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\documentclass[finalversion]{FPSAC2023}
\articlenumber{22}
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%% note that the class file already loads {amsmath, amsthm, amssymb}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}

\input{preamble.tex}
\usepackage{young}
\def\SetTab{\mathsf{SVT}}
\def\ShSetTab{\mathsf{ShSVT}}
\def\ShMSetTab{\mathsf{ShMVT}}
\def\ss{/\hspace{-1mm}/}

\def\BT{\mathsf{BT}}
\def\ShBTQ{\mathsf{ShBT}_Q}
\def\ShBTP{\mathsf{ShBT}_P}

\def\ribbon{\mathsf{ribbon}}
\def\sribbon{\mathsf{ShRibbon}}
\def\cq{\mathsf{q\_corners}}
\def\cp{\mathsf{p\_corners}}

\def\tauShBT{\tau^{\mathrm{ShBT}}}

\def\wtjp{\jp^{\mathrm{comb}}}
\def\wtjq{\jq^{\mathrm{comb}}}
\def\wtgp{\gp^{\mathrm{comb}}}
\def\wtgq{\gq^{\mathrm{comb}}}

\def\cribbonsym{\mathsf{ShRibbons}_Q}
\newcommand{\cribbons}[2]{\cribbonsym(#1\ss#2)}
\newcommand{\bribbons}[2]{\mathsf{ShRibbons}_P(#1\ss#2)}
\def\RC{\mathsf{RC}}

\def\ccoribbons{\mathsf{\mathcal{C}}}
\def\bcoribbons{\mathsf{\mathcal{B}}}

\def\Aast{A}
\def\Bast{B}
\def\Cast{C}
\def\Dast{D}

\def\cp{\mathfrak{p}}
\def\cq{\mathfrak{q}}

\def\cpp{\tilde{\mathfrak{p}}}
\def\cqq{\tilde{\mathfrak{q}}}


\def\cU{\mathcal{U}}

\def\ShRib{\mathsf{ShRibComp}}

\def\PPart{\mathsf{PP}}

\def\yyyletter{\mathfrak{b}}
\def\zzzletter{\mathfrak{c}}
\newcommand{\yyy}[3]{ \yyyletter^{#1}_{#2,(#3)}}
\newcommand{\zzz}[3]{ \zzzletter^{#1}_{#2,(#3)}}

% \def\antipode{\mathrm{S}}

\newcommand{\eric}[1]{\text{\color{red} #1 $\quad$---E.}}


%% define your title in the usual way
\title{Combinatorial formulas for shifted dual stable Grothendieck polynomials}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author%[Optional shorter names]
{Joel Brewster Lewis\thanks{\href{mailto:jblewis@gwu.edu}{jblewis@gwu.edu}. Partially supported by Simons Foundation grant 634530.}\addressmark{1} \and Eric Marberg\thanks{\href{mailto:emarberg@ust.hk}{emarberg@ust.hk}.  Partially supported by Hong Kong RGC grant GRF 16306120.}\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, George Washington University \\ \addressmark{2}Department of Mathematics, Hong Kong University of Science and Technology}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{The $K$-theoretic Schur $P$- and $Q$-functions $\GP_\lambda$ and $\GQ_\lambda$ may be concretely defined 
as weight generating functions for semistandard shifted set-valued tableaux.
These symmetric functions are the shifted analogues of stable Grothendieck polynomials, and were introduced by Ikeda and Naruse 
for applications in geometry. 
Nakagawa and Naruse specified families of dual $K$-theoretic Schur $P$- and $Q$-functions 
$\gp_\lambda$ and $\gq_\lambda$ via a Cauchy identity involving  $\GP_\lambda$ and $\GQ_\lambda$.
They conjectured that the dual power series are weight generating functions for certain shifted plane partitions. 
We prove this conjecture. We also derive a related generating function formula for the images of $\gp_\lambda$ and $\gq_\lambda$
under the $\omega$ involution of the ring of symmetric functions. This confirms a conjecture of Chiu and the second author.
Using these results, we verify a conjecture of Ikeda and Naruse that the $\GQ$-functions are a basis for a ring. }

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

%% put your keywords here, or comment this out if you don't have them yet
\keywords{$K$-theoretic Schur $P$- and $Q$-functions, Cauchy identities, shifted tableaux, set-valued tableaux, plane partitions}

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

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{fpsac.bib}
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%\bibitem{GunnZinnJustin} A. Gunna and P. Zinn-Justin. ``Vertex models for canonical Grothendieck polynomials and their duals''. Preprint (2020), {\tt arXiv:2009.13172}.
% 





\begin{document}

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


\section{Introduction}

The main results of this extended abstract are explicit combinatorial generating functions
for certain families of ``dual'' power series that were originally defined indirectly by Cauchy identities.
The formulas that we establish were conjectured in \cite{ChiuMarberg,NakagawaNaruse}.
We start by giving a summary of the power series involved and
the generating functions derived. We then explain an application of our formulas 
to resolve a conjecture of Ikeda and Naruse from \cite{IkedaNaruse}. We also provide a comparison with the work of Lam and Pylyavskyy in the unshifted case \cite{LamPyl}.  For the full version of these results, with proofs, see \cite{LM2022,M2022}.

\section{Shifted set-valued generating functions}

Throughout this section, $\lambda = (\lambda_1 > \lambda_2   > \dots> \lambda_k>0)$
denotes a strict integer partition.
The \emph{shifted Young diagram}
of $\lambda$
is 
$\SD_\lambda := \{ (i,i+j-1) \in [k]\times \ZZ : 1 \leq j \leq \lambda_i\}$ where $[k] := \{1,2,\dots,k\}$.
Elements of $\SD_\lambda$ are called \emph{positions} or \emph{boxes}.
A \emph{shifted set-valued tableau} of shape $\lambda$
is a filling  $T $ of $\SD_{\lambda}$ by finite, nonempty subsets of $\tfrac{1}{2}\ZZ$.
Throughout, we let
$ i' := i - \tfrac{1}{2}$  for $i \in \ZZ$
and refer to half-integers as \emph{primed numbers}. 

Let $T_{ij}$ denote the entry assigned by  $T$ to box  $(i,j) \in \SD_{\lambda}$,
and write $(i,j) \in T$ when $(i,j) $ is in the domain of $T$.
The \emph{diagonal positions} of  $T$ are the boxes $(i,j) \in T$ with $i=j$.
%
A shifted set-valued tableau $T$ is \emph{semistandard} if all of the following conditions hold:
\begin{itemize}
\item[(S1)] its entries $T_{ij}$ are nonempty finite subsets of $\{1'<1<2'<2<\dots\}$,
\item[(S2)] $\max(T_{ij}) \leq \min(T_{i+1,j})$ and $\max(T_{ij}) \leq \min(T_{i,j+1})$ for all relevant $(i,j) \in T$,
\item[(S3)] no unprimed number appears in different boxes within the same column, and
\item[(S4)] no primed number appears in different boxes within the same row.
\end{itemize}
We draw shifted tableaux in French notation; for example, both
\[
\ytableausetup{boxsize=0.7cm,aligntableaux=center}
 \begin{ytableau}
\none & \none & 345 & \none\\
\none &  2'  & 3' & \none\\
1 & 2' & 2 & 3'3
\end{ytableau}
\quand
 \begin{ytableau}
\none & \none & 5 & \none\\
\none &  3'  & 3 & \none\\
12 & 2 & 23' & 34
\end{ytableau}
\]
are semistandard shifted set-valued tableaux of shape $(4, 2, 1)$.

