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\title{Colored five-vertex models and Lascoux polynomials and atoms}

\author{
Valentin Buciumas\thanks{\href{mailto:valentin.buciumas@gmail.com}{valentin.buciumas@gmail.com}. Partially supported by the Australian Research Council DP180103150.}\addressmark{1},
Travis Scrimshaw\thanks{\href{mailto:tcscrims@gmail.com}{tcscrims@gmail.com}. Partially supported by the Australian Research Council DP170102648.}\addressmark{1},
\and
Katherine Weber\thanks{\href{mailto:webe0629@umn.edu}{webe0629@umn.edu}. }\addressmark{2}}

\address{\addressmark{1}School of Mathematics and Physics, 
The University of Queensland, 
St.\ Lucia, 
Australia
\\
\addressmark{2}School of Mathematics,
University of Minnesota,
Minneapolis, MN,
USA}

\keywords{Lascoux polynomial, Lascoux atom, Grothendieck polynomial, colored lattice model, five-vertex model}
%\subjclass[2010]{05E05, 82B23, 14M15, 05A19}

%% put the date of submission here
\received{\today}

\abstract{
We construct an integrable colored five-vertex model whose partition function is a Lascoux atom based on the five-vertex model of Motegi and Sakai and the colored five-vertex model of Brubaker, the first author, Bump, and Gustafsson.
We then modify this model in two different ways to construct a Lascoux polynomial, yielding the first known proven combinatorial interpretation of a Lascoux polynomial and atom.
Using this, we prove a conjectured combinatorial interpretation in terms of set-valued tableaux of a Lascoux polynomial and atom due to Pechenik and the second author.
We also prove the combinatorial interpretation of the Lascoux atom using set-valued skyline tableaux of Monical.
}

\begin{document}

\maketitle

\section{Introduction}
\label{sec:intro}

Solvable lattice models are often models for simplified physical systems such as water molecules, but are known to have applications to a diverse number of mathematical fields.
By tuning the Boltzmann weights, special functions can be expressed as the partition function of the lattice model.
Then the Yang--Baxter equation can be used on the model in order to prove functional equations for the partition function, often simplifying intricate combinatorial or algebraic arguments.
For example, this approach was applied by Kuperberg in counting the number of alternating sign matrices using a six-vertex model~\cite{Kuperberg96}.
Similar techniques have also been used to study probabilistic models such as the (totally) asymmetric simple exclusion process, \textit{e.g.},~\cite{MS13}.

We will be focusing on the five-vertex model of Motegi and Sakai~\cite{MS13,MS14} (with a gauge transformation on the Boltzmann weights), whose partition function is a \emph{(symmetric $\beta$-)Grothendieck polynomial}~\cite{FK94,LS82,LS83}. This was used to establish a Cauchy identity and skew decomposition for Grothendieck polynomials.
Grothendieck polynomials arise from the study of the connective K-theory of the \emph{Grassmannian}, the space of $k$-dimensional subspaces of $\CC^n$, where they correspond to the push-forward of the class for any Bott--Samelson resolution of a Schubert variety.
These form a basis for the connective K-theory ring of the Grassmannian and are indexed by partitions that fit inside a $k \times (n-k)$ rectangle.
Thus, they are the K-theory analog of Schur functions, which are recovered by setting $\beta = 0$.
Grothendieck polynomials have been well-studied with a combinatorial interpretation using set-valued tableaux and a Littlewood--Richardson rule~\cite{Buch02}.
Recently, a crystal structure was applied to set-valued tableaux~\cite{MPS18}.
The equivariant K-theory of the Grassmannian was studied using integrable systems by Wheeler and Zinn-Justin~\cite{WZJ16}, yielding a construction of double Grothendieck polynomials.

There is a refinement of Schur functions that are known as \emph{key polynomials} given in terms of divided difference operators~\cite{Lascoux01}.
Key polynomials are also known as Demazure characters as they can be interpreted as characters of Demazure modules, which also have crystal bases and an explicit combinatorial description~\cite{K93} and a geometric construction~\cite{Andersen85,LMS79}.
The K-theory analog of key polynomials are the so-called \emph{Lascoux polynomials}~\cite{Lascoux01}, which despite recent attention~\cite{Kirillov:notes,Monical16,MPS18,MPS18II,PS19,RY15}, do not have any known geometric or representation theoretic interpretation and have many conjectural combinatorial interpretations~\cite{Kirillov:notes,Monical16,MPS18,PS19,RY15}, some of which are known to be equivalent~\cite{Monical16,MPS18II}.

The goal of this paper is to modify the five-vertex model so that the partition function is a Lascoux polynomial.
To do this, we need an even smaller piece, the \emph{Lascoux atom}~\cite{Monical16}, which is essentially the new terms that appear when taking a larger Lascoux polynomial and has a description in terms of divided difference operators.
On the solvable lattice model side, we employ the idea of Borodin and Wheeler of using a \emph{colored} lattice model~\cite{BorodinWheelerColored}, where one can then study the atoms of special functions.
%Borodin and Wheeler used a colored vertex model to study nonsymmetric spin Hall--Littlewood polynomials and nonsymmetric Macdonald polynomials~\cite{BorodinWheelernsMac}.
%Brubaker, Bump, the first author, and Gustafsson studied Iwahori Whittaker functions on $p$-adic groups\footnote{Here Iwahori Whittaker functions should be considered as atoms for the spherical Whittaker function; the latter is given, up to a quantum factor, by a Schur polynomial by the work of Shintani, Casselman, and Shalika} using colored lattice models~\cite{BBBGIwahori}.
Indeed by modifying the colored five-vertex by Brubaker, Bump, the first author, and Gustafsson~\cite{BBBG19} using the Motegi--Sakai weights, our main result is the construction of an integrable colored five-vertex model whose partition function is a Lascoux atom.
Then by a suitable modification of our model, we obtain a Lascoux polynomial.
In fact, we provide two such modifications and show they are naturally in bijection.

