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\articlenumber{10}
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 %Cara's
 
 \newcommand\complexes{{\mathbb C}}

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\newcommand{\qcolor}[1]{\textcolor{violet}{#1}}
\newcommand{\scolor}[1]{\textcolor{orange}{#1}}
\newcommand{\pcolor}[1]{\textcolor{Green}{#1}}

%% define your title in the usual way
\title{$K$-theoretic polynomials}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Cara Monical\addressmark{1}, Oliver Pechenik\thanks{\href{mailto:pechenik@umich.edu}{pechenik@umich.edu}.}\addressmark{2}, \and Dominic Searles\addressmark{3}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Dept.\ of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA \\ \addressmark{2}Dept.\ of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA \\ \addressmark{3}Dept.\ of Mathematics and Statistics, University of Otago, Dunedin 9016, New Zealand}

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\received{\today}

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%% 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{We contribute to modern symmetric function theory, introducing two new bases of the polynomial ring and studying their relations to known bases.  The \emph{quasiLascoux} basis is a $K$-theoretic deformation of the quasikey basis that is also a nonsymmetric lift of quasiGrothendieck polynomials. We give positive expansions of this new basis into the glide and Lascoux atom bases, as well as of the Lascoux basis into this new basis. These results include the first proof that quasiGrothendieck polynomials expand positively in the multifundamental quasisymmetric polynomials of Lam--Pylyavskyy.
The \emph{kaons} are $K$-theoretic deformations of fundamental particles.  We give positive expansions of the glide and Lascoux atom bases into this new kaon basis.
Throughout, we explore parallels between these $K$-analogues and their cohomological counterparts.}

%% 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{Demazure character, Demazure atom, Lascoux polynomial, Lascoux atom, Grothendieck polynomial, quasiLascoux polynomial, kaon}

%% 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, ie. [xx, yy] in the text

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%\addbibresource{FPSAC.bib}

\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}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:introduction}

Considerations in representation theory and algebraic geometry yield a number of interesting and important bases of $\Poly_n \coloneqq \mathbb{Z}[x_1, \dots, x_n]$. This extended abstract summarizes  \cite{Monical.Pechenik.Searles}, contributing two new bases and studying their relations to known bases; we find that our new bases exhibit well-behaved structure and fill natural holes in the previously developed theory. This study is part of a general program to develop a combinatorial theory of $\Poly_n$ that mirrors the rich classical theory of symmetric functions.

Foremost among known bases of $\Poly_n$ are the celebrated \emph{Schubert polynomials} $\{ \schub_a \}$ of Lascoux--Sch\"{u}tzenberger \cite{Lascoux.Schutzenberger}.
Let $X = \flags_m(\mathbb{C} )$ be the parameter space of complete flags 
of nested vector subspaces of $\mathbb{C}^m$. The \emph{Schubert varieties} of $X$ are naturally indexed by weak compositions $a= (a_1,\ldots,a_m)$ (i.e., sequences of nonnegative integers) and the corresponding \emph{Schubert classes} $\{ \sigma_a \}$ form a $\mathbb{Z}$-linear basis for the Chow ring $A^\star(X)$. The Schubert polynomials are polynomial representatives for Schubert classes in the sense that one has (up to truncation)
\[\schub_a \cdot \schub_b = \sum_c C_{a,b}^c \; \schub_c \qquad \mbox{ if and only if } \qquad \sigma_a \cdot \sigma_b = \sum_c C_{a,b}^c \; \sigma_c.\]
Despite explicit formulas for Schubert polynomials, it remains a major open problem to give a positive combinatorial formula for the \emph{Schubert structure constants} $C_{a,b}^c \in \mathbb{Z}_{\geq 0}$.

The (type A) \emph{Demazure characters} $\{ \key_a \}$ form another basis of $\Poly_n$, important in representation theory \cite{Demazure}. Remarkably, Demazure characters \emph{refine} Schubert polynomials:
\[
\schub_a = \sum_b E^a_b \key_b\text{, for some \emph{nonnegative} integers $E^a_b \in \mathbb{Z}_{\geq 0}$ \cite{Lascoux.Schutzenberger:key,Reiner.Shimozono}.}
\]

Letting the symmetric group $S_n$ act on $\Poly_n$ by permuting variables, the $S_n$-invariants are the \emph{symmetric polynomials} $\Sym_n \subset \Poly_n$. Remarkably, $\{\schub_a \}$ and $\{ \key_a \}$ each contains (as a subset) the \emph{Schur basis} $\{ \schur_\lambda \}$ of $\Sym_n$; in fact,
$\{\schub_a \} \cap \Sym_n = \{ \key_a \} \cap \Sym_n = \{ \schur_\lambda \}.$
Thus, both $\{\schub_a \}$ and $\{ \key_a \}$ are \emph{lifts} of $\{ \schur_\lambda \}$ to $\Poly_n$. The Schur basis, moreover, has important refinements into the \emph{quasiSchur polynomials} $\{ \qschur_\alpha \}$ of \cite{HLMvW11:QS, HLMvW11:LRrule} and further into the \emph{fundamental quasisymmetric polynomials} $\{ \fundamental_\alpha \}$ of \cite{Gessel}, two bases of the quasisymmetric polynomials $\QSym_n \subset \Poly_n$. (A polynomial $f \in \Poly_n$ is \emph{quasisymmetric} if the coefficient of $x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_k^{\alpha_k}$ equals the coefficient of $x_{j_1}^{\alpha_1} x_{j_2}^{\alpha_2} \cdots x_{j_k}^{\alpha_k}$ for every sequence $j_1 < j_2 < \cdots < j_k$.) 

While a rich combinatorial theory of $\Sym_n$ and $\QSym_n$ exists, the analogous theory for $\Poly_n$ is more sparse. For example, unlike for $\{ \schub_a \}$, many positive combinatorial formulas are known for the structure constants of $\{\schur_\lambda \}$. A natural program, championed by Lascoux \cite{Lascoux:polynomials}, is to develop analogous combinatorial theory for $\Poly_n$ by lifting known bases and relationships to $\Poly_n$ from the better-understood subrings $\Sym_n$ and $\QSym_n$, and by 
developing uniform combinatorial models for these lifted bases and for the relations among them, extending models from $\Sym_n$ and $\QSym_n$. 
This ``modern symmetric function theory'' aims to eventually bear dividends on major problems involving polynomials, such as the Schubert problem mentioned above.

Recent work in this area has provided lifts to $\Poly_n$ of the quasiSchur and fundamental bases of $\QSym_n$: respectively, the \emph{quasikey polynomials} $\{ \qkey_a \}$ of \cite{Assaf.Searles:2} and the \emph{(fundamental) slide polynomials} $\{ \slide_a \}$ of \cite{Assaf.Searles}. (Note that quasikey polynomials are not quasisymmetric, despite the name.) These families provide further refinements of Schubert polynomials: Each Demazure character is a positive sum of quasikeys, each of which is a positive sum of slides \cite{Assaf.Searles:2}. The slide basis even has positive structure constants; in fact, one even has a Littlewood-Richardson-type rule for multiplying slide polynomials \cite{Assaf.Searles}.

