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%\usepackage[utf8]{inputenc}
%\usepackage[colorlinks]{hyperref}
\usepackage[colorinlistoftodos]{todonotes}
\usepackage{pgf}
\usepackage{tikz}
\usepackage{tikz-cd}
\usetikzlibrary{positioning,shapes,shadows,arrows}
\usepackage{ytableau}

\newcommand{\Mark}[1]{\todo[size=\tiny,inline,color=green!30]{#1
\\ \hfill --- Mark}}
\newcommand{\Tianyi}[1]{\todo[size=\tiny,inline,color=blue!30]{#1
\\ \hfill --- Tianyi}}

\newtheorem{prop}{Proposition}[section]
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\newcommand{\col}{\mathrm{col}}
\newcommand{\CR}{\mathrm{CR}}
\newcommand{\CSS}{\mathrm{CSS}}
\newcommand{\Dec}{\mathrm{Dec}}
\newcommand{\df}{:=}
\newcommand{\extra}{\mathrm{ex}}
\newcommand{\fG}{\mathfrak{G}}
\newcommand{\fGb}{\fG^{(\beta)}}
\newcommand{\fL}{\mathfrak{L}}
\newcommand{\fLb}{\fL^{(\beta)}}
\newcommand{\fmin}{f_{\mathrm{min}}}
\newcommand{\fmax}{f_{\mathrm{max}}}
\newcommand{\fS}{\mathfrak{S}}
\newcommand{\Gb}{G^{(\beta)}}
\newcommand{\HD}{\xrightarrow{H}}
\newcommand{\HDR}{\xleftarrow{H,>}}
\newcommand{\HeckeMonoid}{\mathcal{H}}
\newcommand{\Hequiv}{\equiv_H}
\newcommand{\HI}{\xrightarrow{H,<}}
\newcommand{\HIR}{\xleftarrow{H,<}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\Inc}{\mathrm{Inc}}
\newcommand{\Insert}{\mathrm{Insert}}
\newcommand{\JNW}{J^\nwarrow}
\newcommand{\JSE}{J^\searrow}
\newcommand{\Kequiv}{\equiv_K}
\newcommand{\KK}{K^{\mathrm{Kjdt}}}
\newcommand{\key}{\mathrm{key}}
\newcommand{\la}{\lambda}
\newcommand{\lastcol}{\col_{\mathrm{last}}}
\newcommand{\myref}[1]{\href{#1}{#1}}
\newcommand{\pab}{\partial^{(\beta)}}
\newcommand{\pib}{\pi^{(\beta)}}
\newcommand{\rev}{\mathrm{rev}}
\newcommand{\RevInsert}{\mathrm{RevInsert}}
\newcommand{\RSVT}{\mathrm{RSVT}}
\newcommand{\RSSYT}{\mathrm{RSSYT}}
\newcommand{\SSYT}{\mathrm{SSYT}}
\newcommand{\SVT}{\mathrm{SVT}}
\newcommand{\switch}{\mathrm{switch}}
\newcommand{\VNW}{V^\nwarrow}
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\newcommand{\word}{\mathrm{word}}
\newcommand{\wt}{\mathrm{wt}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\Ct}{\widetilde{\mathcal{C}}}
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\newcommand{\Tt}{\widetilde{\mathcal{T}}}

%% define your title in the usual way
\title{Grothendieck-to-Lascoux Expansions}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[M.~Shimozono and T.~Yu]{Mark Shimozono\thanks{\href{mailto:mshimo@math.vt.edu}{mshimo@math.vt.edu}}\addressmark{1} 
\and
Tianyi Yu\thanks{\href{mailto:tiy059@ucsd.edu}{tiy059@ucsd.edu}}\addressmark{2} 
}


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\address{\addressmark{1}Department of Mathematics, Virginia Tech\\ \addressmark{2}Department of Mathematics, University of California, San Diego}

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\abstract{We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the $K$-theoretic analogue of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. We extend our expansions to flagged Grothendieck polynomials. }

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%\resume{\lipsum[2]}

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\keywords{Grothendieck polynomial, Demazure character, Lascoux polynomial, set-valued tableau, Hecke insertion} 

