
correspond to skew shapes. 
We call our crystal $\star$-crystal. It is local in the sense that the crystal operators
$f^\star_i$ and $e^\star_i$ only act on the $i$-th and $(i+1)$-th factors of the decreasing factorization. It generalizes the crystal of 
Morse and Schilling~\cite{MorseSchilling.2016} for Stanley symmetric functions (or 
equivalently reduced decreasing factorizations of $w$) in the $321$-avoiding case. We show that the $\star$-crystal and the 
crystal  on set-valued tableaux intertwine under the residue map (see \cref{thm: star}). We also show that the residue 
map and the Hecke insertion~\cite{BKSRY.2008} are related (see \cref{thm: res}), thereby 
resolving~\cite[Open Problem 5.8]{MPS.2018} in the $321$-avoiding case.
In addition, we provide a new insertion algorithm, which we call $\star$-insertion, from decreasing factorizations on 
$321$-avoiding elements in the 0-Hecke monoid to pairs of (conjugates of) semistandard Young tableaux of the same shape (see 
\cref{def: new_insertion} and \cref{theorem.star insertion bijection}), which intertwines with crystal 
operators (see \cref{thm: star-crystal.star-insertion}). This recovers the Schur expansion of 
$\mathfrak{G}_w$ of Fomin and Greene~\cite{FominGreene.1998} when $w$ is $321$-avoiding, stating that
\[
 	\mathfrak{G}_w = \sum_\mu \beta^{|\mu|-\ell(w)} g_w^{\mu} s_\mu,
\]
where $g_w^{\mu} = |\{ T \in \mathsf{SSYT}^n(\mu') \mid w_C(T) \equiv w\} |$ and
$w_C(T)$ is the column reading word of $T$ (see \cref{remark.schur expansion}). We also show that the 
composition of the residue map with the $\star$-insertion is related to the uncrowding algorithm~\cite{Buch.2002} (see 
\cref{theorem.uncrowding}).

The paper is organized as follows. In \cref{section.star crystal}, we introduce the $\star$-crystal
on decreasing factorizations in the 0-Hecke monoid and show that it intertwines with the crystal on semistandard 
set-valued tableaux~\cite{MPS.2018} under the residue map.
In \cref{section.insertion}, we discuss two insertion algorithms for decreasing factorizations. 
The first is the Hecke insertion introduced by Buch et al.~\cite{BKSRY.2008} and the second is the
$\star$-insertion. In \cref{section.properties}, properties of the $\star$-insertion are discussed.
In particular, we state that it intertwines with the crystal operators and that it relates to the uncrowding algorithm.
Proofs of all statements appear in the long version of this extended abstract~\cite{MPPS.2019}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The $\star$-crystal}
\label{section.star crystal}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, we define the $K$-theoretic generalization of the crystal on decreasing factorizations by
Morse and Schilling~\cite{MorseSchilling.2016} when the associated word is $321$-avoiding. The underlying
combinatorial objects are decreasing factorizations in the 0-Hecke monoid introduced in 
\cref{section.decreasing factorizations}. The $\star$-crystal on these decreasing factorizations
is defined in \cref{section.crystal}. We review the crystal structure on set-valued tableaux 
introduced by Monical, Pechenik and Scrimshaw~\cite{MPS.2018} in \cref{section.SVT}. The residue
map and the theorem that it intertwines the $\star$-crystal and the crystal on set-valued tableaux is given
in \cref{section.residue map}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Decreasing factorizations in the 0-Hecke monoid}
\label{section.decreasing factorizations}

The \defn{symmetric group} $\mathbb{S}_n$ is generated by the simple transpositions $s_i$ for $1\leqslant i<n$
with relations $s_i s_j =s_j s_i$ for $|i-j|>1$, $s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}$, and $s^2_i =1$.
A \defn{reduced expression} for an element $w\in \mathbb{S}_n$ is a word $a_1 a_2 \ldots a_\ell$ with 
$a_i \in [n-1]:=\{1,2,\ldots,n-1\}$ such that $w=s_{a_1}\cdots s_{a_\ell}$
and $\ell$ is minimal among all words representing $w$. In this case, $\ell$ is called the length of $w$.

\begin{definition}
The \defn{$0$-Hecke monoid} $\mathcal{H}_0(n)$, where $n\geqslant 1$ is an integer, is a monoid of finite words generated 
by positive integers in the alphabet $[n-1]$ subject to the relations
\begin{equation}
\label{equation.relations H0}
	pq \equiv qp \quad \text{ if } |p-q|>1,\qquad pqp \equiv qpq \quad \text{ for all } p,q,\qquad
	pp \equiv p \text{ for all } p.
\end{equation}
\end{definition}