Let $\beta, x_1,x_2,x_3,\dots$ be commuting indeterminates. 
Define $|T| := \sum_{(i,j) \in T} |T_{ij}|$ and ${\bf x}^T := \prod_i x_i^{a_i+b_i}$
 where $a_i$ and $b_i$ are the number of times that 
$i$ and $i'$ appear in $T$, respectively. 
Our examples above both have $|T| = 10$ and ${\bf x}^T = x_1 x_2^3 x_3^4 x_4 x_5$. 

\begin{definition}
Let $\ShSetTab_Q(\lambda)$ 
be the  set of all semistandard shifted set-valued tableaux of shape $\lambda$ and let
 $\ShSetTab_P(\lambda)$ be the subset of such  tableaux with
no \textbf{primed} numbers in any diagonal positions.  
The \emph{$K$-theoretic Schur $P$- and $Q$-functions} indexed by  $\lambda$ are the formal power series
$ \GP_{\lambda} := \sum_{T \in \ShSetTab_P(\lambda)} \beta^{|T|-|\lambda|} {\bf x}^{T}$
and
$\GQ_{\lambda} := \sum_{T \in \ShSetTab_Q(\lambda)} \beta^{|T|-|\lambda|} {\bf x}^{T}$.
 \end{definition}
 
 
The functions $\GP_\lambda$ and $\GQ_\lambda$  are sometimes defined  
by these formulas with $\beta=-1$.
This loses no generality since we can recover  
$ \GP_{\lambda} $ from $ \sum_{T \in \ShSetTab_P(\lambda)} (-1)^{|T|-|\lambda|} {\bf x}^{T}$
by substituting $x_i\mapsto  -\beta x_i$ then dividing by $\beta^{|\lambda|}$.
Similar comments apply to $\GQ_\lambda$. % and various other power series in this paper involving $\beta$.


 
Both $\GP_\lambda$ and $\GQ_\lambda$ belong to ring $\ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket  $.
If $\deg \beta=-1$  then $\GP_\lambda$ and $\GQ_\lambda$ are   homogeneous of degree $|\lambda|$,
but if $\deg \beta=0$ then the power series have unbounded degree.
Both power series are symmetric in the $x_i$ variables  
 \cite[Thm.~9.1]{IkedaNaruse}.
%
Setting $\beta=0$ turns $\GP_\lambda$ and $\GQ_\lambda$ into the classical \emph{Schur $P$- and $Q$-functions} $P_\lambda$ and $Q_\lambda$.
It follows that as
 $\lambda$ ranges over all strict partitions,
the sets  $\{ \GP_\lambda\}$ and $\{\GQ_\lambda\}$ are linearly independent.
While $Q_\lambda=2^{\ell(\lambda) }P_\lambda$,
each $\GQ_\lambda$ is a  more complicated (though still finite) $\ZZ[\beta]$-linear combination of $\GP_\mu$'s \cite[Thm.~1.1]{ChiuMarberg}.


Ikeda and Naruse introduced $\GP_\lambda$ and $\GQ_\lambda$   in \cite{IkedaNaruse}
for applications in geometry.
Specializations of these symmetric functions represent the structure sheaves of Schubert varieties in the torus-equivariant $K$-theory of the maximal isotropic Grassmannians of orthogonal and symplectic types \cite[Cor.~8.1]{IkedaNaruse}; see also  \cite{NakagawaNaruse2017,NakagawaNaruse,Naruse}. 
These power series further appear as ``stable limits'' of $K$-theory classes of
 certain orbit closures in \cite{MP2020,MP2021}.

\section{Dual functions via Cauchy identities}\label{cauchy-intro-sect}
 
 Our main results concern the following dual forms of $\GP_\lambda$ and $\GQ_\lambda$.
%In the following definition, let $\overline{x} := \frac{-x}{1+\beta x}$ for any expression $x$.

\begin{definition}
The \emph{dual $K$-theoretic Schur $P$- and $Q$-functions} $\gp_\lambda$ and $\gq_\lambda$
are the unique elements of $\ZZ[\beta]\llbracket  x_1,x_2,\dots\rrbracket  $ indexed by strict partitions $\lambda$ 
satisfying the Cauchy identities
 \be\label{cauchy-eq} \sum_\lambda \GQ_\lambda({\bf x}) \gp_\lambda({\bf y}) = \sum_\lambda \GP_\lambda({\bf x}) \gq_\lambda({\bf y}) = \prod_{i,j \geq 1} \frac{ 1 - \overline{x_i} y_j}{1-x_iy_j}
 \quad\text{where }\overline{x} := \frac{-x}{1+\beta x}
 .\ee
  \end{definition}

The power series
$\gp_{\lambda}$
and
$\gq_{\lambda}$
are special cases of Nakagawa and Naruse's \emph{dual universal factorial Schur $P$- and $Q$-functions},
which are defined via a more general
version of \eqref{cauchy-eq} \cite[Def.~3.2]{NakagawaNaruse}.
Both $\{\gp_\lambda\}$ and $\{\gq_\lambda\}$ are families  of  linearly independent functions that are symmetric in the $x_i$ variables \cite[Thm.~3.1]{NakagawaNaruse} and  
homogeneous if $\deg \beta = 1$ \cite[Prop.~6.1]{ChiuMarberg}. These properties let one define
the following conjugate symmetric functions, which were first considered in \cite{ChiuMarberg}.

  
 \begin{definition}
   Write $\omega$ for the $\ZZ[\beta]$-linear involution of the ring of symmetric functions acting on Schur functions as $\omega(s_\mu) = s_{\mu^\top}$.
 The \emph{conjugate dual $K$-theoretic Schur $P$- and $Q$-functions} of a strict partition $\lambda$ are given by
 $ \jp_\lambda:= \omega(\gp_\lambda)$ and $ \jq_\lambda:= \omega(\gq_\lambda).$
 \end{definition}
 

 






 


