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\usepackage[utf8]{inputenc}
\usepackage{mathtools}
\usepackage{amsfonts}
\usepackage{fancyhdr}
\usepackage{xcolor}
\usepackage{tikz}
\usetikzlibrary{cd}
\usepackage{ytableau}
\usepackage{makecell}
\usepackage{cleveref}
\usepackage{bm}

\newcommand{\defn}[1]{{\color{darkred}\emph{#1}}} % emphasis of a definition
\definecolor{darkblue}{rgb}{0.0,0,0.7} % darkblue color
\definecolor{darkred}{rgb}{0.7,0,0} % darkred color
\definecolor{darkgreen}{rgb}{0, .6, 0} % darkgreen color

\newcommand{\ZZ}{\mathbb{Z}}
\newcommand\BFT{\mathsf{BFT}}
\newcommand\EXQ{\mathsf{EXQ}}
\newcommand\ET{\operatorname{ET}}
\newcommand\ml{{\mu/\lambda}}
\newcommand{\II}{\mathcal{I}}
\newcommand{\va}{\bm{a}}
\newcommand{\vb}{\bm{b}}

\newcommand{\uncrowd}{\mathsf{uncrowd}}
\newcommand{\row}{\mathsf{row}}
\newcommand{\col}{\mathsf{col}}
\newcommand{\HVT}{\mathsf{HVT}}
\newcommand{\TAB}{\mathsf{TAB}}
\newcommand{\FMT}{\mathsf{FMT}}
\newcommand{\MVT}{\mathsf{MVT}}
\newcommand{\SVT}{\mathsf{SVT}}
\newcommand{\SSYT}{\mathsf{SSYT}}
\newcommand{\RPP}{\mathsf{RPP}}
\newcommand{\ev}{\mathsf{ev}}
\newcommand{\sort}{\mathsf{sort}}
\newcommand{\rd}{\mathsf{rd}}
\newcommand{\word}{\mathsf{word}}
\newcommand{\sfA}{\mathsf{A}}
\newcommand{\sfL}{\mathsf{L}}
\newcommand{\sfH}{\mathsf{H}}
\newcommand{\shape}{\mathsf{shape}}
\newcommand{\wt}{\mathsf{wt}}

\newcommand{\U}{\mathcal{U}}
\newcommand{\A}{\mathcal{A}}
\renewcommand{\L}{\mathcal{L}}
\newcommand{\Aunc}{\U_{\A}}
\newcommand{\Lunc}{\U_{\L}}
\newcommand{\Abump}{\mathcal{A}_{b}}
\newcommand{\Lbump}{\mathcal{L}_{b}}
\newcommand{\shuff}{\mathsf{shuff}}
\newcommand{\rev}{\mathsf{rev}}
\newcommand{\jdt}{\mathsf{jdt}_{\mathrm{GG}}}

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\theoremstyle{definition}
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% \title[Hook-valued tableaux uncrowding and tableau switching]{Hook-valued tableaux uncrowding and tableau switching}
% \title{Hook-valued tableaux uncrowding and tableau switching}
\title{A hidden symmetry of refined canonical stable Grothendieck polynomials}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[J.~Jang, J.~S.~Kim, J.~Pan, J.~Pan, A.~Schilling]{\texorpdfstring{Jihyeug Jang\thanks{\href{mailto:jihyeugjang@gmail.com}{jihyeugjang@gmail.com}.}\addressmark{1}, Jang Soo Kim \addressmark{1}, Jianping Pan\addressmark{2}, Joseph Pappe\addressmark{3}, \and \\
  Anne Schilling\thanks{A. Schilling was partially supported by NSF grant DMS--2053350.}\addressmark{4}}{Jihyeug Jang, Jang Soo Kim , Jianping Pan, Joseph Pappe, and
  Anne Schilling}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics,
Sungkyunkwan University (SKKU), Suwon, South Korea \\
  \addressmark{2}Department of Mathematics, Arizona State University, Tempe, AZ, USA\\
  \addressmark{3}Department of Mathematics, Colorado State University, Fort Collins, CO, USA\\
  \addressmark{4}Department of Mathematics, UC Davis, One Shields Ave., Davis, CA, USA\\
  }

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\abstract{
  Refined canonical stable Grothendieck polynomials were introduced by
  Hwang, Jang, Kim, Song, and Song. There exist two combinatorial
  models for these polynomials: one using hook-valued tableaux and
  the other using pairs of a semistandard Young tableau and (what we
  call) an exquisite tableau. An uncrowding algorithm on hook-valued
  tableaux was introduced by Pan, Pappe, Poh, and Schilling. In this
  paper, we discover a novel connection between the two models via the
  uncrowding and Goulden--Greene's jeu de taquin algorithms, using a classical result of
  Benkart, Sottile, and Stroomer on tableau switching. This
  connection reveals a hidden symmetry of the uncrowding algorithm
  defined on hook-valued tableaux. As a corollary, we obtain another
  combinatorial model for the refined canonical stable Grothendieck
  polynomials in terms of biflagged tableaux, which naturally appear 
  in the characterization of the image of the uncrowding map.
}

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% \resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Grothendieck polynomials, hook-valued tableaux, flagged tableaux, uncrowding algorithm, tableau switching, jeu de taquin}

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\begin{document}

\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{section.intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Refined canonical stable Grothendieck polynomials were introduced in~\cite{HJKSS1}, generalizing and unifying many of the previous variants
of Grothendieck polynomials. They encompass the Grassmannian Grothendieck polynomials introduced by Lascoux and Sch\"utzenberger~\cite{LS.1982}, 
the stable Grassmannian $\beta$-Grothendieck polynomials of Fomin and Kirillov~\cite{FK.1996}, the canonical stable Grothendieck polynomials of 
Yeliussizov~\cite{Yeliussizov.2017}, flagged Grothendieck polynomials of Matsumura~\cite{Mat.2019}, 
and the refined stable Grothendieck polynomials of Chan and Pflueger~\cite{CP.2021}.
% The refined canonical stable Grothendieck polynomials were further studied by Iwao, Motegi, and Scrimshaw \cite{Iwao2024}
% in terms of vertex operators.