We may form an equivalence relation $\equiv_{\mathcal{H}_0}$ on all words in the alphabet $[n-1]$ based on the 
relations~\eqref{equation.relations H0}. The equivalence classes are infinite since the last relation changes the length
of the word. We say that a word $a = a_1 a_2\ldots a_\ell$ is \defn{reduced} if $\ell\geqslant 0$ 
is the smallest among all words in $\mathcal{H}_0(n)$ equivalent to $a$. In this case $\ell$ is the length of $a$.
Note that $\mathcal{H}_0(n)$ is in bijection with $\mathbb{S}_n$ by identifying the reduced word $a_1 a_2\ldots a_\ell$ 
in $\mathcal{H}_0(n)$ with $s_{a_1} s_{a_2}\cdots s_{a_\ell}\in \mathbb{S}_n$. 
We say $w \in \mathcal{H}_0(n)$ or $\mathbb{S}_n$ is \defn{$321$-avoiding} if
none of the reduced words equivalent to $w$ contain a consecutive subword of the form $i\; i+1\; i$ 
for any $i\in [n-1]$.

\begin{definition}\label{def: decr_fact}
A \defn{decreasing factorization} of $w \in \mathcal{H}_0(n)$ into $m$ \defn{factors} is a product of the form 
$\mathbf{h}=h^m \ldots h^2 h^1$, where the sequence in each factor $h^i = h^i_1h^i_2 \ldots h^i_{\ell_i}$
is either empty (meaning $\ell_i=0$) or strictly decreasing (meaning $h^i_1>h^i_2>\cdots>h^i_{\ell_i}$) for each 
$1\leqslant i\leqslant m$ and $\mathbf{h}\equiv_{\mathcal{H}_0} w$ in $\mathcal{H}_0(n)$.
The set of all possible decreasing factorizations into $m$ factors is denoted by $\mathcal{H}^m$.
We call $\ext(\mathbf{h})=\len(\mathbf{h})-\ell$ the \defn{excess} of $\mathbf{h}$, where $\len(\mathbf{h})$ is the number of
letters in $\mathbf{h}$ and $\ell$ is the length of $w$. We say $\mathbf{h}$ is $321$-avoiding if $w$ is $321$-avoiding.
\end{definition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The $\star$-crystal}
\label{section.crystal}

Let $\mathcal{H}^{m,\star}$ be the set of $321$-avoiding decreasing factorizations in $\mathcal{H}^m$.
We introduce a type $A_{m-1}$ crystal structure on $\mathcal{H}^{m,\star}$, which we call the $\star$-crystal.
This generalizes the crystal for Stanley symmetric functions~\cite{MorseSchilling.2016} (see also~\cite{Lenart.2004}).

\begin{definition}\label{def: star}
For any $\mathbf{h} = h^m\dots h^2h^1 \in \mathcal{H}^{m,\star}$, we define \defn{crystal 
operators} $e^\star_i$ and $f^\star_i$ for $i\in [m-1]$ and a \defn{weight function}
$\wt(\mathbf{h}) = (\len(h^1),\len(h^2),\ldots,\len(h^m))$.
We begin with a pairing process:
	\begin{itemize}
		\item Start with the largest letter $b$ in $h^{i+1}$, pair it with the smallest $a\geqslant b$ in $h^{i}$. 
		If there is no such $a$, then $b$ is unpaired.
		\item The pairing proceeds in decreasing order on elements of $h^{i+1}$ and with each iteration, 
		previously paired letters of $h^{i}$ are ignored.
	\end{itemize}
If all letters in $h^i$ are paired, then $f^\star_i$ annihilates $\mathbf{h}$. Otherwise, let $x$ be 
the largest unpaired letter in $h^i$. The crystal operator $f^\star_i$ acts on $\mathbf{h}$ in either of the following ways:
	\begin{enumerate}
		\item If $x+1 \in h^i \cap h^{i+1}$, then remove $x+1$ from $h^i$, add $x$ to $h^{i+1}$.
		\item Otherwise, remove $x$ from $h^i$ and add $x$ to $h^{i+1}$.
	\end{enumerate}
The operator $e_i^\star$ is defined as the partial inverse of $f_i^\star$, that is, $e_i^\star(\mathbf{h})$ is the unique 
$\mathbf{h}'$ such that $f_i^\star(\mathbf{h}')=\mathbf{h}$ if it exists and zero otherwise.
\end{definition}

\begin{example}
Let $\mathbf{h} = (7532)(621)(6)$, then $f^\star_1(\mathbf{h})=0$ and $f^\star_2(\mathbf{h})= (75321)(61)(6)$.
\end{example}

\begin{proposition}
Let $\mathbf{h} =h^m \ldots h^1\in \mathcal{H}^{m,\star}$ such that $e_i^\star(\mathbf{h})\neq 0$.
Then $e_i^\star(\mathbf{h}) \in \mathcal{H}^{m,\star}$, $e_i^\star(\mathbf{h}) \equiv_{\mathcal{H}_0} \mathbf{h}$, and
$\ext(e_i^\star(\mathbf{h})) = \ext(\mathbf{h})$. Furthermore, the $j$-th factor in $e_i^\star(\mathbf{h})$ and $\mathbf{h}$
agrees for $j\notin \{i,i+1\}$. Analogous statements hold for $f_i^\star$.
\end{proposition}

It will be shown in \cref{section.residue map} that $\mathcal{H}^{m,\star}$ is indeed a Stembridge
crystal of type $A_{m-1}$ (for an introduction to crystal and terminology, see~\cite{BumpSchilling.2017}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The crystal on set-valued tableaux}
\label{section.SVT}

