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\newcommand{\mj}{\mathbf{j}^+} \newcommand{\mJ}{\widehat{\mathbf{J}}^+} \newcommand{\mJP}{\widehat{\mathbf{JP}}{}^+} \newcommand{\mJQ}{\widehat{\mathbf{JQ}}{}^+} \newcommand{\mg}{\mathbf{g}^+} \newcommand{\mgp}{\mathbf{gp}^+} \newcommand{\mgq}{\mathbf{gq}^+} \newcommand{\ytab}[1]{ \ytableausetup{boxsize = .4cm,aligntableaux=center} {\small\begin{ytableau} #1 \end{ytableau}} } \definecolor{darkred}{rgb}{0.7,0,0} % darkred color \newcommand{\defn}[1]{{\color{darkred}\emph{#1}}} % emphasis of a definition %% The title of the paper: amsart's syntax. \title %% The optionnal argument is the short version for headings. %[A \textup{(}useful?\textup{)} guide for authors] %% The mandatory argument is for the title page, summaries, headings %% if optionnal void. {Shifted combinatorial Hopf algebras from \texorpdfstring{$K$}{K}-theory} %% Authors according to amsart's syntax + distinction between Given %% and Proper names: \author[\initial{E.} Marberg]{\firstname{Eric} \lastname{Marberg}} %% Do not include any other information inside \author's argument! %% Other author data have special commands for them: \address{Department of Mathematics\\ HKUST\\ Clear Water Bay\\ Hong Kong} %% Current address, if different from institutionnal address %\curraddr{Coll\`ege Royal Henri-Le-Grand\\ %Prytan\'ee National Militaire\\ %72000 La Fl\`eche, Sarthe (France)} %% e-mail address \email{emarberg@ust.hk} %% possibly home page URL (not encouraged by journal's style) %\urladdr{https://en.wikipedia.org/wiki/Marin\_Mersenne} %% Acknowledgements are not a footnote in %% \author, but are given apart: \thanks{The author was partially supported by grant GRF 16306120 from the Hong Kong Research Grants Council.} %% If co-authors exist, add them the same way, in alaphabetical order %\author{\firstname{Joseph} \lastname{Fourier}} %\address{Universit\'e de Grenoble\\ % Institut Moi-m\^eme\\ % BP74, 38402 SMH Cedex (France)} %\email{fourier@fourier.edu.fr} % Key words and phrases: \keywords{combinatorial Hopf algebras, Malvenuto-Reutenauer Hopf algebra, peak algebra, peak quasisymmetric functions, K-theoretic symmetric functions, shifted tableaux} %% Mathematical classification (2010) %% This will not be printed but can be useful for database search \subjclass{05E05, 16T30} \setcounter{localmathalphabets}{0} \datepublished{2024-09-03} \begin{document} \newtheorem{problem}[cdrthm]{Problem} %% Abstracts must be placed before \maketitle \begin{abstract} In prior joint work with Lewis, we developed a theory of enriched set-valued $P$-partitions to construct a $K$-theoretic generalization of the Hopf algebra of peak quasisymmetric functions. Here, we situate this object in a diagram of six Hopf algebras, providing a shifted version of the diagram of $K$-theoretic combinatorial Hopf algebras studied by Lam and Pylyavskyy. This allows us to describe new $K$-theoretic analogues of the classical peak algebra. We also study the Hopf algebras generated by Ikeda and Naruse's $K$-theoretic Schur $P$- and $Q$-functions, as well as their duals. Along the way, we derive several product, coproduct, and antipode formulas and outline a number of open problems and conjectures. \end{abstract} \maketitle \section{Introduction} There is a classical diagram of Hopf algebras \be\label{classical-diagram} \begin{tikzcd} \Sym \arrow[d, dash] & \NSym \arrow[l, twoheadrightarrow] \arrow[d, dash] \arrow[r, hook] & \MPR \arrow[d, dash] \\ \Sym \arrow[r,hook] & \QSym & \MPR \arrow[l,twoheadrightarrow] \end{tikzcd} \ee in which the vertical lines are dualities, the $\hookrightarrow$ arrows are inclusions, and the $\twoheadrightarrow$ arrows are their adjoints. The self-dual object $\Sym$ on the left is the familiar Hopf algebra of bounded degree symmetric functions \cite[\S2]{GrinbergReiner}, which has an orthonormal basis given by the Schur functions $s_\lambda$. The self-dual object $\MPR$ on the right is the \defn{Malvenuto-Reutenauer Hopf algebra} of permutations from \cite{AguiarSottile,Malvenuto}. In the middle, we have the dual pair of \defn{quasisymmetric functions} $\QSym$ and \defn{noncommutative symmetric functions} $\NSym$, as described in \cite[\S5]{GrinbergReiner}. In \cite{LamPyl}, Lam and Pylyavskyy study a ``$K$-theoretic'' generalization of \eqref{classical-diagram} given by \be\label{k-diagram} \begin{tikzcd} \MSym \arrow[d, dash] & \MNSym \arrow[l, twoheadrightarrow] \arrow[d, dash] \arrow[r, hook] & \MMPR \arrow[d, dash] \\ \mSym \arrow[r,hook] & \mQSym & \mMPR \arrow[l,twoheadrightarrow] \end{tikzcd}. \ee The objects here are modules over $\ZZ[\beta]$ rather than $\ZZ$, where $\beta$ is a formal parameter. Setting $\beta=0$ turns \eqref{k-diagram} into \eqref{classical-diagram}. The objects $\mSym$ and $\mQSym$ consist of the symmetric and quasisymmetric functions over $\ZZ[\beta]$ of unbounded degree. Their duals $\MSym$ and $\MNSym$ are isomorphic to $\Sym$ and $\NSym$ but with scalars extended to $\ZZ[\beta]$. The objects $\mMPR$ and $\MMPR$, finally, are two different generalizations of the Malvenuto-Reutenauer Hopf algebra $ \MPR$. Besides the Schur functions $\{s_\lambda\}$, the Hopf algebras $\mSym$ and $\MSym$ have another pair of dual bases provided by the \defn{stable Grothendieck polynomials} $G^{(\beta)}_\lambda$ and the \defn{dual stable Grothendieck polynomials} $g^{(\beta)}_\lambda$. These power series are relevant to $K$-theory since the $G^{(\beta)}_\lambda$ functions are symmetric limits of connective $K$-theory classes of structure sheaves of Schubert varieties; see \cite{Buch2002,FominKirillov94}. The goal of this article is to investigate two shifted analogues of \eqref{k-diagram}. To motivate this, let us first discuss the shifted versions of \eqref{classical-diagram}. On one hand, we have a diagram \be\label{shifted-diagram} \begin{tikzcd} \Sym_P \arrow[d, dash] & \Peak_P \arrow[l, twoheadrightarrow] \arrow[d, dash] \arrow[r, hook] & \MPR \arrow[d, dash] \\ \Sym_Q \arrow[r,hook] & \QSymQ & \MPR \arrow[l,twoheadrightarrow] \end{tikzcd} \ee in which the vertical lines are again dualities, the $\hookrightarrow$ arrows are inclusions, and the $\twoheadrightarrow$ arrows are their adjoints. Here $\Sym_P$ and $\Sym_Q$ are the subalgebras of $\Sym$ spanned by the \defn{Schur $P$-functions} $P_\lambda$ and \defn{Schur $Q$-functions} $Q_\lambda$, which are indexed by all \defn{strict} integer partitions $\lambda$. These subalgebras are dual Hopf algebras relative to the bilinear form with $[ P_\lambda, Q_\mu] = \delta_{\lambda\mu}$, which is different from the usual Hall inner product on $\Sym$; see \cite[Appendix A]{Stembridge1997a}. In the middle column, $\QSymQ$ is the Hopf algebra of \defn{peak quasisymmetric functions} $\QSymQ$ from \cite{Stembridge1997a} while $\Peak_P$ is the \defn{peak algebra} from \cite{Nyman,Schocker}. There is another version of \eqref{shifted-diagram} in which the roles of $\Sym_P$ and $\Sym_Q$ are interchanged: \be\label{shifted-diagram2} \begin{tikzcd} \Sym_Q \arrow[d, dash] & \Peak_Q \arrow[l, twoheadrightarrow] \arrow[d, dash] \arrow[r, hook] & \MPR \arrow[d, dash] \\ \Sym_P \arrow[r,hook] & \QSymP & \MPR \arrow[l,twoheadrightarrow] \end{tikzcd}. \ee Here $\QSymP$ is a slightly larger version of $\QSymQ$ (namely, the intersection of $\QSymQ_\QQ := \QQ \otimes_\ZZ \QSymQ$ with $\QSym$ \cite[\S3]{Stembridge1997a}) while its dual $\Peak_Q$ is a Hopf subalgebra of $\Peak_P$. The diagrams \eqref{shifted-diagram} and \eqref{shifted-diagram2} coincide if we work over $\QQ$ rather than $\ZZ$. Work of Ikeda and Naruse \cite{IkedaNaruse} identifies $K$-theoretic versions $\bGP_\lambda$ and $\bGQ_\lambda$ of the classical Schur $P$- and $Q$-functions. Whereas $P_\lambda$ and $Q_\lambda$ are generating functions for \defn{(semistandard) shifted tableaux}, $\bGP_\lambda$ and $\bGQ_\lambda$ are generating functions for \defn{(semistandard) shifted set-valued tableaux} \cite[Thm.~9.1]{IkedaNaruse}. These symmetric functions represent the structure sheaves of Schubert varieties in the connective $K$-theory ring of the maximal isotropic Grassmannians of orthogonal and symplectic types \cite[Cor.~8.1]{IkedaNaruse}. Later results of Nakagawa and Naruse \cite{NN2018} construct two additional families of ``dual'' $K$-theoretic Schur $P$- and $Q$-functions $\bgp_\lambda$ and $\bgq_\lambda$. As we will explain in Section~\ref{main-sect}, these power series are $\ZZ[\beta]$-bases for two Hopf subalgebras $\MSymP$ and $\MSymQ$ of $\MSym$, whose respective duals $\mSymQ$ and $\mSymP$ are the completions of the algebras $\ZZ[\beta]\spanning\{\bGQ_\lambda\}$ and $\ZZ[\beta]\spanning\{\bGP_\lambda\}$. These four objects fit into the pair of diagrams \be\begin{aligned}\label{shk-diagram} \begin{tikzcd} \MSymP \arrow[d, dash] & \MPeakP \arrow[l, twoheadrightarrow] \arrow[d, dash] \arrow[r, hook] & \MMPR \arrow[d, dash] \\ \mSymQ \arrow[r, hook] & \mQSymQ & \mMPR \arrow[l, twoheadrightarrow] \end{tikzcd} \\[-8pt] \\ \begin{tikzcd} \MSymQ \arrow[d, dash] & \MPeakQ \arrow[l, twoheadrightarrow] \arrow[d, dash] \arrow[r, hook] & \MMPR \arrow[d, dash] \\ \mSymP \arrow[r, hook] & \mQSymP & \mMPR \arrow[l, twoheadrightarrow] \end{tikzcd} \end{aligned}\ee which specialize to \eqref{shifted-diagram} and \eqref{shifted-diagram2} when $\beta=0$, and which coincide if the scalar ring $\ZZ[\beta]$ is extended to $\QQ[\beta]$. These diagrams are the primary subject of this article. Our main results, building off related work in \cite{ChiuMarberg,LamPyl,LM2022,LM2019}, will provide distinguished bases for all of the objects shown here, explicitly identify the pairings that give the dualities indicated by the vertical lines, and describe the remaining inclusions and their adjoint surjections. We can summarize our new theorems and outline the rest of this paper as follows. One way to motivate \eqref{k-diagram} is through the perspective of \defn{combinatorial Hopf algebras} as defined in \cite{ABS}. However, some