In analogy to Schur functions, which are generating functions of semistandard Young tableaux, Buch~\cite{Buch.2002} showed that
stable Grothendieck polynomials are generating functions of semistandard set-valued tableaux. Each cell in a set-valued tableau contains
a set instead of just a natural number. Combinatorial models for the (refined) canonical stable Grothendieck polynomials are described in terms of hook-valued 
tableaux~\cite{Yeliussizov.2017,HJKSS2}, which contain a semistandard Young tableau of a hook shape in each cell. These tableaux
models are intimately related to the monomial expansions of the different versions of the Grothendieck polynomials.

An important question is to find the Schur expansion of the various Grothendieck polynomials.
Lenart~\cite{Lenart.2000} gave the Schur expansion of the symmetric stable Grothendieck polynomials, whose monomial expansion
is given in terms of set-valued tableaux.

Buch~\cite[Theorem 6.11]{Buch.2002} developed an uncrowding algorithm on set-valued
tableaux to give a bijective proof of Lenart's Schur expansion. The uncrowding algorithm on a set-valued tableau produces a semistandard 
Young tableau (using the RSK bumping algorithm to uncrowd cells that contain more than one integer) and a flagged increasing 
tableau~\cite{Lenart.2000} (also known as an elegant filling~\cite{LP.2007, BM.2012, Pat.2016}), which serves as a recording tableau.
This uncrowding algorithm was generalized by Chan and Pflueger~\cite{CP.2021}, by Reiner, Tenner and Yong~\cite{RTY.2018}, and by Pan, Pappe, 
Poh, and Schilling~\cite{PPPS}.

Hwang et al. \cite{HJKSS1} found the Schur expansion for refined canonical Grothendieck polynomials,
which we rephrase in terms of ``exquisite'' tableaux. 
Hence, it is a natural question to relate the combinatorial model for the refined canonical Grothendieck polynomials in terms of hook-valued tableaux
with the combinatorial model in terms of exquisite tableaux by giving a bijection between hook-valued tableaux and pairs of a semistandard tableau and
an exquisite tableau. In this paper, we find such a bijection (see \Cref{thm:2bij}) using two types of uncrowding maps by combining the 
uncrowding algorithm due to Pan et al.~\cite{PPPS} with Goulden--Greene's jeu de taquin~\cite{GG}.

The uncrowding algorithm on hook-valued tableaux in~\cite{PPPS} uncrowds
the entries in the arms of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. Subsequently applying the uncrowding
algorithm by Buch~\cite{Buch.2002} on the set-valued tableau yields a semistandard Young tableau and a recording tableau. It was proved
in~\cite{PPPS} that this uncrowding operator intertwines with the crystal operators of Hawkes and Scrimshaw~\cite{HS}.
Let us denote this sequence of uncrowding operations by $\U_{\L^\infty \A^\infty}$, which indicates that first arm and then leg uncrowding is performed.
In this paper, we also consider other orderings of leg and arm uncrowding, in particular $\U_{\A^\infty \L^\infty}$ which first performs leg and then arm
uncrowding. We relate the two orderings using tableau switching in the sense of Benkart, Sottile and Stroomer~\cite{BSS}.
This yields a characterization of the recording tableaux under the uncrowding algorithm in terms of biflagged tableaux (see \Cref{corollary.image}).
To connect to the combinatorial model of  \cite{HJKSS1} in terms of exquisite tableaux we use the jeu de taquin algorithm due to Goulden and 
Greene~\cite{GG}, which we call GG-jdt. This map
was further studied by Krattenthaler \cite{Kra}.
We show that  GG-jdt is a partial tableau switching procedure (see \Cref{prop.perforated}).
This yields the bijection between the combinatorial models for the refined canonical stable Grothendieck polynomials (see \Cref{thm:2bij}).
Our proof reveals a hidden symmetry of the uncrowding algorithm on hook-valued tableaux
when interchanging the order of arm and leg uncrowding (see \Cref{thm.shuff.q}). This shows the equivalence of
three combinatorial models for the refined canonical stable Grothendieck polynomials (see \Cref{corollary.grothendieck}).

For the long version containing all proofs, see \cite{Jang2024a}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
\label{section.background}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Refined canonical stable Grothendieck polynomials}
\label{subsec.grothendieck}

Yeliussizov~\cite{Yeliussizov.2017} introduced the \defn{canonical stable Grothendieck polynomial} $G^{(\alpha,\beta)}_\lambda(\bm{x})$
indexed by a partition $\lambda$ and two parameters $\alpha$ and $\beta$. It is a formal power series generalizing the 
\defn{Grassmannian Grothendieck polynomial} $G^{(\beta)}_\lambda(\bm{x})$ with the property
\[
\omega(G^{(\alpha,\beta)}_\lambda(\bm{x})) = G^{(\beta,\alpha)}_{\lambda'}
(\bm{x}),
\]
where $\omega$ is the involution that sends the Schur function
$s_{\lambda}(\bm{x})$ to $s_{\lambda'}(\bm{x})$ indexed by the transpose partition $\lambda'$. 
The canonical stable Grothendieck polynomials have recently been studied in~\cite{HS, PPPS}.

The \defn{refined canonical stable Grothendieck polynomials}
$G_\lambda(\bm{x}, \bm{\alpha},\bm{\beta})$, introduced by Hwang, Jang, Kim, Song, and Song \cite{HJKSS1}, are refinements of
$G^{(\alpha,\beta)}_\lambda(\bm{x})$ with infinite sets of parameters
\( \bm{\alpha}= (\alpha_1,\alpha_2,\dots) \) and
\( \bm{\beta}= (\beta_1,\beta_2,\dots) \).
Here, we replaced every \( \beta_i \) with \( -\beta_i \) in the
original definition \cite[Definition~1.1]{HJKSS1} in order to make
\( G_\lambda(\bm{x}, \bm{\alpha},\bm{\beta}) \) a formal power
series with positive coefficients. We
can set $\alpha_i = \alpha$ and $\beta_i = \beta$, for all $i$, in the
refined version $G_\lambda(\bm{x}, \bm{\alpha},\bm{\beta})$ to get the
original version $G^{(\alpha,\beta)}_\lambda(\bm{x})$.
Combinatorially, both $G^{(\alpha,\beta)}_\lambda(\bm{x})$ and
$G_\lambda(\bm{x}, \bm{\alpha},\bm{\beta})$ are the generating
functions for hook-valued tableaux~\cite{Yeliussizov.2017,HJKSS2}. 