\section{Shifted plane partition generating functions}

Our first main result is a generating function formula for $\gp_\lambda$ and $\gq_\lambda$ 
that was predicted in \cite{NakagawaNaruse}.
A \emph{shifted plane partition} of strict partition shape $\lambda$
is
a filling 
of $\SD_{\lambda}$ by elements of $ \{1'<1<2'<2<\dots\}$ with weakly increasing rows and columns. Examples include 
\[
\ytableausetup{boxsize=0.5cm,aligntableaux=center}
\begin{ytableau}
\none &\none & \none & 3  \\
\none &\none & 2 & 3 \\
\none & 1& 2' & 2 \\
1' &1'  & 1& 1 & 5' 
\end{ytableau}
\quand
\begin{ytableau}
\none &\none & \none & 3'  \\
\none &\none & 2' & 2 \\
\none &1 & 2' & 2 \\
1  & 1 & 1&1& 5 
\end{ytableau}
\]
which both have shape $(5,3,2,1)$.
Given such a filling $T$, let
 $c_i$ be the number of distinct columns of $T$ containing $i$
and let $r_i$ be the number of distinct rows of $T$ containing  $i'$. Then define
$
|\weight_\RPP(T)| = c_1 + r_1 + c_2 + r_2 + \dots
$
and
$
\textstyle{\bf x}^{\weight_\RPP(T)} :=   \prod_{i\geq 1} x_i^{c_i + r_i}.
$
Both examples above
  have $|\weight_\RPP(T)| =9$ and ${\bf x}^{\weight_\RPP(T)} = x_1^4 x_2^3x_3x_5$.

\begin{theorem}\label{main-thm1}
Let $\MRPP_Q(\lambda)$ be the set of all
shifted plane partitions of shape $\lambda$,
and  let
$\MRPP_P(\lambda)$ 
be the subset of  such fillings
with no \textbf{unprimed} diagonal entries. Then
\[
\gp_{\lambda} = \sum_{T\in \MRPP_P(\lambda)} (-\beta)^{|\lambda| - |\weight_\RPP(T)|} {\bf x}^{\weight_\RPP(T)} 
\text{ and }
\gq_{\lambda} = \sum_{T\in \MRPP_Q(\lambda)} (-\beta)^{|\lambda| -  |\weight_\RPP(T)|} {\bf x}^{\weight_\RPP(T)} .
\]
\end{theorem}



 This result was conjectured by Nakagawa and Naruse as \cite[Conj.~5.1]{NakagawaNaruse}.  
%We will actually prove a more general formula for the skew versions of $\gp_\lambda$ and $\gq_\lambda$; see \cref{gp-gq-thm}.
These formulas make it possible to compute $\gp_\lambda$ and $\gq_\lambda$,
which is not straightforward from \eqref{cauchy-eq}.

\begin{example}\label{main-thm1-ex}
If $\lambda =(2,1)$  then $ \MRPP_P(\lambda)$ consists of 
\[ 
\begin{ytableau}
\none  &c' \\
a' & b 
\end{ytableau}\; ,
\quad
\begin{ytableau}
\none  &c' \\
a' & b' 
\end{ytableau}\; ,
\quad
\begin{ytableau}
\none & b' \\
a' & a 
\end{ytableau}\; ,
\quad
\begin{ytableau}
\none  &b' \\
a' & b'
\end{ytableau}\; ,
\quad
\begin{ytableau}
\none  & a' \\
a' & a' 
\end{ytableau}\; ,
\quand
\begin{ytableau}
\none  & b' \\
a' & a' 
\end{ytableau}
\]
for all positive integers $a<b<c$, so \cref{main-thm1} asserts that
\[
\ba
\gp_{21} &= 2\sum_{a<b<c} x_a x_b x_c + \sum_{a<b} (x_a^2x_b + x_ax_b^2) - \beta \sum_a x_a^2 - \beta \sum_{a<b} x_ax_b
 = s_{21} - \beta s_2.
\ea\]
When $\lambda =(2,1)$, adding primes to the diagonal is a weight-preserving 4-to-1 map $ \MRPP_Q(\lambda)\to \MRPP_P(\lambda)$
so \cref{main-thm1} also tells us that $\gq_{21} =4s_{21} - 4\beta s_2$.
\end{example}

\section{Shifted bar tableaux generating functions}

Our second main result is a generating function formula for $\jp_\lambda$ and $\jq_\lambda$ 
that was predicted in \cite{ChiuMarberg}.
Continue to let $\lambda$ be a strict integer partition.
%Recall that a \emph{set partition} of a collected of disjoint nonempty sets.
%A \emph{partition} of a set $S$ is a set $\Pi$ of disjoint nonempty blocks $B \subseteq S$ with $S = \bigsqcup_{B \in \Pi} B$.
Suppose $V$ is a shifted tableau\footnote{That is, a shifted set-valued tableau whose entries are all singleton sets.} of shape $\lambda$
with no unprimed entries repeated in any column and no primed entries repeated in any row.
Let $\Pi$ be a partition of the diagram $\SD_{\lambda}$
into (disjoint, nonempty) subsets of adjacent boxes containing the same entry in $V$. 
Each block of $ \Pi$ 
is a contiguous ``bar'' of positions in the same row or  column,
and we refer to the pair $T=(V,\Pi)$ as
a \emph{shifted bar tableau} of shape $\lambda$.

If $V$ is semistandard in the sense of having weakly increasing rows and columns,
then we say that $T$ is also \emph{semistandard}. 
We draw shifted bar tableaux as pictures like
\[%\label{colors-eq}
\begin{young}[15.5pt][c] 
, & ]=![cyan!75]2  & =]![cyan!75]  \ynobottom & ![pink!75]3'\ynobottom  \\ 
]=![red!75]1 & =]![red!75]   & ]=]![blue!60] 1 & ]=]![pink!75] \ynotop & ![magenta!75]3 
\end{young}
\quad\text{to represent}\quad
T=(V,\Pi) =  \(\hs \begin{young}[15.5pt][c]
 , &  2 &  2 &  3'\\
1 & 1 &   1 &   3' &   3
\end{young}
\; ,\hs \begin{young}[15.5pt][c] 
, & ]= \cdot & =] \cdot \ynobottom & \cdot\ynobottom  \\ 
]= \cdot & =]\cdot & ]=]\cdot & ]=] \cdot \ynotop &  \cdot
\end{young}\hs \).
\]
These objects are the shifted analogues of what are called \emph{valued-set tableaux} in \cite{LamPyl}.
The word ``valued-set'' is just a formal transposition of ``set-valued'';
we believe that the name ``bar tableau'' is more intuitive and descriptive.