care must be taken to make this rigorous as the objects in the bottom row of \eqref{k-diagram} are certain completions of Hopf algebras rather than actual Hopf algebras. Section~\ref{prelim-sect} provides a brief survey of the technical background needed to address these issues. Section~\ref{k-sect} reviews the construction of the objects and morphisms in \eqref{k-diagram}. What we present is a very mild generalization of what is studied by Lam and Pylyavskyy, and involves a parameter $\beta$ that is implicitly set to $\beta=1$ in \cite{LamPyl}. The algebras $\mQSymP\supset \mQSymQ$ are spanned by $K$-theoretic generalizations of peak quasisymmetric functions studied previously in \cite{LM2019}. In Section~\ref{multipeak-sect} we review the definition of these algebras, and prove that $\mQSymQ$ arises of the image of a canonical morphism of combinatorial Hopf algebras $\mMPR \to \mQSym$; see Theorem~\ref{peak-tl-thm}. In Section~\ref{MPeak-sect} we construct the duals of $\mQSymP$ and $\mQSymQ$ as explicit Hopf subalgebras $\MPeakQ\subset \MNSym$ and $\MPeakP\subset \MNSym$. This gives two $K$-theoretic generalizations of the classical peak algebra. Neither appears to have been considered in previous literature. We also derive (co)product formulas for the distinguished bases of $\MPeakQ$ and $\MPeakP$, and we identify the adjoint maps $\mMPR \to \mQSymQ$ and $\mMPR \to \mQSymP$ in \eqref{shk-diagram}. Sections~\ref{shifted-sym-sect} and \ref{pq-sect2} discuss the Hopf algebras $\mSymP\supset \mSymQ$ and their duals $\MSymQ\subset \MSymP$. We prove an identity relating the pairings $\MSymQ\times \mSymP \to \ZZ[\beta]$ and $\MSymP\times \mSymQ \to \ZZ[\beta]$ to a surjective morphism $\ttheta : \mQSym \to \mQSymQ$; see Theorem~\ref{ttheta-thm}. We also identify the adjoint maps $\MPeakP \to \MSymP$ and $\MPeakQ \to \MSymQ$ in \eqref{shk-diagram}; see Theorem~\ref{last-adj-thm}. These results rely on conjectures from \cite{LM2019,NN2018} proved in \cite{ChiuMarberg,LM2022}. Section~\ref{antipode-sect} discusses antipode formulas for the objects in \eqref{shk-diagram}. Building off recent work in \cite{LM2022}, we show that the respective (finite) $\ZZ[\beta]$-linear spans of all $\bGP$- and $\bGQ$-functions are sub-bialgebras of $\mSym$ that are not Hopf algebras; see Proposition~\ref{bialgebras-thm}. Finally, Section~\ref{open-sect} provides a survey of related open problems and positivity conjectures. Computer calculations indicate that the coefficients appearing in many different expansions of the distinguished bases for the objects in \eqref{shk-diagram} are always positive. In several special cases, it is an open problem to find combinatorial interpretations of these numbers. \section{Preliminaries}\label{prelim-sect} Throughout, we write $\ZZ$ for the set of integers and let $[n] := \{i \in \ZZ : 0 w_{i+1}\}$. We write $\cC(w)$ for the composition of $\ell(w)$ with $I(\cC(w)) = \Des(w)$. The as yet unmotivated definition of $L^{(\beta)}_\alpha$ is algebraically natural in view of the following: \begin{theorem}%[{\cite{LamPyl}}] \label{tl-thm} The continuous linear map with $[w]^{(\beta)}_\m \mapsto L^{(\beta)}_{\cC(w)}$ for all $w \in \Sm_\infty$ is the unique morphism of combinatorial LC-Hopf algebras $(\mMPR,\zeta_{<}) \to (\mQSym,\zetaq)$. \end{theorem} \begin{proof} The claim that this map is a morphism of LC-bialgebras (and therefore also of LC-Hopf algebras) is equivalent to \cite[Thm.~5.11]{LamPyl}. Choose $w \in \Sm_\infty$ and set $\alpha := \cC(w)$ and $N := |\alpha| = \ell(w)$. In view of Theorem~\ref{abs-thm}, we just need to check that $\zeta_{<}([w]^{(\beta)}_\m) = \zetaq(L^{(\beta)}_{\alpha})$. As $\zetaq$ sends $x_1\mapsto t$ and $x_i\mapsto 0$ for $i>0$, we either have $\zetaq(L^{(\beta)}_{\alpha}) = t^N$ if the $N$-tuple $S = (\{1\} \preceq \{1\} \preceq \dots \preceq \{1\})$ satisfies the conditions in Definition~\ref{lbeta-def}, or else $\zetaq(L^{(\beta)}_{\alpha}) =0$. This means that $\zetaq(L^{(\beta)}_{\alpha}) = t^N = \zeta_{<}([w]^{(\beta)}_\m)$ if $I(\alpha) = \Des(w)$ is empty and otherwise $\zetaq(L^{(\beta)}_{\alpha}) = 0 = \zeta_{<}([w]^{(\beta)}_\m)$ as needed. \end{proof} \subsection{Noncommutative symmetric functions} We now review the construction from \cite{LamPyl} of the \defn{multi-noncommutative symmetric functions} $\MNSym$ in the top row of \eqref{k-diagram}. % The \defn{descent set} of a set composition $A = A_1 A_2\cdots A_m \in \SetComp_n$ is \[\Des(A) := \{ i \in [n-1]: i+1 \in A_j\text{ and }i \in A_k\text{ for any indices }j}:\mWQSym \to \ZZ[\beta]\llbracket t\rrbracket $ for the continuous linear map whose value at $w=w_1\cdots w_n \in \PWords$ is \be \zeta_>(w_1w_2\cdots w_n) := \zeta_<(w_n \cdots w_2w_1) = \begin{cases} t^n &\text{if }w_1>w_2>\dots> w_m \\ 0&\text{otherwise}.\end{cases} \ee This is an algebra morphism with $\zeta_{>}([w]^{(\beta)}_\m) = \zeta_{>}(w)$ for $w \in \Sm_\infty$. By Theorem~\ref{abs-thm} there is a unique morphism of combinatorial LC-Hopf algebras $(\mMPR,\zeta_{>}) \to (\mQSym,\zetaq)$. Although this map is different from the one in Theorem~\ref{tl-thm}, it also sends $\{ [w]^{(\beta)}_\m : w \in \Sm_\infty\}$ to the pseudobasis of multifundamental quasisymmetric functions $\{ L^{(\beta)}_\alpha\}$; see the proof of \cite[Prop.~6.3]{LM2019}. To construct something new, we consider the \defn{convolution} of the maps $\zeta_{>}$ and $\zeta_{<}$ defined by the formula $ \zeta_{>|<} := \nabla_{\ZZ[\beta]} \circ (\zeta_> \htimes \zeta_<)\circ \Delta : \mWQSym \to \ZZ\llbracket t\rrbracket . $ This is a continuous algebra morphism $ \mWQSym \to \ZZ\llbracket t\rrbracket $. If $w=w_1w_2\cdots w_n \in \PWords$ then \be \zeta_{>|<}(w)= \begin{cases} 2 t^n &\text{if $w_1>\dots > w_i <\dots < w_n$ for some $i \in [n]$} \\ t^n &\text{if $w_1>\dots > w_i = w_{i+1} < \dots < w_n$ for some $i \in [n-1]$} \\ 1 & \text{if }n=0\\ 0&\text{otherwise}.\end{cases} \ee It follows that if $w=w_1w_2\cdots w_n \in \Sm_\infty$ then \be \zeta_{>|<}([w]^{(\beta)}_\m) = \begin{cases} t^n(2+\beta t) &\text{if $w_1>\dots > w_i < \dots < w_n$ for some $i \in [n]$} \\ 1 &\text{if } n=0\\ 0&\text{otherwise}. \end{cases} \ee For any composition $\alpha\vDash n$, let $\Lambda(\alpha)$ be the unique peak composition of $n$ satisfying $ I(\Lambda(\alpha)) = \{ i \in I(\alpha) : 0 < i-1 \notin I(\alpha)\}.$ For example, one has \[\Lambda((1,2,1,1,1,3,1)) = (3,6,1).\] Then define $\ttheta : \mQSym \to \mQSymQ$ to be the continuous linear map with \be\label{ttheta-eq} \ttheta(L^{(\beta)}_\alpha) = K^{(\beta)}_{\Lambda(\alpha)} \quad\text{for all compositions $\alpha$.} \ee By \cite[Cor.~4.22]{LM2019}, this map is a surjective morphism of LC-Hopf algebras. The \defn{peak set} of a word $w=w_1w_2\cdots w_n$ is \[\PeakSet(w) := \{1 w_{i+1}\}.\] Let $\cD(w)$ be the unique peak composition $\alpha\vDash\ell(w)$ with $I(\alpha) = \PeakSet(w)$. If $w \in \Sm_\infty$ then \be\label{peakset-eq} \PeakSet(w) = \{ i \in \Des(w) : 0 < i-1 \notin \Des(w)\}\quad\text{so}\quad \Lambda(\cC(w))= \cD(w),\ee and we have $ \zeta_{>|<}([w]^{(\beta)}_\m) \neq 0$ if and only if $\PeakSet(w) = \varnothing$, in which case $\ell(\cD(w))\leq 1$. The multipeak quasisymmetric functions are motivated algebraically by this analogue of Theorem~\ref{tl-thm}: \begin{theorem}\label{peak-tl-thm} The continuous linear map with $[w]^{(\beta)}_\m \mapsto K^{(\beta)}_{\cD(w)}$ for $w \in \Sm_\infty$ is the unique morphism of combinatorial LC-Hopf algebras $(\mMPR,\zeta_{>|<}) \to (\mQSym, \zetaq)$. \end{theorem} \begin{proof} If $w \in \Sm_\infty$ then $\cD(w) = \Lambda(\cC(w))$ since \[ \ba I( \Lambda(\cC(w))) &= \{ i \in I(\cC(w)) : 0 < i-1 \notin I(\cC(w))\}\\& = \{ i \in \Des(w) : 0 < i-1 \notin \Des(w)\}= \PeakSet(w) .\ea\] Our map $\Psi : \mMPR \to \mQSym$ is thus the composition of $\Phi : (\mMPR,\zeta_{<}) \to (\mQSym, \zetaq)$ from Theorem~\ref{tl-thm} and $\ttheta : \mQSym \to \mQSymQ$, so $\Psi$ is at least a morphism of LC-Hopf algebras. It remains to check that $\zeta_{>|<} = \zetaq\circ \Psi$. For this, it suffices to show that if $\alpha\vDash N$ is a peak composition then $ \zetaq(K^{(\beta)}_\alpha) = t^{|\alpha|}(2+\beta t)$ if $I(\alpha) = \varnothing$ and $ \zetaq(K^{(\beta)}_\alpha) =0$ otherwise. Recall that $\zetaq$ corresponds to setting $x_1=t$ and $x_i=0$ for $i>1$. Thus $ \zetaq(K^{(\beta)}_\alpha) = \sum_{S} \beta^{|S|-N} t^{|S|} $ where the sum is over all weakly increasing $N$-tuples of sets $S= (S_1\preceq S_2 \preceq \cdots \preceq S_N)$ with $\varnothing\neq S_i \subseteq \{1'<1\}$, $S_i \cap S_{i+1} \subseteq \{1'\}$ for $i \in I(\alpha)$, and $S_i \cap S_{i+1} \subseteq \{1\}$ for $i \notin I(\alpha)$. If $I(\alpha) \neq \varnothing$, then $\alpha_1 \geq 2$ since $\alpha$ is a peak composition, so there are no such tuples $S$ since we must have $S_{i-1} \cap S_i \subset \{1\}$ and $S_i \cap S_{i+1} \subset \{1'\}$ for $i \in I(\alpha)$. Thus $\zetaq(K^{(\beta)}_\alpha) = 0$ if $I(\alpha) \neq \varnothing $ as claimed. % On the other hand, if $I(\alpha) = \varnothing$, then we have $S_i \cap S_{i+1} \subseteq \{1\}$ for all $i \in [N-1]$, so there are only three possibilities for $S$, given by $( \{1\}, \{1\},\dots,\{1\})$, $( \{1'\}, \{1\},\dots,\{1\})$, and $( \{1',1\}, \{1\},\dots,\{1\})$, so we have $ \zetaq(K^{(\beta)}_\alpha) = t^{|\alpha|}(2+\beta t)$ as needed. \end{proof} At this point it is useful to describe the product and coproduct in $\mMPR$ more explicitly. Recall the definition of $\leq_\m$ from Section~\ref{wqsym-sect}. Given a word $w$ and a set $S$, define $w\cap S$ to be the subword of $w$ formed by omitting all letters not in $ S$. Then, as explained in \cite[\S4]{LamPyl}, one has \be\label{mMPR-prod-eq} \nabla([w']^{(\beta)}_\m\otimes [w'']^{(\beta)}_\m) = \sum_w \beta^{\ell(w) - \ell(w')-\ell(w'')}[w]^{(\beta)}_\m \quad\text{for $w' \in \Sm_m$ and $w'' \in \Sm_n$,}\ee where the sum is over all $w \in \Sm_{m+n}$ such that $w' \leq_\m w \cap [m] $ and $w'' \uparrow m \leq_\m w \cap (m+[n])$. If we fix $w =w_1w_2\cdots w_n \in \Sm_\infty$ and define $\llbracket v\rrbracket ^{(\beta)}_\m := [\st(v)]^{(\beta)}_\m$ for an arbitrary word $v$, then \be\label{mMPR-coprod-eq} \ba \Delta([w]^{(\beta)}_\m) = &\sum_{i=0}^n \llbracket w_1\cdots w_i\rrbracket ^{(\beta)}_\m \otimes \llbracket w_{i+1}\cdots w_n\rrbracket ^{(\beta)}_\m \\&+ \beta \sum_{i=1}^n \llbracket w_1\cdots w_i\rrbracket ^{(\beta)}_\m \otimes \llbracket w_i\cdots w_n\rrbracket ^{(\beta)}_\m. \ea\ee These formulas follow directly from the definitions of $\mWQSym$ and $[w]^{(\beta)}_\m$. Using Theorem~\ref{form2-thm}, one can translate these identities by duality to product and coproduct formulas for $\MMPR$; see \cite[\S7.1]{LamPyl}. On the other hand, invoking Theorem~\ref{peak-tl-thm} leads to the following formulas for $\mQSymQ$: \begin{proposition}\label{K-prod-prop} Suppose $\alpha' $ and $\alpha'' $ are peak compositions. Choose any $w' ,w'' \in \Sm_\infty$ with $\cD(w') = \alpha'$ and $\cD(w'') =\alpha''$, and set $m=\max(\{0\} \cup w')$ and $n=\max(\{0\} \cup w'')$. Then \[ K^{(\beta)}_{\alpha'} K^{(\beta)}_{\alpha''} = \sum_w \beta^{\ell(w) - |\alpha'| - |\alpha''|} K^{(\beta)}_{\cD(w)} \] where the sum is over all $w \in \Sm_{m+n}$ with $w' \leq_\m w \cap [m]$ and $w'' \uparrow m \leq_\m w \cap (m + [n])$. % If $\alpha$ is a peak composition and $w=w_1w_2\cdots w_\ell \in\Sm_\infty$ has $\cD(w) = \alpha$, then \[ \Delta(K^{(\beta)}_{\alpha}) = \sum_{i=0}^\ell K^{(\beta)}_{\cD(w_1\cdots w_i)} \otimes K^{(\beta)}_{\cD(w_{i+1}\cdots w_\ell)} + \beta \sum_{i=1}^\ell K^{(\beta)}_{\cD(w_1\cdots w_i)} \otimes K^{(\beta)}_{\cD(w_{i}\cdots w_\ell)}. \] \end{proposition} \begin{proof} Apply the morphism in Theorem~\ref{peak-tl-thm} to both sides of \eqref{mMPR-prod-eq} and \eqref{mMPR-coprod-eq}. \end{proof} In principle one can also compute products and coproducts in the pseudobasis $\{ \bK^{(\beta)}_\alpha\}$ by combining the formulas in Proposition~\ref{K-prod-prop} with the change-of-basis identities in \eqref{417-eq}. \begin{example}\label{K-prod-ex} If $\alpha' = \alpha'' = (1)$ then $K^{(\beta)}_{\alpha'} K^{(\beta)}_{\alpha''} $ is the sum $ \sum_{w} \beta^{\ell(w) - 2} K^{(\beta)}_{\cD(w)}$ over all words $w\in \{ 12, 21, 121, 212, 1212, 2121, 12121,21212, \dots\},$ so there is an infinite product expansion \[K^{(\beta)}_{(1)} K^{(\beta)}_{(1)} = 2 K^{(\beta)}_{(2)} + \beta K^{(\beta)}_{(2,1)} + \beta K^{(\beta)}_{(3)}+ \beta ^2 K^{(\beta)}_{(2,2)}+ \beta^2 K^{(\beta)}_{(3,1)}+ \beta^3 K^{(\beta)}_{(2,2,1)}%+\beta^3 K^{(\beta)}_{(3,2)} + \cdots . \] However, there is a finite coproduct expansion \[ \ba \Delta(K^{(\beta)}_{(2)}) &= \Delta(K^{(\beta)}_{\cD(12)}) \\& = 1 \otimes K^{(\beta)}_{(2)} + K^{(\beta)}_{(1)}\otimes K^{(\beta)}_{(1)} + K^{(\beta)}_{(2)}\otimes 1 + \beta\left( K^{(\beta)}_{(1)}\otimes K^{(\beta)}_{(2)} + K^{(\beta)}_{(2)}\otimes K^{(\beta)}_{(1)}\right).\ea \] \end{example} \subsection{Multipeak noncommutative symmetric functions}\label{MPeak-sect} We now consider the duals of $\mQSymQ$ and $\mQSymP$. % Fix a set composition $A = A_1 A_2\cdots A_m \in \SetComp_n$, so that the union of the blocks of $A$ is $[n]$. Recall that $i \in [n-1]$ belongs to $\Des(A)$ if and only if the block of $A$ containing $i$ is after the block containing $i+1$. The \defn{peak set} of $A$ is $\PeakSet(A) := \{11$, where $\tpeakR_{(n,0)} := 0$. Likewise, if $2\leq i < n$ then $ \tpeakR_{(i, n-i)} +\cI = \tpeakR_{i}\tpeakR_{n-i}- \tpeakR_{n}-\tpeakR_{(i+1,n-i-1)} + \cI $. We obtain the lemma by successively expanding the right hand side of the first identity using the second. \end{proof} \begin{proposition}\label{MPeak-free-prop} The set $\left\{\tpeakR_n: n =1,3,5,\dots\right\}$ freely generates $\MPeakP$ as a $\ZZ[\beta]$-algebra. \end{proposition} \begin{proof} Suppose $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_m)$ is a composition and let \[\bXi_\alpha := \tpeakR_{\alpha_1} \tpeakR_{\alpha_2} \cdots \tpeakR_{\alpha_m}.\] We say that $\alpha$ is \defn{odd} if $\alpha_i$ is an odd integer for each $i \in [m]$. It suffices to show that the elements $\bXi_\alpha$ form a $\ZZ[\beta]$-basis for $\MPeakP$ when $\alpha$ ranges over all odd compositions. % First let $<_{\mathrm{revlex}}$ be the partial order on compositions with $\gamma <_{\mathrm{revlex}} \alpha$ if $|\gamma| < |\alpha|$ or if $|\gamma| = |\alpha|$ and $\gamma$ exceeds $\alpha$ in lexicographic order. It follows from Proposition~\ref{MPeakP-prod-prop} that if $\alpha$ is any peak composition then $\bXi_\alpha \in \tpeakR_\alpha + \ZZ[\beta]\spanning\left\{ \tpeakR_\gamma : \gamma <_{\mathrm{revlex}} \alpha\right\}$. Thus the elements $\bXi_\alpha$ form a basis for $\MPeakP$ at least when $\alpha$ ranges over all peak compositions. If $\alpha$ is a peak composition then let $\mathsf{odd}(\alpha)$ be the odd composition formed by replacing each even part $\alpha_i$ by two consecutive parts $(1,\alpha_i-1)$. For example, $\mathsf{odd}((3,6,3,4,2)) = (3,1,5,3,1,3,1,1)$. Let $<_{\mathrm{lex}}$ be the partial order on compositions with $\gamma <_{\mathrm{lex}} \alpha$ if $|\gamma| < |\alpha|$ or if $|\gamma| = |\alpha|$ and $\gamma$ precedes $\alpha$ lexicographically. It follows by induction on $\ell(\alpha)$ using Lemma~\ref{peak-free-lem} that $\bXi_{\mathsf{odd}(\alpha)} \in \bXi_\alpha + \ZZ[\beta]\spanning\left\{ \bXi_\gamma : \gamma <_{\mathrm{lex}} \alpha\right\}$. Since $\mathsf{odd}$ is a bijection from peak compositions to odd compositions, we deduce that $\left\{\bXi_\alpha : \alpha\text{ is an odd composition}\right\}$ is another $\ZZ[\beta]$-basis for $\MPeakP$ as desired. \end{proof} To describe the dual of $\mQSymP$ we need a variant of $\SM_n$. Define $\barSM_n$ to consist of the set compositions $A = A_1 A_2\cdots A_m \in \SetComp_n$ such that if $\{i,i+1\} \subseteq A_j$ for $i \in [n-1]$ and $j \in [m]$ then the union $A_1 \sqcup A_2 \sqcup \dots \sqcup A_{j}$ contains neither $i-1$ nor $i+2$. Then $\SM_n \subseteq \barSM_n$, and for $A \in \barSM_n$ it still holds that $i \in \PeakSet(A)$ if and only if $12$ or $\gamma=(2)$.} and otherwise $r=0$ (respectively, $s=0$). \end{proposition} \begin{proof} For each $x,y\in\NN$ define \[ (\alpha\mid x] := \sum_{\substack{\delta \in \{0,1\}^{m} \\ \delta_m=0}} 2^{m-1-|\delta|} \beta^{|\delta|} \tpeakR_{(\alpha_1,\dots,\alpha_{m-1}, x)-\delta} \] and \[ [y\mid \gamma) := \sum_{\substack{\delta \in \{0,1\}^{n} \\ \delta_1=0 }} 2^{n-1-|\delta|} \beta^{|\delta|} \tpeakR_{(y, \gamma_2, \dots,\gamma_n)-\delta} \] so that $\opeakR_\alpha= 2\cdot (\alpha\mid \alpha_m]+ \beta\cdot (\alpha\mid \alpha_m-1]$ and $\opeakR_\gamma= 2\cdot [\gamma_1\mid \gamma)+ \beta\cdot [\gamma_1-1\mid \gamma)$. Also let \[ \ba (\alpha\mid x \mid \gamma) &:= \sum_{\substack{\delta \in \{0,1\}^{m+n-1} \\ \delta_m=0}} 2^{m+n-2-|\delta|} \beta^{|\delta|} \tpeakR_{(\alpha_1,\dots,\alpha_{m-1}, x,\gamma_2,\dots,\gamma_n)-\delta}, \\ (\alpha\mid x \mid y \mid \gamma) &:= \sum_{\substack{\delta \in \{0,1\}^{m+n} \\ \delta_m=\delta_{m+1}=0 }} 2^{m+n-2-|\delta|} \beta^{|\delta|} \tpeakR_{(\alpha_1,\dots,\alpha_{m-1}, x, y, \gamma_2, \dots,\gamma_n)-\delta}. \ea \] Our conventions mean that these summations are zero if $x=0$ or $y=0$. By Proposition~\ref{MPeakP-prod-prop} \[ \ba (\alpha\mid x]\cdot [y\mid \gamma) = (\alpha\mid x+1 \mid y-1 \mid \gamma) &+ (\alpha\mid x + y \mid \gamma) + (\alpha\mid x \mid y \mid \gamma) \\&+ \beta\cdot (\alpha\mid x + y-1 \mid \gamma) + \beta \cdot (\alpha\mid x \mid y-1 \mid \gamma) \ea \] for any positive integers $x$ and $y$. The desired formula follows by using this identity to expand the right side of \[ \opeakR_\alpha \opeakR_\gamma= \left(2\cdot (\alpha\mid \alpha_m]+ \beta\cdot (\alpha\mid \alpha_m-1]\right) \left(2\cdot [\gamma_1\mid \gamma)+ \beta \cdot [\gamma_1-1\mid \gamma)\right) \] and then combining terms. There are a large number of terms and a few different cases to consider (according to whether $\alpha_m=1$ or $\gamma_1= 1$), but this is all straightforward algebra. \end{proof} \begin{example} It holds that {\[ \ba \opeakR_{(3,2,5,2)} \opeakR_{(4,2)} = \opeakR_{(3,2,5,3,3,2)} &+ 2\cdot\opeakR_{(3,2,5,6,2)} + \opeakR_{(3,2,5,2,4,2)} \\&+3\beta\cdot \opeakR_{(3,2,5,5,2)} +\beta \cdot\opeakR_{(3,2,5,2,3,2)} + \beta^2\cdot \opeakR_{(3,2,5,4,2)} \ea \] }while $ \opeakR_{(3,2,5,1)} \opeakR_{(4,2)} = \opeakR_{(3,2,5,2,3,2)} +2\cdot\opeakR_{(3,2,5,5,2)} +2\beta \cdot\opeakR_{(3,2,5,4,2)} . $ \end{example} As above, for $n \in \PP$ we set \be\label{opeak-Rn} \opeakR_{n}:=\opeakR_{(n)} = \begin{cases} 2\cdot \tpeakR_n + \beta\cdot \tpeakR_{n-1}&\text{if }n>1 \\ 2\cdot \tpeakR_n &\text{if }n=1 \end{cases} \quand \opeakR_0 := 1.\ee The following identities suffice to compute coproducts in $\MPeakP$ and $\MPeakQ$: \begin{proposition}\label{peak-coproduct-prop} If $n\in \PP$ then $ \Delta(\opeakR_n) = \sum_{i=0}^{n} \opeakR_i \otimes \opeakR_{n-i}$ and \[ \Delta(\tpeakR_n) = 1\otimes \tpeakR_{n} + \sum_{i=1}^{n} \tpeakR_i \otimes \opeakR_{n-i} = \tpeakR_{n}\otimes 1 + \sum_{i=0}^{n-1} \opeakR_i \otimes \tpeakR_{n-i} .\] \end{proposition} \begin{proof} Let $\alpha'$ and $\alpha''$ be peak compositions and choose $u \in \Sm_p$ and $v \in \Sm_q$ with $\cD(u) = \alpha'$ and $\cD(v) =\alpha''$. Keeping in mind the product formula in Proposition~\ref{K-prod-prop}, suppose $w \in \Sm_{p+q}$ has $u \leq_\m w \cap [p]$ and $v \uparrow p \leq_\m w \cap (p + [q])$. Write $m:=\ell(v) = |\alpha''|$. The only way that we can have $\PeakSet(w) = \varnothing$ is if $\PeakSet(u)=\PeakSet(v) = \varnothing$ and $w$ is either \[\tilde v_1 \cdots \tilde v_i \cdot u\cdot \tilde v_{i+1} \cdots \tilde v_m \quord \tilde v_1 \cdots \tilde v_{i-1} \cdot u\cdot \tilde v_{i} \cdots \tilde v_m \quord \tilde v_1 \cdots \tilde v_i \cdot u\cdot \tilde v_{i} \cdots \tilde v_m\] where $i \in [m]$ is the index of the smallest letter of $v$ and $\tilde v_j := v_j+p$. Thus by Proposition~\ref{K-prod-prop} we have \[ [\Delta(\tpeakR_n), K^{(\beta)}_{\alpha'} \otimes K^{(\beta)}_{\alpha''}] = [\tpeakR_n, K^{(\beta)}_{\alpha'} K^{(\beta)}_{\alpha''} ] \in \{0,1,2,\beta\}.