As an application, we prove~\cite[Conjecture 6.1]{PS19}, thus establishing the first combinatorial interpretation of Lascoux polynomials and atoms by using a notion of a K-key tableau of a set-valued tableau.
We do this by refining the bijection between Gelfand--Tsetlin patterns and states of our five-vertex model to allow markings in certain places, as in~\cite{MPS18}, in order to obtain a bijection with set-valued tableaux.
To make this weight preserving, we need to also twist by the Lusztig involution (an action of the long element of the symmetric group), which utilizes the crystal structure on set-valued tableaux from~\cite{MPS18}.
Another application is proving the conjectured combinatorial interpretation of~\cite[Conjecture 5.2]{Monical16}.
We do this by (1) showing our model is in bijection with reverse set-valued tableaux; (2) noting the bijection from~\cite[Theorem~2.4]{Monical16} is governed by the semistandard case of~\cite{Mason08} and adding the so-called free entries (which are just markings on certain vertices the state); and (3) using that the semistandard case is known to give Demazure atoms~\cite{Mason08,Mason09}.

This extended abstract is organized as follows.
In \cref{sec:background}, we provide the necessary background on tableaux combinatorics, Grothendieck polynomials, and Lascoux polynomials/atoms. 
In \cref{sec:coloredmodels} we introduce a new colored lattice model and prove by using a Yang--Baxter equation, that its partition function is equal to a Lascoux atom. 
In \cref{sec:proofofconjecture} we prove~\cite[Conjecture 6.1]{PS19} and~\cite[Conjecture 5.2]{Monical16} by using our main result.

This is an extended abstract of~\cite{BSW19}, where we refer the reader for additional details.







%===============================================================================
\section{Background}
\label{sec:background}

Fix a positive integer $n$.
Let $\zz = (z_1, z_2, \dotsc, z_n)$ be a finite number of indeterminates.
%For any sequence $(\alpha_1, \dotsc, \alpha_n) \in \ZZ^n$ of length $n$, denote $\zz^{\alpha} := z_1^{\alpha_1} z_2^{\alpha_2} \cdots z_n^{\alpha_n}$.
Let $\sym{n}$ denote the symmetric group on $n$ elements with simple transpositions $(s_1, \dotsc, s_{n-1})$.
For $w \in \sym{n}$, let $\ell(w)$ denote the \defn{length} of $w$: the minimal number of simple transpositions whose product equals $w$.
We denote by $w_0$ the longest element in  $S_n$. 
Let $\leq$ denote the (strong) Bruhat order on $\sym{n}$.
For more information on the symmetric group, we refer the reader to~\cite{Sagan01}.
%For any $w \in \sym{n}$, define $w\zz = (z_{w(1)}, z_{w(2)}, \dotsc, z_{w(n)})$.
Let $\lambda = (\lambda_1, \lambda_2, \dotsc, \lambda_n)$ be a \defn{partition}, a sequence of weakly decreasing nonnegative integers. Let $\ell(\lambda)$ be the \defn{length}, the number of non-zero entries of $\lambda$.
We use English convention for our tableaux.


%%%%%%%%%%%%%%%%%%%%
%\subsection{Tableaux combinatorics}

A \defn{(semistandard) set-valued tableau of shape $\lambda$} is a filling of the boxes of the Young diagram of $\lambda$ with finite non-empty sets of positive integers that satisfy
\[
\ytableausetup{boxsize=1.5em}
\ytableaushort{XY,Z}\ \ \text{ implies } \max X \leq \min Y \text{ and } \max X < \min Z.
\]
Let $\svt^n(\lambda)$ denote the set of all set-valued tableaux of shape $\lambda$ such that the maximum integer appearing is $n$.
Let $\ssyt^n(\lambda)$ denote the set of \defn{semistandard (Young) tableaux}, where every entry has size $1$, of shape $\lambda$ and maximum entry $n$.
Define the \defn{$\beta$-weight} of a set-valued tableau $T \in \svt^n(\lambda)$ to be
\[
\wt_{\beta}(T) := \beta^{|T|-|\lambda|} z_1^{m_1} \dotsc z_n^{m_n}
\]
where $m_i$ are the number of $i$'s occurring in $T$ and $|T| = m_1 + \cdots + m_n$.

A semistandard Young tableau is called a \defn{key tableau} if the entries of column $i + 1$ are a subset of the entries of column $i$ for all $1 \leq i < \lambda_1$.
We define a left $\sym{n}$-action on key tableau $K$ with maximum entry $n$ by applying $w \in \sym{n}$ to each entry of $K$ and sorting columns to be strictly increasing.
Let $K_{w\lambda}$ denote the key tableau by applying $w$ to the key tableau of shape $\lambda$ with every entry of row $i$ filled by $i$.

Let $T$ be a set-valued tableau.
For a semistandard Young tableau $S$, let $k(S)$ denote the (right) key tableau associated to $S$ (see, \textit{e.g.},~\cite{Willis13,BBBG19} for algorithms to compute this).
Let $\min(T)$ denote the semistandard Young tableau formed by taking the minimum of each entry in $T$.
Let $T^*$ denote the \defn{Lusztig involution} on $T$ using the crystal structure from~\cite{MPS18} (see~\cite[Eq.~(6.2)]{PS19}).
We do not require the exact definition of the Lusztig involution, only that $\wt(T^*) = w_0 \wt(T)$.
From~\cite[Sec.~6]{PS19}, we define the \defn{(right) K-key tableau} of $T$ to be
\[
K(T) := k(\min(T^*)^*).
\]

