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%% define your title in the usual way
\title{Refined canonical stable Grothendieck polynomials and their duals}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[B.-H. Hwang, J. Jang, J. S. Kim, M. Song, and U-K. Song]{
  Byung-Hak Hwang\thanks{\href{mailto:byunghakhwang@gmail.com}{byunghakhwang@gmail.com}.}\addressmark{1},
Jihyeug Jang\thanks{\href{mailto:4242ab@gmail.com}{4242ab@gmail.com}. Supported by NRF grants \#2022R1A2C101100911.}\addressmark{1},
Jang Soo Kim\thanks{\href{mailto:jangsookim@skku.edu}{jangsookim@skku.edu}. Supported by NRF grants \#2022R1A2C101100911 and \#2016R1A5A1008055.}\addressmark{1},\\
Minho Song\thanks{\href{mailto:smh3227@skku.edu}{smh3227@skku.edu}. Partially supported by NRF-2019R1I1A1A01058099.}\addressmark{1}, 
\and U-Keun Song\thanks{\href{mailto:sukeun319@gmail.com}{sukeun319@gmail.com}. Supported by NRF grants \#2022R1A2C101100911 and \#2016R1A5A1008055.}\addressmark{1}}

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\address{\addressmark{1}Department of Mathematics, Sungkyunkwan University, Suwon,
South Korea}

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\received{\today}

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\abstract{In this extended abstract we introduce refined canonical stable Grothendieck polynomials
and their duals with two infinite sequences of parameters. These
polynomials unify several generalizations of Grothendieck polynomials
including canonical stable Grothendieck polynomials due to Yeliussizov,
refined Grothendieck polynomials due to Chan and Pflueger, and refined dual
Grothendieck polynomials due to Galashin, Liu, and Grinberg. We give
Jacobi--Trudi-like formulas, combinatorial models, Schur expansions, Schur
positivity, and dualities of these polynomials. We also consider flagged
versions of Grothendieck polynomials and their duals with skew shapes.}

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\keywords{Grothendieck polynomials, symmetric functions, Jacobi--Trudi formulas, Schur
positivity}

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\begin{document}

\maketitle
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\section{Introduction}
\label{sec:introduction}

\subsection{History of Grothendieck polynomials}
\label{sec:hist-groth-polyn}


Grothendieck polynomials were introduced
by Lascoux and Sch\"{u}tzen\-berger \cite{LS1982}
for \linebreak studying the Grothendieck ring of vector bundles on a flag variety.
When focusing on the Grothendieck ring of vector bundles on Grassmannians rather
than that of flag varieties, the stable Grothendieck polynomials are indexed by
partitions. These stable Grothendieck polynomials form a basis for the
(connective) \( K \)-theory ring of Grassmannians, see
\cite[Section~2.3]{Monical2020} and references therein. In this context,
Lenart~\cite{Lenart2000} gave the Schur expansion of \( G_\lambda(\vx) \) and
showed that \( \{G_\lambda(\vx) : \mbox{\( \lambda \) is a partition}\} \) is a
basis for (a completion of) the space of symmetric functions.
Buch \cite{Buch2002} showed that 
the stable Grothendieck polynomial \( G_\lambda(\vx) \) is equal to a generating
function for semistandard set-valued tableaux of shape \( \lambda \).

After Buch's work~\cite{Buch2002}, many generalizations of \( G_\lambda(\vx) \)
have been studied from various viewpoints.
In \cite{LP2007}, Lam and Pylyavskyy introduced
the dual stable Grothendieck polynomials \( g_\lambda(\vx) \),
and found a combinatorial interpretation for them
in terms of reverse plane partitions.
Similar to the Grothendieck polynomial case, the dual Grothendieck polynomials
form a basis for the space of symmetric functions.

Yeliussizov \cite{Yeliussizov2017} introduced
the canonical stable Grothendieck polynomials
\( G^{(\alpha,\beta)}_\lambda(\vx) \) with two parameters \( \alpha \) and \(
\beta \),
and defined the dual canonical stable Grothendieck polynomials
\( g^{(\alpha,\beta)}_\lambda(\vx) \) by the relation
\(
  \langle G^{(-\alpha,-\beta)}_\mu(\vx), g^{(\alpha,\beta)}_\lambda(\vx) \rangle
  = \delta_{\lambda,\mu},
\)
where \( \langle -,- \rangle \) is the Hall inner product.
Yeliussizov \cite{Yeliussizov2017} showed that these polynomials
behave as nicely as Schur functions under the involution \( \omega \).
Moreover, he found combinatorial interpretations for
\( G^{(\alpha,\beta)}_\lambda(\vx) \) and \( g^{(\alpha,\beta)}_\lambda(\vx) \).

There are also generalizations of Grothendieck polynomials and their duals with
infinite parameters; see~\cite{CP21:grothendieck, GGL2016, Kim_JT22, Motegi2021}.
In this extended abstract, motivated by these previous works, we introduce refined canonical
stable Grothendieck polynomials \( G_\lambda(\vx_n;\va,\vb) \) and their duals
\( g_\lambda(\vx_n;\va,\vb) \) with infinite parameters \(
\va=(\alpha_1,\alpha_2,\dots) \) and \( \vb=(\beta_1,\beta_2,\dots) \), and
furthermore their flagged versions with skew shapes. Our generalization unifies all 
generalizations of Grothendieck polynomials mentioned above.