In this section, we review the type $A$ crystal structure on set-valued tableaux introduced in~\cite{MPS.2018}.
In fact, in~\cite{MPS.2018} the authors only considered the crystal structure on straight-shaped set-valued
tableaux. Here we consider the slight generalization to skew shapes, see \cref{theorem.SVT crystal}.
We use French notation for partitions $\lambda=(\lambda_1,\lambda_2,\ldots)$ with $\lambda_1\geqslant \lambda_2 
\geqslant \cdots \geqslant 0$, that is, in the Ferrers diagram for $\lambda$, the largest part $\lambda_1$ is at the bottom.

\begin{definition}[\cite{Buch.2002}]
A \defn{semistandard set-valued tableau} $T$ is the filling of a skew shape $\lambda/\mu$
with nonempty subsets of positive integers such that 
(1) for all adjacent cells $A$, $B$ in the same row with $A$ to the left of $B$, $\max(A)\leqslant \min(B)$,
(2) for all adjacent cells $A$, $C$ in the same column with $A$ below $C$, $\max(A)< \min(C)$.
We denote the set of all semistandard set-valued tableaux of shape $\lambda/\mu$ by \defn{$\SVT(\lambda/\mu)$}. 
If the maximum entry in the tableaux is $m$, the set is denoted by $\SVT^m(\lambda/\mu)$.
\end{definition}

We now review the crystal structure on semistandard set-valued tableaux given in~\cite{MPS.2018}.
We state the definition on skew shapes rather than just straight shapes.

\begin{definition} \label{def: sksvt}
Let $T\in \SVT^m(\lambda/\mu)$. We employ the following pairing rule for letters $i$ and $i+1$.
Assign $-$ to every column of $T$ containing an $i$ but not an $i+1$. Similarly, assign $+$ to 
every column of $T$ containing an $i+1$ but not an $i$. Then, successively pair each $+$ that is adjacent to a $-$, 
removing all paired signs until nothing can be paired.

The \defn{operator $f_i$} changes the $i$ in the rightmost 
column with an unpaired $-$ (if this exists) to $i+1$, except if the 
cell $b$ containing that $i$ has a cell to its right, denoted $b^\rightarrow$, that contains both $i$ and $i+1$. In that 
case, $f_i$ removes $i$ from $b^\rightarrow$ and adds $i+1$ to $b$. Finally, if no unpaired $-$ exists, then $f_i$ 
annihilates $T$. The \defn{operator $e_i$} is the partial inverse of $f_i$.

The \defn{weight $\wt(T)$} of $T$ is the integer vector whose $i$-th component 
counts the number of $i$'s that occur in $T$. 
\end{definition}

The above described operators $e_i$ and $f_i$ define a type $A_{m-1}$ crystal structure on 
$\SVT^m(\lambda)$~\cite[Theorem 3.9]{MPS.2018}. Their proof also holds for skew shapes.

\begin{theorem}
\label{theorem.SVT crystal}
The crystal $\SVT^m(\lambda /\mu)$ of \cref{def: sksvt} is a Stembridge crystal of type $A_{m-1}$.
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The residue map}
\label{section.residue map}

In this section, we define the residue map from set-valued tableaux of skew shape to $321$-avoiding decreasing factorizations
in the 0-Hecke monoid.  We then show in \cref{thm: star} that the residue map intertwines with the 
crystal operators, proving that $\mathcal{H}^{m,\star}$ is indeed a crystal of type $A_{m-1}$ (see 
\cref{corollary.crystal}).

\begin{definition}
Given $T\in \SVT^m(\lambda/\mu)$, we define the \defn{residue map} $\res: \SVT^m(\lambda/\mu)\to\mathcal{H}^{m}$ 
as follows. Label all cells $(i,j)$ in $\lambda/\mu$ with $\ell(\lambda)+j-i$, where $\ell(\lambda)$ is the number of 
parts in $\lambda$.
Produce a decreasing factorization $\mathbf{h}=h^m h^{m-1}\ldots h^2 h^1$ by declaring $h^i$ to be the (possibly empty)
sequence formed by taking the labels of all cells in $T$ containing $i$ and then arranging these labels in decreasing 
order. This defines $\res(T):=\mathbf{h}$. 
\end{definition}

\begin{example}
Let $T$ be the set-valued tableau of shape $(2,2)/(1)$
\[
\ytableausetup{notabloids}
T = 
\scalebox{0.8}{
\begin{ytableau}
23 & 3\\
\none & 12
\end{ytableau}}\,.
\qquad
\text{We label the cells of $T$ as follows:} \qquad
\scalebox{0.8}{
\ytableausetup{notabloids} 
\begin{ytableau}
23_1 & 3_2\\
\none & 12_3
\end{ytableau}}\,.
\]
To read off the third factor, we search for all cells containing 3; these cells have  labels 1 and 2, 
so we have 21 in the third factor. Altogether, we obtain $\res(T)=(21)(31)(3)$. 
\end{example}