We use French notation for partitions and tableaux throughout the paper.

\begin{definition}\label{def:HVT}
A \defn{hook-valued tableau} is a filling of a partition shape satisfying:
\begin{enumerate}
  \item Each box contains a semistandard Young tableau of hook shape, i.e.,
  \[\begin{ytableau}
    L_\ell\\
    \vdots\\
    L_1\\
    h & A_1 & \cdots & A_k
  \end{ytableau}\]
  where $h<L_1<\dots<L_\ell$ and $h\leqslant A_1 \leqslant \dots \leqslant A_k$ are positive integers. The entry $h$ is called the \defn{hook entry}, the $L_i$'s the \defn{leg entries} and the $A_i$'s the \defn{arm entries}. 
  \item Each row is weakly increasing, i.e., any entry in a box is weakly smaller than any entry in the box directly to the right of it.
  \item Each column is strictly increasing, i.e., any entry in a box is strictly smaller than any entry in the box directly above it.
\end{enumerate}
We denote by $\HVT(\lambda)$ the set of hook-valued tableaux of shape
$\lambda$. We write $\HVT$ for the set of hook-valued tableaux of any
shape. The \defn{weight} \( \wt(T) \) of $T \in \HVT$ is
\[
  \wt(T) = \prod_{i\geqslant 1}
  \alpha_i^{(\# \text{ of arm entries in column } i)}
  \beta_i^{(\# \text{ of leg entries in row } i)}
  x_i^{(\# \text{ of $i$'s in } T)}.
\]
\end{definition}

\begin{example}
\[
\text{Let} \qquad 
{\def\mc#1{\makecell[lb]{#1}}
T_1 =\,{\begin{array}[lb]{*{3}{|l}|}\cline{1-2}
\mc{6\\4\\335}&\mc{67}\\\cline{1-3}
\mc{2\\11}&\mc{4\\334}&\mc{9\\445}\\\cline{1-3}
\end{array}} \qquad \text{and} \qquad
T_2 =\,{\begin{array}[lb]{*{4}{|l}|}\cline{1-2}
\mc{7\\445}&\mc{789}\\\cline{1-4}
\mc{4\\3\\122}&\mc{5\\345}&\mc{8\\7\\567}&\mc{7}\\\cline{1-4}
\end{array}\,.}
}
\]
  The tableau $T_1$ is a hook-valued tableau of shape $(3,2)$, and
  $T_2$ is not a hook-valued tableau because the first row of \( T_2 \) is not weakly
  increasing (the first column of \( T_2 \) is not strictly increasing either). Furthermore,
\(\wt(T_1) = \alpha_1^3\alpha_2^3\alpha_3^2 \beta_1^3\beta_2^2
x_1^2x_2x_3^4x_4^5x_5^2x_6^2x_7x_9 \).
\end{example}

\begin{theorem}\label{thm:HVT} \cite{HJKSS2}
  We have
\[
  G_\lambda(\bm{x}, \bm{\alpha},\bm{\beta}) = \sum_{T\in \HVT(\lambda)} \wt(T).
\]
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Various mixed tableaux}


\begin{definition}\label{def:MT}
A \defn{mixed tableau} of shape \( \ml \) is a filling \( T \) of the
cells of \( \ml \) with elements in
\( \{\alpha_k \mid k\in \ZZ_{>0}\}\cup \{\beta_k \mid k\in \ZZ\} \).
The \defn{weight} \( \wt(T) \) of a mixed tableau \( T \) is the
product of its entries.
\end{definition}

\begin{definition}\label{def:FMT}
  A \defn{flagged-mixed tableau} is a mixed tableau satisfying the
  following conditions: if \( T(i,j) = \alpha_k \), then \( 0<k<j \), and
  if \( T(i,j) = \beta_k \), then \( 0<k<i \).
  Here, $T(i,j)$ denotes the entry in the $i$\textsuperscript{th} row and $j$\textsuperscript{th} column.
  We denote by \( \FMT \) the set of all flagged-mixed tableaux.
\end{definition}

\begin{definition}\label{def:strict}
  Let \( T \) be a mixed tableau. For
  \( \gamma\in \{\alpha,\beta\} \), we say that \( T \) is
  \defn{\( \gamma \)-column-strict}
  (resp.~\defn{\( \gamma \)-row-strict}) if the following
  conditions hold:
  \begin{enumerate}
  \item If \(\gamma_i\) and \(\gamma_j\) are entries in \(T\) such that
    \( \gamma_i \) is weakly southwest of \( \gamma_j \), then
    \( i\geqslant j \).
  \item If \(\gamma_i\) and \(\gamma_j\) are entries in \(T\) in the same
    column (resp.~row), then \( i\ne j \).
\end{enumerate}

We also say that \( T \) is \defn{totally column-strict} if the following
conditions hold:
\begin{enumerate} 
\item All indices of \( \alpha \) and \( \beta \) in each column are strictly decreasing.
  More precisely, if \( T(i,j)=\gamma_r \) and
  \( T(i+1,j)=\delta_s \) with
  \( \gamma,\delta\in \{\alpha,\beta\} \), then \( r> s \).
\item All indices of \( \alpha \) and \( \beta \) in each row are
  weakly decreasing. More precisely, if \( T(i,j)=\gamma_r \) and
  \( T(i,j+1)=\delta_s \) with
  \( \gamma,\delta\in \{\alpha,\beta\} \), then \( r\geqslant s \).
\end{enumerate}
\end{definition}

\begin{definition}\label{def:sorted}
  Let \( T \) be a mixed tableau of shape \( \ml \). Let \( A \)
  (resp.~\( B \)) be the set of cells in \( T \) containing
  \( \alpha_k \) (resp.~\( \beta_k \)) for any \( k\in \ZZ \). We say
  that \( T \) is \defn{\( (\alpha,\beta) \)-sorted} if \( A \) and
  \( B \) form skew shapes \( \nu/\lambda \) and \( \mu/\nu \),
  respectively, for some partition \( \nu \) with
  \( \lambda\subseteq\nu\subseteq\mu \).
  Similarly, we say that \( T \) is \defn{\( (\beta,\alpha) \)-sorted} if \( B \) and
  \( A \) form skew shapes \( \nu/\lambda \) and \( \mu/\nu \),
  respectively, for some partition \( \nu \) with
  \( \lambda\subseteq\nu\subseteq\mu \).
\end{definition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Exquisite tableaux}
\label{subsec.exquisite}