\] Specifically, the value of the form is zero if $\ell(\alpha') > 1$ or $\ell(\alpha'')>1$ as then $\PeakSet(u)$ or $ \PeakSet(v)$ is nonempty. If this does not occur, then the value of the form is $\ell(\alpha') + \ell(\alpha'')$ when $n= |\alpha'| + |\alpha''|$, or $\beta$ when $\ell(\alpha') = \ell(\alpha'')=1$ and $n = |\alpha'| + |\alpha''| + 1$, or else zero. We conclude by the definition of $ [\cdot,\cdot ]$ that \[ \Delta(\tpeakR_n) = 1\otimes \tpeakR_{n} + \tpeakR_{n}\otimes 1 + 2\sum_{i=1}^{n-1} \tpeakR_i \otimes \tpeakR_{n-i} + \beta \sum_{i=1}^{n-2} \tpeakR_i \otimes \tpeakR_{n-i-1}.\] This identity is equivalent to the displayed equation for $ \Delta(\tpeakR_n)$ via \eqref{opeak-Rn}. It follows that \[\Delta(\opeakR_1) = 2\cdot \Delta(\tpeakR_1) = 1\otimes \opeakR_{n} + \opeakR_{n}\otimes 1 \] and \[ \Delta(\opeakR_n) =2\cdot \Delta( \tpeakR_n) + \beta\cdot \Delta(\tpeakR_{n-1}) \] when $n>1$. The expression on the right expands to \[\ba 2\otimes \tpeakR_{n} &+ \tpeakR_{n}\otimes 2 + \beta\otimes \tpeakR_{n-1} + \tpeakR_{n-1}\otimes \beta \\&+ 2\sum_{i=1}^{n-1} \tpeakR_i \otimes \opeakR_{n-i} + \beta\sum_{i=1}^{n-2} \tpeakR_i \otimes \opeakR_{n-i - 1}\ea \] and one can check that this is equal to $ \sum_{i=0}^{n} \opeakR_i \otimes \opeakR_{n-i} $. \end{proof} There is no $Q$-version of Proposition~\ref{MPeak-free-prop}. Over $\ZZ[\beta]$, the set $\{\opeakR_n: n=1,3,5,\dots\}$ generates a proper subalgebra of $\MPeakQ$ which contains $ 2\cdot \opeakR_n$ but not $\opeakR_n$ for even $n \in \PP$. This set does freely generate $\QQ[\beta]\otimes_{\ZZ[\beta]}\MPeakP=\QQ[\beta]\otimes_{\ZZ[\beta]}\MPeakQ$ as a $\QQ[\beta]$-algebra. The classical peak algebra is also freely generated by a countable set \cite[Thm.~3]{Schocker}, so after an appropriate extension scalars it is isomorphic to \[\QQ[\beta]\otimes_{\ZZ[\beta]}\MPeakP=\QQ[\beta]\otimes_{\ZZ[\beta]}\MPeakQ\] as an algebra. By duality, the tensor product $\QQ[\beta]\otimes_{\ZZ[\beta]}\mQSymQ= \QQ[\beta]\otimes_{\ZZ[\beta]}\mQSymP$ is isomorphic as a coalgebra to the completion of the peak quasisymmetric functions $\mathbf{\Pi}_\QQ$ from \cite{BMSW,Stembridge1997a}. The (co)algebra isomorphisms that come from these observations do not extend to isomorphisms of Hopf algebras. This is different from the unshifted case \eqref{k-diagram}, where both $ L^{(\beta)}_{\alpha}$ and $ L_\alpha := L^{(0)}_{\alpha}$ span $\mQSym$, and we have $\MNSym \cong \NSym$ as Hopf algebras (after an appropriate extension of scalars) \cite[Prop.~8.5]{LamPyl}. In principle, our shifted objects might still be isomorphic to their $\beta=0$ specializations by some other maps; determining whether such maps exist is an open problem. \subsection{Shifted symmetric functions}\label{shifted-sym-sect} Finally, we turn to the shifted analogues of symmetric functions that arise in $K$-theory. Recall that a partition is strict if its nonzero parts are all distinct. Choose strict partitions $\mu \subseteq \lambda$ and write \[ \SD_{\lambda/\mu} := \{(i,i+j-1) \in \PP\times \PP : \mu_i< j \leq \lambda_i\}\] for the \defn{shifted Young diagram}. We often refer to the positions in this diagram as \defn{boxes}. A \defn{shifted set-valued tableau} of shape $\lambda/\mu$ is a map $T $ assigning nonempty finite subsets of $\{1'<1<2'<2<\dots\}$ to the boxes in $\SD_{\lambda/\mu}$. We write $(i,j) \in T$ when $(i,j) \in \SD_{\lambda/\mu}$ and let $T_{ij}$ denote the set assigned by $T$ to box $(i,j)$. A shifted set-valued tableau has \defn{weakly increasing rows and columns} if $ \max(T_{ij}) \leq \min(T_{i+1,j}) $ and $\max(T_{ij}) \leq \min(T_{i,j+1})$ for all relevant $(i,j)\in T$. A shifted set-valued tableau $T$ with this property is \defn{semistandard} if no primed number occurs in multiple boxes of $T$ in the same row and no unprimed number occurs in multiple boxes of $T$ in the same column. % For example, \[ \ytableausetup{boxsize=0.65cm,aligntableaux=center} \begin{ytableau} \none & \none & 345 & \none\\ \none & 12 & 23' & 7 & \none\\ \none[\cdot] & \none[\cdot] & 1' & 2'3 \end{ytableau} \quand \begin{ytableau} \none & \none & 3'4 & \none\\ \none & 2' 2 & 3' & 7'& \none\\ \none[\cdot] & \none[\cdot] & 1'1& 235 \end{ytableau} \] are semistandard shifted set-valued tableaux of shape $(4,3,1)/(2)$ drawn in French notation. Given such a tableau $T$, we let \[ |T| := \sum_{(i,j) \in T} |T_{ij}| \quand x^T := \prod_{(i,j) \in T} \prod_{k \in T_{ij}} x_{\lceil k\rceil }.\] Our examples have $|T| = 11$ and $x^T = x_1^2 x_2^3 x_3^3 x_4x_5 x_7$. Also set $|\lambda/\mu| := |\SD_{\lambda/\mu|}$. The following definitions originate in work of Ikeda and Naruse \cite[\S9]{IkedaNaruse}: \begin{definition} Let $\ShSetTab_Q(\lambda/\mu)$ denote the set of all 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. The \defn{$K$-theoretic Schur $P$- and $Q$-functions} are the formal power series \[ \bGP_{\lambda/\mu} := \sum_{T \in \ShSetTab_P(\lambda/\mu)} \beta^{|T|-|\lambda/\mu|} x^T \quand \bGQ_{\lambda/\mu} := \sum_{T\in \ShSetTab_Q(\lambda/\mu)} \beta^{|T|-|\lambda/\mu|} x^T. \] When $\mu=\emptyset$ is the empty partition let $\bGP_{\lambda} := \bGP_{\lambda/\emptyset}$ and $\bGQ_{\lambda} := \bGQ_{\lambda/\emptyset}.$ \end{definition} If $\deg(\beta) =0$ and $ \deg(x_i)=1$, then $\bGP_{\lambda/\mu}$ and $\bGQ_{\lambda/\mu}$ have unbounded degree, but their lowest degree terms are the Schur $P$- and Schur $Q$-functions $P_{\lambda/\mu}$ and $Q_{\lambda/\mu} = 2^{\ell(\lambda)-\ell(\mu)} P_{\lambda/\mu}$. As $\{P_\lambda\}$ and $\{Q_\lambda\}$ are bases for subalgebras of $\MSym$, the sets $\{\bGP_\lambda\}$ and $\{\bGQ_\lambda\}$ are linearly independent. \begin{definition} Define $\mSymP $ and $\mSymQ $ to be the linearly compact $\ZZ[\beta]$-modules with the sets $\{\bGP_\lambda\}$ and $\{\bGQ_\lambda\}$ (where $\lambda$ ranges over all strict partitions) as respective pseudobases. \end{definition} If one sets $\deg(\beta) =-1$ and $ \deg(x_i)=1$ then $\bGP_{\lambda/\mu}$ and $\bGQ_{\lambda/\mu}$ are homogeneous of degree $|\lambda/\mu|$. Both $\mSymP $ and $\mSymQ $ are LC-Hopf subalgebras of $\mSym$ \cite[Thm.~5.11]{LM2019} and if $\mu \subseteq \lambda$ are strict partitions then $\bGP_{\lambda/\mu}\in \mSymP$ and $\bGQ_{\lambda/\mu} \in \mSymQ$ \cite[Cor.~5.13]{LM2019}. It remains to identify the dual Hopf algebras $\MSymP $ and $\MSymQ $ in \eqref{shk-diagram}. Continue to assume $\mu \subseteq \lambda$ are strict partitions. A \defn{shifted reverse plane partition} of shape $\lambda/\mu$ is an assignment of numbers from $\{1'<1<2'<2<\dots\}$ to the boxes in $\SD_{\lambda/\mu}$ such that rows and columns are weakly increasing. If $T$ is a shifted reverse plane partition, then we let \[\wtmrpp(T) := (a_1+b_1, a_2+b_2,\dots) \quand |\wtmrpp(T) | := a_1+b_1+ a_2+b_2 + \dots\] where $a_i$ is the number of distinct columns of $T$ containing $i$ and $b_i$ is the number of distinct rows of $T$ containing $i'$. For example, if $\lambda = (4,3,2)$ and $\mu=(2)$ then $T$ could be either of \[ \ytableausetup{boxsize=0.45cm,aligntableaux=center} \begin{ytableau} \none & \none & 5' & 5'\\ \none & 3 & 3 & 5'\\ \none[\cdot] & \none[\cdot] & 1' & 1' \end{ytableau} \quand \begin{ytableau} \none & \none & 5 & 5\\ \none & 3' & 3' & 3\\ \none[\cdot] & \none[\cdot] & 1 & 3 \end{ytableau} \] and we would have $\wtmrpp(T) = (1,0,2,0,2)$ and $|\wtmrpp(T)| = 5$ in both cases. \begin{definition}\label{bgpq-def} Let $\MRPP_Q(\lambda/\mu)$ be the set of shifted reverse plane partitions of shape $\lambda/\mu$, and let $\MRPP_P(\lambda/\mu)$ be the subset of $T \in \MRPP_Q(\lambda/\mu)$ whose diagonal entries are all primed. The \defn{dual $K$-theoretic Schur $P$- and $Q$-functions} are \[ \bgp_{\lambda/\mu} := \sum_{T\in \MRPP_P(\lambda/\mu)} (-\beta)^{|\lambda/\mu| - |\wtmrpp(T)|} x^{\wtmrpp(T)} \] and \[ \bgq_{\lambda/\mu} := \sum_{T\in \MRPP_Q(\lambda/\mu)} (-\beta)^{|\lambda/\mu| - |\wtmrpp(T)|} x^{\wtmrpp(T)} . \] When $\mu=\emptyset$ is the empty partition we write $\bgp_{\lambda} := \bgp_{\lambda/\emptyset}$ and $\bgq_{\lambda} := \bgq_{\lambda/\emptyset}.$ We will adopt a similar convention for all later notations indexed by skew shapes $\lambda/\mu$. \end{definition} By \cite[Thm.~1.4]{LM2022}, $\bgp_{\lambda}$ and $\bgq_{\lambda}$ are special cases of the \defn{dual universal factorial Schur $P$- and $Q$-functions} that Nakagawa and Naruse characterize in \cite[Def.~3.2]{NN2018} via a general Cauchy identity. The skew versions $\bgp_{\lambda/\mu} $ and $\bgq_{\lambda/\mu} $ of these functions were first considered in \cite[\S6]{ChiuMarberg}. If we set $\deg(\beta) = \deg(x_i)=1$, then $\bgp_{\lambda/\mu}$ and $\bgq_{\lambda/\mu}$ are both homogeneous of (bounded) degree $|\lambda/\mu|$. If instead $\deg(\beta) = 0$ and $\deg(x_i)=1$, then the terms of highest degree in $\bgp_{\lambda/\mu}$ and $\bgq_{\lambda/\mu}$ are $P_{\lambda/\mu}$ and $Q_{\lambda/\mu} $, so $\{\bgp_\lambda\}$ and $\{\bgq_\lambda\}$ are linearly independent over $\ZZ[\beta]$. \begin{definition} Define $\MSymP $ and $\MSymQ $ to be the free $\ZZ[\beta]$-modules with the sets $\{\bgp_\lambda\}$ and $\{\bgq_\lambda\}$ (where $\lambda$ ranges over all strict partitions) as respective bases. \end{definition} The functions $\bgp_{\lambda/\mu}$ and $\bgq_{\lambda/\mu}$ satisfy the following Cauchy identity. Let $\mathbf x = (x_1,x_2,\dots)$ and $\mathbf y = (y_1,y_2,\dots)$ be commuting variables and set $\overline{x}_i = \frac{-x_i}{1+\beta x_i}$. Then \be\label{cauchy-eq} \ds\sum_{\lambda} \bGP_{\lambda}(\mathbf x) \bgq_{\lambda}(\mathbf y) = \sum_{\lambda} \bGQ_{\lambda}(\mathbf x) \bgp_{\lambda}(\mathbf y)= \prod_{i,j\geq 1} \tfrac{1-\overline{x}_i y_j}{1-x_iy_j} \ee by \cite[Thm.~1.4]{LM2022} via \cite[Conj. 5.1]{NN2018}.