The image of the residue map $\res$ is $\mathcal{H}^{m,\star}$, the set of 
$321$-avoiding decreasing factorizations into $m$ factors. In fact, $\res$ is a bijection from
skew set-valued tableaux on the alphabet $[m]$ to $\mathcal{H}^{m,\star}$ up to shifts in the skew shape.
For this purpose, let us describe the inverse of the residue map.
Let $\mathbf{h}=h^m h^{m-1}\ldots h^2 h^1\in \mathcal{H}^{m,\star}$.
Begin by filling $m$ along the diagonals labeled by the letters that appear in $h^m$. As the resulting
$T$ is supposed to be of skew shape, the cells containing $m$ along increasing diagonals need to go weakly down from 
left to right. If these diagonals are consecutive, then the cells have to be in the same row of $T$ since $T$ is semistandard. 
Continue the procedure above by filling $i$ along the diagonals specified by $h^i$ for all $i=m-1,m-2,\ldots,1$, applying the 
condition that the resulting filling should be semistandard. 

\begin{proposition} \label{prop.res.image}
If $\mathbf{h} \in\mathcal{H}^{m,\star}$, then the above algorithm is well-defined up to shifts
along diagonals. It produces a skew semistandard set-valued tableau $T$ such that $\res(T)=\mathbf{h}$.
\end{proposition}

If the skew shape $\lambda/\mu$ of the tableau $T$ is known, then one may simplify the procedure above noting that 
the filling of $i$ specified by letters in $h^i$ must occur along a horizontal strip for all $i=m,m-1,\ldots,1$. In this case, 
the recovered tableau $T$ is unique.

\begin{example}\label{eg: inverse_res1}
Let $\mathbf{h}=(61)(752)(75)(762)$.
In the procedure to determine a suitable skew tableau whose residue map is $\mathbf{h}$, after filling 4's along the 
diagonals with labels 1 and 6 respectively, due to semistandardness, the 3 in diagonal 2 is below the 4 in diagonal 1, 
while the 3's in diagonals 5 and 7 are respectively to the left and below the 4 in diagonal 6.
Continuing with the remaining fillings, we have two possibilities:
\[
T_1=
\scalebox{0.6}{
\raisebox{1.2cm}{
\ytableausetup{notabloids, boxsize=2em} 
\begin{ytableau}
4_{\color{red}{1}}\\
13_{\color{red}{2}}\\
\none & \none & 23_{\color{red}{5}} & 4_{\color{red}{6}}\\
\none & \none & 1_{\color{red}{6}} & 123_{\color{red}{7}}\\
\end{ytableau}}}
\scalebox{0.8}{
$\in\SVT^4((4,4,1,1)/(2,2))$}
\text{ or }
T_2=
\scalebox{0.6}{
\raisebox{1.6cm}{
\ytableausetup{notabloids, boxsize=2em} 
\begin{ytableau}
4_{\color{red}{1}}\\
13_{\color{red}{2}}\\
\none\\
\none & 23_{\color{red}{5}} & 4_{\color{red}{6}}\\
\none & 1_{\color{red}{6}} & 123_{\color{red}{7}}\\
\end{ytableau}}} 
\scalebox{0.8}{$\in\SVT^4((3,3,1,1,1)/(1,1,1))$.}
\]
\end{example}

\begin{theorem}
\label{thm: star}
The crystal on set-valued tableaux $\SVT^m(\lambda/\mu)$ and the crystal on decreasing factorizations $\mathcal{H}^{m,\star}$
intertwine under the residue map. That is, the following diagram commutes:
\[
	\begin{tikzcd}
	\SVT^m(\lambda/\mu) \arrow["f_k",d] \arrow[r,"\res"] & \mathcal{H}^{m,\star} \arrow[d,"f^{\star}_k"] \\
	\SVT^m(\lambda/\mu) \arrow[r,"\res"]& \mathcal{H}^{m,\star}.
	\end{tikzcd}
\]
\end{theorem}

\begin{corollary}
\label{corollary.crystal}
The set $\mathcal{H}^{m,\star}$, together with crystal operators $e^{\star}_i$ and $f^{\star}_i$ for $1\leqslant i<m$ 
and weight function $\wt$ defined in \cref{def: star}, is a Stembridge crystal.
\end{corollary}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Insertion algorithms}
\label{section.insertion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, we discuss two insertion algorithms for decreasing factorizations in $\mathcal{H}^m$ (resp. 
$\mathcal{H}^{m,\star}$).
The first is the Hecke insertion introduced by Buch et al.~\cite{BKSRY.2008}, which we review in
\cref{section.hecke insertion}. We prove a relationship between Hecke insertion and the residue map 
(see \cref{thm: res}). In particular, this proves~\cite[Open Problem 5.8]{MPS.2018} for $321$-avoiding
permutations. The second insertion is a new insertion, which we call $\star$-insertion, introduced in
\cref{section.star insertion}. It goes from $321$-avoiding decreasing factorizations in the 0-Hecke monoid to pairs of 
(transposes of) semistandard tableaux of the same shape and is well-behaved with respect to the crystal operators.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hecke insertion}
\label{section.hecke insertion}