The \defn{content} \( c(i,j) \) of the cell \( (i,j) \) is defined by
$c(i,j) = j-i$.

\begin{definition}\label{def:c+-}
  Let \( T \) be a mixed tableau. We define \( c_\beta^+(T) \)
  (resp.~\( c_\beta^-(T) \)) to be the tableau obtained from \( T \)
  by replacing every \( \beta_r \) by \( \beta_{r+c} \)
  (resp.~\( \beta_{r-c} \)), where \( c \) is the content of the cell
  containing \( \beta_r \).
\end{definition}

\begin{definition}\label{def:exq}
 An \defn{exquisite tableau} is a flagged-mixed tableau $E$ such that
  \( c_\beta^+(E) \) totally column-strict. Let \(\EXQ(\ml)\) denote
  the set of all exquisite tableaux of shape \(\ml\).
\end{definition}

\begin{example}\label{exa:1}
Let \(\lambda = (2,1)\) and \(\mu = (3,3,1)\). Then
  \[
   \ytableausetup{boxsize = 1.3em}
\EXQ(\ml) = \left\{
\raisebox{1em}{\begin{ytableau}
\beta_2 \\
*(gray) & \beta_1 & \alpha_1 \\
*(gray) & *(gray) & \alpha_2
\end{ytableau}}\,,
\raisebox{1em}{\begin{ytableau}
\beta_1 \\
*(gray) & \beta_1 & \alpha_1 \\
*(gray) & *(gray) & \alpha_2
\end{ytableau}}\,,
\raisebox{1em}{\begin{ytableau}
\beta_2 \\
*(gray) & \alpha_1 & \alpha_1 \\
*(gray) & *(gray) & \alpha_2
\end{ytableau}}\,,
\raisebox{1em}{\begin{ytableau}
\beta_1 \\
*(gray) & \alpha_1 & \alpha_1 \\
*(gray) & *(gray) & \alpha_2
\end{ytableau}}
\right\}\,.
\]
\end{example}

\begin{theorem}\cite[Corollary~4.5]{HJKSS1}
\label{thm.exq}
We have
\[
  G_\lambda(\bm{x}, \bm{\alpha},\bm{\beta}) = \sum_{\mu \supseteq \lambda} s_\mu(\bm{x}) \sum_{E \in \EXQ(\ml)} \wt(E).
\]
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Uncrowding algorithm and tableau switching}
\label{section.uncrowding}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Uncrowding algorithms for hook-valued tableaux}
\label{subsec.uncrowding}

We define the uncrowding algorithms for hook-valued
tableaux following~\cite{PPPS}. 
\begin{definition}
  The \defn{arm-uncrowding bumping} \( \A_b \colon \HVT\to \HVT \) is defined
  in \cite[Definition~3.2]{PPPS}.
  The \defn{leg-uncrowding bumping} \( \L_b\colon \HVT\to \HVT \) is defined
  in \cite[Definition~3.33]{PPPS}.
\end{definition}


\begin{definition}
  The \defn{single-arm-uncrowding map} \( \A \colon \HVT\to \HVT \) is
  defined as follows. Let \( T\in \HVT \). If \( T \) has no arms, or
  equivalently, if \( \A_b(T) = T \), then define \( \A(T) = T \).
  Otherwise, we define \( \A(T) = \A_b^m(T) \), where \( m \) is the
  smallest integer such that the shape of \( \A_b^m(T) \) is larger
  than that of \( T \).
\end{definition}

\begin{definition}
  The \defn{single-leg-uncrowding map} \( \L \colon \HVT\to \HVT \) is
  defined as follows. Let \( T\in \HVT \). If \( T \) has no legs,
  then \( \L(T) = T \).
  Otherwise, we define \( \L(T) = \L_b^m(T) \), where \( m \) is the
  smallest integer such that the shape of \( \L_b^m(T) \) is larger
  than that of \( T \).
\end{definition}

\begin{definition}\label{defn.uncrowd}
  Let \( \U = \U_{f_n\cdots f_1} \), where \( f_n\cdots f_1 \) is a
  word in the alphabet \( \{\A,\L\} \).
  Then the \defn{uncrowding map}
  \( \U \colon \HVT\to \HVT\times \FMT \) is defined as follows.

  Let \( T\in \HVT \). We construct two tableaux \( P \) and
  \( Q \). For each \( i=0,1,2,\dots,n \), let
  \( T_i = f_i \circ \cdots \circ f_1 (T) \), where \( T_0=T \), and
  let \( \lambda^{(i)} \) be the shape of \( T_i \). First, we define
  \( P = T_n \). Now we define a flagged-mixed tableau \( Q \) of
  shape \( \lambda^{(n)}/\lambda^{(0)} \) as follows. For each
  \( i=1,2,\dots,n \), there are two cases.

  \begin{description}
  \item[Case 1] \( f_i = \A \). 
    Then
    \( T_{i} = \A(T_{i-1}) \). If \( T_{i-1} \) has arms, suppose
    that \( (r,c) \) is the cell that contains the largest arm entry in the
    rightmost column containing an arm. Then fill the unique cell
    \( \lambda^{(i)}/\lambda^{(i-1)} \) in \( Q \) with
    \( \alpha_c \). (If \( T_{i-1} \) has no arms, nothing happens.)
  \item[Case 2] \( f_i = \L \). 
    Then
    \( T_{i} = \L(T_{i-1}) \). If \( T_{i-1} \) has legs, suppose
    that \( (r,c) \) is the cell that contains the largest leg entry in the
    topmost row containing a leg. Then fill the unique cell
    \( \lambda^{(i)}/\lambda^{(i-1)} \) in \( Q \) with \( \beta_r \).
    (If \( T_{i-1} \) has no legs, nothing happens.)
  \end{description}
  