 
 
Given a shifted bar tableau $T=(V,\Pi)$ let
$ |T| := |\Pi|$ and $ {\bf x}^T := \prod_{i\geq 1} x_i^{b_i}$
where $b_i$ is the number of blocks in $\Pi$ containing $i$ or $i'$.
In our example, $|T| = 5$ and ${\bf x}^T = x_1^2 x_2 x_3^2$. 


\begin{theorem}\label{main-thm2}
Let $\ShBTQ(\lambda)$ be the set of all semistandard shifted bar tableaux of shape $\lambda$.
Let $\ShBTP(\lambda)$
be the subset of such tableaux with no \textbf{primed} diagonal entries.
Then
\[
\ba
 \jp_{\lambda} = \sum_{T\in \ShBTP(\lambda)} (-\beta)^{|\lambda|-|T|} {\bf x}^{T} 
 \quand
\jq_{\lambda} = \sum_{T\in \ShBTQ(\lambda)} (-\beta)^{|\lambda| -|T|} {\bf x}^{T} .
\ea
\]
\end{theorem}

This result was conjectured by Chiu and the second author as \cite[Conj.~7.2]{ChiuMarberg}.
%We will actually prove a more general formula for the skew versions of $\jp_\lambda$ and $\jq_\lambda$; see Theorems~\ref{jq-thm} and \ref{jp-thm}.

 
\begin{example}
Suppose $\lambda =(2,1)$. Then $  \ShBTP(\lambda)$ consists of 
\[ 
\begin{young}[14pt][c] 
, & ]=]![cyan!75]c    \\ 
]=]![red!75]a & ]=]![yellow!75]b    \end{young}\; ,
\quad
\begin{young}[14pt][c] 
, & ]=]![cyan!75]c    \\ 
]=]![red!75]a & ]=]![yellow!75]b'    \end{young}\; ,
\quad
\begin{young}[14pt][c] 
, & ]=]![cyan!75]b    \\ 
]=]![red!75]a & ]=]![yellow!75]a    \end{young}\; ,
\quad
\begin{young}[14pt][c] 
, & ]=]![cyan!75]b    \\ 
]=]![red!75]a & ]=]![yellow!75]b'    \end{young}\; ,
\quand
\begin{young}[14pt][c] 
, & ]=]![cyan!75]b    \\ 
]=![red!75]a & =]![red!75]    \end{young}
\]
for all positive integers $a<b<c$, so \cref{main-thm2} asserts that
\[
\ba
\jp_{21} &= 2\sum_{a<b<c} x_a x_b x_c + \sum_{a<b} (x_a^2x_b + x_ax_b^2) - \beta \sum_{a<b} x_ax_b
 = s_{21} - \beta s_{11} =\omega(\gp_{21}).
\ea\]
As in \cref{main-thm1-ex}, there is a weight-preserving 4-to-1 map $ \ShBTQ(\lambda)\to \ShBTP(\lambda)$;
this is given by either removing all diagonal primes or applying the map
 \[ 
 \left\{\ \begin{young}[14pt][c] 
, & ]=]![cyan!75]b    \\ 
]=![red!75]a & =]![red!75]    \end{young}\; ,\
\begin{young}[14pt][c] 
, & ]=]![cyan!75]b'    \\ 
]=![red!75]a & =]![red!75]    \end{young}\; ,\
 \begin{young}[14pt][c] 
, & ]=]![cyan!75]b'  \ynobottom  \\ 
]=]![red!75]a & ]=]![cyan!75] \ynotop    \end{young}\; ,\
 \begin{young}[14pt][c] 
, & ]=]![cyan!75]b'  \ynobottom  \\ 
]=]![red!75]a' & ]=]![cyan!75] \ynotop    \end{young}\ \right\}
\mapsto
\left\{\ \begin{young}[14pt][c] 
, & ]=]![cyan!75]b    \\ 
]=![red!75]a & =]![red!75]    \end{young}\ \right\}.\]
Thus \cref{main-thm2} also tells us that
$\jq_{21} =4s_{21} - 4\beta s_{11}=\omega(\gq_{21})$.
\end{example}


\section{Application to conjectures of Ikeda and Naruse}\label{app-sect}



Consider the modules %of $\ZZ[\beta]\llbracket  x_1,x_2,\dots\rrbracket  $
consisting of all infinite $\ZZ[\beta]$-linear combinations of 
the functions $\{ \GP_\lambda\}$ and $\{\GQ_\lambda\}$, with $\lambda$ ranging over all strict partitions.
Ikeda and Naruse proved that these modules are both rings \cite[Props.~3.4 and 3.5]{IkedaNaruse}.
This means concretely that  $\GP_\lambda \GP_\mu$ (respectively, $\GQ_\lambda \GQ_\mu$)
always expands as a (possibly infinite) $\ZZ[\beta]$-linear combination of $\GP_\nu$'s (respectively, $\GQ_\nu$'s).

Ikeda and Naruse conjectured that these expansions are actually finite \cite[Conj.~3.1 and 3.2]{IkedaNaruse},
meaning that the \textbf{finite} linear spans 
\be\label{GPGQ-eq} \mathbf{GP} := \ZZ[\beta]\spanning\{\GP_\lambda\}
\quand 
\mathbf{GQ} := \ZZ[\beta]\spanning\{\GQ_\lambda\}
\ee
(with $\lambda$ ranging over all strict partition)
are also rings. 
The fact that $\mathbf{GP}$ is a ring follows from results of Clifford, Thomas, and Yong in \cite{CTY}; for other proofs see 
\cite[\S4]{HKRWZZ}, 
\cite[\S1.2]{M2021}, and \cite[\S8]{PechenikYong}.
%; see the discussion in \cite[Rem.~3.4]{IkedaNaruse}.
The problem of showing $\mathbf{GQ}$ is a ring appears to still be open, however.


Building off  \cite{ChiuMarberg}, 
we are able to resolve this problem.
Specifically, \cite[Cor.~7.5]{ChiuMarberg}  is exactly the assertion that the ring property for 
the $\GQ$-functions
follows from \cref{main-thm2} (or more precisely from its skew version, to be given as \cref{skew-thm}). It follows that:

\begin{theorem}\label{consequence-thm}
The $\ZZ[\beta]$-module $\mathbf{GQ}$ is a subring of $\mathbf{GP}$. Thus, each product  $\GQ_\lambda \GQ_\mu$
is a \textbf{finite} $\ZZ[\beta]$-linear combination of $\GQ_\nu$'s.
\end{theorem}

%\begin{proof}
%As noted, the ring property for the span of the $\GP$-functions follows from \cite{CTY}.
%Meanwhile,
%\cite[Cor.~7.5]{ChiuMarberg}  is exactly the assertion that the ring property for 
%the span of the $\GQ$-functions
%follows from \cref{main-thm2} (or more precisely from its skew version, which is \cref{skew-thm}).
%\end{proof}


%The reason why our generating function for $\jp_\lambda$ implies the ring property for $\ZZ[\beta]\spanning\{\GQ_\lambda\}$ in this proof
%is a bit technical. The key observation is that if $\jp_\lambda$ is nonzero when restricted to the variables $x_1,x_2,\dots,x_n$ 
%(meaning that we set $x_{n+1} = x_{n+2}=\dots=0$), then $\lambda$ cannot have more than $n$ parts. \joel{This feels like it should be part of the proof?} This claim 
%is only evident given \cref{main-thm2}, and does not apply to $\gp_\lambda$. 