\ytableausetup{boxsize=1.5em}
Hecke insertion was first introduced in~\cite{BKSRY.2008} as column insertion. Here we state the row insertion version 
as in \cite{PP.2016}. In this section, we represent a decreasing factorization $\mathbf{h} = h^m h^{m-1} \cdots h^1$,
where $h^i = h^i_1 h^i_2 \ldots h^i_{\ell_i}$, by a \defn{decreasing Hecke biword}
\[
\begin{bmatrix}
\mathbf{k}\\ 
\mathbf{h}
\end{bmatrix}
=
\begin{bmatrix}
m & \ldots & m & \ldots & 1 & \ldots &1 \\
h^m_1 & \ldots & h^m_{\ell_m} & \ldots & h^1_1 & \ldots & h^1_{\ell_1}
\end{bmatrix}.
\]
We say that $[\mathbf{k},\mathbf{h}]^t$ is \defn{$321$-avoiding} if $\mathbf{h}$ is $321$-avoiding.

\begin{example}\label{example.biword}
Consider the decreasing Hecke factorization $\mathbf{h}=(2)(3)(31)(2)$. Then the corresponding biword 
$[\mathbf{k},\mathbf{h}]^t$ is
\[
\begin{bmatrix}
\mathbf{k}\\ 
\mathbf{h}
\end{bmatrix}
=
\begin{bmatrix}
4 & 3 & 2 & 2 & 1 \\
2 & 3 & 3 & 1 & 2
\end{bmatrix}.
\]
\end{example}

\begin{definition}
\label{definition.hecke insertion}
Starting with a decreasing Hecke biword $[\mathbf{k},\mathbf{h}]^t$, we define \defn{Hecke row insertion} from the right. 
The insertion sequence is read from right to left. Suppose there are $n$ columns in $[\mathbf{k},\mathbf{h}]^t$.
Start the insertion with $(P_0,Q_0)$ being both empty tableaux. We recursively construct $(P_{i+1},Q_{i+1})$ from 
$(P_i,Q_i)$. Suppose the $(n-i)$-th column in $[\mathbf{k},\mathbf{h}]^t$ is $[y,x]^t$.

We describe how to insert $x$ into $P_i$, denoted  $P_i \leftarrow x$, by describing how to insert $x$ into a row $R$. 
The insertion may modify the row and may produce an output integer, which will be inserted into the next row. First, 
we insert $x$ into the first row $R$ of $P_i$ following the rules below:
\begin{enumerate}
	\item If $x\geqslant z$ for all $z\in R$, then the insertion terminates in either of the following ways:
		\begin{enumerate}
			\item If we can append $x$ to the right of $R$ and obtain an increasing tableau, the result $P_{i+1}$ 
			is obtained by doing so; form $Q_{i+1}$ by adding a box with $y$ in the same position where $x$ is 
			added to $P_i$.
			\item Otherwise $P_{i+1}=P_i$. Form $Q_{i+1}$ by adding $y$ to the existing corner of $Q_i$ whose 
			column contains the rightmost box of row $R$.
		\end{enumerate}
	\item Otherwise, there exists a smallest $z$ in $R$ such that $z>x$.
		\begin{enumerate} 
			\item If replacing $z$ with $x$ results in an increasing tableau, then do so. Let $z$ be the output integer 
			to be inserted into the next row.
			\item Otherwise, row $R$ remains unchanged. Let $z$ be the output integer to be inserted into the next row.
		\end{enumerate}
\end{enumerate}
The entire Hecke insertion terminates  at $(P_n,Q_n)$ after we have inserted every letter from the Hecke biword. 
The resulting insertion tableau $P_n$ is an increasing tableau.
If $\mathbf{k}=(n,n-1,\ldots,1)$, the recording tableau $Q_n$ is a standard set-valued tableau.
\end{definition}

\begin{example}\label{eg: hecke_insertion}
Take $[\mathbf{k},\mathbf{h}]^t$ 
from \cref{example.biword}. Following the Hecke row insertion, we compute its insertion tableau and recording tableau:
\[
\begin{aligned}
\emptyset
& \rightarrow &
\scalebox{0.7}{
\ytableausetup{notabloids}
\begin{ytableau}
2
\end{ytableau}}
& \rightarrow &
\scalebox{0.7}{
\ytableausetup{notabloids}
\begin{ytableau}
2 \\
1
\end{ytableau}}
& \rightarrow &
\scalebox{0.7}{
\ytableausetup{notabloids}
\begin{ytableau}
2& \none \\
1 &3
\end{ytableau}}
& \rightarrow &
\scalebox{0.7}{
\ytableausetup{notabloids}
\begin{ytableau}
2 & \none \\
1 & 3
\end{ytableau}}
& \rightarrow &
\scalebox{0.7}{
\ytableausetup{notabloids}
\begin{ytableau}
2 & 3 \\
1 & 2
\end{ytableau}} = P,\\
\emptyset
& \rightarrow &
\scalebox{0.7}{
\ytableausetup{notabloids}
\begin{ytableau}
1
\end{ytableau}}
& \rightarrow &
\scalebox{0.7}{
\ytableausetup{notabloids}
\begin{ytableau}
2 \\
1
\end{ytableau}}
& \rightarrow &
\scalebox{0.7}{
\ytableausetup{notabloids}
\begin{ytableau}
2 & \none \\
1 & 2
\end{ytableau}}
& \rightarrow &
\scalebox{0.7}{
\ytableausetup{notabloids}
\begin{ytableau}
2 & \none \\
1 & 23
\end{ytableau}}
& \rightarrow &
\scalebox{0.7}{
\ytableausetup{notabloids}
\begin{ytableau}
2 & 4 \\
1 & 23
\end{ytableau}} = Q.
\end{aligned}
\]
\end{example}