A more classical approach to Demazure characters is to study their refinement, not into slides, but into \emph{Demazure atoms} $\{ \atom_a \}$ \cite{Lascoux.Schutzenberger:key,Mason:atom}. While Demazure atoms refine quasikeys \cite{Searles}, as do slides, Demazure atoms have no known direct relation to slides. The \emph{(fundamental) particles} $\{ \particle_a \}$ are a common refinement of $\{ \atom_a \}$ and $\{ \slide_a \}$ \cite{Searles}. \cref{fig:H_bases} illustrates the relations among these nine families of `cohomological' polynomials.

\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}
\node[Kplaque, rounded corners] (Schubert) {\pcolor{$\schub_a$}};
\node[right=1.8 of Schubert,Kplaque, rounded corners] (key) {\pcolor{$\key_a$}};
\node[right=1.8 of key,Kplaque, rounded corners] (qkey) {\pcolor{$\qkey_a$}};
\node[above right=0.6 and 1.8 of qkey,Kplaque, rounded corners] (atom) {\pcolor{$\atom_a$}};
\node[below right=0.6 and 1.8 of qkey,Kplaque, rounded corners] (slide) {\pcolor{$\slide_a$}};
\node[right=4 of qkey,Kplaque, rounded corners] (pion) {\pcolor{$\mathfrak{P}_a$}};
\node[below=2.3 of key,Kplaque, rounded corners] (Schur) {\scolor{$\schur_\lambda$}};
\node[below=2.3 of qkey,Kplaque, rounded corners] (qSchur) {\qcolor{$\qschur_\alpha$}};
\node[below=0.8 of slide,Kplaque, rounded corners] (fundamental) {\qcolor{$F_\alpha$}};
%\node[below right= 5 and 2 of Schubert, Kplaque, rounded corners] (Grothendieck) {\pcolor{$\groth_w$}};
%\node[right=2 of Grothendieck,Kplaque, rounded corners] (Lascoux) {\pcolor{$\lascoux_a$}};
%\node[right=2 of Lascoux,newplaque] (qLascoux) {\pcolor{$\qlascoux_a$}};
%\node[above right=1 and 2 of qLascoux,Kplaque, rounded corners] (Lascouxatom) {\pcolor{$\lascouxatom_a$}};
%\node[below right=1 and 2 of qLascoux,Kplaque, rounded corners] (glide) {\pcolor{$\glide_a$}};
%\node[right=4 of qLascoux,newplaque] (kaon) {\pcolor{$\kaon_a$}};
%\node[below=3 of Lascoux,Kplaque, rounded corners] (symGroth) {\scolor{$Ks_\lambda$}};
%\node[below=3 of qLascoux,Kplaque, rounded corners] (qGroth) {\qcolor{$\qgroth_\alpha$}};
%\node[right=1.88 of qGroth,Kplaque, rounded corners] (multifundamental) {\qcolor{$KF_\alpha$}};
  \begin{scope}[nodes = {draw = none}]
    \path (Schubert) edge[posex]   (key)
    (key) edge[posex]  (qkey)
    (qkey) edge[posex] (atom)
    (qkey) edge[posex] (slide)
    (atom) edge[posex] (pion)
    (slide) edge[posex] (pion)
    (Schur) edge[lift] (Schubert)
    (Schur) edge[lift] (key)
    (qSchur) edge[lift] (qkey)
    (fundamental) edge[lift] (slide)
    (Schur) edge[posex] (qSchur)
    (qSchur) edge[posex] (fundamental)
%    (Grothendieck) edge[posex] (Lascoux)
%    (Lascoux) edge[posex,red]  (qLascoux)
%    (qLascoux) edge[posex,red] (Lascouxatom)
%    (qLascoux) edge[posex,red] (glide)
%    (Lascouxatom) edge[posex,red] (kaon)
%    (glide) edge[posex,red] (kaon)
%    (symGroth) edge[lift] (Grothendieck)
%    (symGroth) edge[lift] (Lascoux)
%    (qGroth) edge[lift] (qLascoux)
%    (multifundamental) edge[lift] (glide)
%    (symGroth) edge[posex] (qGroth)
%    (qGroth) edge[posex,red] (multifundamental)
%    (Schubert) edge[beta] (Grothendieck)
%    (key) edge[beta] (Lascoux)
%    (qkey) edge[beta] (qLascoux)
%    (atom) edge[beta] (Lascouxatom)
%    (slide) edge[beta] (glide)
%    (pion) edge[beta] (kaon)
%    (Schur) edge[beta] (symGroth)
%    (qSchur) edge[beta] (qGroth)
%    (fundamental) edge[beta] (multifundamental)
      ;
  \end{scope}
\end{tikzpicture}
\end{center}
\caption{The nine cohomological families of polynomials considered here. Bases of $\Sym_n$ are shown in \scolor{orange}, bases of $\QSym_n$ in \qcolor{purple}, and bases of $\Poly_n$ in \pcolor{green}. The $\hookrightarrow$ arrows denote that the tail basis is a subset of the head basis. The thick $\rightarrow$ arrows denote that the head basis refines the tail basis.}
\label{fig:H_bases}
\end{figure}

We study \emph{$K$-theoretic analogues} of these bases. A theme of modern Schubert calculus is the study of $\flags_n$ (and generalized flag varieties) through complex oriented cohomology theories. To avoid difficulty in defining appropriate Schubert classes, 
we restrict to (specializations of) \emph{connective $K$-theory}. Complex oriented cohomology theories are determined by their formal group laws. For connective $K$-theory, this law is
\begin{equation}\label{eq:fgl}
c_1(L \otimes M) = c_1(L) + c_1(M) + \beta c_1(L) c_1(M),
\end{equation}
where $\beta$ is a formal parameter and $L,M$ are complex line bundles. Cohomology is recovered at $\beta = 0$, and ordinary $K$-theory is recovered by fixing $\beta \in \mathbb{C}^\star$.
In connective $K$-theory, the \emph{$\beta$-Grothendieck polynomials} $\{ \groth_a \}$ \cite{Fomin.Kirillov} represent a Schubert basis \cite{Hudson}. These form an inhomogeneous basis of $\Poly_n[\beta]$, where $\beta$ is the parameter from (\ref{eq:fgl}). At $\beta =0$, one recovers the Schubert basis $\{ \schub_a \}$.  The usual Grothendieck polynomials \cite{Lascoux.Schutzenberger} are realized at $\beta = -1$. (For clarity, we denote the connective $K$-analogue of each basis of \cref{fig:H_bases} by attaching an `overbar' to its notation.) 

Intersecting $\{ \groth_a \}$ with $\Sym_n[\beta]$ yields the basis $\{ \sgroth_\lambda \}$ of \emph{symmetric Grothendieck polynomials}, representing connective $K$-theory Schubert classes on Grassmannians. Like Schur polynomials, these have quasisymmetric refinements; each $\sgroth_\lambda$ expands positively in the \emph{quasiGrothendieck} basis $\{ \qgroth_\alpha \}$ of $\QSym_n[\beta]$, the $K$-analogue of $\{ \qschur_\alpha \}$ \cite{Monical}.