  Finally, we define \( \U(T) = (P,Q) \). We call \( P \) and \( Q \)
  the \defn{insertion tableau} and \defn{recording tableau} of
  \( \U(T) \), respectively. 
\end{definition}

\begin{example}\label{ex:uncrowding}
  Consider the following hook-valued tableaux
  \( T, \A(T), \A\circ\A(T), \L\circ\A\circ\A(T) \), and \( \L\circ \L\circ\A\circ\A(T) \):
\[{\def\mc#1{\makecell[lb]{#1}}
    \def\asep{\,\substack{\A\\ \rightarrow}\,\,}
    \def\lsep{\,\substack{\L\\ \rightarrow}\,\,}
\,{T=\begin{array}[lb]{*{4}{|l}|}\cline{1-1}
\mc{4}\\\cline{1-2}
\mc{3}&\mc{6\\57}\\\cline{1-2}
\mc{2}&\mc{24}\\\cline{1-4}
\mc{1}&\mc{1}&\mc{1}&\mc{5\\3}\\\cline{1-4}
\end{array}\,} \asep
\,{\begin{array}[lb]{*{4}{|l}|}\cline{1-1}
\mc{4}\\\cline{1-2}
\mc{3}&\mc{6\\5}\\\cline{1-3}
\mc{2}&\mc{24}&\mc{7}\\\cline{1-4}
\mc{1}&\mc{1}&\mc{1}&\mc{5\\3}\\\cline{1-4}
\end{array}\,} \asep
\,{\begin{array}[lb]{*{4}{|l}|}\cline{1-1}
\mc{4}\\\cline{1-2}
\mc{3}&\mc{6\\5}\\\cline{1-4}
\mc{2}&\mc{2}&\mc{4}&\mc{7}\\\cline{1-4}
\mc{1}&\mc{1}&\mc{1}&\mc{5\\3}\\\cline{1-4}
\end{array}\,} \lsep
\,{\begin{array}[lb]{*{4}{|l}|}\cline{1-2}
\mc{4}&\mc{6}\\\cline{1-2}
\mc{3}&\mc{\phantom{6}\\5}\\\cline{1-4}
\mc{2}&\mc{2}&\mc{4}&\mc{7}\\\cline{1-4}
\mc{1}&\mc{1}&\mc{1}&\mc{5\\3}\\\cline{1-4}
\end{array}\,} \lsep
\,{\begin{array}[lb]{*{4}{|l}|}\cline{1-2}
\mc{4}&\mc{6}\\\cline{1-3}
\mc{3}&\mc{\phantom{6}\\5}&\mc{7}\\\cline{1-4}
\mc{2}&\mc{2}&\mc{4}&\mc{5}\\\cline{1-4}
\mc{1}&\mc{1}&\mc{1}&\mc{\phantom{7}\\3}\\\cline{1-4}
\end{array}}
}\,.
\]
This shows that \( \U_{\L\L\A\A}(T) = (P,Q) \), where
\[
  \ytableausetup{boxsize = 1.3em}
  P = \raisebox{2em}{
  \begin{ytableau}
4 & 6\\
3 & 5 & 7\\
2 & 2 & 4 & 5\\
1 & 1 & 1 & 3
\end{ytableau}\,}, \qquad 
Q = \raisebox{2em}{
\begin{ytableau}
*(gray) & \beta_3\\
*(gray) & *(gray) & \beta_1\\
*(gray) & *(gray) & \alpha_2 & \alpha_2\\
*(gray) & *(gray) & *(gray) & *(gray)
\end{ytableau}\,}.
\]
\end{example}

Note that if a hook-valued tableau \( T \) has \( a \) arms and
\( \ell \) legs, then \( \A^{a}(T) = \A^{a+1}(T) = \cdots \) and
\( \L^{\ell}(T) = \L^{\ell+1}(T) = \cdots \). Thus,
\( \U_{\L^N\A^M}(T) \) is the same for all \( N\geqslant \ell \) and
\( M\geqslant a \). We will write the result as
\( \U_{\L^\infty\A^\infty}(T) \). We define
\( \U_{\A^\infty\L^\infty}(T) \) similarly.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Tableau switching}
\label{subsec.switching}

We recall results from a paper by Benkart, Sottile, and Stroomer~\cite{BSS} on tableau switching.

\begin{definition}\label{def:switching}
  Let \( T \) be an \( \alpha \)-column-strict and
  \( \beta \)-row-strict mixed tableau. Let \( u=T(i,j) \) be an entry
  of \( T \) and let \( v \) be the entry \( T(i,j+1) \) or
  \( T(i+1,j) \). Suppose \( u=\alpha_r \) and \( v=\beta_s \). Let
  \( T' \) be the mixed tableau obtained from \( T \) by interchanging
  \( u \) and \( v \). If \( T' \) is \( \alpha \)-column-strict and
  \( \beta \)-row-strict, such a process is called a \defn{switch}. If
  there is no possible switch, we say that \( T \) is \defn{fully
    switched}.
\end{definition}

Note that in a switch, we always move an \( \alpha \)-entry to the
north or east, and a \( \beta \)-entry to the south or west. Hence, if
we keep applying switches to a tableau, it eventually becomes fully
switched.

\begin{theorem} \cite[Theorem~2.2]{BSS}
\label{thm.switch}
Let \( T \) be an \( \alpha \)-column-strict, \( \beta \)-row-strict,
and \( (\alpha,\beta) \)-sorted mixed tableau. We apply switches to
\( T \) until it is fully switched. Then, the resulting tableau is
\( \alpha \)-column-strict, \( \beta \)-row-strict, and
\( (\beta,\alpha) \)-sorted. Furthermore, it is independent of the sequence of
switches that produced it.
\end{theorem}

\begin{corollary}\label{cor:switch}
\mbox{}
\begin{enumerate}
\item Let \( T \) be an \( \alpha \)-column-strict,
  \( \beta \)-row-strict, and \( (\alpha,\beta) \)-sorted mixed
  tableau. Let \( X_1 \) and \( X_2 \) be fully switched tableaux obtained from
  \( T \) by some sequences of switches.
  Then \( X_1=X_2 \).
\item Let \( T_1 \) and \( T_2 \) be \( \alpha \)-column-strict,
  \( \beta \)-row-strict, and \( (\alpha,\beta) \)-sorted mixed
  tableaux. Suppose that \( X \) is a fully switched tableau obtained from both
  \( T_1 \) and \( T_2 \) by some
  sequences of switches. Then \( T_1=T_2 \).
\end{enumerate}