From~\cite[Sec.~4]{MPS18}, a \defn{marked Gelfand--Tsetlin (GT) pattern} is a sequence of partitions $\Lambda = (\lambda^{(j)})_{j=0}^n$, called a \defn{Gelfand--Tsetlin (GT) pattern}, such that $\lambda^{(0)} = \emptyset$ and the skew shape $\lambda^{(j)} / \lambda^{(j-1)}$ does not contain a vertical domino (\textit{i.e.}, is a horizontal strip),\footnote{This is equivalent to the usual interlacing condition on GT patterns.} with a set $M$ of entries that are ``marked,'' where the entry $(i, j)$, for $2 \leq j \leq n$ and $1 \leq i < \ell(\lambda^{(j)})$, is allowed to be marked if and only if $\lambda^{(j)}_{i+1} < \lambda^{(j-1)}_i$.
In particular, an entry $(i, j)$ cannot be marked if the entry to the right equals the entry to the southeast.
We depict a marked GT pattern as a triangular array with the top-row corresponding to $\lambda^{(n)}$ and the bottom row $\lambda^{(1)}$ and a marked entry $(i,j)$ as a box around the entry $\lambda^{(j)}_i$.

Next, we recall the bijection $\phi$ between marked GT patterns and set-valued tableaux, which is defined recursively as follows.
Consider a marked GT pattern $(\Lambda, M)$.
Start with $T_0 = \emptyset$.
Suppose we are at step $j$, where the set-valued tableau is $T_{j-1}$ that has entries in $1, \dotsc, j-1$.
For each marked entry $(i,j)$, we add $j$ to the rightmost entry of $i$-th row of $T_{j-1}$, and denote this $T'_j$.
Then we consider the horizontal strip $\lambda^{(j)} / \lambda^{(j-1)}$ with all entries being $\{j\}$, which we add to $T'_j$ to obtain a set-valued tableau $T_j$ of shape $\lambda^{(j)}$.
We repeat this for every row of $\Lambda$ and the result is $\phi(\Lambda, M)$.
We define the weight of a marked GT pattern $\wt(\Lambda, M) = \wt\bigl( \phi(\Lambda, M) \bigr)$.

We also require one additional combinatorial object from~\cite{Monical16}, where we use the description given in~\cite{MPS18II}.
For a permutation $w \in \sym{n}$, define the \defn{(semistandard) skyline diagram} $w\lambda$ to be the Young diagram of $\lambda$ but the rows permuted by $w$.
In particular, we have $w \lambda = (\lambda_{w(1)}, \dotsc, \lambda_{w(n)})$.
A \defn{set-valued skyline tableau of shape $w\lambda$} is a filling of a skyline diagram $w\lambda$ with finite nonempty sets of positive integers such that entries do not repeat in a column; rows weakly decrease in the set-valued sense; \defn{anchors}, the largest entry in a box, satisfy the triple conditions of~\cite[Sec.~2.1]{Mason08}; the \defn{free} entries, the non-anchor entries in a box, are in the top-most row possible; and anchors in the first column equal their row index.
Let $\skyline(w\lambda)$ denote the set of set-valued skyline tableaux of shape $w\lambda$.
Define the $\beta$-weight for a set-valued skyline tableau the same as for a set-valued tableau.

The combinatorial definition of a \defn{Grothendieck polynomial} is due to Buch~\cite[Theorem~3.1]{Buch02} and is generating function of semistandard set-valued tableaux:
\[
\G_{\lambda}(\zz; \beta) = \sum_{T \in \svt^n(\lambda)} \wt_{\beta}(T).
\]
Using $\phi$, a Grothendieck polynomial is also equal to the generating function over marked GT patterns~\cite[Prop.~4.5]{MPS18}. A  definition as a ratio of determinants was given by Ikeda and Naruse~\cite{IN13}. 
We will consider another algebraic definition of the Grothendieck polynomials using the \defn{Demazure--Lascoux operator} $\varpi_i$, which defines an action of the $0$-Hecke algebra on $\ZZ[\beta][\zz]$ by
\[
\varpi_i f(\zz; \beta) = \frac{(z_i  + \beta z_i z_{i+1}) f(\zz; \beta) -  (z_{i+1}  + \beta z_i z_{i+1}) f(s_i \zz; \beta)}{z_i - z_{i+1}}.
\] 
In particular, the Demazure--Lascoux operators satisfy the braid relations and $\varpi_i^2 = \varpi_i$.
Hence, for any permutation $w \in \sym{n}$, one may define $\varpi_w := \varpi_{i_1} \varpi_{i_2} \dotsm \varpi_{i_{\ell}}$ for any choice of reduced expression $w = s_{i_1} s_{i_2} \dotsm s_{i_{\ell}}$.
Thus, we can write $\G_{\lambda}(\zz; \beta) = \varpi_{w_0} z_1^{\lambda_1} \dotsm z_n^{\lambda_n}$~\cite{IN13,IS14,LS82}
When $\beta = 0$, we obtain the \defn{Schur function} $s_{\lambda}(\zz)$.

Next, following~\cite{Monical16}, we define the \defn{Demazure--Lascoux atom operator} $\overline{\varpi}_i := \varpi_i - 1$, which satisfies the braid relations and $\overline{\varpi}_i^2 = -\overline{\varpi}_i$.
We define the \defn{Lascoux polynomials}~\cite{Lascoux01} and \defn{Lascoux atoms}~\cite{Monical16} by
\[
L_{w\lambda}(\zz; \beta) := \varpi_w z_1^{\lambda_1} \dotsm z_n^{\lambda_n},
\qquad\qquad
\overline{L}_{w\lambda}(\zz; \beta) := \overline{\varpi}_w z_1^{\lambda_1} \dotsm z_n^{\lambda_n},
\]
which satisfies $\G_{\lambda}(\zz; \beta) = L_{w_0\lambda}(\zz; \beta)$ and ~\cite[Theorem~5.1]{Monical16}:
\begin{equation}
\label{eq:Lascoux_into_atoms}
L_{w\lambda}(\zz; \beta) = \sum_{u \leq w} \overline{L}_{u\lambda}(\zz; \beta).
\end{equation}
It is an open problem to find a geometric or representation-theoretic interpretation for general Lascoux polynomials.
We prove the following conjectured combinatorial interpretations of Lascoux polynomials and atoms in this extended abstract