Our first new result is that quasiGrothendiecks refine further into the multifundamental quasisymmetric polynomials $\{ \multifundamental_\alpha \}$ of \cite{Lam.Pylyavskyy,Pechenik.Searles}, the $K$-analogues of $\{ \fundamental_\alpha \}$.

\begin{theorem}\label{thm:qGroth}
Each quasiGrothendieck polynomial $\qgroth_\alpha \in \QSym[\beta]$ is a positive sum of multifundamental quasisymmetric polynomials. That is,
\[
\qgroth_\alpha = \sum_\gamma J_\gamma^\alpha \; \multifundamental_\gamma, \text{ where $J_\gamma^\alpha \in \mathbb{Z}_{\geq 0}[\beta].$}
\]
\end{theorem}

The basis $\{ \multifundamental_\alpha \}$ of $\QSym_n[\beta]$ lifts to the \emph{glide basis} $\{\glide_a\}$ of $\Poly_n[\beta]$  \cite{Pechenik.Searles},  a $\beta$-deformation of $\{ \slide_a \}$. An analogous deformation of Demazure characters has been studied in \cite{Kirillov:notes,Lascoux:transition,Monical,Monical.Pechenik.Scrimshaw,Ross.Yong}. We call these \emph{Lascoux polynomials} $\{ \lascoux_a \}$. They can be approached via the \emph{Lascoux atom} basis $\{ \lascouxatom_a \}$ of $\Poly_n[\beta]$ \cite{Monical}, deformations of Demazure atoms. 
%(See \cite{Ross.Yong} for discussion of a conjectural formula for decomposing Grothendieck polynomials $\{\groth_a\}$ into Lascoux polynomials $\{ \lascoux_a \}$. )

We introduce the \emph{kaons} $\{ \kaon_a \}$, as a $\beta$-deformation of particles, yielding a common refinement of $\{ \glide_a\}$ and $\{ \lascouxatom_a \}$, for which we give explicit positive formulas.

\begin{theorem} \label{thm:kaon_properties}
The kaons $\{ \kaon_a \}$ are a basis of $\Poly_n[\beta]$. They deform the fundamental particles, in that $\kaon_a$ recovers $\particle_a$ at $\beta = 0$. Kaons  refine both glides and Lascoux atoms; that is,  
\[\glide_a = \sum_b P_b^a \; \kaon_b \,\,\,\,\, \text{ and } \,\,\,\,\, \lascouxatom_a = \sum_b Q_b^a \; \kaon_b, \,\,\,\, \text{ where $P_b^a, Q_b^a \in \mathbb{Z}_{\geq 0}[b]$.}\] 
\end{theorem}

Finally, we introduce the \emph{quasiLascoux polynomials} $\{ \qlascoux_a \}$, both deforming quasikeys and lifting quasiGrothendiecks to $\Poly_n[\beta]$.  QuasiLascoux polynomials are a coarsening of glides and of Lascoux atoms. We give explicit positive formulas for these refinements.

\begin{theorem} \label{thm:qlascoux_properties}
The quasiLascoux polynomials $\{ \qlascoux_a \}$ are a basis of $\Poly_n[\beta]$. They lift the quasiGrothendieck basis in that 
$
\{ \qlascoux_a \} \cap \QSym_n[\beta] = \{ \qgroth_\alpha \}.
$
Moreover, they deform the quasikeys, in that $\qlascoux_a$ recovers $\qkey_a$ at $\beta = 0$.
Finally, the quasiLascoux polynomials refine Lascoux polynomials and are refined by both glides and by Lascoux atoms; that is, 
\[
\lascoux_a = \sum_b L_b^a \; \qlascoux_b, \,\,\,\,\, \;
\qlascoux_a = \sum_b M_b^a \; \glide_b, \,\,\,\, \text{ and }  \,\,\,\,
\qlascoux_a = \sum_b N_b^a \; \lascouxatom_b, \,\,\,\, \text{ where $L_b^a, M_b^a, N_b^a \in \mathbb{Z}_{\geq 0}[\beta]$.}\]
\end{theorem}

\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}
\node[Kplaque, rounded corners] (Grothendieck) {\pcolor{$\groth_a$}};
\node[right=1.8 of Grothendieck,Kplaque, rounded corners] (Lascoux) {\pcolor{$\lascoux_a$}};
\node[right=1.8 of Lascoux,newplaque] (qLascoux) {\pcolor{$\qlascoux_a$}};
\node[above right=.6 and 1.6 of qLascoux,Kplaque, rounded corners] (Lascouxatom) {\pcolor{$\lascouxatom_a$}};
\node[below right=.55 and 1.6 of qLascoux,Kplaque, rounded corners] (glide) {\pcolor{$\glide_a$}};
\node[right=3.5 of qLascoux,newplaque] (kaon) {\pcolor{$\kaon_a$}};
\node[below=2.2 of Lascoux,Kplaque, rounded corners] (symGroth) {\scolor{$\sgroth_\lambda$}};
\node[below=2.1 of qLascoux,Kplaque, rounded corners] (qGroth) {\qcolor{$\qgroth_\alpha$}};
\node[below=.8 of glide,Kplaque, rounded corners] (multifundamental) {\qcolor{$\multifundamental_\alpha$}};
%\node[below right= 5.1 and 2 of Schubert, Kplaque, rounded corners] (Grothendieck) {\pcolor{$\groth_w$}};
%\node[right=2 of Grothendieck,Kplaque, rounded corners] (Lascoux) {\pcolor{$\lascoux_a$}};
%\node[right=2 of Lascoux,newplaque] (qLascoux) {\pcolor{$\qlascoux_a$}};
%\node[above right=1 and 1.9 of qLascoux,Kplaque, rounded corners] (Lascouxatom) {\pcolor{$\lascouxatom_a$}};
%\node[below right=1 and 1.9 of qLascoux,Kplaque, rounded corners] (glide) {\pcolor{$\glide_a$}};
%\node[right=3.6 of qLascoux,newplaque] (kaon) {\pcolor{$\kaon_a$}};
%\node[below=3 of Lascoux,Kplaque, rounded corners] (symGroth) {\scolor{$\sgroth_\lambda$}};
%\node[below=3 of qLascoux,Kplaque, rounded corners] (qGroth) {\qcolor{$\qgroth_\alpha$}};
%\node[right=1.88 of qGroth,Kplaque, rounded corners] (multifundamental) {\qcolor{$\multifundamental_\alpha$}};
  \begin{scope}[nodes = {draw = none}]
    \path %(Schubert) edge[posex]   (key)
    (Grothendieck) edge[posex,dotted] (Lascoux)
    (Lascoux) edge[posex,red]  (qLascoux)
    (qLascoux) edge[posex,red] (Lascouxatom)
    (qLascoux) edge[posex,red] (glide)
    (Lascouxatom) edge[posex,red] (kaon)
    (glide) edge[posex,red] (kaon)
    (symGroth) edge[lift] (Grothendieck)
    (symGroth) edge[lift] (Lascoux)
    (qGroth) edge[lift,red] (qLascoux)
    (multifundamental) edge[lift] (glide)
    (symGroth) edge[posex] (qGroth)
    (qGroth) edge[posex,red] (multifundamental)
      ;
  \end{scope}
%  \begin{scope}[nodes = {draw = none}]
%  \path (qkey) edge[beta] (qLascoux)
%  ;
%  \end{scope}
\end{tikzpicture}
\end{center}
\caption{The $K$-theoretic analogues of the nine polynomials families of \cref{fig:H_bases}.
The arrows and colors of bases are as in \cref{fig:H_bases}.
 Those polynomials and relations that are new are marked in \textcolor{red}{red}. The dotted arrow is conjectural; see \cite{Ross.Yong}.}\label{fig:K_bases}
\end{figure}