\subsection{Refined canonical Grothendieck polynomials and their duals}
\label{sec:def_G,g}

Let \( \vx = (x_1,x_2,\dots) \), \( \vx_n = (x_1,\dots,x_n) \) be sequences of variables,
and \( \va = (\alpha_1,\alpha_2,\dots) \), \( \vb = (\beta_1,\beta_2,\dots) \)
sequences of parameters.

We generalize the canonical stable Grothendieck polynomials as follows. 
See \cref{sec:preliminary} for precise definitions of the notations.

\begin{defn}\label{def:bi_alt_G}
  For a partition \( \lambda \) with at most \( n \) parts, 
  the \emph{refined canonical stable Grothen\-dieck polynomial}
  \( G_\lambda(\vx_n;\va,\vb) \) is defined by
  \begin{equation}\label{eq:defG}
    G_\lambda(\vx_n;\va,\vb) 
    = \frac{\det
      \left(x_j^{\lambda_i+n-i}
      \dfrac{(1-\beta_1 x_j)\cdots (1-\beta_{i-1} x_j)}
            {(1-\alpha_1 x_j)\cdots (1-\alpha_{\lambda_i} x_j)}
      \right)_{1\le i,j\le n}}
      {\prod_{1\le i<j\le n}(x_i-x_j)}.
  \end{equation}
\end{defn}

Using the notation \( \ominus \) we can rewrite \eqref{eq:defG} as follows:
\begin{equation}
  G_\lambda(\vx_n;\va,\vb)
  = \frac{\det
    \left( h_{\lambda_i+n-i}[x_j \ominus(A_{\lambda_i}-B_{i-1})]
    \right)_{1\le i,j\le n}}
  {\prod_{1\le i<j\le n}(x_i-x_j)},
\end{equation}
where \( A_{k}=\alpha_1+\dots+\alpha_{k} \) and
\( B_{k} = \beta_1+\dots+\beta_{k} \) for \( k\ge1 \),
and \( A_{k}=B_{k}=0 \) for \( k \le 0 \).

We define our generalization of dual Grothendieck polynomials via
the following bialternant formula.

\begin{defn} \label{def:bi_alt_g}
  For a partition \( \lambda \) with at most \( n \) parts, 
  the \emph{refined dual canonical stable Grothendieck polynomial}
  \( g_\lambda(\vx_n;\va,\vb) \) is defined by 
  \begin{equation}\label{eq:defg}
    g_\lambda(\vx_n;\va,\vb)
    = \frac{\det(h_{\lambda_i+n-i}[x_j-A_{\lambda_i-1}+B_{i-1}])_{1\le i,j\le n}}
      {\prod_{1\le i<j\le n}(x_i-x_j)}.
  \end{equation}
\end{defn}

Our generalizations \( G_\lambda(\vx;\va,\vb) \) and \( g_\lambda(\vx;\va,\vb) \)
generalize several well-studied variations of Grothendieck polynomials as follows.
Here \( \bm{0} = (0,0,\dots) \), \( \bm{1}=(1,1,\dots) \),
\( \va_0 = (\alpha,\alpha,\dots) \) and \( \vb_0 = (\beta,\beta,\dots) \).

\begin{center}
  \begin{tabular}{|c|c|c|}
    \hline
    Variations of \( G \) or \( g \) & introduced in & how to specialize \\ \hhline{|=|=|=|}
    \( G_{\nu/\lambda}(\vx) \) & Buch~\cite{Buch2002} &
      \( G_{\nu/\lambda}(\vx;\bm{0},\bm{1}) \)  \\ \hline
    \( g_{\lambda/\mu}(\vx) \) & Lam--Pylyavskyy~\cite{LP2007} &
      \( g_{\lambda/\mu}(\vx;\bm{0},\bm{1}) \) \\  \hline
    \( G^{(\alpha,\beta)}_\lambda(\vx) \) & Yeliussizov~\cite{Yeliussizov2017} &
        \( G_{\lambda}(\vx;\va_0,-\vb_0) \) \\  \hline
    \( g^{(\alpha,\beta)}_\lambda(\vx) \) & Yeliussizov~\cite{Yeliussizov2017} &
        \( g_{\lambda}(\vx;-\va_0,\vb_0) \) \\  \hline
    \( RG_\sigma(\vx;\vb) \) & Chan--Pflueger~\cite{CP21:grothendieck} &
      \( G_\sigma(\vx;\bm{0},\vb) \) \\  \hline
    \( \tg_{\lambda/\mu}(\vx;\vb) \) & Galashin--Grinberg--Liu~\cite{GGL2016} &
      \( g_{\lambda/\mu}(\vx;\bm{0},\vb) \) \\  \hline
    \( G_{\lambda/\mu,f/g}(\vx) \) & Matsumura~\cite{Matsumura_2018} &
      \( G^{\row(g,f)}_{\lambda/\mu}(\vx;\bm{0},-\vb_0) \) \\  \hline
    \( \tg^{\row(\vr,\vs)}_{\lambda/\mu}(\vx;\vb) \) & Grinberg \cite{Grinberg_private}, Kim~\cite{Kim_JT22} &
      \( g^{\row(\vr,\vs)}_{\lambda/\mu}(\vx;\bm{0},\vb) \) \\  \hline
  \end{tabular}
\end{center}