\begin{conjecture}[{\cite[Conjecture 6.1]{PS19} and \cite[Conjecture 5.2]{Monical16}}]
\label{conj:svt_Lascoux}
We have
\[
\overline{L}_{w\lambda}(\zz; \beta) = \sum_{\substack{T \in \svt^n(\lambda) \\ K(T) = K_{w\lambda}}} \wt_{\beta}(T) = \sum_{S \in \skyline(w\lambda)} \wt_{\beta}(S),
\qquad
L_{w\lambda}(\zz; \beta) = \sum_{\substack{T \in \svt^n(\lambda) \\ K(T) \leq K_{w\lambda}}} \wt_{\beta}(T).
\]
\end{conjecture}

Note that the second Lascoux atom formula is also the K-theoretic analog of Demazure characters being described by skyline tableaux~\cite{Mason08,Mason09}.%, which come from nonsymmetric Macdonald polynomials at $t = q = 0$.












%===============================================================================
\section{Colored lattice models and Lascoux atoms}\label{sec:coloredmodels}

We will build colored models that represent Lascoux atoms, generalizing the work in~\cite{BBBG19}, whose partition function was a Demazure atom.
The model we consider is a colored version of the lattice model of Motegi and Sakai~\cite{MS13}, whose partition function is $\G_{\lambda}(\zz; \beta)$.


\begin{figure}
\[
\begin{array}{c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c}
\toprule
  \tt{a}_1&\tt{a}_2&\tt{b}_1&\tt{b}^{\dagger}_1&\tt{b}^{\circ}_1&\tt{b}_2&\tt{c}_1\\
\midrule
\begin{tikzpicture}
\coordinate (a) at (-.75, 0);
\coordinate (b) at (0, .75);
\coordinate (c) at (.75, 0);
\coordinate (d) at (0, -.75);
\coordinate (aa) at (-.75,.5);
\coordinate (cc) at (.75,.5);
\draw (a)--(0,0);
\draw (b)--(0,0);
\draw (c)--(0,0);
\draw (d)--(0,0);
\draw[fill=white] (a) circle (.25);
\draw[fill=white] (b) circle (.25);
\draw[fill=white] (c) circle (.25);
\draw[fill=white] (d) circle (.25);
\node at (0,1) { };
\node at (a) {$0$};
\node at (b) {$0$};
\node at (c) {$0$};
\node at (d) {$0$};
\path[fill=white] (0,0) circle (.2);
\node at (0,0) {$z$};
\end{tikzpicture}
%%%%%%%
& \begin{tikzpicture}
\coordinate (a) at (-.75, 0);
\coordinate (b) at (0, .75);
\coordinate (c) at (.75, 0);
\coordinate (d) at (0, -.75);
\coordinate (aa) at (-.75,.5);
\coordinate (cc) at (.75,.5);
\draw (a)--(0,0);
\draw (b)--(0,0);
\draw[line width=0.5mm, brown] (c)--(0,0);
\draw[line width=0.5mm, brown] (d)--(0,0);
\draw[fill=white] (a) circle (.25);
\draw[fill=white] (b) circle (.25);
\draw[line width=0.5mm,brown, fill=white] (c) circle (.25);
\draw[line width=0.5mm,brown, fill=white] (d) circle (.25);
\node at (0,1) { };
\node at (a) {$0$};
\node at (b) {$0$};
\node at (c) {$d$};
\node at (d) {$d$};
\path[fill=white] (0,0) circle (.2);
\node at (0,0) {$z$};
\end{tikzpicture}
%%%%%%%
& \begin{tikzpicture}
\coordinate (a) at (-.75, 0);
\coordinate (b) at (0, .75);
\coordinate (c) at (.75, 0);
\coordinate (d) at (0, -.75);
\coordinate (aa) at (-.75,.5);
\coordinate (cc) at (.75,.5);
\draw[line width=0.5mm, blue] (a)--(0,0);
\draw[line width=0.5mm, blue] (b)--(0,0);
\draw[line width=0.5mm, red] (c)--(0,0);
\draw[line width=0.5mm, red] (d)--(0,0);
\draw[line width=0.5mm, blue,fill=white] (a) circle (.25);
\draw[line width=0.5mm, blue,fill=white] (b) circle (.25);
\draw[line width=0.5mm, red, fill=white] (c) circle (.25);
\draw[line width=0.5mm, red, fill=white] (d) circle (.25);
\node at (0,1) { };
\node at (a) {$c_j$};
\node at (b) {$c_j$};
\node at (c) {$c_i$};
\node at (d) {$c_i$};
\path[fill=white] (0,0) circle (.2);
\node at (0,0) {$z$};
\end{tikzpicture}
%%%%%%%
& \begin{tikzpicture}
\coordinate (a) at (-.75, 0);
\coordinate (b) at (0, .75);
\coordinate (c) at (.75, 0);
\coordinate (d) at (0, -.75);
\coordinate (aa) at (-.75,.5);
\coordinate (cc) at (.75,.5);
\draw[line width=0.5mm, blue] (a)--(0,0);
\draw[line width=0.6mm, red] (b)--(0,0);
\draw[line width=0.5mm, blue] (c)--(0,0);
\draw[line width=0.6mm, red] (d)--(0,0);
\draw[line width=0.5mm, blue,fill=white] (a) circle (.25);
\draw[line width=0.5mm, red,fill=white] (b) circle (.25);
\draw[line width=0.5mm, blue, fill=white] (c) circle (.25);
\draw[line width=0.5mm, red, fill=white] (d) circle (.25);
\node at (0,1) { };
\node at (a) {$c_j$};
\node at (b) {$c_i$};
\node at (c) {$c_j$};
\node at (d) {$c_i$};
\path[fill=white] (0,0) circle (.2);
\node at (0,0) {$z$};
\end{tikzpicture}
%%%%%%%
& \begin{tikzpicture}
\coordinate (a) at (-.75, 0);
\coordinate (b) at (0, .75);
\coordinate (c) at (.75, 0);
\coordinate (d) at (0, -.75);
\coordinate (aa) at (-.75,.5);
\coordinate (cc) at (.75,.5);
\draw[line width=0.5mm, brown] (a)--(0,0);
\draw[line width=0.6mm, brown] (b)--(0,0);
\draw[line width=0.5mm, brown] (c)--(0,0);
\draw[line width=0.6mm, brown] (d)--(0,0);
\draw[line width=0.5mm, brown,fill=white] (a) circle (.25);
\draw[line width=0.5mm, brown,fill=white] (b) circle (.25);
\draw[line width=0.5mm, brown, fill=white] (c) circle (.25);
\draw[line width=0.5mm, brown, fill=white] (d) circle (.25);
\node at (0,1) { };
\node at (a) {$d$};
\node at (b) {$d$};
\node at (c) {$d$};
\node at (d) {$d$};
\path[fill=white] (0,0) circle (.2);
\node at (0,0) {$z$};
\end{tikzpicture}
%%%%%%%
& \begin{tikzpicture}
\coordinate (a) at (-.75, 0);
\coordinate (b) at (0, .75);
\coordinate (c) at (.75, 0);
\coordinate (d) at (0, -.75);
\coordinate (aa) at (-.75,.5);
\coordinate (cc) at (.75,.5);
\draw[line width=0.5mm, brown] (a)--(0,0);
\draw(b)--(0,0);
\draw[line width=0.5mm, brown] (c)--(0,0);
\draw (d)--(0,0);
\draw[line width=0.5mm,brown,fill=white] (a) circle (.25);
\draw[fill=white] (b) circle (.25);
\draw[line width=0.5mm,brown,fill=white] (c) circle (.25);
\draw[fill=white] (d) circle (.25);
\node at (0,1) { };
\node at (a) {$d$};
\node at (b) {$0$};
\node at (c) {$d$};
\node at (d) {$0$};
\path[fill=white] (0,0) circle (.2);
\node at (0,0) {$z$};
\end{tikzpicture}
%%%%%%%
& \begin{tikzpicture}
\coordinate (a) at (-.75, 0);
\coordinate (b) at (0, .75);
\coordinate (c) at (.75, 0);
\coordinate (d) at (0, -.75);
\coordinate (aa) at (-.75,.5);
\coordinate (cc) at (.75,.5);
\draw[line width=0.5mm, brown] (a)--(0,0);
\draw[line width=0.6mm, brown] (b)--(0,0);
\draw (c)--(0,0);
\draw (d)--(0,0);
\draw[line width=0.5mm,brown,fill=white] (a) circle (.25);
\draw[line width=0.5mm,brown,fill=white] (b) circle (.25);
\draw[fill=white] (c) circle (.25);
\draw[fill=white] (d) circle (.25);
\node at (0,1) { };
\node at (a) {$d$};
\node at (b) {$d$};
\node at (c) {$0$};
\node at (d) {$0$};
\path[fill=white] (0,0) circle (.2);
\node at (0,0) {$z$};
\end{tikzpicture}
%%%%%%%
\\ 
\midrule
1 & 1+\beta z & 1 &1 & 1 & z & 1\\
\bottomrule
\end{array}
\]
\caption{The colored Boltzmann weights with ${\color{red} c_i} > {\color{blue} c_j}$ and ${\color{brown} d}$ being any color.}
\label{fig:colored_weights}
\end{figure}