\begin{example}
\label{example.destandardize}
Note that the recording tableau for the Hecke insertion of \cref{definition.hecke insertion} is not
always a semistandard set-valued tableau. For example, for $\mathbf{h}=(21)(41)$ we have
\[
	P= \scalebox{0.7}{\begin{ytableau} 4\\ 1&2 \end{ytableau}} \qquad \text{and} \qquad
	Q= \scalebox{0.7}{\begin{ytableau} 22\\ 1&1 \end{ytableau}}\;.
\]
However, \cref{thm: res} below states that in certain cases it is.
\end{example}

\begin{theorem}\label{thm: res}
Let $T \in \SVT(\lambda)$ and $[\mathbf{k},\mathbf{h}]^t= \res(T)$. 
Apply Hecke row insertion from the right on $[\mathbf{k},\mathbf{h}]^t$ to obtain the pair of tableaux $(P,Q)$. 
Then $Q = T$.
\end{theorem}

\begin{remark}
Combining \cref{thm: res,thm: star} shows that Hecke insertion from right to left
(as opposed to left to right as in~\cite{MPS.2018})
intertwines the crystal on set-valued tableaux and the $\star$-crystal, even though in general
it is not always well-defined (see \cref{example.destandardize}).
This resolves~\cite[Open Problem 5.8]{MPS.2018} when the decreasing factorizations are $321$-avoiding.
For more details see~\cite{MPPS.2019}.
\end{remark} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The $\star$-insertion}
\label{section.star insertion}

We define the new $\star$-insertion from $321$-avoiding decreasing Hecke biwords
$[\mathbf{k},\mathbf{h}]^t$ to pairs of tableaux $P$ and $Q$, denoted by $\star([\mathbf{k},\mathbf{h}]^t)=(P,Q)$, 
as follows.

\begin{definition}
\label{def: new_insertion}
Fix a $321$-avoiding decreasing Hecke biword $[\mathbf{k},\mathbf{h}]^t$. The insertion is done by reading the columns of
this biword from right to left.
Begin with $(P_0, Q_0)$ being a pair of empty tableaux.
For every integer $i\geqslant 0$, we recursively construct $(P_{i+1}, Q_{i+1})$ from $(P_i, Q_i)$ as follows.
Let $[q,x]^t$ be the $i$-th column (from the right) of $[\mathbf{k},\mathbf{h}]^t$.
Suppose that we are inserting $x$ into row $R$ of $P_i$.
\begin{description}
	\item [Case 1] If $R$ is empty or $x>\max(R)$, form $P_{i+1}$ by appending $x$ to row $R$ and form 
	$Q_{i+1}$ by adding $q$ in the corresponding position to $Q_i$. Terminate and return $(P_{i+1},Q_{i+1})$.
	\item [Case 2] Otherwise, if $x\notin R$, locate the smallest $y$ in $R$ with $y>x$. Bump $y$ with $x$ and insert 
	$y$ into the next row of $P_i$.
	\item [Case 3] Otherwise, $x\in R$, so locate the smallest $y$ in $R$ with $y\leqslant x$ and interval $[y,x]$
	contained in $R$. Row $R$ remains unchanged and $y$ is to be inserted into the next row of $P_i$.
\end{description}
Set $(P,Q)=( P_\ell,Q_\ell)$ if $[\mathbf{k},\mathbf{h}]^t$ has length $\ell$. Define the \defn{$\star$-insertion}
by $\star([\mathbf{k},\mathbf{h}]^t)=(P,Q)$.

Furthermore, denote by $P\leftarrow x$ the tableau obtained by inserting $x$ into $P$.
The collection of all cells in $P\leftarrow x$, where insertion or bumping has occurred is called the \defn{insertion path} 
for $P\leftarrow x$. In particular, in Case 1 the newly added cell is in the insertion path, in Case 2 the cell containing the
bumped letter $y$ is in the insertion path, and in Case 3 the cell containing the same entry as the inserted letter
is in the insertion path.
\end{definition}