\medskip

The remainder of the extended abstract is organized as follows.
In \cref{sec:preliminary}, we briefly recall
necessary definitions.
In \cref{sec:schur_expansion}, we give Schur expansions and
Jacobi--Trudi-like formulas for \(
G_\lambda(\vx_n;\va,\vb) \) and \( g_\lambda(\vx_n;\va,\vb) \) and prove their duality.
In \cref{sec:lattice}, we give tableau models for \( G_\lambda(\vx;\va,\vb)
\) and \( g_\lambda(\vx;\va,\vb) \) and obtain 
Schur positivity of \( G_\lambda(\vx;\va,-\vb) \) and \( g_\lambda(\vx;-\va,\vb) \).
In \cref{sec:flagged G,sec:flagged g}, we extend
refined canonical stable Grothendieck polynomials and their duals
to skew shapes with certain flag conditions.
In \cref{sec:omega_involution}, we give skew Schur expansions
for \( G_{\lambda/\mu}(\vx;\va,\vb) \) and \( g_{\lambda/\mu}(\vx;\va,\vb) \),
and they behave nicely under 
the involution \( \omega \).
The full version of this extended abstract is \cite{HJKSS}, where the missing proofs can be found.


\section{Preliminaries}
\label{sec:preliminary}

In this section, we set up notations and give the necessary background.
Throughout this extended abstract, we denote by \( \ZZ \) (resp.~\( \ZZ_{+} \)) the set of integers (resp.~positive integers).

\subsection{Partitions and tableaux}

A \emph{partition} is a weakly decreasing sequence \(
\lambda=(\lambda_1,\dots,\lambda_\ell) \) of positive integers. Each \(
\lambda_i \) is called a \emph{part}, and the \emph{length}
\( \ell(\lambda)=\ell \) of \( \lambda \) is the number of parts in \( \lambda \).
We denote by \( \Par_n \) the set of partitions with at most \( n \) parts,
and by \( \Par \) the set of all partitions.

We will identify  a partition \(\lambda\) with its Young diagram.
Each element \((i,j) \in \lambda\) is called a \emph{cell} of \(\lambda\).
The Young diagram of \( \lambda \) is visualized by placing a square
\( (i,j) \) for each cell \( (i,j) \)
in \( \lambda \) using the matrix coordinates.
The \emph{transpose} \( \lambda' \) of \( \lambda \) is the partition
whose Young diagram is obtained from that of \( \lambda \)
by reflecting across the diagonal.
The \emph{content} \( c(i,j) \) of a cell \( (i,j)\in\lambda \) is \( j-i \).


For two partitions \(\lambda,\mu \in \Par\) with \(\mu \subseteq \lambda\),
the \emph{skew shape} \(\lambda/\mu\) is defined to be
\(\lambda-\mu=\{ (i,j): 1\leq i \leq \ell(\lambda), \mu_i < j \leq \lambda_i \}\).
We denote by \( |\lm| \) the number of cells in \( \lm \).


A \emph{semistandard Young tableau} of shape \(\lambda/\mu \) is
a filling \(T\) of the cells in \(\lambda/\mu\) with positive integers
such that the integers are weakly increasing in each row and
strictly increasing in each column.
Let \(\SSYT(\lambda/\mu)\) be the set of semistandard Young tableaux
of shape \(\lambda/\mu\).
A \emph{reverse plane partition} of shape \(\lambda/\mu\) is
a filling \(T\) of the cells in \(\lambda/\mu\) with positive integers 
such that the integers are weakly increasing in each row and
in each column.
Let \(\RPP(\lambda/\mu)\)  be the set of reverse plane partitions
of shape \(\lambda/\mu\).


\subsection{Symmetric functions}
\label{sec:symmetric_function}

Let \( \Lambda \) be the \( \QQ
\)-algebra of all symmetric functions in \( \QQ[[x_1,x_2,\dots]] \). 
The \emph{completion} \(
\hat{\Lambda} \) of \( \Lambda \) is the set of symmetric functions with
possibly unbounded degree.
For \( n\ge 1 \), the \emph{complete homogeneous symmetric function}
\( h_n(\vx) \) and the \emph{elementary symmetric function} \( e_n(\vx) \) are defined to be
\[
    h_n(\vx) = \sum_{i_1\le\dots\le i_n} x_{i_1} \cdots x_{i_n}, \qand e_n(\vx) = \sum_{i_1<\dots<i_n} x_{i_1} \cdots x_{i_n}.
\]
Define an endomorphism 
\( \omega:\Lambda \longrightarrow \Lambda \) of the \(
\QQ \)-algebra \( \Lambda \) by \( \omega(h_n(\vx)) = e_n(\vx). \)

An important basis is the Schur function basis.
For a partition \( \lambda\in\Par_n \), the \emph{Schur polynomial}
\( s_\lambda(\vx_n) \) of shape \( \lambda \) is defined by
\[
  s_\lambda(\vx_n)
  = \frac{\det (x_j^{\lambda_i+n-i})_{1\le i,j\le n}}
    {\prod_{1\le i<j\le n}(x_i-x_j)}.
\]
The \emph{Schur function} \( s_\lambda(\vx) \) is defined to be the
coefficient-wise limit of \( s_\lambda(\vx_n) \) as \( n\to\infty \).
The Schur function \( s_{\lambda/\mu}(\vx) \) of shape \( \lambda/\mu \) is
defined by
\(
  s_{\lambda/\mu}(\vx) = \sum_{T\in\SSYT(\lambda/\mu)} \vx^T,
\)
where \( \vx^T=x_1^{m_1} x_2^{m_2} \cdots \) and
\( m_i \) is the number of appearances of \( i \) in \( T \).