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\begin{document}

\maketitle
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\section{Introduction}
The Grothendieck polynomials $\fG_w$ of Lascoux and Sch\"utzenberger~\cite{LS:Groth} are explicit polynomial representatives of the $K$-classes of structure sheaves of Schubert varieties in flag varieties. Reiner and Yong~\cite{ReY} conjectured an explicit combinatorial expansion of Grothendieck polynomials into the basis of Lascoux polynomials $\fL_\alpha$~\cite{Las}.
Our first main theorem (Theorem~\ref{T:first main theorem}) 
rediscovers a combinatorial formula for the Lascoux polynomials, which is implicit in~\cite{BSW}. This is used to prove our
second main theorem (Theorem~\ref{T:second main theorem}) which
establishes the Reiner--Yong conjecture.

\section{Four expansions}
The Grothendieck-to-Lascoux expansion fits into a 
family of four related expansions.
The polynomials to be expanded are the cohomological and K-theoretic Schubert bases given by the Schubert polynomials $\fS_w$ and the Grothendieck polynomials $\fG_w$ respectively, and their symmetrized or stable versions, known as the Stanley symmetric functions $F_w$ and Grothendieck symmetric functions (also known as stable Grothendieck polynomials) $G_w$. These are respectively expanded into type A Demazure characters (also called key polynomials) $\kappa_\alpha$, Lascoux polynomials $\fL_\alpha$, Schur functions $s_\la$, and Grassmannian Grothendieck symmetric functions $G_\la$.

\[
\begin{matrix}
\begin{tikzcd}
\fS_w \arrow[rr,"\mathrm{symmetrize}"] \arrow[d,"\mathrm{expand}"] \arrow[d,swap,"(a)"]  && F_w \arrow[d,swap,"\mathrm{expand}"] \arrow[d,"(b)"] \\
\kappa_\alpha\arrow[rr,swap,"\mathrm{symmetrize}"] && s_\la
\end{tikzcd} \\ \\
\text{cohomology}
\end{matrix}
\qquad
\begin{matrix}
\begin{tikzcd}
\fG_w \arrow[rr,"\mathrm{symmetrize}"] \arrow[d,"\mathrm{expand}"]\arrow[d,swap,"(c)"] && G_w \arrow[d,swap,"\mathrm{expand}"] \arrow[d,"(d)"]\\
\fL_\alpha\arrow[rr,swap,"\mathrm{symmetrize}"] && G_\la
\end{tikzcd} \\ \\
\text{$K$-theory}
\end{matrix}
\]
Using the formalism of connective $K$-theory (equivalently, introducing a harmless grading parameter $\beta$ into the Grothendieck polynomial), as we do in this article, all expansions specialize to their $K$-theoretic or cohomological counterparts by setting $\beta$ to $-1$ or $0$ respectively. 

In chronological order, expansion (b) was established by Edelman and Greene \cite{EG} via a Schensted-type insertion algorithm for reduced words. The expansion (a) was found by Lascoux and Sch\"utzenberger and proved in \cite{RS}. Expansion (d) was established by Buch, Kresch, Shimozono, Tamvakis, and Yong \cite{BKSTY} via Hecke insertion, which takes Hecke words as input. Expansion (c) is the topic of this article.

The expansion coefficients have geometric significance.
The Stanley-to-Schur coefficients of the expansion (b) coincide with large rank affine Stanley to affine Schur coefficients \cite[Proposition~9.17]{LLS}, which in turn coincide with Gromov--Witten invariants for the flag variety via Peterson's Quantum Equals Affine Theorem \cite{LS:QHGr, Pet}, \cite[Part 3, Section~10]{LLMSSZ}. Specializing $w$ to a Zelevinsky permutation, 
(a) and (b) give the expansion of cohomology classes of equioriented type A quiver loci \cite[Theorem 7.14]{KMS}, the latter being shown by Buch and Fulton \cite{BF} to specialize to virtually all known variants of type A Schubert polynomials.
Expansions (c) and (d) give analogous expansions in K-theory \cite{B,Mi}.