\end{corollary}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main results}
\label{section.proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Tableau switching on the recording tableau}
\label{subsec.armleg}
We study the effects of first performing leg-uncrowding and then
arm-uncrowding on a hook-valued tableau in comparison to \cite{PPPS}.
To characterize the changes to the recording tableaux, we 
define a particular sequence of tableau switches on an
$(\alpha, \beta)$-tableau.

\begin{definition} 
\label{def:shuffle} 
Let $Q$ be a flagged-mixed tableau that is \( \alpha \)-column-strict,
\( \beta \)-row-strict, and \( (\alpha,\beta) \)-sorted. The (jdt) \defn{shuffle} of $Q$, denoted $\shuff(Q)$, is the tableau
obtained by:
\begin{enumerate}
\item Find elements with weight $\alpha$ having an element of weight $\beta$ above or to the right.
\item Among them, choose the rightmost element that has the smallest index, say $\alpha_i$.
\item Continue the following `switching process' until the cells above and to the right of $\alpha_i$ have a weight $\alpha$ or are empty. If only one of the cells directly above or to the right of $\alpha_i$ has weight $\beta$, then switch $\alpha_i$ with this element. Otherwise, let $\beta_k$ and $\beta_j$ be the entries above and to the right of $\alpha_i$, respectively, and perform one of the following switches
 \[
\ytableausetup{boxsize = 1.3em}
    \begin{ytableau}
      \beta_k & \none \\
      \alpha_i & \beta_j
    \end{ytableau}
    \mapsto
    \begin{ytableau}
      \alpha_i & \none \\
      \beta_k & \beta_j
    \end{ytableau}
    \quad\mbox{if \( k > j \)},\qquad
    \begin{ytableau}
     \beta_k & \none \\
     \alpha_i & \beta_j
    \end{ytableau}
    \mapsto
    \begin{ytableau}
     \beta_k & \none \\
     \beta_j & \alpha_i
    \end{ytableau}
    \quad\mbox{if \( k \leqslant j \).}
  \] 
\item Repeat steps (1)-(3) until there is no cell having weight $\alpha$ with an element of weight $\beta$ directly above or to its right.
\end{enumerate}
\end{definition}

\begin{example}\label{exa:shuffle}
  The following shows the process of the shuffle, where boxes
  containing an \( \alpha \) are colored in yellow:
  \begin{align*}
  \ytableausetup{boxsize = 1.3em}
  Q= \begin{ytableau}
       \beta_8 & \beta_6 & \beta_5 & \beta_2 \\
      *(yellow)\alpha_2 & *(yellow)\alpha_1 & \beta_6 & \beta_2 & \beta_1 \\
      *(gray) & *(yellow)\alpha_2 & *(yellow)\alpha_2 & *(yellow)\alpha_1 & \beta_5 & \beta_1
    \end{ytableau}
    &\Longrightarrow    
    \begin{ytableau}
       \beta_8 & \beta_6 & \beta_5 & \beta_2 \\
      *(yellow)\alpha_2 & *(yellow)\alpha_1 & \beta_6 & \beta_2 & \beta_1 \\
      *(gray) & *(yellow)\alpha_2 & *(yellow)\alpha_2 & \beta_5 & \beta_1 & *(yellow)\alpha_1
    \end{ytableau}
    \Longrightarrow    
    \begin{ytableau}
       \beta_8 & \beta_6 & \beta_2 & *(yellow)\alpha_1 \\
      *(yellow)\alpha_2 & \beta_6  & \beta_5 & \beta_2 & \beta_1 \\
      *(gray) & *(yellow)\alpha_2 & *(yellow)\alpha_2 & \beta_5 & \beta_1 & *(yellow)\alpha_1
    \end{ytableau} \\
    \Longrightarrow    
    \begin{ytableau}
       \beta_8 & \beta_6 & \beta_2 & *(yellow)\alpha_1 \\
      *(yellow)\alpha_2 & \beta_6  & \beta_5 & \beta_1 & *(yellow)\alpha_2 \\
      *(gray) & *(yellow)\alpha_2 & \beta_5 & \beta_2 & \beta_1 & *(yellow)\alpha_1
    \end{ytableau}
    &\Longrightarrow    
    \begin{ytableau}
       \beta_8 & \beta_2 & *(yellow)\alpha_2 & *(yellow)\alpha_1 \\
      *(yellow)\alpha_2 & \beta_6 & \beta_5 & \beta_1 & *(yellow)\alpha_2 \\
      *(gray) &  \beta_6 & \beta_5 & \beta_2 & \beta_1 & *(yellow)\alpha_1
    \end{ytableau}
    \Longrightarrow    
    \begin{ytableau}
       \beta_2 & *(yellow)\alpha_2 & *(yellow)\alpha_2 & *(yellow)\alpha_1 \\
       \beta_8& \beta_6 & \beta_5 & \beta_1 & *(yellow)\alpha_2 \\
      *(gray) &  \beta_6 & \beta_5 & \beta_2 & \beta_1 & *(yellow)\alpha_1
    \end{ytableau}= \shuff(Q).
  \end{align*}
\end{example}



\begin{theorem} \label{thm.shuff.q} Let $T$ be a hook-valued tableau
  and set $(P_1, Q_1) = \U_{\L^\infty\A^\infty} (T)$ and
  $(P_2, Q_2) = \U_{\A^\infty\L^\infty} (T)$. Then $P_2 = P_1$ and
  $Q_2 = \shuff(Q_1)$.
\end{theorem}

The above theorem reveals a hidden symmetry of hook-valued tableaux
which leads to a characterization of the image for the uncrowding
algorithm and the equivalence of multiple combinatorial models of
\( G_\lambda(\bm{x}, \bm{\alpha},\bm{\beta}) \), which is discussed in
\Cref{subsec.image}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Goulden-Greene jeu de taquin}
\label{subsec.ggjdt}

 We define the Goulden--Greene jeu de taquin
algorithm~\cite{GG} (a process they refer to as the modified jeu de taquin) using our notation.
This map was also studied by Krattenthaler \cite{Kra}.
We then show that it can be realized as a tableau switching.

\begin{definition}[GG-jdt slides]\label{def:GG-jdt-slides}
  Let \( T \) be a \( \alpha \)-column-strict and
  \( \beta \)-row-strict mixed tableau. We say that the entry
  \( T(i,j) \) is \defn{out of order} if at least one of the following
  conditions holds:
  \begin{enumerate}
  \item \( T(i,j) = \alpha_r \) and \( T(i,j+1) = \beta_s \) for some \( r \) and \( s \) with \( r<s + (j+1) - i \).
  \item \( T(i,j) = \alpha_r \) and \( T(i+1,j) = \beta_t \) for some \( r \) and \( t \) with \( r\leqslant t + j - (i+1) \).
  \end{enumerate}
  If Condition (1) (resp. Condition (2)) holds, we define the
  \defn{horizontal slide} (resp. \defn{vertical slide}) at \( (i,j) \)
  to be the operation that swaps \( \alpha_r \) and \( \beta_s \). 