The \emph{Hall inner product} is the inner product on \( \Lambda \) defined by
\(
  \langle s_\lambda(\vx), s_\mu(\vx) \rangle = \delta_{\lambda,\mu}.
\)
This inner product is naturally extended to \( \hat{\Lambda}\times \Lambda \)
because every element \( f\in \hat{\Lambda} \) can be written as \(
f=\sum_{\lambda\in \Par}c_\lambda s_\lambda(\vx) \).

The Jacobi--Trudi formula and its dual give expansions of Schur functions
in terms of \( h_n(\vx) \) and \( e_n(\vx) \) as determinants:
for partitions \( \mu\subseteq\lambda \),
\begin{align*}
  s_{\lambda/\mu}(\vx)
  = \det \left( h_{\lambda_i-\mu_j-i+j}(\vx) \right)_{i,j=1}^{\ell(\lambda)}
  \qand
  s_{\lambda'/\mu'}(\vx)
  = \det \left( e_{\lambda_i-\mu_j-i+j}(\vx) \right)_{i,j=1}^{\ell(\lambda)}.
\end{align*}

In this extended abstract, we consider the completion of \( \Lambda_{\QQ[\va,\vb]}=\QQ[\va,\vb]\otimes\Lambda \)
instead of \( \Lambda \).


\subsection{Plethystic substitutions}
\label{sec:plethystic}
We define plethystic substitution via the \emph{power sum symmetric functions} \( p_n(\vx) \).
For a formal power series \( Z\in\QQ[[z_1,z_2,\dots]] \),
the \emph{plethystic substitution \( p_n[Z] \) of \( Z \) into \( p_n(\vx) \) } is
defined to be the formal power series obtained from \( Z \)
by replacing each \( z_i \) by \( z_i^n \).
By definition, \( f[z_1+z_2+\cdots] = f(z_1,z_2,\dots). \)
For more properties of plethystic substitution,
we refer the reader to \cite{Loehr_2011}.

For integers \( r,s \), and \( n \), we define
\[
  X=x_1+x_2+\cdots, \qquad
  X_n=x_1+x_2+\dots+x_n, \qquad
  X_{[r,s]}=x_r+x_{r+1}+\dots+x_s,
\]
\[
  A_n = \alpha_1+\alpha_2+\dots+\alpha_n,\qquad
  B_n = \beta_1+\beta_2+\dots+\beta_n,
\]
where \( X_n=A_n=B_n=0 \) if \( n\le 0 \) and \( X_{[r,s]}=0 \) if \( r>s \).

In this extended abstract, we only consider the plethystic substitution into
\( h_n(\vx) \) and \( e_n(\vx) \).
We will use the following properties,
which follows from \cite[Theorem~6]{Loehr_2011}.
\begin{prop}\label{prop:h[-Z]}
  For \( n\ge 0 \) and \( Z\in\QQ[[z_1,z_2,\dots]] \), we have
  \[
    h_n[-Z] = (-1)^n e_n[Z] \qand e_n[-Z] = (-1)^n h_n[Z].
  \]
  For \( Z_1,Z_2\in\QQ[[z_1,z_2,\dots]] \),
  \[
    h_n[Z_1+Z_2] = \sum_{a+b=n} h_a[Z_1] h_b[Z_2] \qand
    e_n[Z_1+Z_2] = \sum_{a+b=n} e_a[Z_1] e_b[Z_2].
  \]
\end{prop}

We introduce a novel notation \( \ominus \) as follows.

\begin{defn}
  For \( n\in\ZZ \) and
  \( Z_1,Z_2\in \QQ[[z_1,z_2,\dots]] \) with no constant terms,
  \[
    h_n[Z_1\ominus Z_2] = \sum_{a-b=n} h_a[Z_1] h_b[Z_2] \qand
    e_n[Z_1\ominus Z_2] = \sum_{a-b=n} e_a[Z_1] e_b[Z_2].
  \]
\end{defn}

\section{Schur expansions, Jacobi--Trudi-like formulas, and duality}
\label{sec:schur_expansion}

In this section, we give Schur expansions and Jacobi--Trudi-like formulas for
\(G_{\lambda}(\vx;\va,\vb)\) and \(g_{\lambda}(\vx;\va,\vb)\) using the
Cauchy--Binet theorem. We also give the duality between 
\(G_{\lambda}(\vx;\va,\vb)\) and \(g_{\lambda}(\vx;\va,\vb)\) with respect to
the Hall inner product.

For \( \lambda,\mu\in\Par_n \), define
\begin{align*}
  C_{\lambda,\mu}(\va,\vb) &= \det \left( h_{\mu_i-\lambda_j-i+j}[A_{\lambda_j}-B_{j-1}] \right)_{i,j=1}^{n},  \\
  c_{\lambda,\mu}(\va,\vb) &= \det \left( h_{\lambda_i-\mu_j-i+j}[-A_{\lambda_i-1}+B_{i-1}] \right)_{i,j=1}^{n}.
\end{align*}
We expand \( G_{\lambda}(\vx_n;\va,\vb) \) and \(
g_{\lambda}(\vx_n;\va,\vb) \) into Schur functions as follows.