\begin{example}
Let
\setcounter{MaxMatrixCols}{15}
\[
\begin{bmatrix}
\mathbf{k}\\
\mathbf{h}
\end{bmatrix} =
\begin{bmatrix}
7 & 7 & 7 & 6 & 5 & 5 & 4 & 2 & 2 & 1\\
4 & 2 & 1 & 4 & 3 & 2 & 2 & 5 & 4 & 4
\end{bmatrix}.
\]
Then we have $\star([\mathbf{k},\mathbf{h}]^t)=(P,Q)$, where
\[
P = 
\scalebox{0.6}{
\raisebox{1cm}{
\begin{ytableau}
4\\
4\\
2 & 5\\
2 & 3 & 4\\
1 & 2 & 4
\end{ytableau} }}
\qquad \text{and} \qquad
Q =
\scalebox{0.6}{
\raisebox{1cm}{
\begin{ytableau}
7\\
5\\
4 & 7\\
2 & 5 & 7\\
1 & 2 & 6
\end{ytableau}}}
\;.
\qquad
\text{Furthermore}
\qquad
P\leftarrow 4 \;=\;
\scalebox{0.6}{
\raisebox{2cm}{
\ytableausetup{notabloids}
\begin{ytableau}
*(yellow)4\\
*(yellow)4\\
*(yellow)2\\
*(yellow)2 & 5\\
2 & 3 & *(yellow)4\\
1 & 2 & *(yellow)4
\end{ytableau}}}
\;.
\]
The cells in the insertion path of $P\leftarrow 4$ are highlighted in yellow.
\end{example}

\begin{lemma}\label{lem: star.insertion}
Let $[\mathbf{k},\mathbf{h}]^t$ be a $321$-avoiding decreasing Hecke biword.
Suppose that $\star([\mathbf{k},\mathbf{h}]^t)=(P,Q)$. 
Then: (1) $P^t$ is semistandard and $Q$ has the same shape as $P$.
(2) Let $x$ be an integer such that $x \mathbf{h}$ is $321$-avoiding. Then the insertion path for $P\leftarrow x$ goes 
weakly to the left. 
\end{lemma}

\begin{theorem}
\label{theorem.star insertion bijection}
The $\star$-insertion is a bijection from the set of all $321$-avoiding decreasing Hecke biwords to the set of all pairs of 
tableaux $(P,Q)$ of the same shape, where both $P^t$ and $Q$ are semistandard and $\row(P)$ is 
$321$-avoiding.
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Properties of the $\star$-insertion}
\label{section.properties}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, we show that the $\star$-insertion intertwines with the crystal operators. 
More precisely, the insertion tableau remains invariant on connected crystal components under the $\star$-insertion by 
employing certain micro-moves, see~\cite{MPPS.2019}.
The $\star$-crystal on $\mathcal{H}^{m,\star}$ intertwines with the usual crystal operators on semistandard 
tableaux on the recording tableaux under the $\star$-insertion.
We relate the $\star$-insertion to the uncrowding operation.

\begin{proposition}
For $\mathbf{h}\in \mathcal{H}^{m,\star}$ such that $f^{\star}_k(\mathbf{h})\neq 0$ for some $1\leqslant k< m$, the 
$\star$-insertion tableau for $\mathbf{h}$ equals the $\star$-insertion tableau for $f^{\star}_k(\mathbf{h})$.
\end{proposition}

\begin{proposition}\label{prop: res.star-insertion}
Let $T\in \SSYT(\lambda)$ and $(P,Q) = \star \circ \res(T)$. Then $Q = T$.
\end{proposition}

\begin{theorem}\label{thm: star-crystal.star-insertion}
For $\mathbf{h}\in \mathcal{H}^{m,\star}$ let $(P^\star(\mathbf{h}),Q^\star(\mathbf{h})) = \star(\mathbf{h})$ be the insertion 
and recording tableaux under the
$\star$-insertion of \cref{def: new_insertion}. Then 
(1) $f_i^\star(\mathbf{h})$ is defined if and only if $f_i(Q^\star(\mathbf{h}))$ is defined and (2) 
if $f_i^\star(\mathbf{h})$ is defined, then $Q^\star(f_i^\star(\mathbf{h})) = f_i Q^\star(\mathbf{h})$.
In other words, the following diagram commutes:
\[
	\begin{tikzcd}
	\mathcal{H}^{m,\star} \arrow[r,"Q^\star"] \arrow[d,"f^{\star}_i"] & \SSYT \arrow[d,"f_i"] \\
	\mathcal{H}^{m,\star}\arrow[r,"Q^\star"] & \SSYT.
	\end{tikzcd}
\]
\end{theorem}

\begin{remark}
\label{remark.schur expansion}
\cref{thm: star-crystal.star-insertion} and an analysis of the lowest weight elements provide another 
proof via $\star$-insertion, in the case where $w$ is $321$-avoiding, of the Schur positivity of $\mathfrak{G}_w$ 
of~\cite{FominGreene.1998} $\mathfrak{G}_w = \sum_\mu \beta^{|\mu|-\ell(w)} g_w^{\mu} s_\mu$,
where $g_w^{\mu} = |\{ T \in \mathsf{SSYT}^n(\mu') \mid w_C(T) \equiv w\} |$.
\end{remark}

Let $\lambda$, $\mu$ be partitions such that $\lambda\subseteq\mu$ and $\lambda_1=\mu_1$. 
A \defn{flagged increasing tableau} of shape $\mu/\lambda$ is a tableau of shape $\mu$ 
with fillings by positive integers in the skew shape $\mu/\lambda$ 
such that for all $1\leqslant i\leqslant \ell(\mu)$ all entries in the $i$-th row of the tableau are at most $i-1$ 
and such that this filling is both row strict and column strict. In particular, the bottom row is empty.
Denote the set of all flagged increasing tableaux of shape $\mu/\lambda$ by $\mathcal{F}_{\mu/\lambda}$.
Flagged increasing tableaux are also called \defn{elegant fillings} by some authors~\cite{LamPylyavskyy.2007,
BandlowMorse.2012}.