  If \( T(i,j) \) is out of order, the \defn{GG-jdt slide} at
  \( (i,j) \) is the unique available operation between the horizontal
  slide and the vertical slide at \( (i,j) \) such that the resulting
  tableau is still \( \beta \)-row-strict. More precisely, if only
  Condition (1) (resp.~(2)) holds, then the GG-jdt slide at
  \( (i,j) \) is the horizontal (resp.~vertical) slide at \( (i,j) \).
  If both Conditions (1) and (2) hold, then \( T(i,j) = \alpha_r \),
  \( T(i,j+1) = \beta_s \), and \( T(i+1,j) = \beta_t \) for some
  \( r \), \( s \), and \( t \) with \( r<s + (j+1) - i \) and
  \( r\leqslant t + j - (i+1) \). In this case, the GG-jdt slide at
  \( (i,j) \) is the horizontal slide at \( (i,j) \) if \( t\leqslant s \)
  and the vertical slide at \( (i,j) \) if \( t > s \):
  \ytableausetup{boxsize = 1.3em}
  \[
    \begin{ytableau}
      \beta_t \\
      \alpha_r & \beta_s
    \end{ytableau}
    \mapsto
    \begin{ytableau}
      \beta_t \\
      \beta_s & \alpha_r
    \end{ytableau}
    \quad\mbox{if \( t \leqslant s \)},\qquad
    \begin{ytableau}
      \beta_t \\
      \alpha_r & \beta_s
    \end{ytableau}
    \mapsto
    \begin{ytableau}
      \alpha_r \\
      \beta_t & \beta_s
    \end{ytableau}
    \quad\mbox{if \( t > s \)}.
  \]
\end{definition}

\begin{definition}[GG-jdt map]\label{def:GG-jdt}
  Let \( T \) be a mixed tableau that is \( \alpha \)-column-strict,
  \( \beta \)-row-strict, and \( (\alpha,\beta) \)-sorted. The
  \defn{GG-jdt map} is the map \( \jdt \) sending \( T \) to the
  tableau \( \jdt(T) \) obtained as follows:
\begin{enumerate}
\item Find the smallest \( r \) such that \( \alpha_r \) is out of
  order in \( T \). Let \( (i,j) \) be the rightmost cell containing
  such an \( \alpha_r \).
\item Apply the GG-jdt slide to \( T \) at \( (i,j) \).
\item Repeat (1)-(2) until no entries are out of order.
\end{enumerate}
\end{definition}

Observe that the GG-jdt slide is exactly a switch defined in
\Cref{def:switching} except that it does not require the resulting
tableau to be \( \alpha \)-column-strict and
\( \beta \)-column-strict. However, we will see that this condition
holds automatically. Hence, the GG-jdt map is a sequence of switches.

\begin{example}\label{exa:GGjdt}
  The GG-jdt map applied to \( Q \) proceeds as follows, where we
  truncate the first two columns and the first six rows of the
  tableau:
  \begin{align*}
  \ytableausetup{boxsize = 1.5em}
   Q = \begin{ytableau}
      \beta_8 & \beta_6 & \beta_5 & \beta_2 \\
      *(yellow)\alpha_2 & *(yellow)\alpha_1 & \beta_6 & \beta_2 & \beta_1 \\
      *(gray) & *(yellow)\alpha_2 & *(yellow)\alpha_2 & *(yellow)\alpha_1 & \beta_5 & \beta_1
    \end{ytableau}
    \xrightarrow{\jdt}
    \begin{ytableau}
      *(yellow)\alpha_2 & \beta_6 & *(yellow)\alpha_1 & \beta_2 \\
      \beta_8 & *(yellow)\alpha_2 & \beta_5 & \beta_2 & \beta_1 \\
      *(gray) & \beta_6 & \beta_5 &  *(yellow)\alpha_2 & \beta_1 & *(yellow)\alpha_1
    \end{ytableau}
        \xrightarrow{c_{\beta}^+}
\begin{ytableau}
      *(yellow)\alpha_2 & \beta_1 & *(yellow)\alpha_1 & \beta_{-1} \\
      \beta_3 & *(yellow)\alpha_2 & \beta_2 & \beta_0 & \beta_0 \\
      *(gray) & \beta_3 & \beta_3 &  *(yellow)\alpha_2 & \beta_1 & *(yellow)\alpha_1
    \end{ytableau}.
  \end{align*}

  We can compare this with the tableau switching in
  \Cref{exa:shuffle}. Note that this process is a sequence of
  switches. By the uniqueness of the fully switched tableau, see
  \Cref{cor:switch} (1), if we keep applying switches to
  \( \jdt(Q) \), then we get the same resulting tableau
  \( \shuff(Q) \) at the end of \Cref{exa:shuffle}.
\end{example}


The following properties of the GG-jdt map were stated in \cite[Section~3]{GG}
without proof. Krattenthaler \cite[Lemma~1]{Kra} gave complete proofs
of these results.

\begin{proposition}\label{pro:GGjdt}\
  \begin{enumerate}
  \item   The GG-jdt map is a bijection from the set of
  \( \alpha \)-column-strict, \( \beta \)-row-strict, and
  \( (\alpha,\beta) \)-sorted tableaux to the set of tableaux \( E \)
  such that \( c_\beta^+(E) \) is a totally column-strict tableau.
\item Each iteration of the GG-jdt slide results in an
  \( \alpha \)-column-strict and \( \beta \)-row-strict tableau.
  \end{enumerate}
\end{proposition}

By \Cref{pro:GGjdt}~(2), the GG-jdt map can be formulated as a
tableau switching procedure as defined in
\Cref{subsec.switching}.