\begin{thm}
  \label{thm:Schur_exp G g for Par_n}
  Let \(\lambda \in \Par_n\). We have
  \begin{align*}
    G_{\lambda}(\vx_n;\va,\vb) &= \sum_{\mu \supseteq \lambda}C_{\lambda,\mu}(\va,\vb) s_{\mu}(\vx_n),  \\
    g_{\lambda}(\vx_n;\va,\vb) &= \sum_{\mu \subseteq \lambda}c_{\lambda,\mu}(\va,\vb) s_{\mu}(\vx_n).
  \end{align*}
\end{thm}

 We can apply the Cauchy--Binet theorem again to the Schur expansions in
 \cref{thm:Schur_exp G g for Par_n} to obtain Jacobi--Trudi-like formulas.

\begin{thm}\label{thm:JT_ab}
  For \( \lambda\in\Par_n \), we have
  \begin{align*}
    G_\lambda(\vx_n;\va,\vb)
    &= \det\left(  h_{\lambda_i-i+j}[X_n\ominus (A_{\lambda_i}-B_{i-1})]\right)_{i,j=1}^n,\\
    g_\lambda(\vx_n;\va,\vb)
    &= \det\left(h_{\lambda_i-i+j}[X_n-A_{\lambda_i-1}+B_{i-1}]\right)_{i,j=1}^n.
  \end{align*}
\end{thm}


Using the Schur expansions of \( G_\lambda(\vx;\va,\vb) \) and \( g_\lambda(\vx;
\va,\vb) \), we show the following duality with respect to the Hall inner product, 
which justifies the name ``refined
dual canonical stable Grothendieck polynomial'' for \( g_\lambda(\vx;\va,\vb) \).

\begin{thm}\label{thm:vec dual}
  For \( \lambda,\mu\in\Par \), we have
  \[
    \langle G_\lambda(\vx;\va,\vb), g_\mu(\vx;\va,\vb)  \rangle
  = \delta_{\lambda,\mu}.
\]
\end{thm}

\section{Combinatorial models and Schur positivity}
\label{sec:lattice}

In this section we give combinatorial models for the Schur coefficients
\(C_{\lambda,\mu}(\va,\vb) \) and \(c_{\lambda,\mu}(\va,\vb)\) of
\(G_{\lambda}(\vx;\va,\vb)\) and \(g_{\lambda}(\vx;\va,\vb)\). As a consequence we show that \(G_{\lambda}(\vx;\va,-\vb)\)
and \(g_{\lambda}(\vx;-\va,\vb)\) are Schur-positive.
In order to state our results we need the following tableaux, which are modified
versions of elegant tableaux \cite{LP2007,Lenart2000}.

\begin{defn}
  A \emph{\( \ZZ \)-elegant tableau} of shape $\lm$ is a filling $T$ of the
  cells in $\lm$ with integers such that the rows are weakly increasing, the
  columns are strictly increasing, and
  \begin{equation*}
    \min(i-\mu_i,1) \le T(i,j) < i \qquad\mbox{for all $(i,j)\in \lm$.}
  \end{equation*}
Let $\ET_{\ZZ}(\lm)$ denote the set of all $\ZZ$-elegant
  tableaux of shape $\lm$.
\end{defn}

\begin{defn}
  A \emph{\( \ZZ \)-inelegant tableau} of shape $\ml$ is a filling $T$ of the
  cells in $\ml$ with integers such that the rows are weakly decreasing, the
  columns are strictly decreasing, and
  \begin{equation*}
    \min(\mu_i-i,0) < T(i,j) \le \lambda_i \qquad\mbox{for all $(i,j)\in \ml$.}
  \end{equation*}
  Let $\IET_{\ZZ}(\ml)$ denote the set of all $\ZZ$-inelegant tableaux of shape
  $\ml$.
\end{defn}

We now give combinatorial interpretations for the Schur coefficients
\(C_{\lambda,\mu}(\va,\vb)\) and \(c_{\lambda,\mu}(\va,\vb)\) of
\(G_{\lambda}(\vx;\va,\vb)\) and \(g_{\lambda}(\vx;\va,\vb)\) in
terms of tableaux.


\begin{thm}\label{thm:C_lm&c_lm}
  Let \(\lambda,\mu \in\Par\) with \(\lambda\subseteq\mu\).
  We have 
  \begin{align*}
    C_{\lambda,\mu}(\va,\vb) &= \sum_{T\in\IET_\ZZ(\ml)}
    \prod_{(i,j)\in\ml} (\alpha_{T(i,j)} - \beta_{T(i,j)-c(i,j)}),\\
    c_{\lambda,\mu}(\va,\vb) &= \sum_{T\in\ET_\ZZ(\lm)}
    \prod_{(i,j)\in\lm} (-\alpha_{T(i,j)+c(i,j)} + \beta_{T(i,j)}),
  \end{align*}
  where \( \alpha_m=\beta_m=0 \) for \( m\le 0 \). 
  In particular, $G_\lambda(\vx;\va,-\vb)$ and $g_\lambda(\vx;-\va,\vb)$ are Schur-positive, that is,
  $C_{\lambda,\mu}(\va,-\vb)$ and $c_{\lambda,\mu}(-\va,\vb)$ are polynomials in $\va$ and $\vb$ with
  nonnegative integer coefficients. 
\end{thm}

For the proof of \cref{thm:C_lm&c_lm}, we express \(C_{\lambda,\mu}(\va,\vb) \) and
\(c_{\lambda,\mu}(\va,\vb)\) in terms of nonintersecting lattice paths using the
Lindstr\"om--Gessel--Viennot lemma. The nonintersecting lattice paths are then easily
transformed into the desired tableaux.