The nonsymmetric expansions are subtle refinements of their symmetric counterparts. In the symmetric expansions there is a set of tableaux in which each tableau $T$ in the set, gives a copy of $s_\la$ or $G_\la$ where $\la$ is the shape of $T$. There is a corresponding term $\kappa_\alpha$ or $\fL_\alpha$ in the nonsymmetric expansion, but an additional datum must be supplied: a composition or extremal weight $\alpha$ in the symmetric group orbit of $\la$; see \eqref{E:L sym} through \eqref{E:S sym}. Such constructions assigning a composition to a tableau go by the general name of \emph{key}. In the crystal graph of semistandard Young tableaux of shape $\la$, the left and right keys of the tableau $T$ of shape $\la$ are given by the final and initial directions of the corresponding Littelmann path whose highest weight vector is the directed line segment from the origin to $\la$. The initial direction indicates the smallest Demazure crystal containing the given tableau.

\section{Grothendieck and Lascoux polynomials}
The group $S_+=\bigcup_{n\ge1} S_n$ acts on $R=\Z[\beta][x_1,x_2,\dotsc]$ by permuting the variables: for $i\ge1$, let $s_i$ exchange $x_i$ and $x_{i+1}$. We define the following operators on $R$, where an element $f\in R$ (or its fraction field) denotes the operator of left multiplication by $f$.
\begin{align}
\partial_i &:= (x_i-x_{i+1})^{-1}(1 - s_i), \\
\pi_i &:= \partial_i x_i, \\
\pab_i &:= \partial_i (1+\beta x_{i+1}), \\
\pib_i &:= \pab_i x_i.
\end{align}
All satisfy the braid relations for $S_+$.

Let $w_0^{(n)}\in S_n$ be the long element and $\rho^{(n)}=(n-1,n-2,\dotsc,1,0)$. For $w\in S_n$ the $\beta$-Grothendieck polynomial is defined by \cite{LS:Groth}
\begin{align}
\fGb_w &:=
\begin{cases}
x^{\rho^{(n)}} &\text{if $w=w_0^{(n)}$,} \\
\pab_i \fGb_{ws_i} &\text{if $ws_i>w$.}
\end{cases}
\end{align}
Since the $\pab_i$ satisfy the braid relations, $\fGb_w$ is well-defined for $w\in S_n$. It is also well-defined for $w\in S_+$, that is, unchanged under the standard embedding $S_n\to S_{n+1}$ for all $n\ge1$.
The Schubert $\fS_w$ and Grothendieck polynomials $\fG_w$ are defined by
\begin{align}
\fS_w := \fGb_w|_{\beta=0}, \\ 
\fG_w := \fGb_w|_{\beta=-1}.
\end{align}

Let $\alpha=(\alpha_1,\alpha_2,\dotsc)$ be a composition (sequence of nonnegative integers, almost all $0$). The Lascoux polynomial $\fLb_\alpha$
is defined by \cite{Las}
\begin{align}
\fLb_\alpha = \begin{cases}
x^\alpha & \text{if $\alpha$ is a partition,} \\
\pib_i \fLb_{s_i\alpha} &\text{if $\alpha_i<\alpha_{i+1}$.}
\end{cases}
\end{align}
The Demazure character $\kappa_\alpha$ is defined by
\begin{align}
\kappa_\alpha = \fLb_{\alpha}|_{\beta=0}.
\end{align}

Given a composition $\alpha=(\alpha_1,\dotsc,\alpha_n)\in\Z_{\ge0}^n$ let $\alpha^+$ be the unique partition in the $S_n$-orbit of $\alpha$. For $w\in S_n$ and $w_0\in S_n$ the long element we have the symmetrizations
\begin{align}
\label{E:L sym}
\pib_{w_0}(\fL_\alpha) &= \Gb_{\alpha^+}(x_1,\dotsc,x_n), \\
\label{E:kappa sym}
\pi_{w_0}(\kappa_\alpha) &= s_{\alpha^+}(x_1,\dotsc,x_n), \\
\label{E:G sym}
\pib_{w_0}(\fG_w(x)) &= G_w(x_1,\dotsc,x_n), \\
\label{E:S sym}
\pi_{w_0}(\fS_w(x)) &= F_w(x_1,\dotsc,x_n).
\end{align}

\section{Tableau formula for Lascoux polynomials}
\label{SS:new Lascoux formula}
Given the definition of a certain kind of tableau which involves entries in a totally ordered set, 
we say ``reverse'' to mean the same definition but with the total order reversed.
Thus, a \emph{reverse semistandard Young tableau} (RSSYT) is a tableau where 
entries weakly decrease along rows from left to right 
and strictly decrease along columns from top to bottom.