The following definition is based on the uncrowding operator introduced by Reiner, Tenner and 
Yong~\cite[Definition 3.8]{RTY.2018}.

\begin{definition} \label{def: uncrowd}
Let $T\in\SVT(\lambda)$. Define an \defn{uncrowding operation} on $T$ as follows.
Identify the topmost row in $T$ that contains cells with more than one letter and let $x$ be the largest letter
in this row in a cell containing more than one letter. Remove $x$ from this cell and perform RSK row bumping with $x$ 
into the rows above. The resulting tableau $T'$ is the output of this operation.
The \defn{uncrowding map}, denoted $\uncrowd$, is defined as follows. Let $T\in\SVT(\lambda)$ with 
$\mathsf{ext}(T)=\ell$. 
\begin{itemize}
\item Start with $\widetilde{P}_0=T$ and $\widetilde{Q}_0=F$, where $F$ is the unique flagged increasing tableau of 
shape $\lambda/\lambda$. 
\item For each $1\leqslant i\leqslant \ell$, $\widetilde{P}_i$ is obtained by performing the uncrowding operator on 
$\widetilde{P}_{i-1}$. Suppose that cell $C$ was added to form $\widetilde{P}_i$ by removing the largest entry in cell 
$B$ in $\widetilde{P}_{i-1}$. Add a cell with entry $k$ to $\widetilde{Q}_{i-1}$ at the same position as $C$, where $k$ is 
the difference in the row indices of cells $B$ and $C$.
\item Terminate and return $(\widetilde{P},\widetilde{Q})=(\widetilde{P}_\ell,\widetilde{Q}_\ell)$.
\end{itemize}
\end{definition}

\begin{example}
Let $T$ be the semistandard set-valued tableau
\[
\ytableausetup{mathmode,boxsize=1.8em}
T = 
\scalebox{0.6}{
\raisebox{1.6cm}{
\begin{ytableau}
5\\
4 & 4 & 5\\
2 & 23 & 3\\
1 & 1 & 1 & 12 & 234 & 5 
\end{ytableau}}}\;.
\quad
\text{Perform an uncrowding operation to obtain}
\quad
\scalebox{0.6}{
\raisebox{2.2cm}{
\begin{ytableau}
5\\
4\\
3 & 4 & 5\\
2 & 2 & 3\\
1 & 1 & 1 & 12 & 234 & 5 
\end{ytableau}}}\;.
\]
Proceeding with uncrowding the remainder of the excess entries and recording the changes, we have
$\uncrowd(T)=(\widetilde{P},\widetilde{Q})$, where
\[
\widetilde{P} = 
\scalebox{0.6}{
\raisebox{2.2cm}{
\begin{ytableau}
5 & 5\\
4 & 4\\
3 & 3 & 4\\
2 & 2 & 2 & 3\\
1 & 1 & 1 & 1 & 2 & 5 
\end{ytableau}}}
\qquad \text{and} \qquad
\widetilde{Q} =
\scalebox{0.6}{
\raisebox{2.2cm}{
\begin{ytableau}
3 & 4\\
*(gray) & 3\\
*(gray) & *(gray) & *(gray)\\
*(gray) & *(gray) & *(gray) & 1\\
*(gray) & *(gray) & *(gray) & *(gray) & *(gray) & *(gray) 
\end{ytableau}}}\;.
\]
\end{example}

We remark that $\uncrowd \colon \SVT^m(\lambda)\to\bigsqcup_{\mu}\SSYT^m(\mu)\times \mathcal{F}_{\mu/\lambda}$, 
where the disjoint union is taken over partitions $\mu$ with $\lambda\subseteq\mu$ and $\mu_1=\lambda_1$, 
is a bijection with inverse given by the crowding map $\mathsf{crowd}$. 
A proof of this fact can be found in \cite{Monical.2016} (see also~\cite{RTY.2018}). 
In addition, Monical, Pechenik and Scrimshaw in \cite[Theorem 3.12]{MPS.2018} proved that the crystal operators on 
$\SVT^m(\lambda)$ intertwine with those on $\SSYT^m(\mu)$ under $\uncrowd$. 
Here we relate uncrowding with the $\star$-insertion.

\begin{theorem}
\label{theorem.uncrowding}
Let $T\in\SVT^m(\lambda)$, $(\tilde{P},\tilde{Q})=\uncrowd(T)$, $(P,Q)= \star \circ \res(T)$. Then $Q = \tilde{P}$.
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgments
{We are grateful to Travis Scrimshaw and Zach Hamaker for discussions.
This material is based upon work supported by the National Security Agency under Grant No. H98230-19-1-0119, 
The Lyda Hill Foundation, The McGovern Foundation, and Microsoft Research, while JM and AS were in residence at
the Mathematical Sciences Research Institute in Berkeley, California, during the summer of 2019.}

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