\section{Flagged Grothendieck polynomials}
\label{sec:flagged G}

In this section we give a combinatorial model for \( G_\lambda(\vx;\va,\vb) \) using marked
multiset-valued tableaux. 
More generally, we consider two flagged versions
of \( G_\lambda(\vx;\va,\vb) \) and extend the partition \( \lambda \) to a skew
shape.
To begin with, we introduce the following definition.

\begin{defn}
A \emph{marked multiset-valued tableau} of shape $\lm$ is a filling $T$ of $\lm$
with multisets such that
\begin{itemize}
\item $T(i,j)$ is a nonempty (finite) multiset $\{a_1\le \dots\le a_k\}$ of positive integers, in
  which each integer $a_i$ may be marked if $i\ge2$ and $a_{i-1}<a_i$, and
\item $\max(T(i,j)) \le \min(T(i,j+1))$ and $\max(T(i,j)) < \min(T(i+1,j))$
  if \( (i,j), (i,j+1)\in \lm \) and \( (i,j), (i+1,j)\in \lm \), respectively.
\end{itemize}
Let $\MMSVT(\lm)$ denote the set of marked multiset-valued tableaux of shape
$\lm$.

For $T\in\MMSVT(\lm)$, let \( \vx^{T(i,j)}=x_1^{m_1}x_2^{m_2}\cdots\), where \(
m_k \) is the total number of appearances of \( k \) and \( k^* \) in \( T(i,j) \), and let \( \unmarked(T(i,j)) \)
(resp.~\( \marked(T(i,j)) \)) denote the number of unmarked (resp.~marked)
integers in \( T(i,j) \). We define
\[
  \wt(T) =  \prod_{(i,j)\in\lm} \vx^{T(i,j)} \alpha_j^{\unmarked(T(i,j))-1}
  (-\beta_i)^{\marked(T(i,j))}.
\]
See \cref{fig:mmsvt}.
\end{defn}

\begin{figure}
    \centering
\ytableausetup{boxsize=3.3em}
\begin{ytableau}
  \none & 1,2^*,2 & 2,2,4^* \\
  1 & 3,3
\end{ytableau}
\caption{An example of \( T\in\MMSVT((3,2)/(1)) \),
  where the marked integers are indicated with \( * \).  The weight of \( T \) is given by 
  \( \wt(T)=x_1^2x_2^4x_3^2x_4\alpha_2^2\alpha_3(-\beta_1)^2 \).}
\label{fig:mmsvt}
\end{figure}

The main goal of this section is to prove the following combinatorial
interpretation for \( G_\lambda(\vx;\va,\vb) \).

\begin{thm}\label{thm:main_G_sec6}
  For a partition \( \lambda \), we have
\[
  G_\lambda(\vx;\va,\vb) = \sum_{T\in \MMSVT(\lambda)} \wt(T).
\]
\end{thm}

Our strategy for proving \cref{thm:main_G_sec6} is to introduce a flagged version which
allows us to use induction. We note that a row-flagged version with partition
shape is sufficient to prove this theorem, but for completeness we also consider a
column-flagged version and we generalize the partition shape to any skew shape.

\begin{defn}\label{defn:Grow}
  Let \( \vr=(r_1,r_2,\dots),\ \vs=(s_1,s_2,\dots)\) be in \(\ZZ_{+}^n \).
  Let $\MMSVT^{\row(\vr, \vs)}(\lm)$ denote the set of $T\in \MMSVT(\lm)$ such
  that $r_i \le \min(T(i,j))$ and $\max(T(i,j))\le s_i$ for all $(i,j)\in \lm$.
  Similarly, $\MMSVT^{\col(\vr, \vs)}(\lm)$ denotes the set of $T\in \MMSVT(\lm)$
  such that $r_j \le \min(T(i,j))$ and $\max(T(i,j))\le s_j$ for all $(i,j)\in
  \lm$. We define the \emph{row-flagged} and \emph{column-flagged
    refined canonical stable Grothendieck polynomials} by
  \begin{align*}
    G_\lm^{\row(\vr, \vs)}(\vx;\va,\vb)
    &=\sum_{T\in\MMSVT^{\row(\vr, \vs)}(\lm)} \wt(T),\\
    G_\lm^{\col(\vr, \vs)}(\vx;\va,\vb)
    &=\sum_{T\in\MMSVT^{\col(\vr, \vs)}(\lm)} \wt(T).
  \end{align*}
\end{defn}


Now we give Jacobi--Trudi-like formulas for \( G_\lm^{\row(\vr,
  \vs)}(\vx;\va,\vb) \) and \( G_\lm^{\col(\vr, \vs)}(\vx;\va,\vb) \).

\begin{thm}
  \label{thm:main_G_row}
  Let \( \lambda,\mu\in\Par_n,\vr,\vs\in\ZZ_{+}^n \) with \( \mu\subseteq\lambda \).
  If $r_i\le r_{i+1}$ and $s_i\le s_{i+1}$ whenever
  $\mu_i<\lambda_{i+1}$ for \( 1\le i\le n-1 \), then
  \begin{align}
    \label{eq:G_row}
    G_\lm^{\row(\vr, \vs)}(\vx;\va,\vb) 
    &= C \det\left(   h_{\lambda_i-\mu_j-i+j}
      [X_{[r_j,s_i]}\ominus(A_{\lambda_i}-A_{\mu_j}-B_{i-1} + B_{j})]\right)_{i,j=1}^n,
  \end{align}
where
\(
C = \prod_{i=1}^n \prod_{l=r_i}^{s_i} (1-\beta_i x_l).
\)
\end{thm}