Consider a rectangular grid with $n$ horizontal lines and $m$ vertical lines (we consider a crossing of the lines to be vertices).
We also fix an $n$-tuple of colors $\cc = (c_1 > c_2 > \cdots > c_n > 0)$.
Let $w \in \sym{n}$, and let $w \cc = (c_{w(1)}, c_{w(2)}, \dotsc, c_{w(n)})$ be the colors permuted by $w$.
We label the bottom and right (half) edges by $0$, the left (half) edges by $w w_0 \cc$ from top to bottom, and the top edges by $\lambda$: the $i$-th $1$ in the $\{0,1\}$-sequence of $\lambda$, counted from the left, is labeled by $c_i$.
An \defn{admissible state} is an assignment of labels on the interior edges of the grid such that all vertices are of the form given in \cref{fig:colored_weights}, which also specifies their \defn{Boltzmann weights}.
Let $\states_{\lambda}$ denote the set of all possible admissible states of the model.
The \defn{(Boltzmann) weight} $\wt(S)$ of an admissible state $S \in \states_{\lambda}$ is the product of all of the Boltzmann weights of all vertices with $z = z_i$ in the $i$-th row numbered starting from top.
Let $\overline{\states}_{\lambda,w}$ denote the set of all possible admissible states for this model.
The \defn{partition function} of a model $\model$
\[
Z(\model; \zz; \beta) := \sum_{S \in \model} \wt(S)
\]
is the sum of the Boltzmann weights of all possible admissible states of $\model$.

Using the colors and the admissible configurations, we can think of an admissible state in $\states_{\lambda,w}$ as corresponding to a wiring diagram of $w$, where the different strands are represented by different colors.
Indeed, we can think of $\tt{a}_2$, $\tt{b}_2$, and $\tt{c}_1$ as a single strand passing through the vertex (possibly turning), $\tt{b}_1^{\dagger}$ as two strands crossing at the vertex (thus corresponding to a simple transposition), and $\tt{b}_1$ as two strands both passing near the vertex but not crossing.