The relations among these nine $K$-theoretic families are illustrated in \cref{fig:K_bases}.
Except for $\{ \groth_a \}$ and its subset $\{ \sgroth_\lambda \}$, the geometric significance of these $K$-analogues is currently obscure. While, for example, the glide basis has positive structure constants and appears useful in the study of $\beta$-Grothendieck polynomials (and thereby of the connective $K$-theory of $\flags_n$), it is unknown how to interpret individual glide polynomials geometrically. We conclude with a conjecture that appears to suggest geometric meaning. While it might be proved by combinatorics, the conjecture seems to have the flavor of Euler characteristic calculations. 
The conjecture is fundamentally $K$-theoretic, with no cohomological analogue.
For weak compositions $a$ and $b$, let $M^a_b(\beta)$ be the coefficient of $\glide_b$ in the glide expansion of $\qlascoux_a$ and let $Q^a_b(\beta)$ be the coefficient of $\kaon_b$ in the kaon expansion of $\lascouxatom_a$. 
\begin{conjecture}\label{conj:euler}
For a weak composition $a$, 
$
\sum_b M^a_b(-1) \in \{ 0, 1 \}$ and $\sum_b Q^a_b(-1) \in \{0, 1\},
$
where both sums are over all weak compositions $b$. \qed
\end{conjecture}

For example, for $a = (0,6,6,2)$, we have $\sum_b M_{b}^{a}(\beta) = 16\beta^3 + 75\beta^2 + 94\beta + 36 $  and 
$ \sum_b Q^a_b(\beta) = 16\beta^3 + 66\beta^2 + 80\beta + 31. $
In both cases, substituting $\beta = -1$ yields $1$, as predicted.  
We have checked \cref{conj:euler} for all $a$ with at most 3 zeros and $|a| \leq 7$.

\begin{conjecture}\label{conj:multiplying_lascouxs}
For $a, b$ weak compositions, $\lascoux_a^{(\beta)} \cdot \lascoux_b^{(\beta)}$ is a positive sum of Lascoux atoms.
\end{conjecture}

We have checked \cref{conj:multiplying_lascouxs} for all $a,b$ such that $|a|, |b| \leq 5$ and $a$ and $b$ have at most three $0$s.  Specializing at $\beta=0$ recovers an old conjecture of Reiner--Shimozono.

%This abstract is organized as follows. Section~\ref{sec:defn} recalls necessary combinatorics.  Section~\ref{sec:mesons} introduces kaons and their key properties, while Section~\ref{sec:lascoux} does the same for quasiLascoux polynomials. 

\section{Combinatorial models for $K$-theoretic bases}
\label{sec:defn}

For a weak composition $a$, the {\bf positive part} of $a$ is the (strong) composition $a^+$ obtained by deleting all $0$s from $a$. 
For $a,b$ weak compositions, say $b$ {\bf dominates} $a$, denoted $b \geq a$, if
$
  b_1 + \cdots + b_i \ge a_1 + \cdots + a_i
$
for all $i$. 
 As in \cite{Pechenik.Searles}, a {\bf weak komposition} is a weak composition whose positive integers are colored arbitrarily black or \textcolor{red}{red}.
The {\bf excess} $\ex(b)$ of a weak komposition $b$ is its number of red entries. 
A weak komposition $b$ is a {\bf glide} of $a$ if $b$ can be obtained from $a$ by the following local moves on the colored word:
\begin{itemize}
\item[(m.1)] $0p \Rightarrow p0$, (for $p \in \mathbb{Z}_{>0}$);
\item[(m.2)] $0p \Rightarrow qr$ (for $p,q,r \in \mathbb{Z}_{>0}$ with $q+r = p)$;
\item[(m.3)] $0p \Rightarrow q\red{r}$ (for $p,q,r \in \mathbb{Z}_{>0}$ with $q+r = p+1$).
\end{itemize}
For a weak composition $a$ of length $n$, the {\bf glide (polynomial)} $\glide^{(\beta)}_a = \glide^{(\beta)}_a(x_1, \dots, x_n)$ is 
\[ \glide^{(\beta)}_a = \sum_b \beta^{\ex(b)} x_1^{b_1} \cdots x_n^{b_n}, \] where the sum is over all glides of $a$ \cite{Pechenik.Searles}. 
%As for $\groth^{(\beta)}_w$, we may drop $\beta$ from the notation.
The {\bf fundamental slide polynomials} $\slide_a$ of \cite{Assaf.Searles} are given by
$\slide_a = \glide^{(0)}_a$ \cite{Pechenik.Searles}; we take this as definitional for fundamental slides.

%\begin{example}\label{ex:glidepoly} We have
%  \[\glide_{0201} = \mathbf{x}^{0201} + \mathbf{x}^{1101} + \mathbf{x}^{0210} + \mathbf{x}^{1110} + \mathbf{x}^{2001} + \mathbf{x}^{2010} + \mathbf{x}^{2100} + \] \[\beta\mathbf{x}^{0211} + \beta\mathbf{x}^{1111} + \beta\mathbf{x}^{1201}+\beta\mathbf{x}^{1210} + \beta\mathbf{x}^{2011} + 2\beta\mathbf{x}^{2101} +  2\beta\mathbf{x}^{2110} + \]
%  \[\beta^2\mathbf{x}^{1211} + 2\beta^2\mathbf{x}^{2111},\]
%  where $\mathbf{x}^b = x_1^{b_1}\ldots x_n^{b_n}$.
%\end{example}