We prove \cref{thm:main_G_row} by showing that 
the both sides of \eqref{eq:G_row}
satisfy the same recurrence relations and initial conditions.
 The idea is based on a proof of the Jacobi--Trudi-like 
 formula for flagged Schur functions due to Wachs \cite{Wachs_1985}.
 We prove the following theorem in a similar way.

\begin{thm}
  \label{thm:main_G_col}
  Let \( \lambda,\mu\in\Par_n,\vr,\vs\in\ZZ_{+}^n \) with \( \mu\subseteq\lambda \).
If \( r_i-\mu_i\le r_{i+1}-\mu_{i+1} \) and \( s_i-\lambda_i\le s_{i+1}-\lambda_{i+1}+1 \) 
whenever \( \mu_i<\lambda_{i+1} \) for \( 1\le i\le n-1 \), then  
  \begin{align*}
    G_\lmc^{\col(\vr, \vs)}(\vx;\va,\vb) 
    = D \det\left(   e_{\lambda_i-\mu_j-i+j}
      [X_{[r_j,s_i]}\ominus(A_{i-1}-A_{j}-B_{\lambda_i} + B_{\mu_j})]\right)_{i,j=1}^n,
  \end{align*}
where
\(
D = \prod_{i=1}^n \prod_{l=r_i}^{s_i} (1-\alpha_i x_l)^{-1}.
\)
\end{thm}

\section{Flagged dual Grothendieck polynomials}
\label{sec:flagged g}

In this section we give a combinatorial model for the refined dual canonical
stable Grothendieck polynomials \( g_\lambda(\vx;\va,\vb) \) using marked
reverse plane partitions. To this end we introduce a flagged version of \(
g_\lambda(\vx;\va,\vb) \) using marked reverse plane partitions and prove a
Jacobi--Trudi-like formula for this flagged version. More generally, we consider two flagged versions
of \( g_\lambda(\vx;\va,\vb) \) and extend the partition \( \lambda \) to a skew
shape.

\begin{defn}  \label{def:left MRPP}
  A \emph{left-marked reverse plane partition} (or simply \emph{marked reverse plane partition}) of shape $\lm$ is a
  reverse plane partition $T$ of shape $\lm$ in which every entry $T(i,j)$ with $T(i,j)=T(i,j+1)$
  may be marked. We denote by $\MRPP(\lm)$ the set of
  marked reverse plane partitions of shape $\lm$. For $T\in\MRPP(\lm)$, define
  \[
    \wt(T) = \prod_{(i,j)\in\lm} \wt(T(i,j)),
  \]
  where
  \begin{align*}
    \wt(T(i,j)) =
    \begin{cases}
      -\alpha_{j} & \mbox{if $T(i,j)$ is marked,}\\
      \beta_{i-1} & \mbox{if $T(i,j)$ is not marked and $T(i,j)= T(i-1,j)$,}\\
      x_{T(i,j)} & \mbox{if $T(i,j)$ is not marked and $T(i,j)\ne T(i-1,j)$.}
    \end{cases}
  \end{align*}
  Here, the equality $T(i,j)= T(i-1,j)$ means that their underlying integers are equal.
  See \cref{fig:MRPP'}.
\end{defn}

\begin{figure}
    \centering
  \ytableausetup{mathmode, boxsize=1.7em}
\ytableaushort{\none\none12{4^*}4,\none{1^*}135,\none11,{3^*}{3^*}3}
\caption{An example of \( T\in\MRPP(\lm)\), where \( \lambda=(6,5,3,3) \) and 
\( \mu=(2,1,1) \). We have the weight \( \wt(T)=x_1x_2x_3^2x_4x_5
(-\alpha_1)(-\alpha_2)^2(-\alpha_5)\beta_1\beta_2^2 \). }
  \label{fig:MRPP'}
\end{figure}

The main goal of this section is to prove the following combinatorial
interpretation for \( g_\lambda(\vx;\va,\vb) \).

\begin{thm} 
\label{thm:MRPP}
We have
\begin{equation}\label{eq:MRPP_l}
  g_\lambda(\vx;\va,\vb) = \sum_{T\in\MRPP(\lambda)} \wt(T).
\end{equation} 
\end{thm}

Similar to our approach in the previous section, in order to prove
\cref{thm:MRPP} we introduce a flagged version. For completeness we
again consider row- and column-flagged versions on skew shapes.

\begin{defn}
  Let \( \vr=(r_1,r_2,\dots)\in\ZZ_{+}^n \) and \( \vs=(s_1,s_2,\dots)\in\ZZ_{+}^n \).
  We denote by $\MRPP^{\row(\vr, \vs)}(\lm)$ (resp.~$\MRPP^{\col(\vr, \vs)}(\lm)$) the set of $T\in \MRPP(\lm)$ such
  that $r_i \le T(i,j)\le s_i$ (resp.~$r_j \le T(i,j)\le s_j$) for all $(i,j)\in \lm$.
  We define
  \emph{row-flagged} and \emph{column-flagged
   refined dual canonical stable Grothendieck polynomials} by
  \begin{align*}
    g_\lm^{\row(\vr, \vs)}(\vx;\va,\vb)
    &=\sum_{T\in\MRPP^{\row(\vr, \vs)}(\lm)} \wt(T),\\
    g_\lm^{\col(\vr, \vs)}(\vx;\va,\vb)
    &=\sum_{T\in\MRPP^{\col(\vr, \vs)}(\lm)} \wt(T).
  \end{align*}
\end{defn}