\begin{proposition}\label{prop.perforated}
  Let \( T \) be a flagged-mixed tableau that is
  \( \alpha \)-column-strict, \( \beta \)-row-strict, and
  \( (\alpha,\beta) \)-sorted. Then
  \( \jdt(T) \) is obtained from
  \( T \) by a sequence of switches.
\end{proposition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Bijections and the image of the uncrowding algorithm}
\label{subsec.image}

In this subsection, we find a bijection relating hook-valued
tableaux, exquisite tableaux, and a new class of tableaux called
biflagged tableaux.
We also show that GG-jdt is a bijection between
biflagged tableaux and exquisite tableaux.
As a corollary, we characterize the image of the
uncrowding algorithm \( \U_{\L^\infty\A^\infty} \) defined on
hook-valued tableaux.

\begin{definition}\label{def:BFT}
  A \defn{biflagged tableau} is a mixed tableau \( T \)
  satisfying:
  \begin{enumerate}
  \item \( T \) is \( \alpha \)-column-strict, \( \beta \)-row-strict, and \( (\alpha,\beta) \)-sorted.
  \item Both \( T \) and \( \shuff(T) \) are flagged-mixed tableaux.
  \end{enumerate}
We denote by \( \BFT(\ml) \)
the set of biflagged tableaux of shape \( \ml \).
\end{definition}


\begin{example}\label{eg.biflagged}
When $\lambda = (2,1)$ and $\mu = (3,3,1)$, we have
\ytableausetup{boxsize = 1.4em}
\[
\BFT(\ml) = \left\{
\raisebox{1em}{\begin{ytableau}
\beta_2 \\
*(gray) & \alpha_1 & \beta_1 \\
*(gray) & *(gray) & \alpha_2
\end{ytableau}}\,,
\raisebox{1em}{\begin{ytableau}
\beta_1 \\
*(gray) & \alpha_1 & \beta_1 \\
*(gray) & *(gray) & \alpha_2
\end{ytableau}}\,,
\raisebox{1em}{\begin{ytableau}
\beta_2 \\
*(gray) & \alpha_1 & \alpha_1 \\
*(gray) & *(gray) & \alpha_2
\end{ytableau}}\,,
\raisebox{1em}{\begin{ytableau}
\beta_1 \\
*(gray) & \alpha_1 & \alpha_1 \\
*(gray) & *(gray) & \alpha_2
\end{ytableau}}
\right\}\,.
\]
\end{example}

We will show that GG-jdt is a bijection
from the set of biflagged tableaux \( \BFT(\ml) \) 
to the set of exquisite tableaux $\EXQ(\ml)$ defined in
\Cref{def:exq}. We denote by $\SSYT(\mu)$ the set of semistandard
Young tableaux of shape $\mu$. As usual, the weight \( \wt(P) \) of a
semistandard Young tableau \( P \) is the product of \( x_i \) for all
entries \( i \) in \( P \).


\begin{lemma}
\label{lemma.PQ}
  Let \(T\in \HVT(\lambda)\) and \( \U_{\L^\infty\A^\infty} (T) = (P,Q) \).
  Then \(P \in \SSYT(\mu)\) and \(Q\in \BFT(\ml)\) for a partition \(\mu\) with \(\lambda \subseteq \mu\). Furthermore, $\wt(T) = \wt(P)\wt(Q)$.
\end{lemma}


\begin{lemma}\label{lem:BFT-EXQ}
Let \( Q\in \BFT(\ml) \). Then \( \jdt(Q)\in \EXQ(\ml) \).
\end{lemma}


By \Cref{lemma.PQ,lem:BFT-EXQ}, we can define the following map.

\begin{definition}\label{def:Phi}
We define the map
 \[
\Phi\colon \HVT(\lambda) \to \bigsqcup_{\mu \supseteq \lambda} \left( \SSYT(\mu)\times \EXQ(\ml) \right)
\]
as the composition of the following two maps:
\begin{equation}\label{eq:bijections}
  \HVT(\lambda)
  \xrightarrow{\U_{\L^\infty\A^\infty}}
 \displaystyle  \bigsqcup_{\mu \supseteq \lambda} \left( \SSYT(\mu)\times \BFT(\ml) \right)
  \xrightarrow{\mathrm{id}\times\jdt}
 \displaystyle \bigsqcup_{\mu \supseteq \lambda} \left( \SSYT(\mu)\times \EXQ(\ml) \right).
\end{equation}
In other words, \( \Phi(T) = (P,E) \), where
\( \U_{\L^\infty\A^\infty} (T) = (P,Q) \) and
\( E = \jdt(Q) \).
\end{definition}

\begin{theorem}\label{thm:2bij}
  The following are weight-preserving bijections:
  \[
    \jdt\colon \BFT(\ml) \to \EXQ(\ml)\qquad \mbox{and}
  \]
 \[
\Phi\colon \HVT(\lambda) \to \bigsqcup_{\mu \supseteq \lambda} \left( \SSYT(\mu)\times \EXQ(\ml) \right).
\]
\end{theorem}


\begin{example}\label{eg.bij}
The map \( \jdt \) sends the biflagged tableaux in \Cref{eg.biflagged} to
  the exquisite tableaux in \Cref{exa:1} in that order.
\end{example}

We now characterize the image of the
uncrowding algorithm.

\begin{corollary}
\label{corollary.image}
  The following is a weight-preserving bijection:
  \[
    \U_{\L^\infty\A^\infty}\colon \HVT(\lambda) \to \bigsqcup_{\mu \supseteq \lambda} \left( \SSYT(\mu)\times \BFT(\ml) \right).
  \]
\end{corollary}

\begin{corollary}
\label{corollary.grothendieck}
We have
\[
G_\lambda(\bm{x}, \bm{\alpha},\bm{\beta}) = \sum_{H\in \HVT(\lambda)}\wt(H)
= \sum_{\mu \supseteq \lambda}s_\mu(\bm{x}) \sum_{E\in \EXQ(\ml)} \wt(E) 
= \sum_{\mu \supseteq \lambda}s_\mu(\bm{x}) \sum_{Q\in \BFT(\ml)} \wt(Q).
\]
\end{corollary}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \acknowledgements{The authors thank FPSAC 2023, AWM, Maria Gillespie,
%  and Travis Scrimshaw.}


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\end{document}