\footnote{\cite[Conj. 5.1]{NN2018} asserts that the power series $\bgq_{\lambda}$ and $ \bgp_{\lambda}$ \emph{defined} by \eqref{cauchy-eq} have the generating function formulas in Definition~\ref{bgpq-def}, and \cite[Thm.~1.4]{LM2022} proves this conjecture.} % Since the right hand side of \eqref{cauchy-eq} is invariant under permutations of the $\bf y$ variables, the power series $\bgp_{\lambda} $ and $ \bgq_{\lambda}$ are symmetric. This implies that \be\label{gp-gq-xy-eq} \ba \bgp_{\lambda} ({\bf x},{\bf y}) &= \sum_{\mu} \bgp_\mu({\bf x}) \bgp_{\lambda/ \mu} ({\bf y})&&\text{and} \\ \bgq_{\lambda} ({\bf x},{\bf y}) &= \sum_{\mu} \bgq_\mu({\bf x}) \bgq_{\lambda/ \mu} ({\bf y}), \ea \ee where $f({\bf x},{\bf y})$ is the power series $f(x_1,y_1,x_2,y_2,\dots)$ for $f \in \ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket $. The sums here are over all strict partitions $\mu$, setting $\bgp_{\lambda/\mu} = \bgq_{\lambda/\mu}=0$ when $\mu\not\subseteq \lambda$. Both sides of \eqref{gp-gq-xy-eq} are symmetric under all permutations of the $\bf y$ variables, so $\bgp_{\lambda/ \mu} $ and $\bgq_{\lambda/ \mu} $ are also symmetric. \begin{remark}\label{coproduct-rmk} The continuous $\ZZ[\beta]$-linear map with $f({\bf x})g({\bf y}) \mapsto f \otimes g$ for all power series $f,g \in \ZZ[x_1,x_2,\dots]$ is a bijection \[\ZZ[\beta]\llbracket x_1,y_1,x_2,y_2,\dots\rrbracket \xrightarrow{\sim} \ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket \htimes \ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket .\] Composing this map with $f \mapsto f({\bf x},{\bf y})$ gives an operation \[\ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket \to \ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket \htimes \ZZ[\beta]\llbracket x_1,x_2,\dots\rrbracket \] which restricts to the coproduct of $\mSym$ and $\MSym$ \cite[\S2.1]{GrinbergReiner}. Thus we can rewrite the formula \eqref{gp-gq-xy-eq} as \be\label{gp-gq-xy-eq2} \Delta(\bgp_{\lambda}) = \sum_{\mu} \bgp_\mu \otimes \bgp_{\lambda/ \mu} \quand \Delta(\bgq_{\lambda}) = \sum_{\mu} \bgq_\mu \otimes \bgq_{\lambda/ \mu} \ee where the sums are over all strict partitions. \end{remark} \subsection{Finite expansions and duality}\label{pq-sect2} To identify the algebraic structure of $\MSymP $ and $\MSymQ $ we need a short digression. Suppose $\lambda\subseteq \nu$ are strict partitions. A box $(i,j) \in \SD_\lambda$ is a \defn{removable corner} of $\lambda$ if $\SD_\lambda - \{(i,j)\} = \SD_\mu$ for a strict partition $\mu \subsetneq \lambda$. Let $\RC(\lambda)$ be the set of all such boxes and define \be\label{ss-eq} \bGP_{\nu\sss \lambda} := \sum_{\mu} \beta^{|\lambda|-|\mu|} \bGP_{\nu/\mu} \quand \bGQ_{\nu\sss \lambda} := \sum_{\mu} \beta^{|\lambda|-|\mu|}\bGQ_{\nu/\mu} \ee where both sums are over all strict partitions $\mu \subseteq \lambda$ with $ \SD_{\lambda/\mu}\subseteq \RC(\lambda)$. For strict partitions $\lambda\not\subseteq\nu$ set $\bGP_{\nu\sss \lambda} = \bGQ_{\nu\sss \lambda} :=0$. These functions arise in the identities \be \ba \bGP_\nu({\bf x},{\bf y}) &= \sum_{\lambda} \bGP_\lambda({\bf x}) \bGP_{\nu\sss \lambda} ({\bf y}), \\ \bGQ_\nu({\bf x},{\bf y}) &= \sum_{\lambda} \bGQ_\lambda({\bf x}) \bGQ_{\nu\sss \lambda} ({\bf y}), \ea \ee which by Remark~\ref{coproduct-rmk} can be restated as the coproduct formulas \be \label{Delta-eq} \Delta(\bGP_\nu) = \sum_{\lambda} \bGP_\lambda \otimes \bGP_{\nu\sss \lambda} \quand \Delta(\bGQ_\nu) = \sum_{\lambda} \bGQ_\lambda\otimes \bGQ_{\nu\sss \lambda}. \ee The sums here are over all strict partitions $\lambda$, but the terms indexed by $\lambda\not\subseteq \nu$ are all zero. Because $\mSymP $ and $\mSymQ $ are LC-Hopf algebras, and because the pseudobases $\{\bGP_\lambda\}$ and $\{\bGQ_\lambda\}$ consist of homogeneous elements if we set $\deg(\beta) = -1$, there are unique integers $a_{\lambda\mu}^\nu, b_{\lambda\mu}^\nu, \widehat a_{\lambda\mu}^\nu, \widehat b_{\lambda\mu}^\nu \in \ZZ$ indexed by strict partitions $\lambda$, $\mu$, $\nu$ with \be\label{ab-eq} \ba \bGP_\lambda \bGP_\mu &= \sum_\nu a_{\lambda\mu}^\nu \beta^{|\nu|-|\lambda|-|\mu|} \bGP_\nu,\\ \bGQ_\lambda \bGQ_\mu &= \sum_\nu b_{\lambda\mu}^\nu \beta^{|\nu|-|\lambda|-|\mu|} \bGQ_\nu, \\ \bGP_{\nu\sss\lambda}&=\sum_\mu \widehat b_{\lambda\mu}^\nu \beta^{|\lambda| + |\mu| - |\nu|} \bGP_\mu, \\ \bGQ_{\nu\sss\lambda}&=\sum_\mu \widehat a_{\lambda\mu}^\nu \beta^{|\lambda| + |\mu| - |\nu|} \bGQ_\mu. \ea \ee % The following result is a special case of \cite[Prop.~3.2]{NN2018} since $\bgp_\lambda$ and $\bgq_\lambda$ are special cases of \cite[Def.~3.2]{NN2018}. We outline a self-contained proof for completeness. \begin{proposition}[{\cite[Prop.~3.2]{NN2018}}] For all strict partitions $\lambda$, $\mu$, $\nu$ it holds that \label{abcd-prop} \[ \ba \bgp_\lambda \bgp_\mu &= \sum_\nu \widehat a_{\lambda\mu}^\nu \beta^{|\lambda|+|\mu|-|\nu|} \bgp_\nu \\ \bgq_\lambda \bgq_\mu &= \sum_\nu \widehat b_{\lambda\mu}^\nu \beta^{|\lambda| + |\mu| - |\nu|} \bgq_\nu \ea \quand \ba \bgp_{\nu/\lambda}&=\sum_\mu b_{\lambda\mu}^\nu \beta^{ |\nu|-|\lambda| - |\mu|} \bgp_\mu \\ \bgq_{\nu/\lambda}&=\sum_\mu a_{\lambda\mu}^\nu \beta^{ |\nu|-|\lambda| - |\mu|} \bgq_\mu. \ea \] \end{proposition} \begin{proof} One can derive these identities from \eqref{cauchy-eq} by introducing a third sequence of variables ${\mathbf z} = (z_1,z_2,\dots)$ and then extracting coefficients. For example, we have \[ \sum_\nu \bGQ_\nu({\bf x},{\bf y}) \bgp_\nu({\bf z}) = \prod_{i,j} \tfrac{1-\overline{x}_i z_j}{1-x_iz_j} \prod_{i,j} \tfrac{1-\overline{y}_i z_j}{1-y_iz_j} .\] The left side is \[\sum_{\lambda,\mu,\nu} \widehat a_{\lambda\mu}^\nu \beta^{|\lambda| + |\mu| - |\nu|} \bGQ_\lambda({\bf x}) \bGQ_\mu({\bf y}) \bgp_\nu({\bf z})\] while the right side is \[ \sum_{\lambda,\mu} \bGQ_\lambda({\bf x}) \bGQ_\mu({\bf y}) \bgp_\lambda ({\bf z}) \bgp_\mu({\bf z}),\] which leads to the first formula. \end{proof} Let $\ell(\lambda)$ be the number of parts in a partition $\lambda$. Given strict partitions $\mu \subseteq \lambda$, define $\columns(\lambda/\mu) = |\{ j : (i,j) \in \SD_{\lambda/\mu}\}|$ to be the number of columns occupied by $\SD_{\lambda/\mu}$. A subset of $\PP\times \PP$ is a \defn{vertical strip} if it contains at most one position in each row. Then it holds by \cite[Thm.~1.1]{ChiuMarberg} that \be\label{GQ-to-GP-eq} \bGQ_\mu = \sum_{\lambda} 2^{\ell(\mu)} (-1)^{\columns(\lambda/\mu)} (-\beta/2)^{|\lambda/\mu|} \bGP_\lambda \ee and by \cite[Cor.~6.2]{ChiuMarberg} that \be\label{gq-to-gp-eq} \bgq_\lambda = \sum_{\mu} 2^{\ell(\mu)} (-1)^{\columns(\lambda/\mu)} (-\beta/2)^{|\lambda/\mu|} \bgp_\mu \ee where both sums are over strict partitions $\lambda\supseteq \mu$ with $\ell(\lambda) =\ell(\mu)$ such that $\SD_{\lambda/\mu}$ is a vertical strip. % For example $\bGQ_{(3,2)} = 4 \bGP_{(3,2)} + 2\beta \bGP_{(4,2)} - \beta^2 \bGP_{(4,3)}$ and $\bgq_{(3,2)} = 4 \bgp_{(3,2)} + 2\beta \bgp_{(3,1)} -\beta^2 \bgp_{(2,1)}.$ We use the following notation from \cite{Iwao} just in the next result. Let $\mSymPQ$ be the linearly compact $\QQ(\beta)$-module with $\{ \bGP_\lambda\}$ as a pseudobasis (where $\lambda$ ranges over all strict partitions) and let $\MSymPQ$ be the free $\QQ(\beta)$-module with $\{ \bgp_\lambda\}$ as a basis. The identities \eqref{GQ-to-GP-eq} and \eqref{gq-to-gp-eq} imply that $\{ \bGQ_\lambda\}$ is another pseudobasis for $\mSymPQ$ while $\{ \bgq_\lambda\}$ is another basis for $\MSymPQ$. \begin{proposition}\label{sh-form-prop} There is a unique bilinear form $[\cdot,\cdot] : \MSymPQ \times \mSymPQ\to \QQ(\beta)$, continuous in the second coordinate, with $[ \bgp_\lambda, \bGQ_\mu] = [ \bgq_\lambda, \bGP_\mu] = \delta_{\lambda\mu}$ for all strict partitions $\lambda$ and $\mu$. \end{proposition} \begin{proof} Let $(\cdot,\cdot)$ be the bilinear form, continuous in the second coordinate, with $(\bgp_\lambda,\bGP_\mu) = \delta_{\lambda\mu}$. Write $\phi$ and $\psi$ for the (continuous) linear maps with \[\phi (\bgq_\lambda) = \bgp_\lambda\quand\psi (\bGQ_\mu) = \bGP_\mu.\] The identities \eqref{GQ-to-GP-eq} and \eqref{gq-to-gp-eq} imply that $(\phi(f), g) = (f, \psi(g))$ for all $f \in \MSymPQ $ and $g\in \mSymPQ$. Then the form $[f,g] :=(\phi(f), g) = (f, \psi(g))$ has the desired properties. \end{proof} \begin{remark*} The module $\mSymPQ$ coincides with the ring considered in \cite[\S5.2]{Iwao}, which Iwao defines by a certain \defn{$K$-theoretic $Q$-cancellation property}; \cite[Thm.~3.1]{IkedaNaruse} shows that the infinite linear span of the $\bGP_\lambda$'s is characterized by the same property. Comparing the Cauchy identity \eqref{cauchy-eq} with the one in \cite[\S8.2]{Iwao} shows that $\MSymPQ$ similarly coincides with the ring defined in \cite[\S8.1]{Iwao}, and that the form in Proposition~\ref{sh-form-prop} is equal to the one in \cite[Eq. (30)]{Iwao}. \end{remark*} Putting everything together leads to this theorem: \begin{theorem}\label{gg-thm} Both $\MSymP $ and $\MSymQ $ are Hopf subalgebras of $\MSym$. In particular, $\MSymP$ (respectively, $\MSymQ$) is the Hopf algebra dual to $\mSymQ$ (respectively, $\mSymP$) via $[\cdot,\cdot]$. \end{theorem} \begin{proof} It is clear from \eqref{gp-gq-xy-eq2} and Proposition~\ref{abcd-prop} that $\MSymP $ and $\MSymQ $ are the sub-bialgebras of $\MSym$ dual to $\mSymQ$ and $\mSymP$ via $[\cdot,\cdot]$. The fact that these bialgebras are preserved by the antipode of $\MSym$ follows by duality as $\mSymQ$ and $\mSymP$ are LC-Hopf subalgebras of $\mSym$. \end{proof} As mentioned in Section~\ref{sym-sect}, the Hopf algebras $\MSym$ and $\mSym$ have another pair of dual bases for $\langle\cdot,\cdot\rangle$ besides the Schur functions, given by the (dual) stable Grothendieck polynomials $\{ g^{(\beta)}_\lambda\}$ and $\{ G^{(\beta)}_\lambda\}$. We need to quote two results involving these functions. First, one has \be\label{gpn-eq} \gp_{(n)} = \sum_{i=1}^n g^{(\beta)}_{(i,1^{n-i})} \ee for all positive integers $n$ by \cite[Prop.