The {\bf skyline diagram} $D(a)$ of a weak composition $a$ has $a_i$ left-justified boxes in row $i$ (for us, row 1 is the lowest).
A {\bf triple} of $D(a)$ is a set of $3$ boxes with $2$ adjacent in a row and either the third box above the right box and the lower row weakly longer, or the third box below the left box and the higher row strictly longer. 
Given a numerical filling of the skyline diagram, a triple is called a {\bf coinversion triple} if $\alpha \le  \gamma \le \beta$ (where $\gamma$ is the label of the third box); otherwise, it is an {\bf inversion triple}.
Following \cite{Monical},
a {\bf set-valued filling} of a skyline diagram is an assignment of a non-empty set of positive integers to each box.
The greatest entry in each box is the {\bf anchor}; other entries are {\bf free}.
A set-valued filling is {\bf semistandard} if
(S.1) entries do not repeat in a column, 
(S.2) rows are weakly decreasing (where sets $A \geq B$ if $\min A \geq \max B$),
(S.3) every triple of anchors is an inversion triple, 
(S.4) each free entry appears with the least anchor in its column such that (S.2) is not violated, and 
(S.5) anchors in column $1$ equal their row indices.
%
%\begin{remark}\label{rmk:svsky}
%The condition (S.5) replaces the equivalent ``basement'' requirement in \cite{Monical}. The condition (S.4) above differs slightly from condition (S4) in \cite[\textsection 1.2]{Monical}, which puts free entries in the lowest possible row such that (S.2) is not violated. These definitions are however equivalent in the sense that there is a simple weight-preserving (and moreover column set-preserving) bijection between these two notions of semistandard set-valued skyline fillings via rearranging the free entries appropriately in each column. The convention in Definition~\ref{def:setSSF} turns out to be more natural in the context of the operations we wish to perform on these fillings.
%\end{remark}

Given a set-valued filling $F$ of shape $a$, the {\bf weight} of $F$ is the weak composition $\wt(F) = (c_1,\ldots,c_n)$ where $c_i$ is the number of $i$'s in $F$.   The {\bf excess} $\ex(F)$ of $F$ is its number of free entries. 
For a weak composition $a$, let $\LASSF(a)$ be the set of semistandard set-valued skyline diagrams of shape $a$.; see \cref{fig:SetSkyFill}. Then, the {\bf Lascoux atom} $\lascouxatom_a^{(\beta)}$ is 
\[ \lascouxatom_a^{(\beta)} = \sum_{F \in \LASSF(a)} \beta^{\ex(F)}\x^{\wt(F)}, \] 
while the {\bf Demazure atom} $\atom_a$ is the $\beta = 0$ specialization:
$
\atom_a = \lascouxatom_a^{(0)}$ \cite{Lascoux.Schutzenberger:key,Mason:atom,Monical}.
Lascoux atoms form a basis of $\Poly[\beta]$ \cite{Monical}. 

\begin{figure}[ht]
\begin{center}
\begin{minipage}{.4\linewidth}
\begin{center}
\scalebox{.8}{
	\begin{ytableau}
	\none & \textbf{4} & \textbf{4}31 \\
	\none  & \textbf{3}2 & \textbf{2} & \textbf{2}\\
	\none \vline \\
	\none  & \textbf{1}  
	\end{ytableau}}
\end{center}
\end{minipage}
\begin{minipage}{.4\linewidth}
\begin{center}
  \scalebox{.8}{
	\begin{ytableau}
	 \none & \textbf{4}2 & \textbf{2}1 \\ 
	 \none & \textbf{3} & \textbf{3} & \textbf{1}\\ 
	 \none \vline \\
	\none & \textbf{1} \\ 
	\end{ytableau}}
\end{center}
\end{minipage}
\end{center}
\caption{Two elements of $\LASSF(1,0,3,2)$ with the anchors drawn in bold.}\label{fig:SetSkyFill}
\end{figure}

For a composition $\alpha$, the {\bf quasiGrothendieck polynomial} $\qgroth_\alpha^{(\beta)}$ in $n$ variables is  
\[\qgroth_\alpha^{(\beta)}(x_1,\ldots , x_n) = \sum_{a^+ = \alpha} \lascouxatom_a, \]
where the sum is over weak compositions of length $n$ \cite{Monical}.
The $\beta=0$ specialization 
is the {\bf quasiSchur polynomial} $\qschur_\alpha$ \cite{HLMvW11:LRrule}.
QuasiGrothendieck polynomials are another basis of $\QSym[\beta]$ \cite{Monical}.
The Lascoux atoms refine the symmetric Grothendieck polynomials:

\begin{theorem}[\cite{Monical}]\label{thm:GLasDecomp}
\[ \sgroth_\lambda^{(\beta)}(x_1, \ldots x_n) = \sum_{\sort(a) = \lambda} \lascouxatom_a^{(\beta)},\]
where the sum is over weak compositions of length $n$ and $\sort(a)$ is the partition formed by sorting the parts of $a$ in weakly decreasing order. 
\end{theorem}

\begin{corollary}[\cite{Monical}] \label{cor:grothtoquasigroth}
\[ \sgroth_\lambda^{(\beta)}(x_1, \ldots , x_n) = \sum_{\sort(\alpha) = \lambda} \qgroth_\alpha^{(\beta)}(x_1,\ldots , x_n).\]
\end{corollary}

%Setting $\beta=0$ in Theorem~\ref{thm:GLasDecomp} and Corollary~\ref{cor:grothtoquasigroth} recovers earlier refinements \cite{HLMvW11:QS} of Schur polynomials into  Demazure atoms and into quasiSchur polynomials.

\section{New bases and their relations}\label{sec:mesons}
We introduce a new ``kaon'' basis of polynomials, simultaneous refinements of both glides and Lascoux atoms. 
Let $a$ be a weak composition with nonzero entries in positions $n_1 < \dots < n_\ell$. 
The weak komposition $b$ is a {\bf mesonic glide} of $a$ if $b$ can be obtained from $a$ by a finite sequence of the local moves (m.1), (m.2), and (m.3) that never applies (m.1) at positions $n_j-1$ and $n_j$ for any $j$.
Note that mesonic glides are special cases of glides.

%\begin{example}\label{ex:mesonic_glide}
%Let $a = (0,3,0,2)$. Then $b = (2,1,1,\red{2})$ is a mesonic glide of $a$. 
%
%On the other hand, while $b' = (3,1,0,\red{2})$ is also a glide of $a$, it is not mesonic. To see this fact, observe that $a_{n_1} = a_2 = 3$, while 
%\[
%b'_{n_{j-1} + 1} + \dots + b'_{n_j} - \ex(b'_{n_{j-1}+1}, \dots, b'_{n_{j}}) = b'_1 + b'_2 - \ex(b'_1, b'_2) = 3 +1 - 0 = 4,
%\]
%in violation of (G.$1'$).
%
%The reader may check that both $b$ and $b'$ can be obtained from $a$ by a finite sequence of the local moves (m.1), (m.2), and (m.3). However, the reader may also check that $b'$ cannot be so obtained without applying (m.1) at positions $1$ and $2$.
%\end{example}

\begin{definition}
Let $a$ be a weak composition. The {\bf kaon} $\kaon_a^{(\beta)}$ is the following generating function for mesonic glides:
\[
\kaon_a^{(\beta)} \coloneqq \sum_b \beta^{\ex(b)} {\bf x}^b, \text{ where the sum is over all mesonic glides of $a$.}
\]
\end{definition}