%There is nothing special about the ``$p$'' and ``$Q$'' versions of our power series, as compared to ``$q$'' and ``$P$'' families.
%The proof of \cite[Cor.~7.5]{ChiuMarberg}, with minor changes, likewise shows that 
%our generating function for $\jq_\lambda$ implies the ring property for  
%$\ZZ[\beta]\spanning\{\GP_\lambda\}$.
Our proof of this theorem could be adapted to give another proof that $\mathbf{GP}$ is a ring, 
but in either case our arguments 
are nonconstructive. 
By contrast, \cite[Thm.~1.2]{CTY}
gives an explicit Littlewood--Richardson  rule for products of $\GP$-functions.
It is an open problem to find such a rule for the $\GQ$-functions,
as well as for the $\gp$- and $\gq$-functions,
which span two other subrings of symmetric functions. 



 \section{Comparison with unshifted versions}

Our main results are shifted analogues of ``classical'' theorems, which we summarize here for comparison.

Throughout this section, let $\lambda = (\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_k>0)$
be an arbitrary integer partition, which is not necessarily strict.
The \emph{(unshifted) diagram} of $\lambda$
is the set of pairs 
$\D_\lambda := \{ (i,j) \in [k]\times\ZZ : 1 \leq j \leq \lambda_i\}.$
A \emph{semistandard (unshifted) set-valued tableau} of shape $\lambda$ is defined in the same way as the analogous shifted object,
except that such a tableau is a filling 
of $\D_\lambda$  %(rather than $\SD_\lambda)$ 
by finite nonempty subsets of $\{1<2<3<\dots\}$. % rather than $\{1'<1<2'<2<\dots\}$.

\begin{definition}\label{G-def}
Let $\SetTab(\lambda)$ 
be the  set of semistandard set-valued tableaux of shape $\lambda$.
The \emph{stable Grothendieck polynomial} of  $\lambda$ is the power series
$ G_{\lambda} := \sum_{T \in \SetTab(\lambda)} \beta^{|T|-|\lambda|} {\bf x}^{T}$.
\end{definition}


\begin{definition}\label{g-def}
The \emph{dual stable Grothendieck polynomials} $g_\lambda$ are the unique 
formal power series in $\ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket  $ indexed by integer partitions $\lambda$ satisfying the Cauchy identity 
\[ \sum_\lambda G_\lambda({\bf x}) g_\lambda({\bf y})
 =
 \sum_\lambda s_\lambda({\bf x}) s_\lambda({\bf y})
  = \prod_{i,j \geq 1} \frac{ 1}{1-x_iy_j}.\]
  \end{definition}
  
   \begin{definition}
 The \emph{conjugate dual stable Grothendieck polynomial} of  $\lambda$ is
 $ j_\lambda:= \omega(g_{\lambda^\top})$.
 \end{definition}

All three families $\{ G_\lambda\}$, $\{g_\lambda\}$, and $\{ j_\lambda\}$ are linearly independent symmetric functions
which coincide with the usual Schur functions $\{ s_\lambda\}$ when $\beta=0$ \cite{Buch,LamPyl}. Our definition of $j_\lambda$ involves a transposition of indices
 compared to \cite[\S9.8]{LamPyl}; this ensures that $j_\lambda|_{\beta=0} = s_\lambda$.


We define   \emph{plane partitions} and \emph{semistandard bar tableaux}
of shape $\lambda$ in the same way as our shifted versions, except the relevant objects are fillings of $\D_\lambda$
by positive integers.
Let $\PPart(\lambda)$ be the set of
 plane partitions of shape $\lambda$. 
 Let $\BT(\lambda)$ be the set of semistandard bar tableaux of shape $\lambda$; these objects are called \emph{valued-set tableaux} in \cite[\S9]{LamPyl}.

\begin{theorem}[{Lam and Pylyavskyy \cite[\S9]{LamPyl}}]
\label{gj-thm}
For all partitions $\lambda$ it holds that
\[
g_{\lambda} = \sum_{T\in \PPart(\lambda)} (-\beta)^{|\lambda| - |\weight_\RPP(T)|} {\bf x}^{\weight_\RPP(T)}
\quand  j_{\lambda} = \sum_{T\in \BT(\lambda)} (-\beta)^{|\lambda|-|T|} {\bf x}^{T} .\]
\end{theorem}



The stable Grothendieck polynomials $G_\lambda$ were introduced in Fomin and Kirillov's paper \cite{FominKirillov}  
as certain limits of Lascoux and Sch\"utzenberger's \emph{Grothendieck polynomials} \cite{LS1982},
which are $K$-theory representatives for Schubert varieties.
Buch \cite[Thm.~3.1]{Buch} derived the set-valued tableaux generating function for $G_\lambda$
given in \cref{G-def}, and also 
proved that the stable Grothendieck polynomials are a $\ZZ[\beta]$-basis for a ring \cite[Cor.~5.5]{Buch}.
For another proof of this ring property, see \cite{Y2019}.

 Lam and Pylyavskyy defined $g_\lambda$ and $j_\lambda$ by the formulas in \cref{gj-thm} with $\beta=-1$.
 They then proved that $\{ g_\lambda\}$ is the basis for the ring of symmetric functions dual to $\{G_\lambda\}$
 via the Hall inner product \cite[Thm.~9.15]{LamPyl}, and showed that $j_\lambda = \omega(g_{\lambda^\top})$ \cite[Prop.~9.25]{LamPyl}.
For other proofs of the Cauchy identity in \cref{g-def}, see \cite{GunnZinnJustin,LascouxNaruse,Y2019}.


\section{Skew versions}



As mentioned in \cref{app-sect}, there are skew generalizations of \cref{main-thm1,main-thm2}.  
We write $\mu\subseteq \lambda$ if $\mu$ and $\lambda$ are partitions with $\mu_i \leq \lambda_i$ for all $i$.
If $\mu \subseteq\lambda$ are strict partitions then the \emph{shifted diagram} of $\lambda/\mu$ is the set difference $\SD_{\lambda/\mu} := \SD_\lambda - \SD_\mu$. 

Assume $\mu\subseteq\lambda$ are strict partitions. We define shifted set-valued tableaux, plane partitions, and bar tableaux of skew shape $\lambda/\mu$
in exactly the same way as above, only now the relevant objects are fillings of $\SD_{\lambda/\mu}$.
The definitions of all related weight statistics like the monomials ${\bf x}^T$ are also unchanged.
 Here is some more relevant notation:
\begin{itemize}[nosep]
%\item Let $\ShSetTab_Q(\lambda/\mu)$ be the set of semistandard shifted set-valued tableaux of shape $\lambda/\mu$, and
%let $\ShSetTab_P(\lambda/\mu)$ be the subset of such tableaux with no primed numbers in diagonal boxes.

\item Let $\MRPP_Q(\lambda/\mu)$ be the set of 
shifted plane partitions of shape $\lambda/\mu$,
and define 
$\MRPP_P(\lambda/\mu)$ 
 to be the subset of  such fillings
with no unprimed diagonal entries.

\item Let $\ShBTQ(\lambda/\mu)$ be the set of  semistandard shifted bar tableaux of shape $\lambda/\mu$,
and let $\ShBTP(\lambda/\mu)$
be the subset of such tableaux with no primed diagonal entries.