For a partition $\la$,
a \emph{reverse set-valued tableau} (RSVT) $T$ of shape $\la$ is 
a filling of the boxes of $\la$ 
by finite nonempty subsets of $\Z_{>0}$ satisfying the following. For the box $s\in \la$ let $T(s)$ be the set which occupies the box $s$ in $T$.
\begin{enumerate}
\item $\min(T(s)) \ge \max(T(t))$ if the box $t$ is immediately right of the box $s$ in $\la$.
\item $\min(T(s))>\max(T(t))$ if the box $t$ is immediately below the box $s$ in $\la$.
\end{enumerate}
This is the reverse of Buch's set-valued tableaux \cite{B}.

Given a RSVT $T$, let $L(T)$ be the RSSYT obtained from $T$ by replacing every entry $T(s)$ by its largest value $\max(T(s))$.

The \emph{weight} $\wt(T)$ of a tableau $T$ is the composition 
whose $i$-th part is the total number of times $i$ appears in $T$.

A \emph{key tableau} (or just key) is a RSSYT of partition shape such that the $j$-th column, 
viewed as a set, contains the $(j+1)$-th for all $j$. 
There is a bijection
$\alpha\mapsto \key(\alpha)$ from compositions to keys where
$\key(\alpha)$ is the unique RSSYT of shape $\alpha^+$ 
and weight $\alpha$. 
Its $j$-th column consists of the numbers $\{i\mid \alpha_i \ge j\}$.

The \emph{left key} $K_-(T)$ of a RSSYT $T$ is a key computed in the following way.
For each $j$,
let $T_{\leq j}$ be the RSSYT we get if we only keep the first $j$ columns of $T$.
We may anti-rectify $T_{\leq j}$ using the jeu-de-taquin (jdt).
Then the leftmost column of the result becomes column $j$ of $K_-(T)$.

Let $|\alpha|=\sum_{i\ge1} \alpha_i$. 
Let $\RSVT_\la$ be the set of reverse set-valued tableaux of shape $\la$.
For $T\in\RSVT_\la$ let $\extra(T)=|\wt(T)|-|\la|$.
Our first main theorem is:

\begin{thm} \label{T:first main theorem}
For any composition $\alpha$
\begin{align}\label{E:Lascoux RSVT}
  \fLb_\alpha = \sum_{\substack{T\in\RSVT_{\alpha^+} \\ K_-(L(T))\le \key(\alpha)}} \beta^{\extra(T)} x^{\wt(T)}.
\end{align}
Here $\le$ indicates entrywise conparison.
\end{thm}

\begin{ex}
The following RSVTs 
contribute to $\fLb_{(1,0,2)}$:
\begin{align*}
\begin{ytableau}
2 & 1\\
1 
\end{ytableau}\,,
\\[1mm]
\begin{ytableau}
2 & 2\\
1 
\end{ytableau}\,,
&&
\begin{ytableau}
2 & 21\\
1 
\end{ytableau}\,,
\\[1mm]
\begin{ytableau}
3 & 1\\
1 
\end{ytableau}\,,
&&
\begin{ytableau}
32 & 1\\
1 
\end{ytableau}\,,
\\[1mm]
\begin{ytableau}
3 & 2\\
1 
\end{ytableau}\,,
&&
\begin{ytableau}
3 & 21\\
1 
\end{ytableau}\,,
&&
\begin{ytableau}
32 & 2\\
1 
\end{ytableau}\,,
&&
\begin{ytableau}
32 & 21\\
1 
\end{ytableau}\,,
\\[1mm]
\begin{ytableau}
3 & 3\\
1 
\end{ytableau}\,,
&&
\begin{ytableau}
3 & 31\\
1 
\end{ytableau}\,,
&&
\begin{ytableau}
3 & 32\\
1 
\end{ytableau}\,,
&&
\begin{ytableau}
3 & 321\\
1 
\end{ytableau}\,.
\end{align*}
Thus, we may write $\fLb_{(1,0,2)}$ as
\begin{equation*}
\begin{split}
&x_1^2x_2 + x_1x_2^2 + x_1^2x_3 
+ x_1x_2x_3 + x_1x_3^2 \\
+ & \beta(x_1^2x_2^2 + 2x_1^2x_2x_3 + 
x_1x_2^2x_3 + x_1^2x_3^2 + x_1x_2x_3^2)\\
+ & \beta^2(x_1^2x_2^2x_3 + x_1^2x_2x_3^2).
\end{split}
\end{equation*}
\end{ex}