%\begin{example}\label{ex:kaon}
%Let $a = (0,3,0,2)$. Then the corresponding kaon is
%\begin{align*}
%\kaon_a^{(\beta)} &= \x^{0302} + \x^{0311} + \x^{1202} + \x^{1211} + \x^{2102} + \x^{2111} \\
% &+ \beta\x^{0312} + \beta\x^{0321} + \beta\x^{1212} + \beta\x^{1221} + \beta\x^{1302} + \beta\x^{1311} \\ 
% &+  \beta\x^{2112} + \beta\x^{2121} + \beta\x^{2202} + \beta\x^{2211} + \beta\x^{3102} + \beta\x^{3111} \\
% &+ \beta^2\x^{1312} + \beta^2\x^{1321} + \beta^2\x^{2212} + \beta^2\x^{2221} + \beta^2\x^{3112} + \beta^2\x^{3121}.
%\end{align*}
%\end{example}

\begin{proposition}\label{prop:glide2kaon}
Every glide $\glide_a^{(\beta)}$ is a positive sum of kaons. Indeed,
\[
\glide_a^{(\beta)} = \sum_{\substack{b \geq a \\ b^+= a^+ }} \kaon_b^{(\beta)}.
\]
\end{proposition}

\begin{theorem}\label{thm:kaon_basis}
The set 
\[
\{ \beta^k \kaon^{(\beta)}_a : k \in \mathbb{Z}_{\geq 0} \text{ and $a$ is a weak composition of length $n$} \} 
\]
is an additive basis of the free $\mathbb{Z}$-module $\Poly_n[\beta]$.
\end{theorem}

The \emph{fundamental particle} basis $\{ \particle_a \}$ of $\Poly_n$ is a common refinement of fundamental slides and Demazure atoms \cite{Searles}. We show that kaons play the analogous role for glides and Lascoux atoms.

\begin{proposition}\label{prop:pions_are_specialized_kaons}
The fundamental particles $\particle_a$ are the kaons at $\beta=0$:  $\particle_a = \kaon_a^{(0)}.$
\end{proposition}

%\begin{remark}
%Proposition~\ref{prop:pions_are_specialized_kaons} motivates our choice of the name `kaon' for these polynomials. In high energy physics, the `$K$' fundamental particles are the \emph{K-mesons} or \emph{kaons}. By analogy, perhaps, the cohomological specialization $\pi_a = \kaon_a^{(0)}$ could be named for the lighter analogue of the $K$-meson, the \emph{$\pi$-meson} or \emph{pion}. These mesons play a role in the structural integrity of atomic nuclei; somewhat analogously, we will show momentarily that the kaon polynomials decompose the Lascoux atoms, thereby controlling their structure.
%\end{remark}

%\begin{remark}
%In light of Proposition~\ref{prop:pions_are_specialized_kaons}, setting $\beta = 0$ in Proposition~\ref{prop:glide2kaon} recovers \cite[Proposition~4.7]{Searles} on the expansion of fundamental slide polynomials into fundamental particles. Note that although the $K$-theoretic deformations of both families of polynomials are significantly larger, the matrix of basis change from Proposition~\ref{prop:glide2kaon} is exactly the same as for fundamental slides into fundamental particles. Taking $p = 0$ in Theorem~\ref{thm:kaon_basis} recovers \cite[Proposition~4.6]{Searles} as a special case.
%\end{remark}

Kaons do not have positive structure coefficients. Nonetheless, we conjecture:

\begin{conjecture}\label{conj:kaon_pos_product}
For any weak compositions $a$ and $b$, the product
$ \kaon_a \cdot \glide_b$
of a kaon and a glide polynomial expands positively in the kaon basis.
\end{conjecture}

We have verified \cref{conj:kaon_pos_product} for all weak compositions $a,b$ with at most 3 zeros and $|a|, |b| \leq 5$. To our knowledge, \cref{conj:kaon_pos_product} is new even in the special $\beta=0$ case of the fundamental particle expansion of the product of a fundamental particle by a fundamental slide polynomial.

\begin{definition}\label{def:meson_highest}
Let $a$ be a weak composition and $T \in \LASSF(a)$. We say $T$ is {\bf meson-highest} if, for every integer $i$ appearing in $T$, either 
\begin{itemize}
\item the leftmost $i$ is in the leftmost column and is an anchor, or 
\item there is a $i^\uparrow$ in some column weakly to the right of the leftmost $i$ and in a different box, where $i^\uparrow$ is the smallest label greater than $i$ appearing in $T$.
\end{itemize}
We write $\latok(a)$ for the set of all meson-highest $T \in \LASSF(a)$.
\end{definition}

\begin{theorem}\label{thm:LascouxAtom2kaon}
For any weak composition $a$, we have
\begin{equation}\label{eq:LascouxAtom2kaon}
\lascouxatom_a^{(\beta)} = \sum_{T \in \latok(a)} \beta^{|T|-|a|}\kaon_{\wt(T)}^{(\beta)}.
\end{equation}
In particular, every Lascoux atom $\lascouxatom_a^{(\beta)}$ is a positive sum of kaons.
\end{theorem}

%\begin{figure}[ht]
%  \begin{center}
%    \begin{displaymath}
%      \begin{array}{c@{\hskip2\cellsize}c@{\hskip2\cellsize}c@{\hskip2\cellsize}c@{\hskip2\cellsize}c}
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{3} \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} \vspace{1cm} &
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{3}2 \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{2} \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%	\begin{ytableau}
%	\none  & \textbf{3}2 & \textbf{2} \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	\\
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{3}1 \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{3}21 \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} &
%
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{2}1 \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%
%	\begin{ytableau}
%	\none  & \textbf{3}2 & \textbf{2}1 \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%      \end{array}
%    \end{displaymath}
%    \caption{The eight elements of $\LASSF(102)$. The set $\latok(102)$ consists of two fillings, specifically the leftmost filling in each of the two rows.  Each of the eight illustrated fillings destandardizes to the leftmost filling in its row.}\label{fig:lascouxatom}
%  \end{center}
%\end{figure}


%
%\begin{remark}
%Setting $\beta=0$ in Theorem~\ref{thm:LascouxAtom2kaon} recovers the expansion of Demazure atoms into fundamental particles from \cite{Searles}. In particular, the meson-highest fillings with no free entries are exactly the ``particle-highest'' fillings of \cite{Searles}.
%\end{remark}

The quasikey basis of \cite{Assaf.Searles:2} is a coarsening of the fundamental slides and Demazure atoms, a refinement of the Demazure characters, and a lift of the quasiSchur basis from $\QSym_n$ to $\Poly_n$. 
We introduce a $K$-analogue of the quasikey basis, a lift of the quasiGrothendiecks from $\QSym[\beta]$ to $\Poly[\beta]$. 