\end{itemize}
We define  these sets to be empty if $\mu \not\subseteq\lambda$.
These sets will index the terms in generating functions for the skew analogues of  
%$\GQ_\lambda$, $\GP_\lambda$, 
$\gq_\lambda$, $\gp_\lambda$, $\jq_\lambda$, and $\jp_\lambda$,
which we now define.

Let $y_1,y_2,\dots$ be another countable set of commuting variables,
and for each $f \in \ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket $
define 
\[f({\bf x}) := f(x_1,x_2,\dots)=f,\ f({\bf y}):=f(y_1,y_2,\dots), \text{ and }
f({\bf x},{\bf y}) := f(x_1,y_1,x_2,y_2,\dots).\]
%, that is, we replace each $x_{2i-1}$ by $x_i$ and each $x_{2i}$ by $y_i$. 
If $f$ is symmetric,
then we think of $f({\bf x},{\bf y})$ as ``$f(x_1,x_2,\dots,y_1,y_2,\dots)$''
although it is not clear how to interpret this notation for an arbitrary power series.
\begin{definition}
For strict partitions $\lambda$ and $\mu$, let 
$\gp_{\lambda/\mu}$ and $\gq_{\lambda/\mu} $ be 
the elements of $ \ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket  $   with
$ %\be\label{gp-xy-eq}
\gp_\lambda({\bf x},{\bf y}) = \sum_{\mu }\gp_\mu({\bf x}) \gp_{\lambda/\mu}({\bf y})
$ and $
\gq_\lambda({\bf x},{\bf y}) = \sum_{\mu }\gq_\mu({\bf x}) \gq_{\lambda/\mu}({\bf y})
$, %\ee
where the sums are over all strict partitions $\mu$.
\end{definition}

Both $\gp_{\lambda/\mu}$ and $\gq_{\lambda/\mu}$ are symmetric and homogeneous if $\deg \beta = 1$,
but the families of such functions are no longer linearly independent. 
These power series are defined for all strict $\lambda $ and $\mu$ but are nonzero
if and only if $\mu \subseteq \lambda$ \cite[Props.~6.4 and 6.6]{ChiuMarberg}.

\begin{definition}
For strict partitions $\lambda$ and $\mu$, let 
$\jp_{\lambda/\mu}$ and $\jq_{\lambda/\mu} $ be 
the elements of $ \ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket  $   with
$ %\be\label{jp-xy-eq}
\jp_\lambda({\bf x},{\bf y}) = \sum_{\mu }\jp_\mu({\bf x}) \jp_{\lambda/\mu}({\bf y})
$ and $
\jq_\lambda({\bf x},{\bf y}) = \sum_{\mu }\jq_\mu({\bf x}) \jq_{\lambda/\mu}({\bf y})
$, %\ee
where the sums are over all strict partitions $\mu$.
\end{definition}

Equivalently, one could define these skew analogues by the following formula:

\begin{proposition}[{\cite[Eq.~(7.4)]{ChiuMarberg}}]
It holds that $\jp_{\lambda/\mu} := \omega(\gp_{\lambda/\mu})$ and $\jq_{\lambda/\mu} := \omega(\gq_{\lambda/\mu})$.
\end{proposition}

Our main theorems extend to this skew setting as follows.

\begin{theorem}\label{skew-thm}
For all strict partitions $\lambda $ and $\mu$, one has
\[
\ba
\gp_{\lambda/\mu} &= \sum_{T\in \MRPP_P(\lambda/\mu)} (-\beta)^{|\lambda/\mu| - |\weight_\RPP(T)|} {\bf x}^{\weight_\RPP(T)} 
,\\
\gq_{\lambda/\mu} &= \sum_{T\in \MRPP_Q(\lambda/\mu)} (-\beta)^{|\lambda/\mu| -  |\weight_\RPP(T)|} {\bf x}^{\weight_\RPP(T)}, \ea
\quad
\ba
 \jp_{\lambda/\mu} &= \sum_{T\in \ShBTP(\lambda/\mu)} (-\beta)^{|\lambda/\mu|-|T|} {\bf x}^{T} 
,\\
\jq_{\lambda/\mu} &= \sum_{T\in \ShBTQ(\lambda/\mu)} (-\beta)^{|\lambda/\mu| -|T|} {\bf x}^{T} .
\ea
\]
\end{theorem}

There are also skew versions of $\GP_\lambda$ and $\GQ_\lambda$, which are symmetric by results in 
\cite{LM2021}.

\section{Proof ideas}

The most difficult part of our proofs of \cref{main-thm1,main-thm2,skew-thm}
is showing that the desired generating functions are symmetric.
We derive this explicitly for $\jp_\lambda$ and $\jq_\lambda$
by constructing Bender--Knuth involutions for semistandard shifted bar tableaux.  We briefly sketch this construction,
which may be of independent interest. 

The $i$th Bender--Knuth involution applied to a semistandard bar tableau $T$ produces a new semistandard bar tableau $U$ in which the number of bars filled with $i$ or $i'$ in $T$ is equal to the number of bars filled with $i + 1$ or $(i + 1)'$ in $U$, and vice versa.  It is enough to consider the case $i = 1$.  The involution proceeds in three steps, using an intermediate object called \emph{sorted bar tableaux}.  

Essentially, these intermediate objects appear to be semistandard bar tableaux if the order on the entries were $1' < 2' < 1 < 2$ (rather than $1' < 1 < 2' < 2$); however, some special configurations that violate this description may occur near the diagonal.  The first step, converting a semistandard shifted bar tableau to a sorted shifted bar tableau, consists of a number of moves of the form
\[
 \begin{young}[15.5pt][c]   
   ![pink!99] 2' \ynobottom \\
    ![pink!99] \ynotop \\ 
    ]=![cyan!99] 1  & ![cyan!99] & ![cyan!99]  \end{young} \mapsto 
     \begin{young}[15.5pt][c]  
        ![pink!99]2' \ynobottom\\ 
        ![pink!99]\ynobottom \ynotop \\ 
        ![pink!99] \ynotop & ]=![cyan!99]1 & =]![cyan!99]\end{young}
\qquord
 \begin{young}[15.5pt][c]    
  ![pink!99] 2' \ynobottom \\ 
  ![pink!99] \ynotop \\ 
  ![cyan!99] 1    \end{young} \mapsto  
  \begin{young}[15.5pt][c]     
  ![cyan!99]1 \\ 
  ![pink!99]2'  \ynobottom\\ 
  ![pink!99] \ynotop \end{young}
\]
and
\[
  \begin{young}[15.5pt][c] 
]=![cyan!99] 1  & =]![cyan!99] & ![pink!99] 2' \ynobottom\\ 
, & , & ![pink!99] \ynotop \ynobottom \\ 
, & , & ![pink!99] \ynotop \end{young} \mapsto  
  \begin{young}[15.5pt][c]
]=![cyan!99] 1 & ![cyan!99] & =]![cyan!99]  \\ 
, & , & ![pink!99]2' \ynobottom \\ 
, & , & ![pink!99] \ynotop\end{young}
\qquord
  \begin{young}[15.5pt][c] 
]=![cyan!99] 1  & =]![cyan!99] & ![pink!99] 2'  \end{young}
 \mapsto  
  \begin{young}[15.5pt][c] ![pink!99] 2' & ]=![cyan!99] 1& =]![cyan!99] \end{young}
\]
in an appropriate order.