Now we give Jacobi--Trudi-like formulas
for the two flagged versions of \( g_\lambda(\vx;\va,\vb) \).

\begin{thm}
\label{thm:main_g}
Let \( \lambda,\mu\in\Par_n, \vr,\vs\in \ZZ_{+}^n \) such that \( r_i\le r_{i+1} \) and \( s_i\le s_{i+1} \)
whenever \( \mu_{i}<\lambda_{i+1} \). Then we have
\begin{align*}
  g_\lm^{\row(\vr, \vs)}(\vx;\va,\vb)
  &= \det\left(   h_{\lambda_i-\mu_j-i+j}
  [X_{[r_j,s_i]}-A_{\lambda_i-1}+A_{\mu_j}+B_{i-1} - B_{j-1}]\right)_{i,j=1}^n,\\
  g_{\lambda'/\mu'}^{\col(\vr, \vs)}(\vx;\va,\vb)
  &= \det\left(   e_{\lambda_i-\mu_j-i+j}
  [X_{[r_j,s_i]}-A_{i-1}+A_{j-1}+B_{\lambda_i-1} - B_{\mu_j}]\right)_{i,j=1}^n.
\end{align*}
\end{thm}

The proof is based on the ideas in \cite{Kim_JT22}.

\section{Schur expansions and the omega involution for skew shapes}
\label{sec:omega_involution}

When we set \( r_i = 1 \) and \( s_i = \infty \) for each \( i \) in \cref{thm:main_G_row,thm:main_G_col,thm:main_g},
we have the refined canonical stable Grothendieck polynomial
\( G_{\lambda/\mu}(\vx;\va,\vb) \) of skew shape \( \lambda/\mu \),
and its dual \( g_{\lambda/\mu}(\vx;\va,\vb)  \).

We define a \emph{generalized partition} of length \(n\) to be a sequence
  \((\lambda_1,\dots,\lambda_n)\in\ZZ^n\) satisfying \(\lambda_1\geq \dots\geq
  \lambda_n\). Denote by \( \GPar_n \) the set of generalized partitions of
  length \( n \).
We expand \( G_\lm(\vx;\va,\vb) \) and \(g_\lm(\vx;\va,\vb)\) in terms of skew Schur functions (up to some factor) using the Cauchy--Binet Theorem.

  \begin{thm} \label{thm:G=CsC}
    For \(\lambda,\mu\in \Par_n\) with \( \mu\subseteq\lambda \), we have
    \begin{align} \label{eq:G=CsC}
      G_\lm(\vx;\va,\vb)= C
      \sum_{\substack{\rho,\nu\in \GPar_n \\ \rho\subseteq\mu\subseteq\lambda\subseteq\nu}}
      C_{\lambda,\nu}(\va,\vb) s_{\nu/\rho}(\vx) C'_{\rho,\mu}(\va,\vb),
    \end{align}
    where \( C = \prod_{i=1}^{n} \prod_{l=1}^{\infty} (1-\beta_i x_l) \) and 
    \begin{align*}
      C_{\lambda,\nu}(\va,\vb) &= \det \left( h_{\nu_i-\lambda_j-i+j}[A_{\lambda_j}-B_{j-1}] \right)_{i,j=1}^n, \\
      C'_{\rho,\mu}(\va,\vb) &= \det \left( h_{\mu_i-\rho_j-i+j}[-A_{\mu_i}+B_i] \right)_{i,j=1}^n.
    \end{align*}
  \end{thm}

\begin{thm} \label{thm:g=csc}
  For \(\lambda,\mu\in \Par_n\), we have
  \begin{align*}
    g_\lm(\vx;\va,\vb) = \sum_{\substack{\rho,\nu\in \Par_n \\ \mu\subseteq\rho\subseteq\nu\subseteq\lambda}} c_{\lambda,\nu}(\va,\vb) s_{\nu/\rho}(\vx) c'_{\rho,\mu}(\va,\vb),
  \end{align*}
  where
  \begin{align*}
    c_{\lambda,\nu}(\va,\vb) &= \det\left( h_{\lambda_i-\nu_j-i+j}[-A_{\lambda_i-1}+B_{i-1}]  \right)_{i,j=1}^n ,  \\
    c'_{\rho,\mu}(\va,\vb) &= \det\left( h_{\rho_i-\mu_j-i+j}[A_{\mu_j}-B_{j-1}]  \right)_{i,j=1}^n.
  \end{align*}
\end{thm}

  Finally, we obtain the effect of  the involution \(\omega\) on \(G_\lm(\vx;\va,\vb)\) and \(g_\lm(\vx;\va,\vb)\).

  \begin{thm}\label{thm:omega_G}
    For \(\lambda,\mu\in \Par\) with \( \mu\subseteq\lambda \), we have
    \begin{align*}
      \omega(G_\lm(\vx;\va,\vb)) &= G_{\lambda'/\mu'}(\vx;-\vb,-\va),\\
      \omega(g_\lm(\vx;\va,\vb)) &= g_{\lambda'/\mu'}(\vx;-\vb,-\va).
    \end{align*}
  \end{thm}

\section*{Acknowledgments}

The authors are grateful to Travis Scrimshaw for many helpful comments and
fruitful discussions. They also thank Darij Grinberg for useful comments.



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