\begin{definition}\label{def:QL}
For a weak composition $a$, the {\bf quasiLascoux polynomial} $\qlascoux_a$ is 
\[\qlascoux_a^{(\beta)} =  \sum_{\stackrel{b\ge a}{b^+=a^+}}\lascouxatom_b^{(\beta)}.\]
\end{definition}

\begin{proposition}\label{prop:qlascoux_basis}
The set 
\[
\{ \beta^k \qlascoux^{(\beta)}_a : k \in \mathbb{Z}_{\geq 0} \text{ and $a$ is a weak composition of length $n$} \} 
\]
is an additive basis of the free $\mathbb{Z}$-module $\Poly_n[\beta]$. 
\end{proposition}

\begin{proposition}\label{prop:generalize_qkey}
The quasikey polynomials are the $\beta=0$ specialization of quasiLascoux polynomials: $\qlascoux_a^{(0)}=\qkey_a$.
\end{proposition}

\begin{proposition}\label{prop:generalize_qGroth}
Suppose that the positions of the nonzero entries in the weak composition $a$ form an interval and that $a_k$ is the last nonzero entry of $a$. Then, 
\[  \qlascoux_a^{(\beta)}  = \qgroth_{a^+}^{(\beta)}(x_1, \ldots , x_k).   \] 
In particular, every quasiGrothendieck polynomial is a quasiLascoux polynomial.
\end{proposition}

Moreover, we have

\begin{proposition}\label{prop:qlascoux_stable_limit}
Let $a$ be a weak composition. Then the stable limit $\lim_{m \to \infty} \qlascoux_{0^m \times a}^{(\beta)}$  of the quasiLascoux polynomial $\qlascoux_a^{(\beta)}$ is the quasiGrothendieck function $\qgroth_{a^+}^{(\beta)}(x_1, x_2, \ldots ).$
\end{proposition}

To give the monomial expansion of a quasiLascoux polynomial directly, we define $\qlascouxSSF(a)$ (the {\bf set-valued quasi-skyline fillings} of $a$) to be all set-valued skyline fillings of shape $a$ satisfying (S.1)--(S.4), as well as
\begin{itemize}
\item[(S.$5'$)] anchors in the first column are at most their row index and decrease from top to bottom.
\end{itemize}

\begin{proposition}\label{prop:LSetSF}
Given a weak composition $a$, we have
\[\qlascoux_a^{(\beta)} = \sum_{S\in \qlascouxSSF(a)}\beta^{|S|-|a|}\x^{\wt(S)}.\]
\end{proposition}

%\begin{example}
%For $a = (1,0,2)$, the set $\qlascouxSSF(a)$ consists of the ten fillings shown in Figure~\ref{fig:qlascoux}. 
%Therefore, we have 
%\[\qlascoux_{102}^{(\beta)} = \x^{102} + \x^{111} + \x^{120} + \beta\x^{112} + \beta\x^{202} + \beta\x^{121} + \beta\x^{211} + \beta\x^{220} + \beta^2\x^{212} + \beta^2\x^{221}.\]
%\end{example}

%\begin{figure}[h]
%  \begin{center}
%    \begin{displaymath}
%      \begin{array}{c@{\hskip2\cellsize}c@{\hskip2\cellsize}c@{\hskip2\cellsize}c@{\hskip2\cellsize}c@{\hskip2\cellsize}c}
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{3} \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} \vspace{1cm} &
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{3}2 \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{3}1 \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{3}21 \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{2} \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	\\
%	\begin{ytableau}
%	\none  & \textbf{3} & \textbf{2}1 \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%	\begin{ytableau}
%	\none  & \textbf{3}2 & \textbf{2} \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%	\begin{ytableau}
%	\none  & \textbf{3}2 & \textbf{2}1 \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%	\begin{ytableau}
%	\none  & \textbf{2} & \textbf{2} \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&
%	\begin{ytableau}
%	\none  & \textbf{2} & \textbf{2}1 \\
%	\none \vline \\
%	\none  & \textbf{1}  
%	\end{ytableau} 	&								
%      \end{array}
%    \end{displaymath}
%    \caption{\label{fig:qlascoux}The ten elements of $\qlascouxSSF(102)$.}
%  \end{center}
%\end{figure}

\begin{definition}\label{def:QY_setvalued_skylines}
Let $a$ be a weak composition and let $S \in \qlascouxSSF(a)$ be a set-valued quasi-skyline filling. We say $S$ is {\bf quasiYamanouchi} if, for every integer $i$ appearing in $S$, either 
\begin{itemize}
\item the leftmost $i$ is an anchor in row $i$ of the leftmost column, or 
\item there is an $i+1$ in some column weakly right of the leftmost $i$ and in a different box.
\end{itemize}
We write $\qltog(a)$ for the set of all quasiYamanouchi $S \in \qlascouxSSF(a)$.
\end{definition}

%\begin{remark}
%The quasiYamanouchi condition (Definition~\ref{def:QY_setvalued_skylines}) for set-valued quasi-skyline fillings is exactly the meson-highest condition (Definition~\ref{def:meson_highest}) for semistandard set-valued skyline fillings, with $i^\uparrow$ replaced by $i+1$.
%\end{remark}
%
%\begin{example}
%The first and third fillings in the top row of Figure~\ref{fig:qlascoux} are quasiYamanouchi. The other fillings in Figure~\ref{fig:qlascoux} are not.
%\end{example}

\begin{theorem}\label{thm:QuasiLascoux2glide}
For any weak composition $a$, we have
\begin{equation*}
\qlascoux_a^{(\beta)} = \sum_{S \in \qltog(a)} \beta^{|S|-|a|}\glide_{\wt(S)}^{(\beta)}.
\end{equation*}
In particular, every quasiLascoux polynomial $\qlascoux_a^{(\beta)}$ is a positive sum of glide polynomials.
\end{theorem}

%\begin{example}
%The first, second, fifth, seventh and ninth fillings of Figure~\ref{fig:qlascoux} destandardize to the first filling; the remaining fillings destandardize to the third filling.
%\end{example}


\begin{remark}
Setting $\beta=0$ in the statement of \cref{thm:QuasiLascoux2glide} yields a positive combinatorial formula for the fundamental slide expansion of a quasikey polynomial in terms of quasiYamanouchi semi-skyline fillings. Such a formula was alluded to in \cite{Searles}, but not stated explicitly.
\end{remark}

\begin{corollary}\label{cor:qG_into_multifunds}
The quasiGrothendieck polynomials expand positively in the basis of multi-fundamental quasisymmetric polynomials.
\end{corollary}

We now study the Lascoux polynomials of \cite{Monical} and their relations to the other polynomials in this paper.
Given a skyline diagram, we augment it on the left with an additional column $0$, called the {\bf basement}. We write $b_i$ for the basement entry in row $i$. A set-valued skyline filling with basement is {\bf semistandard} if it (including the basement) satisfies (S.1), (S.2), (S.3), and (S.4). Basement entries do not count towards the weight $\wt(F)$ of a filling $F$ with basement. 
%In diagrams, we shade the boxes of the basement in \textcolor{gray}{gray} to distinguish them from the ordinary boxes.
For a weak composition $a$, let $\overleftarrow{a}$ denote the weak composition formed by reversing the order of the parts of $a$. 
 Let $\LPSSF(a)$ be the set of semistandard set-valued skyline fillings of shape $\overleftarrow{a}$ with basement $b_i = n-i+1$.

\begin{definition}[{\cite{HLMvW11:LRrule}, \cite{Monical}}]\label{def:L}
Let $a$ be a weak composition.  The {\bf Lascoux polynomial} $\lascoux_a$ is \[ \lascoux_a^{(\beta)} = \sum_{F \in \LPSSF(a)} \beta^{\ex(F)}\x^{\wt(F)}. \] The {\bf Demazure character} is the $\beta=0$ specialization: 
$
\key_a = \lascoux_a^{(0)}.
$
\end{definition}

%Demazure characters are, in fact, characters of certain modules with relation to Schubert calculus \cite{Demazure}; no such representation-theoretic realization of Lascoux polynomials is currently known.