%\begin{remark}
%In contrast to~\cite{BBBG19} we have colored in the left side instead of the right side and we are taking $w w_0 \cc$ (instead of $w \cc$) due to our indexing convention.
%Furthermore, we include this $w_0$ in order to emphasize that taking $\overline{\states}_{\lambda,1}$ corresponds to the wiring diagram of $w_0$ (with every pair of strands crossing exactly once).
%\end{remark}

\begin{figure}%[h]
\[
\begin{array}{c@{\hspace{30pt}}c@{\hspace{30pt}}c@{\hspace{30pt}}c}
\toprule
%  ++++ : z2
\begin{tikzpicture}[scale=0.7]
\draw (0,0) to [out = 0, in = 180] (2,2);
\draw (0,2) to [out = 0, in = 180] (2,0);
\draw[fill=white] (0,0) circle (.35);
\draw[fill=white] (0,2) circle (.35);
\draw[fill=white] (2,0) circle (.35);
\draw[fill=white] (2,2) circle (.35);
\node at (0,0) {$0$};
\node at (0,2) {$0$};
\node at (2,2) {$0$};
\node at (2,0) {$0$};
\path[fill=white] (1,1) circle (.3);
\node at (1,1) {$z_i,z_j$};
\end{tikzpicture}&
%  +RR+ : z2
\begin{tikzpicture}[scale=0.7]
\draw (0,0) to [out = 0, in = 180] (2,2);
\draw (0,2) to [out = 0, in = 180] (2,0);
\draw[fill=white] (0,0) circle (.35);
\draw[line width=0.5mm, brown, fill=white] (0,2) circle (.35);
\draw[line width=0.5mm, brown, fill=white] (2,2) circle (.35);
\draw[fill=white] (2,0) circle (.35);
\node at (0,0) {$0$};
\node at (0,2) {$d$};
\node at (2,2) {$d$};
\node at (2,0) {$0$};
\path[fill=white] (1,1) circle (.3);
\node at (1,1) {$z_i,z_j$};
\end{tikzpicture}&
%  R++R : z1
\begin{tikzpicture}[scale=0.7]
\draw (0,0) to [out = 0, in = 180] (2,2);
\draw (0,2) to [out = 0, in = 180] (2,0);
\draw[fill=white] (0,0) circle (.35);
\draw[line width=0.5mm, brown, fill=white] (0,2) circle (.35);
\draw[fill=white] (2,2) circle (.35);
\draw[line width=0.5mm, brown, fill=white] (2,0) circle (.35);
\node at (0,0) {$0$};
\node at (0,2) {$d$};
\node at (2,2) {$0$};
\node at (2,0) {$d$};
\path[fill=white] (1,1) circle (.3);
\node at (1,1) {$z_i,z_j$};
\end{tikzpicture}&
%  R++R : z2
\begin{tikzpicture}[scale=0.7]
\draw (0,0) to [out = 0, in = 180] (2,2);
\draw (0,2) to [out = 0, in = 180] (2,0);
\draw[line width=0.5mm, brown, fill=white] (0,0) circle (.35);
\draw[fill=white] (0,2) circle (.35);
\draw[fill=white] (2,2) circle (.35);
\draw[line width=0.5mm, brown, fill=white] (2,0) circle (.35);
\node at (0,0) {$d$};
\node at (0,2) {$0$};
\node at (2,2) {$0$};
\node at (2,0) {$d$};
\path[fill=white] (1,1) circle (.3);
\node at (1,1) {$z_i,z_j$};
\end{tikzpicture}\\
   \midrule
 (1 + \beta z_i) z_j & (1+ \beta z_i) z_j & (z_j - z_i) z_j & (1 + \beta z_j) z_j \\
   \midrule
%  BRRB : z1
\begin{tikzpicture}[scale=0.7]
\draw (0,0) to [out = 0, in = 180] (2,2);
\draw (0,2) to [out = 0, in = 180] (2,0);
\draw[line width=0.5mm, blue, fill=white] (0,0) circle (.35);
\draw[line width=0.5mm, red, fill=white] (0,2) circle (.35);
\draw[line width=0.5mm, red, fill=white] (2,2) circle (.35);
\draw[line width=0.5mm, blue, fill=white] (2,0) circle (.35);
\node at (0,0) {$c_j$};
\node at (0,2) {$c_i$};
\node at (2,2) {$c_i$};
\node at (2,0) {$c_j$};
\path[fill=white] (1,1) circle (.3);
\node at (1,1) {$z_i,z_j$};
\end{tikzpicture}&
%  RBBR : z2
\begin{tikzpicture}[scale=0.7]
\draw (0,0) to [out = 0, in = 180] (2,2);
\draw (0,2) to [out = 0, in = 180] (2,0);
\draw[line width=0.5mm, red, fill=white] (0,0) circle (.35);
\draw[line width=0.5mm, blue, fill=white] (0,2) circle (.35);
\draw[line width=0.5mm, blue, fill=white] (2,2) circle (.35);
\draw[line width=0.5mm, red, fill=white] (2,0) circle (.35);
\node at (0,0) {$c_i$};
\node at (0,2) {$c_j$};
\node at (2,2) {$c_j$};
\node at (2,0) {$c_i$};
\path[fill=white] (1,1) circle (.3);
\node at (1,1) {$z_i,z_j$};
\end{tikzpicture}&
%  RBRB : z1 - z2
\begin{tikzpicture}[scale=0.7]
\draw (0,0) to [out = 0, in = 180] (2,2);
\draw (0,2) to [out = 0, in = 180] (2,0);
\draw[line width=0.5mm, red, fill=white] (0,0) circle (.35);
\draw[line width=0.5mm, blue, fill=white] (0,2) circle (.35);
\draw[line width=0.5mm, red, fill=white] (2,2) circle (.35);
\draw[line width=0.5mm, blue, fill=white] (2,0) circle (.35);
\node at (0,0) {$c_i$};
\node at (0,2) {$c_j$};
\node at (2,2) {$c_i$};
\node at (2,0) {$c_j$};
\path[fill=white] (1,1) circle (.3);
\node at (1,1) {$z_i,z_j$};
\end{tikzpicture}&
%  RRRR : z2
\begin{tikzpicture}[scale=0.7]
\draw (0,0) to [out = 0, in = 180] (2,2);
\draw (0,2) to [out = 0, in = 180] (2,0);
\draw[line width=0.5mm, brown, fill=white] (0,0) circle (.35);
\draw[line width=0.5mm, brown, fill=white] (0,2) circle (.35);
\draw[line width=0.5mm, brown, fill=white] (2,2) circle (.35);
\draw[line width=0.5mm, brown, fill=white] (2,0) circle (.35);
\node at (0,0) {$d$};
\node at (0,2) {$d$};
\node at (2,2) {$d$};
\node at (2,0) {$d$};
\path[fill=white] (1,1) circle (.3);
\node at (1,1) {$z_i,z_j$};
\end{tikzpicture}\\
   \midrule
(1 + \beta z_j) z_ i & (1 + \beta z_i) z_j & z_j - z_i & (1 + \beta z_i) z_j \\
   \bottomrule
\end{array}
\]
\caption{The colored $R$-matrix with ${\color{red} c_i} > {\color{blue} c_j}$ and ${\color{brown} d}$ being any color. Note that the weights are not symmetric with respect to color.}
\label{fig:colored_R_matrix}
\end{figure}