~5.3]{NN2018}; see the proof of \cite[Prop.~7.5]{ChiuMarberg} for another derivation. Second, if $\lambda$ is a partition with $k$ parts and $\ttheta: \mQSym \to \mQSymQ $ is the map \eqref{ttheta-eq} then \be\ttheta(G^{(\beta)}_\lambda) = \bGQ_{(\lambda+\delta)/\delta}\quad\text{for $\delta := (k-1,\dots,2,1,0)$, by \cite[\S4.6]{LM2019}.} \ee Taking $k=1$ gives $\ttheta(G^{(\beta)}_{(n)}) = \bGQ_{(n)}$ for all $n>0$. Thus $\ttheta$ restricts to a map $\mSym \to \mSymQ$, which is surjective by \cite[Cor.~5.17]{LM2019}. The following result reduces to \cite[(A.9)]{Stembridge1997a} when $\beta=0$: \begin{theorem}\label{ttheta-thm} If $f \in \MSymPQ$ and $g \in \mSym$ then $\left[f,\ttheta(g)\right] = \langle f,g\rangle$. \end{theorem} \begin{proof} Write $\bgp_n := \bgp_{(n)}$ and $\bgq_n := \bgq_{(n)}$ for $n \in \PP$. Define $G^{(\beta)}_n$ and $\bGQ_n$ analogously. For compositions $\alpha = (\alpha_1,\alpha_2,\dots,\alpha_k)$ let $G(\alpha) :=\prod_{i\in[k]} G^{(\beta)}_{\alpha_i}$, $\GQ(\alpha):=\ttheta(G(\alpha)) = \prod_{i\in[k]} \bGQ_{\alpha_i}$, and $\gp(\alpha):=\prod_{i\in[k]} \bgp_{\alpha_i}$. The Pieri rule in \cite[Thm.~3.4]{Lenart2000} implies that every element of $\mSym$ is possibly infinite $\ZZ[\beta]$-linear combination of $G(\alpha)$'s. The analogous Pieri rule in \cite[Prop.~2.7]{LM2022} with \eqref{gq-to-gp-eq} implies that every element of $\MSymPQ$ is a finite $\QQ(\beta)$-linear combination of $\gp(\alpha)$'s. Thus we just need to check that $[\gp(\alpha), \GQ(\gamma)] = \langle \gp(\alpha),G(\gamma)\rangle$ for all compositions $\alpha$ and $\gamma$. To show this, let $k=\ell(\alpha)$ and $l=\ell(\gamma)$ be the lengths of two arbitrary compositions. Recall that we denote iterated coproducts by $\Delta^{(k)} := (1 \otimes \Delta^{(k-1)})\circ \Delta = ( \Delta^{(k-1)}\otimes 1 )\circ \Delta$ where $\Delta^{(1)}:=\Delta$. Since $[\cdot,\cdot]: \MSymP\times \mSymQ \to \ZZ[\beta]$ and $\langle \cdot,\cdot\rangle : \MSym \times \mSym\to\ZZ[\beta]$ induce Hopf algebra dualities and since coproducts in Hopf algebras are algebra morphisms, we have \[\ba {[\gp(\alpha), \GQ(\gamma)] } &= \left[\Delta^{(l-1)}(\gp(\alpha)), \bigotimes_{j \in[l]} \bGQ_{\gamma_j} \right] \\&= \left[\bigotimes_{i \in[k]} \Delta^{(l-1)}( \bgp_{\alpha_i}) ,\bigotimes_{j \in[l]} \Delta^{(k-1)}( \bGQ_{\gamma_j}) \right] \ea \] and similarly \[ \ba \langle\gp(\alpha), G(\gamma)\rangle &= \left\langle\Delta^{(l-1)}(\gp(\alpha)), \bigotimes_{j \in[l]} G^{(\beta)}_{\gamma_j} \right\rangle \\&= \left\langle\bigotimes_{i \in[k]} \Delta^{(l-1)}( \bgp_{\alpha_i}) ,\bigotimes_{j \in[l]} \Delta^{(k-1)}( G^{(\beta)}_{\gamma_j}) \right\rangle \ea \] where we appropriately reorder the $kl$ tensor factors in $\bigotimes_{i \in[k]} \Delta^{(l-1)}( \bgp_{\alpha_i})$ to evaluate the two rightmost expressions. It is clear from \eqref{gp-gq-xy-eq2} that $\Delta(\bgp_n) =\sum_{i=0}^n \bgp_i\otimes \bgp_{n-i}$ and from \eqref{Delta-eq} that \[\Delta(\bGQ_n) =\sum_{i=0}^n \bGQ_i\otimes \bGQ_{n-i} +\beta \sum_{i=1}^n \bGQ_i \otimes \bGQ_{n+1-i}\] where $\bgp_0 = \bGQ_0 :=1$. It follows similarly from the well-known set-valued tableau generating function for $G_\lambda$ (see e.g. \cite[Eq.\ (3.9)]{LM2019}) that \[\Delta(G^{(\beta)}_n) =\sum_{i=0}^n G^{(\beta)}_i\otimes G^{(\beta)}_{n-i} +\beta \sum_{i=1}^n G^{(\beta)}_i \otimes G^{(\beta)}_{n+1-i}.\] We deduce from these formulas that the equality $[\gp(\alpha), \GQ(\gamma)] = \langle \gp(\alpha),G(\gamma)\rangle$ will hold if we can just show that $[\bgp_m, \bGQ_n] =\delta_{mn}= \langle \bgp_m,G^{(\beta)}_n\rangle$ for all $m,n \in \NN$. This simpler identity is immediate from \eqref{gpn-eq}. \end{proof} Recall the elements $\tpeakR_n \in \MPeakP$ and $\opeakR_n \in \MPeakQ$ for $n \in \PP$ from Section~\ref{MPeak-sect}. \begin{theorem}\label{last-adj-thm} The map $\MPeakP \to \MSymP$ adjoint to $\mSymQ\hookrightarrow \mQSymQ$ relative to the forms $[\cdot,\cdot]$ in Theorems~\ref{mpeakp-thm} and \ref{gg-thm} is the unique algebra morphism with $\tpeakR_n \mapsto \bgp_{n} $ and $\opeakR_n \mapsto \bgq_{n} $. This morphism restricts to the map $\MPeakQ \to \MSymQ$ adjoint to $\mSymP\hookrightarrow \mQSymP$. \end{theorem} \begin{proof} We first claim that $[ \tpeakR_n, \bGQ_\lambda]= \delta_{(n),\lambda}$ for all $n \in \NN$ and strict partitions $\lambda$. This follows from the discussion in \cite[\S4.6]{LM2019} which gives the $K^{(\beta)}_\alpha$-decomposition of $\bGQ_\lambda$. In detail, define a \defn{standard shifted set-valued tableau} of shape $\lambda$ to be a semistandard shifted set-valued tableau of shape $\lambda$ whose entries are pairwise disjoint nonempty sets, never containing any consecutive integers, with union $\{1,2,\dots,N\}$ for some $N \geq |\lambda|$. Suppose $T$ is such a tableau and set $|T| := N$. The \defn{peak set} of $T$ is the set $\PeakSet(T)$ of integers $11$, then every standard shifted set-valued tableau $T$ of shape $\lambda$ has a nonempty peak set, since if $i+1 $ is the smallest number in box $(2,2)$ of $T$ then $i \in \PeakSet(T)$. On the other hand, if $\ell(\lambda)\leq 1$ then there is exactly one standard shifted set-valued tableau $T$ of shape $\lambda$, and this tableau has $|T| = |\lambda|$ and $\PeakSet(T) = \varnothing$. Thus if $\alpha= (n)$ for some $n \in \NN$ so that $I(\alpha) = \varnothing$, then we have $k^{\alpha}_\lambda = \delta_{(n),\lambda}$ and $[ \tpeakR_n, \bGQ_\lambda]= \delta_{(n),\lambda}$ as desired. Our claim shows that $[ \tpeakR_n, \bGQ_\lambda] = [ \bgp_n, \bGQ_\lambda] $ so the adjoint map $\MPeakP \to \MSymP$ must send $\tpeakR_n \mapsto \bgp_n$ for all $n \in \NN$. There is at most one algebra morphism with this property since $\{\tpeakR_n : n=1,3,5,\dots\}$ freely generates $\MPeakP$ by Proposition~\ref{MPeak-free-prop}. The adjoint map also sends $\opeakR_n \mapsto \bgq_{n} $ since we have \[\opeakR_1 = 2 \tpeakR_1\quand \bgq_1 = 2\bgp_1\] as well as \[\opeakR_n = 2 \tpeakR_n + \beta \tpeakR_{n-1}\quand \bgq_n = 2 \bgp_n + \beta \bgp_{n-1}\] for $n>1$ by \eqref{opeak-Rn} and \eqref{gq-to-gp-eq}. \end{proof} The maps in Theorem~\ref{last-adj-thm} are shifted analogues of the morphism $\MNSym \to \MSym$ in \eqref{k-diagram}. As noted earlier, Lam and Pylyavskyy show that the image of $\tR_{\alpha}$ under this map is a specific dual stable Grothendieck polynomial $g^{(\beta)}_{\lambda/\mu}$ \cite[Thm.~9.13]{LamPyl}. We do not know if this result has a shifted version. When $\alpha$ is a peak composition, the images of $\tpeakR_\alpha$ and $\opeakR_\alpha$ under the adjoint maps $\MPeakP \to \MSymP$ and $\MPeakQ \to \MSymQ$ typically are not of the form $\bgp_{\lambda/\mu}$ or $\bgq_{\lambda/\mu}$. \subsection{Antipode formulas}\label{antipode-sect} We gave the general definition of the antipode for a Hopf algebra or LC-Hopf algebra in Section~\ref{prelim-sect}. Here we describe some specific antipode formulas for the objects in \eqref{shk-diagram}. If $\alpha$ is a finite sequence then we write $\alpha^\ttr$ for its reversal. Given $\alpha \vDash n$, let $\alpha^\ttc$ be the unique composition of $n$ with $I(\alpha^\ttc) =[n - 1] \setminus I(\alpha)$, and define $\alpha^\ttt := (\alpha^\ttc)^\ttr = (\alpha^\ttr)^\ttc$. For example, we have $(3,2)^\ttr = (2,3)$, $(3,2)^\ttc = (1,1,2,1)$, and $(3,2)^\ttt = (1,2,1,1)$. % Recall from Remark~\ref{beta-rmk} that the homogeneous functions \[ L_\alpha := L^{(0)}_\alpha = \sum_{{\gamma\vDash |\alpha|, I(\gamma) \supseteq I(\alpha)}} M_{\gamma}\] form another basis for $\QSym$. % Write $\omega : \QSym \to \QSym$ for the linear map with $\omega(L_\alpha) := L_{\alpha^\ttt}$. This map is a Hopf algebra automorphism which preserves $\Sym$, acting on Schur functions as $\omega(s_\lambda) =s_{\lambda^\top}$ where $\lambda^\top$ is the transpose of a partition $\lambda$. % The antipode of $\QSym$ is the linear map $\antipode$ with \be\label{qsym-antipode-eq} \ba \antipode(L_\alpha) &= (-1)^{|\alpha|} L_{\alpha^\ttt} = (-1)^{|\alpha|} \omega(L_{\alpha}) \text{ and } \\ \antipode(s_\lambda) &= (-1)^{|\lambda|} s_{\lambda^\top} = (-1)^{|\lambda|} \omega(s_{\lambda}) \ea \ee for all compositions $\alpha$ and partitions $\lambda$ \cite[\S3.6]{LMW}. We can extend $\omega$ to a continuous automorphism of $\mQSym$. The antipode of $\mQSym$ is the continuous extension of the antipode of $\QSym$. A \defn{multiset} is a set allowing repeated elements. Given a peak composition $\alpha = (\alpha_1,\alpha_2,\dots,\alpha_k)$ let $\alpha^\flat := (\alpha_k+1, \alpha_{k-1},\dots,\alpha_2,\alpha_1-1)$ when $k>1$ and set $\alpha^\flat := \alpha$ if $k\leq 1$. The following statement is equivalent to identities in \cite{LM2019}, and reduces to \cite[Prop.~3.5]{Stembridge1997a} when $\beta=0$. \begin{proposition}[{\cite[Prop 6.5 and Obs. 6.9]{LM2019}}] If $\alpha$ is a peak composition then \[ \antipode\left(K^{(\beta)}_{\alpha^\flat}\right) = \sum_S (-\beta)^{|S| - N} x^S \quand \antipode\left(\bK^{(\beta)}_{\alpha^\flat}\right) = \sum_S (-\beta)^{|S| - N} x^S \] where both sums are over $N$-tuples $S=(S_1 \preceq \cdots \preceq S_N)$ satisfying the same respective conditions \eqref{peak-def-eq1} and \eqref{peak-def-eq2} as in Definition~\ref{peak-def}, but with each $S_i$ a finite nonempty multi-subset of $\MM$. \end{proposition} \begin{proposition} The antipode $\antipode$ of $\MPeakQ\supset \MPeakP$ is the algebra anti-automorphism with \ben \item[(a)] $\ds\antipode\left(\opeakR_{n}\right) = (-1)^n\sum_{k\in[n]} \tbinom{n-1}{k-1} \cdot \beta^{n-k} \cdot\opeakR_{k}$ for all $n\in \PP$, and \item[(b)] $\ds\antipode\left(\tpeakR_{n}\right) = (-1)^n\sum_{k\in[n]} \tbinom{n}{k} \cdot \beta^{n-k} \cdot\tpeakR_{k}$ for all $n\in \PP$. \een \end{proposition} \begin{proof} The antipode of any Hopf algebra is an anti-endomorphism \cite[Prop.~1.4.10]{GrinbergReiner} and is invertible when the Hopf algebra is cocommutative \cite[Rem.