%\begin{figure}[ht]
%\begin{center}
%    \begin{displaymath}
%      \begin{array}{c@{\hskip2\cellsize}c@{\hskip2\cellsize}c@{\hskip2\cellsize}c@{\hskip2\cellsize}c@{\hskip2\cellsize}c}
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{2} & \textbf{1} \\
%\end{ytableau} \vspace*{.5cm} & 
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{2} & \textbf{2} \\
%\end{ytableau} &
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{2} & \textbf{2}1 \\
%\end{ytableau} &
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{3} & \textbf{1} \\
%\end{ytableau} &
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{3} & \textbf{2} \\
%\end{ytableau} \\
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{3} & \textbf{3} \\
%\end{ytableau} \vspace*{.5cm} &
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{3} & \textbf{2}1 \\
%\end{ytableau} &
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{3} & \textbf{3}1 \\
%\end{ytableau} &
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{3} & \textbf{3}2 \\
%\end{ytableau} &
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{3} & \textbf{3}21 \\
%\end{ytableau} \\ &
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{3}2 & \textbf{1} \\
%\end{ytableau} &
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{3}2 & \textbf{2} \\
%\end{ytableau} &
%\begin{ytableau}
%*(gray) 1 & \textbf{1} \\
%*(gray) 2 \\
%*(gray) 3 & \textbf{3}2 & \textbf{2}1 \\
%\end{ytableau} &
%      \end{array}
%    \end{displaymath}
%    \caption{The 13 elements of $\LPSSF(102)$.}\label{fig:lascouxPoly}
%  \end{center}
%\end{figure}

%\begin{example}
%The monomial expansion of the Lascoux polynomial $\lascoux_{102}^{(\beta)}$ is
%\begin{align*}
%       \lascoux_{102}^{(\beta)} &= x_1^2x_2 + x_1x_2^2 + \beta x_1^2x_2^2 + x_1^2x_3 + x_1x_2x_3 \\ 
%       &+ x_1x_3^2 + \beta x_1^2x_2x_3  + \beta x_1^2x_3^2 + \beta x_1x_2x_3^2 + \beta^2 x_1^2x_2x_3^2 \\
%        &+ \beta x_1^2 x_2x_3 + \beta x_1x_2^2x_3 + \beta^2 x_1^2x_2^2x_3. \end{align*} 
%\end{example}

Our last main result is that Lascoux polynomials are positive sums of quasiLascoux polynomials. We first give a positive formula for the Lascoux atom expansion of a Lascoux polynomial. %that $\lascoux_a^{(\beta)}$ is a positive sum of Lascoux atoms.
Given a weak composition $a$, let $w(a)$ be the minimal (Coxeter) length permutation sending $a$ to $\sort(a)$. 

\begin{theorem}\label{thm:lDecomp}
For any weak composition $a$, we have 
\[
\lascoux_a^{(\beta)} = \sum_{\stackrel{\sort(b)=\sort(a)}{w(b)\le w(a)}} \lascouxatom_b^{(\beta)}, \text{ where  $\le$ denotes strong Bruhat order on permutations.}
\]
In particular, every Lascoux polynomial $\lascoux_a^{(\beta)}$ is a positive sum of Lascoux atoms.
\end{theorem}

%\begin{example} Let $a=(0,1,0,3)$. Then $\sort(a)=(3,1,0,0)$ and $w(a)=3241$. Hence,
%\[\lascoux_{0103}^{(\beta)} = \lascouxatom_{0103}^{(\beta)} + \lascouxatom_{1003}^{(\beta)} + \lascouxatom_{0130}^{(\beta)} + \lascouxatom_{1030}^{(\beta)}+ \lascouxatom_{1300}^{(\beta)} + \lascouxatom_{0301}^{(\beta)} + \lascouxatom_{0310}^{(\beta)} + \lascouxatom_{3001}^{(\beta)}+ \lascouxatom_{3010}^{(\beta)} + \lascouxatom_{3100}^{(\beta)}.\]
%\end{example}

Specializing \cref{thm:lDecomp} at $\beta=0$ recovers a known formula for the Demazure atom expansion of a Demazure character (see, e.g., \cite{Mason:atom}, \cite{HLMvW11:LRrule}).


\begin{remark}
In \cite{Lascoux:transition}, Lascoux introduced $K$-theoretic analogues of Demazure characters in terms of divided difference operators. Monical \cite{Monical} conjectured that these are the Lascoux polynomials $\lascoux_a^{(\beta)}$. Similarly, operators give polynomials that conjecturally equal the Lascoux atoms \cite{Monical}.
By \cite{Monical}, the basis change between these two operator-defined bases is also as in \cref{thm:lDecomp}. 
Hence, \cref{thm:lDecomp} proves the equivalence of \cite[Conjecture~5.2]{Monical} and \cite[Conjecture~5.3]{Monical}.   
\end{remark}


%Before continuing with our proofs of the positive expansions of a Lascoux polynomial in the Lascoux atom and quasiLascoux bases, we formulate the following related conjecture, which is in some sense a strengthening of Theorem~\ref{thm:lDecomp}.

The Lascoux polynomials expand positively in the basis of quasiLascoux polynomials. 
Following \cite{Assaf.Searles:2}, define $\Qlswap(a)$ to be all $b\in \lswap(a)$ such that if $c\in \lswap(a)$ and $b^+ = c^+$, then $c \ge b$. 

%\begin{example}
%For the weak compositions $a=(0,1,2)$ and $b = (0,3,1)$, we have
%\[
% \Qlswap(a) = \{(0,1,2), (0,2,1)\}
%\]
%and
%\[ \pushQED{\qed}  \Qlswap(b) = \{ (0,3,1), (1,3,0) \} . \qedhere \popQED \] \let\qed\relax 
%\end{example}

\begin{theorem}\label{LtoQL}
For any weak composition $a$, we have
\[\lascoux_a^{(\beta)} = \sum_{b\in \Qlswap(a)} \qlascoux_b^{(\beta)}.\]
In particular, every Lascoux polynomial $\lascoux_a^{(\beta)}$ is a positive sum of quasiLascoux polynomials.
\end{theorem}

\begin{remark}
The $\beta=0$ specialization of \cref{LtoQL} recovers the expansion of Demazure characters in the quasikey basis \cite{Assaf.Searles:2}.
\end{remark}

\acknowledgements{OP was partially supported by an NSF Postdoctoral Fellowship (\#1703696) and is grateful for conversations with A.~Knutson, E.~Sergel, and A.~Thurston-Shaine.  CM was partially supported by a GAANN Fellowship from the Mathematics Department, UIUC.}

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