In the second stage, one performs a weight-reversing involution on sorted tableaux.  This map is straightforward to describe away from the diagonal.  For example, for a collection of bars filled with $1$s and $2$s forms a configuration like the ones below, we form another two-row-group occupying the same boxes
that has a bar division between columns $j$ and $j+1$ in row $l$ (respectively, $l+1$)
 if and only if 
a bar division occurs between columns $j$ and $j+1$ in row $l+1$ (respectively, $l$)
in the starting group:
\[
 \begin{young}[15.5pt][c] 
]=![cyan!99] 2 & ==![cyan!99]  & ==![cyan!99]  & =]![cyan!99] & ]=]![pink!99]2  & ]=]![blue!60]2  \\
, &  ![lightgray!99]1 &  ]=![yellow!99]1 &  ==![yellow!99] &  ==![yellow!99] & ==![yellow!99] & =]![yellow!99]   \end{young}  \longleftrightarrow
\begin{young}[15.5pt][c]
 ]=![lightgray!99]2 & =]![lightgray!99]2 & ]=![yellow!99] 2 & ==![yellow!99] &  ==![yellow!99] & =]![yellow!99]    \\
, &  ]=![cyan!99]1 &  ==![cyan!99] &  =]![cyan!99] &  ]=]![pink!99]1 & ]=![blue!60]1 & =]![blue!60]
\end{young}\; .
\]
If, instead, a collection of bars filled with $1$s and $2$s forms a configuration like the ones below, then we form another one-row-group by swapping the number of $1$-blocks and $2$-blocks (but leaving the bar divisions untouched):
\[
\begin{young}[15.5pt][c]  ![lightgray!99]1 &  ]=![yellow!99]1 &  =]![yellow!99] &  ![cyan!99]1 & ]=![pink!99]2& ![pink!99] & =]![pink!99]   \end{young}
\longleftrightarrow
\begin{young}[15.5pt][c]  ![lightgray!99]1 &  ]=![yellow!99]2 &  =]![yellow!99] &  ![cyan!99]2 & ]=![pink!99]2& ![pink!99] & =]![pink!99]   \end{young}\; .
\]
One applies similar moves to the column bars filled with $1'$ and $2'$.    These maps obviously reverse the weight, but it is necessary to consider numerous special cases that occur near the diagonal.

Finally, in the third step, one reverses the process from the first step to produce a semistandard shifted bar tableau, whose weight is reversed from the original tableau.


Once symmetry is established, we are able to prove \cref{skew-thm} (and so in particular \cref{main-thm1,main-thm2}) by an inductive algebraic argument.  
The first step in this argument is to verify \cref{skew-thm} when $\lambda=(r)$ has a single nonzero part. This is carried out in \cite[\S7]{ChiuMarberg} by a direct computation.

For the inductive step, it turns out to be sufficient to show that both sides of the identities in
\cref{main-thm2} satisfy the same Pieri rule when multiplied by $\jq_{(r)}$.
The Cauchy identity \eqref{cauchy-eq} lets us derive
Pieri rules expanding $\jp_\lambda \jq_{(r)}$ and $\jq_\lambda \jq_{(r)}$ from
the tableau generating functions for $\GP_\lambda$ and $\GQ_\lambda$.
Our proof that the same rules apply to the desired combinatorial formulas for $\jp_\lambda$ and $\jq_\lambda$
relies on the symmetry of those generating functions: we restrict to a finite number of variables, and interpret a single variable $x$ as recording on one hand the smallest number appearing in the tableaux (corresponding to an inner shape an a skewing operation) but also (on the other hand, by symmetry) as recording the largest number appearing in the tableau (corresponding to a border strip on the outer boundary).  

\section{Hopf algebras}

The power series $\GP_\lambda$, $\GQ_\lambda$, $\gp_{\lambda/\mu}$, $\gq_{\lambda/\mu}$, $\jp_{\lambda/\mu}$
and $\jq_{\lambda/\mu}$   arise when studying 
certain Hopf algebras of symmetric functions.
We discuss these objects here for more context.

%First, some quick general background. 
Let $R$ be a commutative ring. Write $\otimes =\otimes_R$ for the corresponding tensor product.  
We have a familiar notion of an (associative, unital) \emph{$R$-algebra}: this is
an $R$-module $A$ with an $R$-linear map $\nabla: A\otimes A \to A$ (called the \emph{product})
and another $R$-linear map $\iota: R \to A$ (called the \emph{unit}) satisfying a few compatibility axioms.
%The product is often written as concatenation and 
%The unit is completely determined 
%by the \emph{unit element}
%in $A$, which is the image of $1 \in R$.

Dually, an  \emph{$R$-coalgebra} is an $R$-module with an $R$-linear map $\Delta : A \to A \otimes A$,
 (called the \emph{coproduct})
and a linear map $\varepsilon : A \to R$ (called the \emph{counit}) subject to a similar set of axioms; see \cite[\S1]{GrinbergReiner} for the complete definitions.
An $R$-algebra that is also a coalgebra is an \emph{$R$-bialgebra} if the coproduct and counit maps are algebra homomorphisms.

If $A$ is an $R$-bialgebra then the set $\End(A)$ of $R$-linear maps $f:A \to A$ is itself an $R$-algebra for the 
product
$
f \ast g := \nabla \circ (f\otimes g) \circ \Delta
$.
The unit element of this \emph{convolution algebra}
is the composition $ \iota\circ \epsilon$.
The bialgebra $A$ is a \emph{Hopf algebra} if the identity map $ A \to A$ has a
 two-sided inverse 
$\antipode : A \to A$ (called the \emph{antipode}) in $\End(A)$ relative to $\ast$. %When it exists, $\antipode$ is called the \emph{antipode} of $A$.


We specialize to the case when $R=\ZZ[\beta]$. 
The objects of interest to us are bialgebras of formal power series in
 $\ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket  $. 
In this setting, the counit  
$\varepsilon$ will always be the map setting $x_1=x_2=\dots=0$, 
and the product and unit will always be the ones inherited from $\ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket  $.
The only maps left to specify are $\Delta$ and $\antipode$ (if it exists).