~1.4.13]{GrinbergReiner}. One has $\nabla \circ (\antipode\otimes \id) \circ \Delta = \iota\circ\epsilon$ by definition, so by Proposition~\ref{peak-coproduct-prop} we deduce that $ \antipode(\opeakR_n) = -\opeakR_n - \sum_{m=1}^{n-1}\antipode(\opeakR_m) \opeakR_{n-m} $ for all $n \in \PP$. Checking that the formula in part (a) satisfies this recurrence, using Proposition~\ref{opeak-prod-prop}, is a somewhat involved but completely elementary computation, requiring only sum manipulations and the identity $\binom{m}{k} = \binom{m-1}{k-1} + \binom{m-1}{k}$. Part (b) follows from part (a) by \eqref{opeak-Rn}, since one can similarly check that the formula for $\antipode(\tpeakR_{n}) $ is the unique solution to the recurrence with $\antipode(\opeakR_{1})=2\cdot\antipode(\tpeakR_{1}) $ and $\antipode(\opeakR_{n})=2\cdot \antipode(\tpeakR_{n}) + \beta\cdot \antipode(\tpeakR_{n-1}) $ for $n>1$. \end{proof} Patrias \cite[Thms.~33 and 35]{Patrias} computes explicit antipode formulas for $\mQSym$ and $\MNSym$ in the (pseudo)bases $\{ L^{(\beta)}_\alpha\}$ and $\{ \tR_\alpha\}$, using slightly different notation.\footnote{To convert the functions $\tilde L_\alpha$ and $\tilde R_\alpha$ in \cite{Patrias} to our notation, set $\tilde L_\alpha = \beta^{|\alpha|} L^{(\beta)}_\alpha$ and $\beta^{|\alpha|}\tilde R_\alpha = \tR_\alpha$. } We have only given partial formulas for the antipodes of the shifted analogues of these Hopf algebras. It may be possible to extend our results along the lines of \cite{Patrias}. Now we turn to the shifted versions of stable Grothendieck polynomials. Choose strict partitions $\mu \subseteq \lambda$. A \defn{shifted multiset-valued tableau} of shape $\lambda/\mu$ is a map $T $ assigning nonempty finite multi-subsets of $\MM$ to the boxes in $\SD_{\lambda/\mu}$. We write $T_{ij}$ to denote the multiset assigned by $T$ to position $(i,j)$. The definition of a \defn{semistandard} shifted multiset-valued tableau is identical to the set-valued case. Let $\ShMSetTab_Q(\lambda/\mu)$ denote the set of all semistandard shifted multiset-valued tableaux of shape $\lambda/\mu$, and let $\ShMSetTab_P(\lambda/\mu)$ be the subset of such tableaux with no primed numbers appearing in diagonal positions. Then define \be \ba \bJP_{\lambda/\mu} &:= \sum_{T \in \ShMSetTab_P(\lambda/\mu)} (-\beta)^{|T|-|\lambda/\mu|} x^T, \\ \bJQ_{\lambda/\mu} &:= \sum_{T\in \ShMSetTab_Q(\lambda/\mu)} (-\beta)^{|T|-|\lambda/\mu|} x^T , \ea \ee where as usual $ |T| := \sum_{(i,j) \in \SD_{\lambda/\mu}} |T_{ij}|$ and $ x^T := \prod_{(i,j) \in \SD_{\lambda/\mu}} \prod_{k \in T_{ij}} x_{\lceil k\rceil }$. The functions $ \bJP_{\lambda/\mu}$ become the \defn{weak shifted stable Grothendieck polynomials} from \cite[\S3]{HKPWZZ} when $\beta=-1$ and $\mu=\emptyset$. \begin{proposition}[\cite{LM2019}]\label{GP-GQ-S-prop} If $\mu\subseteq\lambda$ are strict partitions then \[ \antipode\left(\bGP_{\lambda/\mu}\right) = (-1)^{|\lambda/\mu|} \bJP_{\lambda/\mu} \quand \antipode\left(\bGQ_{\lambda/\mu}\right) = (-1)^{|\lambda/\mu|} \bJQ_{\lambda/\mu}. \] \end{proposition} \begin{proof} This holds since $\omega\left(\bGP_{\lambda/\mu}\right) = \JP^{(-\beta)}_{\lambda/\mu} $ and $ \omega\left(\bGQ_{\lambda/\mu}\right) = \JQ^{(-\beta)}_{\lambda/\mu}$ by \cite[Cor.~6.6]{LM2019}. \end{proof} A partition of a set $S$ is a set $\Pi$ of disjoint nonempty blocks $B \subseteq S$ with $S = \bigsqcup_{B \in \Pi} B$. Choose strict partitions $\mu \subseteq \lambda$. A \defn{semistandard shifted bar tableau} of shape $\lambda/\mu$ is a pair $T=(V,\Pi)$, where $V$ is a semistandard shifted tableau\footnote{That is, a semistandard shifted set-valued tableau whose entries are all sets with exactly one element. } of shape $\lambda/\mu$ and $\Pi$ is a partition of $\SD_{\lambda/\mu}$ into subsets of adjacent positions containing the same entry in $V$. One might draw this as a picture like \be\label{colors-eq} \vcenter{\hbox{\begin{tikzpicture}[scale=0.45] \filldraw [fill=cyan, draw=black] (1,0) rectangle (4,-1); \filldraw [fill=pink, draw=black] (4,0) rectangle (5,-2); \filldraw [fill=red, draw=black] (2,-1) rectangle (3,-2); \filldraw [fill=yellow, draw=black] (3,-1) rectangle (4,-2); \filldraw [fill=magenta, draw=black] (5,-1) rectangle (6,-2); \draw (1,0) + (.5,-.5) node {$2$}; \draw (4,0) + (.5,-.5) node {$3'$}; \draw (0,-1) + (.5,-.5) node {$\cdot$}; \draw (1,-1) + (.5,-.5) node {$\cdot$}; \draw (2,-1) + (.5,-.5) node {$1$}; \draw (3,-1) + (.5,-.5) node {$1$}; \draw (5,-1) + (.5,-.5) node {$3$}; \end{tikzpicture}}} \ee to represent the pair \[ (V, \Pi) = \Bigg( \vcenter{\hbox{\begin{tikzpicture}[scale=0.45] \foreach \x in {1, 2, 3, 4} { \draw (\x, 0) rectangle (\x+1, -1); } \foreach \x in {2, 3, 4, 5} { \draw (\x, -1) rectangle (\x+1, -2); } \draw (1,0) + (.5,-.5) node {$2$}; \draw (2,0) + (.5,-.5) node {$2$}; \draw (3,0) + (.5,-.5) node {$2$}; \draw (4,0) + (.5,-.5) node {$3'$}; \draw (4,-1) + (.5,-.5) node {$3'$}; \draw (0,-1) + (.5,-.5) node {$\cdot$}; \draw (1,-1) + (.5,-.5) node {$\cdot$}; \draw (2,-1) + (.5,-.5) node {$1$}; \draw (3,-1) + (.5,-.5) node {$1$}; \draw (5,-1) + (.5,-.5) node {$3$}; \end{tikzpicture}}} \; , \vcenter{\hbox{ \begin{tikzpicture}[scale=0.5] \draw (1,0) rectangle (4,-1); \draw (4,0) rectangle (5,-2); \draw (2,-1) rectangle (3,-2); \draw (3,-1) rectangle (4,-2); \draw (5,-1) rectangle (6,-2); \draw (0,-1) + (.5,-.5) node {$\cdot$}; \draw (1,-1) + (.5,-.5) node {$\cdot$}; \end{tikzpicture}}} \ \Bigg) \] when $\lambda = (6,4)$ and $\mu=(2)$. Let $ |T| := |\Pi|$ and $ x^T := \prod_{i\geq 1} x_i^{b_i}$ where $b_i$ is the number of blocks in $\Pi$ containing $i$ or $i'$. Our example \eqref{colors-eq} has $|T| = 5$ and $x^T = x_1^2 x_2 x_3^2$. 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 entries in diagonal positions. Then define \[ \bjp_{\lambda/\mu} := \sum_{T\in \ShBTP(\lambda/\mu)} \beta^{|\lambda/\mu|-|T|} x^{T} \quand \bjq_{\lambda/\mu} := \sum_{T\in \ShBTQ(\lambda/\mu)} \beta^{|\lambda/\mu| -|T|} x^{T} . \] These generating functions were first considered as part of some conjectural formulas in \cite[\S7]{ChiuMarberg}. \begin{proposition}[\cite{LM2022}] If $\mu\subseteq\lambda$ are strict partitions then \[ \antipode\left(\bgp_{\lambda/\mu}\right) = (-1)^{|\lambda/\mu|} \bjp_{\lambda/\mu} \quand \antipode\left(\bgq_{\lambda/\mu}\right) = (-1)^{|\lambda/\mu|} \bjq_{\lambda/\mu}. \] \end{proposition} \begin{proof} This holds since $\omega\left(\bgp_{\lambda/\mu}\right) = \jp^{(-\beta)}_{\lambda/\mu} $ and $ \omega\left(\bgq_{\lambda/\mu}\right) = \jq^{(-\beta)}_{\lambda/\mu}$ by \cite[Thms.~1.4 and 1.5]{LM2022}. \end{proof} There are similar formulas in the unshifted case for $\antipode\left(G^{(\beta)}_{\lambda/\mu}\right)$ and $\antipode\left(g^{(\beta)}_{\lambda/\mu}\right)$; see \cite[\S8]{Patrias}. \begin{corollary} For all strict partitions $\mu\subseteq\lambda$ one has \[\bjp_{\lambda/\mu} \in \MSymP, \quad\bjq_{\lambda/\mu} \in \MSymQ, \quad\bJP_{\lambda/\mu} \in \mSymP, \quand \bJQ_{\lambda/\mu} \in \mSymQ.\] Moreover, the form in Proposition~\ref{sh-form-prop} has $ [ \bjp_\lambda, \bJQ_\mu] = [ \bjq_\lambda, \bJP_\mu] = \delta_{\lambda\mu}$ for all $\lambda$, $\mu$. \end{corollary} \begin{proof} The containments hold since $\MSymP$, $\MSymQ$, $\mSymP$, and $\mSymQ$ are all closed under $\antipode$. The antipode of any commutative Hopf algebra is an involution \cite[Cor.~1.4.12]{GrinbergReiner} so Theorem~\ref{gg-thm} implies \[ \ba (-1)^{|\lambda|+|\mu|} [ \bjp_\lambda, \bJQ_\mu] & = [ \antipode (\bgp_\lambda), \antipode(\bGQ_\mu)] \\&= [ \antipode^2(\bgp_\lambda), \bGQ_\mu] \\&= [ \bgp_\lambda, \bGQ_\mu] = \delta_{\lambda\mu}.\ea\] One derives the identity $[ \bjq_\lambda, \bJP_\mu] = \delta_{\lambda\mu}$ similarly. \end{proof} Let $\mathbf{GP} \subsetneq \mSymP$ and $\mathbf{GQ} \subsetneq \mSymQ$ denote the proper $\ZZ[\beta]$-submodules with $\{ \bGP_\lambda\}$ and $\{\bGQ_\lambda\}$ as bases, rather than pseudobases, where $\lambda$ ranges over all strict partitions. \begin{proposition}\label{bialgebras-thm} Both $\mathbf{GP} $ and $\mathbf{GQ}$ are sub-bialgebras of $\mSym$ but not Hopf algebras. \end{proposition} \begin{proof} We already know that $\{ \bGP_\lambda\}$ and $\{\bGQ_\lambda\}$ are pseudobases for LC-Hopf subalgebras of $\mSym$. This result makes three nontrivial additional claims. % First, the products $\bGP_\lambda \bGP_\mu$ and $\bGQ_\lambda \bGQ_\mu$ always expand as finite linear combinations of $\bGP$- and $\bGQ$-functions. For the $\bGP$-functions, this was first shown in \cite{CTY}; for other proofs, see \cite[\S4]{HKPWZZ}, \cite[\S1.2]{Marberg2021}, or \cite[\S8]{PechenikYong}. For the $\bGQ$-functions, the desired finiteness property is \cite[Thm 1.6]{LM2022}. Second, the coproducts $\Delta(\bGP_\nu)$ and $\Delta(\bGQ_\nu)$ are always finite linear combinations of tensor products of the form $\bGP_\lambda\otimes \bGP_\mu$ and $\bGQ_\lambda\otimes \bGQ_\mu$. By \eqref{Delta-eq} and \eqref{ab-eq} this is equivalent to the numbers $\widehat a_{\lambda\mu}^\nu$ and $\widehat b_{\lambda\mu}^\nu$ being nonzero for only finitely many pairs $(\lambda,\mu)$ when $\nu$ is fixed. This holds since both numbers are zero if $\lambda \not\subseteq \nu$ (by definition) or if $\mu \not\subseteq\nu$ (since $\widehat a_{\lambda\mu}^\nu=\widehat a_{\mu\lambda}^\nu$ and $\widehat b_{\lambda\mu}^\nu=\widehat b_{\mu\lambda}^\nu$ as $\mSym$ is cocommutative). To show that $\mathbf{GP} $ and $\mathbf{GQ}$ are not Hopf algebras, it suffices to check that $\bJP_\lambda =\pm \antipode(\bGP_\lambda)$ and $\bJQ_\lambda = \pm \antipode(\bGQ_\lambda)$ may fail to be finite linear combinations of $\bGP$- and $\bGQ$-functions. This can already be seen for $\lambda =(1)$ by setting $x_i=0$ for all $i>1$. Under this specialization one has $\bJP_{(1)} = \frac{-x_1}{1+\beta x_1}$, $\bJQ_{(1)} = \frac{-2x_1+\beta x_1^2}{1+\beta x_1}$, $\bGP_{(n)} = x_1^n$, and $\bGQ_{(n)} = (2 + \beta x_1)x_1^n$ for all $n \in \PP$, while $\bGP_{\mu} = \bGQ_{\mu} =0$ whenever $\ell(\mu)>1$, so the relevant expansions are clearly infinite. \end{proof} \begin{remark} The span of the stable Grothendieck polynomials $\{G^{(\beta)}_\lambda\}$ is a bialgebra by \cite[Cor.