\begin{rem} \label{R:Lascoux}
There have been a number of conjectural combinatorial formulas for Lascoux polynomials, such as the K-Kohnert move rule of Ross and Yong \cite{RoY} (\cite[Footnote on page~19]{Kir} for the general $\beta$ version),
the set-valued skyline filling formula of Monical \cite{Mon}, and a set-valued tableau (SVT) rule of Pechenik and Scrimshaw \cite{PS}. 
Buciumas, Scrimshaw and Weber \cite{BSW} proved the last two of these rules using 
solvable lattice models. 
In response to a previous version of this article, Travis Scrimshaw kindly informed us that Theorem~\ref{T:first main theorem} is implicit in \cite{BSW}: see the proof of \cite[Theorem 4.4]{BSW}. 
We feel it is worthwhile to state these theorems in their simplest and most explicit form.
We note that the naive nonreversed analogue of the RSVT formula does not yield the Lascoux polynomial.
\end{rem}

\section{Fomin--Kirillov monomial formula}
\label{SS:FK}
Our point of departure is the explicit monomial expansion of Grothendieck polynomials due to Fomin and Kirillov \cite{FK}.

The \emph{$0$-Hecke monoid} $\HeckeMonoid$ is the quotient of the free monoid of words on the alphabet $\Z_{>0}$ by the relations 
\begin{align}
i (i+1) i &\Hequiv (i+1) i (i+1), \\
ii &\Hequiv i, \\
ij &\Hequiv ji \qquad\text{for $|i-j| \ge 2$.}
\end{align}
$\HeckeMonoid$ acts on $S_+$ by 
\begin{align*}
  i * w = \begin{cases}
  s_i w & \text{if $s_i w>w$}, \\
  w &\text{if $s_iw<w$.}
  \end{cases}
\end{align*}
Given a word $u\in \HeckeMonoid$ define its \emph{associated permutation} by
$u * \id\in S_+$. For $w\in S_+$ let $\HeckeMonoid_w$ be the words in 
$\HeckeMonoid$ with associated permutation $w$. The subsets $\HeckeMonoid_w\subset \HeckeMonoid$ are the $\Hequiv$-equivalence classes.

\begin{lem} \label{L:reverse Hecke word} $u\in \HeckeMonoid_w$ if and only if $\rev(u)\in\HeckeMonoid_{w^{-1}}$.
\end{lem}


For $a\in\HeckeMonoid_w$ let $\extra(a) = \text{length}(a) - \ell(w)$, the \emph{excess} of the length of $a$ above the minimum possible, the Coxeter length $\ell(w)$ of $w$.

The following is merely the definition in \cite{BJS} but with both words reversed, which is better suited to our 
use of decreasing tableaux.
\begin{defn}[{\cite{BJS}}] \label{D:compatible pair}
A pair of words $(a,i)$
is \emph{compatible} if
they satisfy
\begin{enumerate}
\item $a,i$ are words of positive
numbers with the same length.
\item $i$ is weakly decreasing
\item $i_j = i_{j+1}$ implies 
$a_j < a_{j+1}$.
\end{enumerate}
A compatible pair $(a,i)$ is \emph{bounded} 
if $i_j \le a_j$ for all $j$.
\end{defn}
Let $\C$ be the set of all compatible pairs, $\C^b$ those that are bounded, $\C_w$ the pairs $(a,i)\in \C$ such that $a\in \HeckeMonoid_w$, and $\C_w^b = \C^b \cap \C_w$.
The following monomial expansion of $\beta$-Grothendieck polynomials is due to Fomin and Kirillov \cite{FK}:
\begin{align}\label{E:Groth compatible pairs}
\fGb_w = \sum_{(a,i)\in \C_{w^{-1}}^b} \beta^{\extra(a)} x^{\wt(i)}.
\end{align}
When $\beta=0$ this is the Billey--Jockusch--Stanley formula for Schubert polynomials~\cite{BJS}.