Our model is amenable to study via the Yang--Baxter equation.
We introduce the $R$-matrix for this model, which we call the \defn{colored $R$-matrix}, and the admissible configurations with their Boltzmann weights are given in \cref{fig:colored_R_matrix}.
Furthermore, we can see that the $R$-matrix generally corresponds to the vertices of the $L$-matrix rotated by $45^{\circ}$ clockwise and the weights of the $L$-matrix take $z = \frac{z_j - z_i}{1 + \beta z_i}$ and are scaled by $(1 + \beta z_i) z_j$.
To distinguish them from the usual vertices given by the $L$ matrix, we draw them tilted on their side.
Together with the previously introduced vertices, they satisfy the $RLL$ version of the Yang--Baxter equation (hence, this model is integrable). It is a finite computation to verify this since it requires at most 3 colors, which can easily be done by computer.

\begin{proposition}
\label{prop:YBE}
Consider the $L$-matrix given in \cref{fig:colored_weights} and $R$-matrix given in \cref{fig:colored_R_matrix}.
The partition function of the following two models
\begin{equation}
\label{eq:RLL_relation}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.8]
  \draw (0,1) to [out = 0, in = 180] (2,3) to (4,3);
  \draw (0,3) to [out = 0, in = 180] (2,1) to (4,1);
  \draw (3,0) to (3,4);
  \draw[fill=white] (0,1) circle (.3);
  \draw[fill=white] (0,3) circle (.3);
  \draw[fill=white] (3,4) circle (.3);
  \draw[fill=white] (4,3) circle (.3);
  \draw[fill=white] (4,1) circle (.3);
  \draw[fill=white] (3,0) circle (.3);
  \node at (0,1) {$a$};
  \node at (0,3) {$b$};
  \node at (3,4) {$c$};
  \node at (4,3) {$d$};
  \node at (4,1) {$e$};
  \node at (3,0) {$f$};
\path[fill=white] (3,3) circle (.3);
\node at (3,3) {$z_i$};
\path[fill=white] (3,1) circle (.3);
\node at (3,1) {$z_j$};
\path[fill=white] (1,2) circle (.3);
\node at (1,2) {$z_i,z_j$};
\end{tikzpicture}\qquad\qquad
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.8]
  \draw (0,1) to (2,1) to [out = 0, in = 180] (4,3);
  \draw (0,3) to (2,3) to [out = 0, in = 180] (4,1);
  \draw (1,0) to (1,4);
  \draw[fill=white] (0,1) circle (.3);
  \draw[fill=white] (0,3) circle (.3);
  \draw[fill=white] (1,4) circle (.3);
  \draw[fill=white] (4,3) circle (.3);
  \draw[fill=white] (4,1) circle (.3);
  \draw[fill=white] (1,0) circle (.3);
  \node at (0,1) {$a$};
  \node at (0,3) {$b$};
  \node at (1,4) {$c$};
  \node at (4,3) {$d$};
  \node at (4,1) {$e$};
  \node at (1,0) {$f$};
\path[fill=white] (1,3) circle (.3);
\node at (1,3) {$z_j$};
\path[fill=white] (1,1) circle (.3);
\node at (1,1) {$z_i$};
\path[fill=white] (3,2) circle (.3);
\node at (3,2) {$z_i,z_j$};
\end{tikzpicture}
\end{equation}
are equal for any boundary conditions $a,b,c,d,e,f \in \{ 0, c_1, \dotsc, c_n \}$. 
\end{proposition}
 
By using the Yang--Baxter equation and the train argument (see~\cite{BSW19} for more information on the train argument), we can derive the following equation for the partition functions of our lattice model. That is to say, we add an $R$-matrix to one row, pass it through to the other side using the Yang--Baxter equation (see \cref{fig:train_argument}), and then obtain a functional equation that corresponds to acting by a Demazure--Lascoux operator.