For example, let $\Sym = \ZZ[\beta]\spanning\{ s_\lambda : \lambda\text{ is any partition}\}$ be the $ \ZZ[\beta]$-algebra of bounded-degree symmetric functions. %Write $\otimes =\otimes_{\ZZ[\beta]}$ for the tensor product over $\ZZ[\beta]$.
Write $c_{\lambda\mu}^\nu \in \NN$ for the Littlewood--Richardson coefficients 
such that $s_\lambda s_\mu = \sum_\nu c_{\lambda\mu}^\nu s_\nu$.
It is well-known \cite[\S2]{GrinbergReiner} that $\Sym$ is a Hopf algebra for the coproduct $\Delta$ and antipode $\antipode$ satisfying
\be\label{ss-S-eq}
\Delta(s_\nu) = \sum_{\lambda,\mu\subseteq \nu}  c_{\lambda\mu}^\nu  s_\mu \otimes s_\lambda
\quand 
\antipode(s_\nu) = (-1)^{|\nu|} s_{\nu^\top} = s_{\nu^\top}(-{\bf x}),
\ee
where in the sum $\lambda$ and $\mu$ range over all partitions% and where $\nu^\top$ denotes the transpose of a partition
.
It also holds that 
$ s_\nu({\bf x},{\bf y}) =  \sum_{\lambda,\mu}  c_{\lambda\mu}^\nu  s_\mu({\bf x})   s_\lambda({\bf y})$, so
the map $\Delta$ is the composition of 
 $f \mapsto f({\bf x},{\bf y})$
 with the isomorphism $\Sym({\bf x }) \otimes \Sym({\bf y}) \xrightarrow{\sim} \Sym\otimes \Sym$.

Define 
$
\textbf{gp} = \ZZ[\beta]\spanning\{ \gp_\lambda\}
$
and
$
\textbf{gq} = \ZZ[\beta]\spanning\{ \gq_\lambda\}
$
where $\lambda $ ranges over all strict partitions.
These are both submodules of $\Sym$, with  $\textbf{gq} \subseteq \textbf{gp}$ by results in \cite{ChiuMarberg}.
The skew and conjugate versions of $\gp_\lambda$ and $\gq_\lambda$
are motivated algebraically by the following:

\begin{theorem}\label{h1-thm}
Both $\textbf{gp} $ and $ \textbf{gq}  $ are Hopf subalgebras of $\Sym$. It holds that
\[ \Delta(\gp_\nu) = \sum_{\lambda\subseteq \nu} \gp_\lambda \otimes \gp_{\nu/\lambda}
\quand \Delta(\gq_\nu) = \sum_{\lambda\subseteq \nu} \gq_\lambda \otimes \gq_{\nu/\lambda},\]
where the sums are over strict partitions.
Also,  $\antipode(\gp_\nu) = \jp_\nu(-{\bf x})$ and  $\antipode(\gq_\nu) = \jq_\nu(-{\bf x})$.
\end{theorem}


Recall the definitions of $\textbf{GP} \subseteq \textbf{GQ}  $ from \eqref{GPGQ-eq}.
Although $\GP_\lambda $ and $\GQ_\lambda $ are infinite $\ZZ[\beta]$-linear combinations
of Schur functions,  the formula for $\Delta$ in \eqref{ss-S-eq} 
extends by ``continuous linearity'' to a well-defined map  $\textbf{GP}\to\textbf{GP}\otimes\textbf{GP}$,
and the following holds:

\begin{theorem}\label{h2-thm}
Both $\textbf{GP} $ and $ \textbf{GQ}  $ are bialgebras, but neither is a Hopf algebra.
\end{theorem}

Finally, let $\langle\cdot,\cdot\rangle : \Sym \otimes \Sym \to \ZZ[\beta]$
be the $\ZZ[\beta]$-linear form with $\langle s_\lambda,s_\mu\rangle = \delta_{\lambda\mu}$
for all partitions $\lambda$ and $\mu$. This extends to a bilinear form on 
$ \Sym\otimes  \Sym$ with $\langle f_1\otimes f_2 , g_1\otimes g_2 \rangle := \langle f_1,g_1 \rangle\langle f_2,g_2\rangle$.
The formulas \eqref{ss-S-eq} imply that $\Sym$ is self-dual with respect to this form,
in the sense that $\langle fg,h\rangle  =\langle f\otimes g,\Delta(h)\rangle $ for all $f,g,h \in \Sym$.
Analogously:

\begin{theorem}
There is a unique $\ZZ[\beta]$-bilinear form $[\cdot,\cdot]  : \textbf{gp} \otimes \textbf{GP} \to \QQ[\beta]$
such that 
\[[\gp_\lambda,\GQ_\mu] = [\gq_\lambda,\GP_\mu]  =\delta_{\lambda\mu},
\ \ 
[fg,H] =[f\otimes g,\Delta(H)],
\ \ \text{and}\ \ 
[f,HK] =[\Delta(f),H\otimes K]
\]
 for all strict partitions $\lambda$, $\mu$ and all elements $f,g \in \textbf{gp}$ and $H,K \in \textbf{GP}$.
\end{theorem}

Despite this result,  $\textbf{GP}$ and $\textbf{GQ}$ are not the duals 
of $\textbf{gq}$ and $\textbf{gp}$, because the 
restricted forms  $[\cdot,\cdot] : \textbf{gq} \otimes \textbf{GP} \to \ZZ[\beta]$
and
$[\cdot,\cdot] : \textbf{gp} \otimes \textbf{GQ} \to \ZZ[\beta]$
 are technically degenerate.
Instead, the duals of $\textbf{gq}$ and $\textbf{gp}$ are the ``completions'' 
$\widehat{\textbf{G}}\textbf{P}$ and $\widehat{\textbf{G}}\textbf{Q}$, which 
respectively
 consist of all infinite $\ZZ[\beta]$-linear combinations of $\GP$- and $\GQ$-functions.

These objects are too large to be Hopf algebras (at least, in the category of $\ZZ[\beta]$-modules),
but are examples of what are called \emph{linearly compact Hopf algebras} in \cite{LM2021,Marberg2018,M2022}.
The reason why $\textbf{GP}$ and $\textbf{GQ}$ are not Hopf algebras is that the adjoints of
 the antipodes of
$ \textbf{gq}$  and $  \textbf{gp}$  under $[\cdot,\cdot]$ only make sense as maps
$\widehat{\textbf{G}}\textbf{P} \to\widehat{\textbf{G}}\textbf{P}$ and $\widehat{\textbf{G}}\textbf{Q} \to\widehat{\textbf{G}}\textbf{Q}$.

There are still many open questions related to the 
symmetric functions in this abstract. For example,
\cref{h1-thm,h2-thm}
imply that products 
of $\gp$-/$\gq$-/$\GP$-/$\GQ$-functions respectively expand as finite $\ZZ[\beta]$-linear combinations of $\gp$-/$\gq$-/$\GP$-/$\GQ$-functions. It is expected
that the coefficients in these expansions are all actually in $\NN[\beta] \subsetneq \ZZ[\beta]$.
At present, however, this is conjectural outside the case of $\GP$-functions; in the $\GP$ case, 
combinatorial interpretations 
for the coefficients expanding $\GP_\lambda \GP_\mu$ in $\bigoplus_\nu \NN[\beta]\GP_\nu$
appear in \cite{CTY,HKRWZZ,PechenikYong}.
For a list of related open problems, see \cite[\S4.6]{M2022}.
\printbibliography



\end{document}