For $w\in S_n$ and a positive integer $N$ let
$1^N\times w$ be the permutation of $S_{n+N}$ obtained by adding $N$ fixed points before $w$. The $\beta$-Grothendieck symmetric function is defined by
\begin{align}
\Gb_w = \lim_{N\to\infty} \fG_{1^N\times w}
\end{align}
It lives in a completion of the ring of symmetric functions over $\Z[\beta]$.
The Stanley and Grothendieck symmetric functions are defined by
\begin{align}
F_w = \Gb_w|_{\beta=0} \\
G_w = \Gb_w|_{\beta=-1}.
\end{align}
It follows from \eqref{E:Groth compatible pairs} and the definitions that
\begin{align}\label{E:GSF compatible pairs}
\Gb_w = \sum_{(a,i)\in \C_{w^{-1}}} \beta^{\extra(a)} x^{\wt(i)}.
\end{align}

\section{\texorpdfstring{$\Gb_w$ to $\Gb_\la$}{Gw to Gla} via Hecke insertion: restriction of compatible pairs according to \texorpdfstring{$w$}{w}}
\label{SS:Grass Groth}
The code $c(w)$ of a permutation $w$ is the sequence $(c_1,c_2,\dotsc)$ such that 
$$c_i = | \{j \mid\text{$1 \le j < w(i)$ and $w^{-1}(j)>i$} \}|.$$
For a partition $\la=(\la_1,\la_2,\dotsc,\la_k)$ the \emph{Grassmannian Grothendieck symmetric function} $\Gb_\la$ is by definition equal to $\Gb_w$ where $w$ is the permutation with code $(\la_k,\dotsc,\la_1,0,0,\dotsc)$.

Buch \cite{B} showed that the $\Z[\beta]$-span of the $\Gb_w$ for $w\in S_+$, has basis given by the $\Gb_\la$ and proved the increasing version of the following:
\begin{align}\label{E:Grass Groth tableau}
\Gb_\la = \sum_{T\in\RSVT_\la} \beta^{\extra(T)} x^{\wt(T)}
\end{align}
For $\beta=0$ this becomes the RSSYT formula for the Schur function $s_\la$.

To find the coefficients of the $\Gb_w$ to $\Gb_\la$ expansion, the \emph{Hecke insertion} algorithm was developed in \cite{BKSTY} in the language of \emph{increasing tableaux}, which strictly increase along rows from left to right and 
strictly increase along columns from top to bottom. 
We use the variant of Hecke insertion for \emph{decreasing tableaux}, which strictly decrease along rows from left to right and strictly decrease along columns from top to bottom.

It was not explicitly stated in \cite{BKSTY} but all ingredients are there to define a Hecke Robinson--Schensted--Knuth (RSK) bijection called Insert (and its inverse bijection RevInsert)
\tikzstyle{start}=[to path={(\tikztostart.#1) -- (\tikztotarget)}]
\[
\begin{tikzcd}[every arrow/.append style={shift left}]
    \C \arrow[rr,"\textrm{Insert}"] && \bigsqcup_\la (\Dec_\la \times \RSVT_\la)=:\T \arrow[ll,"\textrm{RevInsert}"] \\
    (a,i) \arrow[rr] && (P,Q) \arrow[ll]
\end{tikzcd}
\]
where $\Dec_\la$ is the set of decreasing tableaux of shape $\la$. Note that the set $\T$ as defined by the above diagram, consists of pairs $(P,Q)$ of tableaux of the same partition shape with $P$ decreasing and $Q$ reverse set-valued.

Let $(P,Q)=\Insert(a,i)$. 
Let $\word(P)$ be the word we get
if we read entries of $P$ from left to right,
and move from bottom to top within each column. 
Then the bijection satisfies
\begin{align}
\rev(a) \Hequiv \word(P) \quad \text{ and } \quad \wt(Q)=\wt(i).
\end{align}
By Lemma~\ref{L:reverse Hecke word},
the  bijection $\Insert$ restricts to a bijection 
\begin{align}\label{E:w bijection}
\C_{w^{-1}}\leftrightarrow \bigsqcup_\la \left(\Dec_\la^w \times \RSVT_\la\right):=\T_w
\end{align}
where $\Dec_\la^w = \{ T\in \Dec_\la\mid \word(T)\in \HeckeMonoid_w\}$.
Taking the generating function of both sides we obtain
\begin{align}
  \Gb_w = \sum_\la |\Dec_\la^w| \Gb_\la.
\end{align}