~6.7]{Buch2002}. A similar argument shows that this bialgebra is also not a Hopf algebra, as $G^{(\beta)}_{(1)}= \bGP_{(1)} $. \end{remark} It is often of interest to derive cancellation-free antipode formulas. We should point out that the results in this section are mostly not of this form, as we do not know how to expand $\bJP_{\lambda/\mu}$, $\bJQ_{\lambda/\mu}$, $\bjp_{\lambda/\mu}$, and $\bjq_{\lambda/\mu}$ in the respective $\{ \bGP_\nu\}$, $\{ \bGQ_\nu\}$, $\{ \bgp_\nu\}$, and $\{ \bgq_\nu\}$ bases. We have also not discussed the multi-Malvenuto-Reutenauer Hopf algebras $\mMPR$ and $\MMPR$. The problem of finding cancellation-free antipode formulas for these Hopf algebras appears to be open. Progress on this question would give $K$-theoretic generalizations of the results in \cite[\S5]{AguiarSottile}. \subsection{Positivity properties}\label{open-sect} To conclude this article, we collect some open problems and conjectures related to positivity properties of our various symmetric functions. Let $\mG$ and $\mg$ denote the respective (finite) $\NN[\beta]$-linear spans of the stable Grothendieck polynomials $\{G^{(\beta)}_\lambda\}$ and their dual versions $\{g^{(\beta)}_\lambda\}$, with $\lambda$ ranging over all partitions. Buch \cite[Cors. 5.5 and 6.7]{Buch2002} derives Littlewood-Richardson rules for (co)products of stable Grothendieck polynomials, which imply that $G^{(\beta)}_\lambda G^{(\beta)}_\mu \in \mG$ and $\Delta(G^{(\beta)}_{\lambda}) \in \mG\otimes \mG$ for all partitions $\lambda$ and $\mu$. Similarly, let $\mGP$, $\mGQ$, $\mgp$, and $\mgq$ be the respective (finite) $\NN[\beta]$-linear spans of $\{\bGP_\lambda\}$, $\{\bGQ_\lambda\}$, $\{\bgp_\lambda\}$, and $\{\bgq_\lambda\}$, with $\lambda$ ranging over all strict partitions. It is known that $\bGP_\lambda\bGP_\mu \in \mGP$ and $\bGQ_\lambda\bGQ_\mu \in \mGQ$ for all strict partitions $\lambda$ and $\mu$, or equivalently that the integers $a_{\lambda\mu}^\nu$ and $b_{\lambda\mu}^\nu$ in \eqref{ab-eq} are always nonnegative \cite[Thm.~1.6]{LM2022}. By Proposition~\ref{abcd-prop}, this implies that we always have $\bgp_{\lambda/\mu} \in \mgp$ and $\bgq_{\lambda/\mu} \in \mgq$. Computations support some other conjectural positivity properties: \begin{conjecture}\label{co-gp-conj} One has $\bGP_{\lambda\sss \mu} \in \mGP$ and $\bgq_{\lambda}\bgq_\mu \in \mgq$ for all strict partitions $\lambda$, $\mu$. \end{conjecture} \begin{conjecture}\label{co-gq-conj} One has $\bGQ_{\lambda\sss \mu} \in \mGQ$ and $\bgp_{\lambda}\bgp_\mu \in \mgp$ for all strict partitions $\lambda$, $\mu$. \end{conjecture} These conjectures are equivalent to the inequalities $\widehat b_{\lambda\mu}^\nu\geq0$ and $\widehat a_{\lambda\mu}^\nu\geq0$, or via \eqref{Delta-eq} to the coproduct identities $\Delta(\bGP_\lambda) \in \mGP\otimes\mGP$ and $\Delta(\bGQ_\lambda) \in \mGQ\otimes\mGQ$. We do not know how to leverage the geometric interpretation of $\bGP_\lambda$ and $\bGQ_\lambda$ in \cite[\S8.3]{IkedaNaruse} to prove these properties. Littlewood-Richardson rules are known for the coefficients $a_{\lambda\mu}^\nu$ ; see \cite[Thm.~1.2]{CTY} or \cite[\S8]{PechenikYong}. Outside some special cases considered in \cite{BuchRavikumar,LM2022}, the following problem is open: \begin{problem} Find combinatorial interpretations of the integers $b_{\lambda\mu}^\nu$, $\widehat a_{\lambda\mu}^\nu$, and $\widehat b_{\lambda\mu}^\nu$ in \eqref{ab-eq}. \end{problem} \begin{remark*} As noted in \cite[Conj. 5.15]{LM2019}, it also seems to hold that $\bGP_{\lambda/ \mu} \in \mGP$ and $\bGQ_{\lambda/ \mu} \in \mGQ$. By \eqref{ss-eq}, these containments would imply Conjectures~\ref{co-gp-conj} and \ref{co-gq-conj}. The analogous property for stable Grothendieck polynomials $G^{(\beta)}_{\lambda/\mu}$ indexed by skew shapes follows from \cite[Thm.~6.9]{Buch2002}. \end{remark*} The \defn{conjugate (dual) stable Grothendieck polynomials} are given by \[ J^{(\beta)}_\lambda := (-1)^{|\lambda|} \antipode\left(G^{(\beta)}_{\lambda}\right) = \omega\left(G^{(-\beta)}_{\lambda}\right) \quand j^{(\beta)}_\lambda := (-1)^{|\lambda|} \antipode\left(g^{(\beta)}_{\lambda}\right) = \omega\left(g^{(-\beta)}_{\lambda}\right) \] for partitions $\lambda$. The second equalities in these definitions hold by \cite[Thm.~4.6]{Yeliussizov2019}. Setting $\beta=-1$ turns $J^{(\beta)}_\lambda$ into the \defn{weak set-valued tableau} generating function $J_\lambda$ in \cite[\S9.7]{LamPyl}. Setting $\beta=1$ turns $j^{(\beta)}_\lambda$ into the \defn{valued-set tableau} generating function $j_\lambda$ in \cite[\S9.8]{LamPyl}. It follows from \cite[\S9]{LamPyl} that \be\label{Jj-eq} \ba J^{(\beta)}_\lambda &= (-\beta)^{-|\lambda|} J_\lambda(-\beta x_1,-\beta x_2,\dots), \\ j^{(\beta)}_\lambda &= \beta^{|\lambda|} j_\lambda(\beta^{-1} x_1,\beta^{-1} x_2,\dots). \ea \ee The power series $\{J^{(\beta)}_\lambda\}$ and $\{j^{(\beta)}_\lambda\}$ are another pair of dual bases for $\mSym$ and $\MSym$ relative to the form $\langle\cdot,\cdot\rangle$, since this inner product is $\antipode$-invariant. Below, we use the term \defn{Schur positive} to refer to any element of $\mSym$ that can be expressed as a possibly infinite linear combination of Schur functions with coefficients in $\NN[\beta]$. \begin{theorem}[\cite{LamPyl,Lenart2000}] \label{both-thm} For each partition $\lambda$, both $G^{(\beta)}_\lambda$ and $j^{(\beta)}_\lambda$ are Schur positive, while $s_\lambda$ is both a finite $\NN[\beta]$-linear combination of $g^{(\beta)}_\mu$'s and an infinite $\NN[\beta]$-linear combination of $J^{(\beta)}_\mu$'s. \end{theorem} \begin{proof} A few algebraic manipulations are needed to derive this statement from \cite{LamPyl,Lenart2000}. First, \cite[Thm.~2.8]{Lenart2000} expresses $s_\lambda$ as an infinite $\NN$-linear combination of $G^{(-1)}_\mu$ functions. On substituting $x_i \mapsto \beta x_i$, dividing both sides by $\beta^{|\lambda|}$, and applying $\omega$, this becomes a $\NN[\beta]$-linear expansion of $s_\lambda$ into $J^{(\beta)}_\mu$ functions. By duality $j^{(\beta)}_\lambda$ is Schur positive; in view of \eqref{Jj-eq}, this also follows from \cite[Thm.~9.8]{LamPyl}, which gives the Schur expansion of $g_\lambda = \omega(j_\lambda)$. Finally, \cite[Thm.~2.2]{Lenart2000} gives a positive combinatorial interpretation of the Schur expansion of $G^{(\beta)}_\lambda$, and by duality we have $s_\lambda \in \mg$. \end{proof} It is known that $\bGP_{\lambda} $ and $\bGQ_{\lambda} $ are both in $ \mG$ and hence Schur positive, for any strict partition $\lambda$ \cite[Thms.~3.27 and 3.40]{MarScr2023}. Combining \cite[Cor.~4.7]{MP2020} and \cite[Thm.~4.17]{MP2021} with the results in \cite{BKSTY} gives an algorithm to compute the $G^{(\beta)}_\mu$ terms appearing in $\bGP_{\lambda}$. The only known algorithm to do the same for $\bGQ_\lambda$ is to expand the right side of \eqref{GQ-to-GP-eq}, which may involve cancellations. Computations suggest some other instances of Schur positivity: \begin{conjecture} \label{bgq-mq-conj2} If $\lambda$ is a strict partition then $\bjp_\lambda$ and $\bjq_\lambda$ are Schur positive. \end{conjecture} The more interesting open problem implicit in this conjecture is the following: \begin{problem} Find combinatorial interpretations of the coefficients in the expansions of $\bGP_{\lambda}$, $\bGQ_\lambda$, $\bjp_\lambda$, and $\bjq_\lambda$ into Schur functions and stable Grothendieck polynomials. \end{problem} The \defn{canonical (dual) stable Grothendieck functions} $G^{(\alpha,\beta)}_\lambda$ and $g^{(\alpha,\beta)}_\lambda$ are generalizations of stable Grothendieck polynomials introduced in \cite{Yeliussizov2017}. In our notation, they satisfy $G^{(0,\beta)}_\lambda=G^{(\beta)}_\lambda$, $G^{(-\beta,0)}_\lambda=J^{(\beta)}_\lambda$, $g^{(0,-\beta)}_\lambda=g^{(\beta)}_\lambda$, and $g^{(\beta,0)}_\lambda=j^{(\beta)}_\lambda$. Both $G^{(\alpha,\beta)}_\lambda$ and $g^{(\alpha,\beta)}_\lambda$ are Schur positive by \cite[Thm.~4.6]{HawScr} and \cite[Thm.~9.8]{Yeliussizov2017}. This suggests another open problem: \begin{problem} Describe the shifted analogues $\GP^{(\alpha,\beta)}_\lambda$ and $\GQ^{(\alpha,\beta)}_\lambda$ (respectively, $\gp^{(\alpha,\beta)}_\lambda$ and $\gq^{(\alpha,\beta)}_\lambda$) of the power series $G^{(\alpha,\beta)}_\lambda$ (respectively, $g^{(\alpha,\beta)}_\lambda$) and prove similar positivity results. \end{problem} Theorem~\ref{both-thm} implies that $G^{(\beta)}_\lambda$ is an infinite $\NN[\beta]$-linear combination of $J^{(\beta)}_\mu$'s and that $j^{(\beta)}_\lambda$ is a finite $\NN[\beta]$-linear combination of $g^{(\beta)}_\mu$'s. Patrias gives an explicit description of the coefficients in these expansions in \cite[Thm.~59]{Patrias} using the notation $G_\lambda := G^{(-1)}_\lambda $ and $\tilde j_\lambda := j^{(-1)}_\lambda $. There appears to be a shifted analogue of this result. Here, we write $\mJP$ and $\mJQ$ for the respective sets of infinite $\NN[\beta]$-linear combinations of $\bJP$- and $\bJQ$-functions. \begin{conjecture}\label{last1-conj} If $\lambda$ is a strict partition then $\bGP_\lambda\in \mJP$ and $\bjq_\lambda \in \mgq$. \end{conjecture} \begin{conjecture}\label{last2-conj} If $\lambda$ is a strict partition then $\bGQ_\lambda\in \mJQ$ and $\bjp_\lambda \in \mgp$. \end{conjecture} As usual, beyond simply proving these conjectures, the following is of interest: \begin{problem} Find combinatorial interpretations of the coefficients appearing in the positive expansions suggested by Conjectures~\ref{last1-conj} and \ref{last2-conj} \end{problem} \longthanks{We thank Joel Lewis for many helpful discussions and for his hospitality during several productive visits to George Washington University.} %\nocite{*} \bibliographystyle{mersenne-plain} \bibliography{ALCO_Marberg_944} \end{document}