\section{\texorpdfstring{$\fGb_w$ to $\fLb_\alpha$}{Gw to Lal} by Hecke insertion and keys: restriction to bounded compatible pairs}
\label{SS:bounded bijection}
Let $\star$ denote the following right action of the monoid of words with letters in the 
set $\Z_{>0}$, on the set of subsets of $\Z_{>0}$.

Let $S \subseteq \mathbb{Z}_{>0}$
and let $m\in \Z_{>0}$. 
Let $m'$ be the smallest number in $S$ of value
at least $m$.
If $m'$ does not exist, 
we let $S \star m = S \sqcup \{m \}$.
Otherwise,
we define $S \star m = (S - \{ m'\})
\sqcup \{ m \}$.

More generally, 
if $w = w_1 \dots w_n$ is a word of
positive integers, 
we define $S \star w =
(S \star w_1) \star (w_2 \dots w_n)$,
and $S \star w = S$ 
if $w$ is the empty word.

\begin{ex}
We have: 
\begin{align*}
\emptyset \star 3414 & = \{1,4\},\\
\{3,4,7\} \star 3414 & = \{1,4,7\}.
\end{align*}
\end{ex}
For each decreasing tableau $P$,
we define its right key $K_+(P)$ to be the 
RSSYT whose $j$-th column is the column given by 
$\emptyset \star \word(P_{\ge j})$ where $P_{\ge j}$ is the 
decreasing tableau obtained by removing the first $j-1$ columns of $P$.

The \emph{K-jeu-de-taquin} (Kjdt) of Thomas and Yong \cite{TY} may be used (following \cite{ReY} for increasing tableaux) to give another definition of right key of decreasing tableau, 
and two definitions of right key are shown to coincide. 

Let $\T^b$ be the subset of pairs $(P,Q)\in \T$ such that $K_+(P) \ge K_-(L(Q))$.
With $\T_w$ defined as in \eqref{E:w bijection},
let $\T_w^b = \T^b \cap \T_w$.
Then we show:

\begin{thm} \label{T:compatible bounded}
$\Insert$ restricts to a bijection $\C^b \cong \T^b$.
\end{thm}

Intersecting with the bijection \eqref{E:w bijection}, $\Insert$ restricts to a bijection 
\begin{align}\label{E:w bounded bijection}
\C^b_{w^{-1}}\cong \T^b_w \qquad\text{for every $w\in S_+$.} 
\end{align}
Using Theorem~\ref{T:first main theorem} we obtain our second main theorem, 
the Grothendieck-to-Lascoux expansion via decreasing tableaux.

\begin{thm} \label{T:second main theorem} 
\begin{align}\label{E:Groth to Lascoux}
\fGb_w = \sum_\la \sum_{P\in \Dec_\la^w} \fLb_{\wt(K_+(P))}.
\end{align}
\end{thm}

\section{Connecting with the Reiner--Yong conjecture}

The Reiner--Yong conjecture asserts:

\begin{thm}[{\cite{ReY}}]
\begin{align}
\fGb_w = \sum_\la \sum_{P\in \Inc_\la^{w^{-1}}} \fLb_{\wt(K_-(P))},
\end{align}
where $\Inc_\la^w$ is similar to $\Dec_\la^w$ except that the tableaux are increasing and $K_-(P)$ is the left key construction on the increasing tableau $P$ using the Kjdt. 
\end{thm}

\begin{proof}[Proof sketch]
We construct a map
$\Dec_\la\to\Inc_\la: T\mapsto T^\sharp$.
For each $T \in \Dec_\la$,
we anti-rectify it using Kjdt and then rotate
the result by $180^o$.
We show it is a bijection satisfying:
$K_+(T)=K_-(T^\sharp)$ and $\word(T^\sharp)\Kequiv \rev(\word(T))$.
Using Lemma~\ref{L:reverse Hecke word} we see that $T\in \HeckeMonoid_w$ if and only if $T^\sharp\in\HeckeMonoid_{w^{-1}}$. Thus 
the bijection restricts to a bijection $\Dec_\la^w\cong \Inc_\la^{w^{-1}}$ as required.
\end{proof}




\section{Flagged Grothendieck to Lascoux}
In this section we extend our expansion to flagged Grothendieck polynomials.

In the literature there is a definition of flagged Grothendieck polynomial whose generality extends to the 
case of 321-avoiding permutations \cite{Mat}; see \cite{KMY2} for the case of vexillary permutations.
For 321-avoiding permutations there is a monomial tableau formula and a determinantal formula. 

We use a divided difference definition of flagged Grothendieck polynomial from~\cite{LLS:back},
which is valid for any permutation. This flagged Grothendieck polynomial has an explicit monomial expansion given in Proposition~\ref{P:flagged G}.

The main result of this subsection is a Lascoux polynomial expansion of flagged Grothendiecks.

A \emph{flag} is a sequence of integers $f=(f_1,f_2,\cdots,f_n)$ which is weakly increasing, satisfies 
$f_i \ge i$ for all $i$, and $f_n=n$. Let $\fmin=(1,2,\cdots,n)$ and $\fmax=(n,n,\cdots,n)$ be the minimum and maximum flags respectively. Given a flag $f$, define the permutation $\sigma_f\in S_n$ as follows. For the minimum flag $\fmin$ let $\sigma_{\fmin} = \id$. For $f\ne \fmin$ there is an index $j$ such that $f_j > j$; take the minimum such. Define $\sigma_f = s_i \sigma_{f'}$ where $i+1=f_j$ and $f'$ is obtained from $f$ by replacing the $i+1$ by $i$. The \emph{flagged Grothendieck polynomial} is defined by $\fGb_{w,f}=\pib_{\sigma_f}(\fGb_w)$.

The flagged Grothendieck polynomials have the following explicit monomial expansion.

\begin{prop}[{\cite{LLS:back}}]\label{P:flagged G}
	\begin{align}\label{E:flagged G}
		\fGb_{w,f} &= \sum_{\substack{(a,i)\in \C_{w^{-1}} \\ i_k \le f_{a_k}}}
		\beta^{\extra(a)} x^{\wt(i)} .
	\end{align}
\end{prop}

\begin{rem}
Note that only the bound $i_k \le a_k$ in \eqref{E:Groth compatible pairs} has been changed to 
$i_k \le f_{a_k}$. 
\end{rem}

The flagged Grothendieck polynomials interpolate between Grothendieck polynomials and 
their symmetric counterparts.

\begin{cor} \label{C:symmetrize G} For $w\in S_n$, 
let $w_0$ be the long element in $S_n$.
Then
	\begin{align}\label{E:symmetrize G}
		\pib_{w_0}(\fGb_w) = \Gb_w(x_1,\dotsc,x_n).
	\end{align}
\end{cor}
\begin{proof} We have
\begin{align*}
\pib_{w_0}(\fGb_w) 
&= \pib_{w_0}(\fGb_{w,\fmin}) \\
&= \fGb_{w,\fmax} \\
&= \Gb_w(x_1,\dotsc,x_n),
\end{align*}
where the last equality holds by the equality of \eqref{E:flagged G} with
\eqref{E:GSF compatible pairs} with $x_i$ set to $0$ for $i>n$.
\end{proof}

Define the Demazure action $\circ$ of $S_+$ on compositions by
\begin{align}
   s_i \circ \alpha = \begin{cases}
   s_i(\alpha) & \text{if $\alpha_i > \alpha_{i+1}$}, \\
   \alpha & \text{otherwise.}
   \end{cases}
\end{align}

Theorem~\ref{T:second main theorem} implies the following.

\begin{cor} 
\begin{align}
    \fGb_{w,f} = \sum_\lambda \sum_{P\in \Dec_\lambda^w} \fLb_{\sigma_f \circ \wt(K_+(P))}.
\end{align}
\end{cor}

\acknowledgements{The authors thank Alex Yong for helpful conversations and especially for sharing with us the details of his conjecture with Vic Reiner. M. S. thanks Tomoo Matsumura for help related to flagged Grothendieck polynomials. T. Y. thanks Brendon Rhoades for helpful conversations.}
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