\begin{lemma}
\label{lemma:recurrence_eq}
Let $w \in \sym{n}$, and consider $s_i$ be such that $s_i w > w$.
Then we have
\[
Z(\overline{\states}_{\lambda, s_i w}; \zz; \beta) = \frac{(1  + \beta z_i ) z_{i+1} \bigl( Z(\overline{\states}_{\lambda, w}; \zz; \beta) -  Z(\overline{\states}_{\lambda, w}; s_i \zz; \beta) \bigr)}{z_i - z_{i+1}}.
\]
\end{lemma}


\begin{figure}
\begin{tikzpicture}[scale=0.7,baseline=0]
\begin{scope}[shift={(-6,0)}]
  \draw (4,1) to (6,1) to [out = 0, in = 180] (8,3);
  \draw (4,3) to (6,3) to [out = 0, in = 180] (8,1);
  \draw[fill=white] (8.3,3) circle (.55);
  \draw[fill=white] (8.3,1) circle (.55);
  \draw (0,1) to (2,1);
  \draw (0,3) to (2,3);
  \draw (5,0.25) to (5,3.75);
  \draw (1,0.25) to (1,3.75);
  \draw[line width=0.5mm,red,fill=white] (-0.3,1) circle (.55);
  \draw[line width=0.5mm,blue,fill=white] (-0.3,3) circle (.55);

  \node at (3,1) {$\cdots$};
  \node at (3,3) {$\cdots$};

  \draw[densely dashed] (1,3.75) to (1,4.25);
  \draw[densely dashed] (1,0.25) to (1,-0.25);
  \draw[densely dashed] (5,3.75) to (5,4.25);
  \draw[densely dashed] (5,0.25) to (5,-0.25);
  \path[fill=white] (1,3) circle (.5);
  \node at (1,3) {\scriptsize$z_{i+1}$};
  \path[fill=white] (1,1) circle (.4);
  \node at (1,1) {\scriptsize$z_i$};

  \path[fill=white] (5,3) circle (.5);
  \node (a) at (5,3) {\scriptsize$z_{i+1}$};
  \path[fill=white] (5,1) circle (.4);
  \node at (5,1) {\scriptsize$z_i$};

  \path[fill=white] (7,2) circle (.3);
  \node at (7,2) {\scriptsize$z_i,z_{i+1}$};
  \node at (8.3,1) {$0$};
  \node at (8.3,3) {$0$};
  \node at (-0.3,1) {$d_i$};
  \node at (-0.3,3) {$d_{i+1}$};
\end{scope}
\begin{scope}[shift={(6,0)}]
  \draw (0,1) to [out = 0, in = 180] (2,3) to (4,3);
  \draw (0,3) to [out = 0, in = 180] (2,1) to (4,1);
  \draw (3,0.25) to (3,3.75);
  \draw (7,0.25) to (7,3.75);
  \draw (6,1) to (8,1);
  \draw (6,3) to (8,3);
  \draw[line width=0.5mm,red,fill=white] (-0.3,1) circle (.55);
  \draw[line width=0.5mm,blue,fill=white] (-0.3,3) circle (.55);
  \draw[fill=white] (8.3,3) circle (.55);
  \draw[fill=white] (8.3,1) circle (.55);
  \node at (-0.3,1) {$d_i$};
  \node at (-0.3,3) {$d_{i+1}$};
  \node at (5,3) {$\cdots$};
  \node at (5,1) {$\cdots$};
  \draw[densely dashed] (3,3.75) to (3,4.25);
  \draw[densely dashed] (3,0.25) to (3,-0.25);
  \draw[densely dashed] (7,3.75) to (7,4.25);
  \draw[densely dashed] (7,0.25) to (7,-0.25);
  \node at (8.3,1) {$0$};
  \node at (8.3,3) {$0$};
\path[fill=white] (3,3) circle (.4);
\node at (3,3) {\scriptsize$z_i$};
\path[fill=white] (3,1) circle (.5);
\node at (3,1) {\scriptsize$z_{i+1}$};
\path[fill=white] (7,3) circle (.4);
\node at (7,3) {\scriptsize$z_i$};
\path[fill=white] (7,1) circle (.5);
\node at (7,1) {\scriptsize$z_{i+1}$};
\path[fill=white] (1,2) circle (.3);
\node at (1,2) {\scriptsize$z_i,z_{i+1}$};
\end{scope}
\end{tikzpicture}
\caption{Left: The model $\overline{\states}_{\lambda,w}$ with an $R$-matrix attached on the right. Right: The model after using the Yang--Baxter equation in the same model.}
\label{fig:train_argument}
\end{figure}


\begin{theorem}
\label{thm:5vertex_atoms}
We have
$
\overline{L}_{w\lambda}(\zz; \beta) = Z(\overline{\states}_{\lambda,w}; \zz; \beta).
$
\end{theorem}


\begin{example}
A state for the colored system $\overline{\states}_{\lambda,s_2 s_1}$, with $m =8$, $n=3$, and $\lambda = (4,2,1)$:
\[
\begin{tikzpicture}[scale=0.80, font=\small,baseline=0]
    % Draw the grid and weights
    \foreach \y in {1,3,5}
        \draw (0,\y)--(16,\y);
    \foreach \x in {1,3,5,7,9,11,13,15}
      \draw (\x,0)--(\x,6);
    \draw[line width=0.5mm,darkgreen] (13,6) -- (13,5) -- (5,5) -- (5,3) -- (0,3);
    \draw[line width=0.5mm,blue] (7,6) -- (7,3) -- (5,3) -- (5,1) -- (0,1);
    \draw[line width=0.5mm,red] (3,6) -- (3,5) -- (0,5);
    \foreach \x in {0,2,4,6,8,10,12,14,16}
    {
      \draw[fill=white] (\x,5) circle (.35);
      \draw[fill=white] (\x,3) circle (.35);
      \draw[fill=white] (\x,1) circle (.35);
    }
    \foreach \x in {1,3,5,7,9,11,13,15}
    {
      \draw[fill=white] (\x,0) circle (.35);
      \draw[fill=white] (\x,2) circle (.35);
      \draw[fill=white] (\x,4) circle (.35);
      \draw[fill=white] (\x,6) circle (.35);
      \path[fill=white] (\x,1) circle (.2);
      \node at (\x,1) {$z_3$};
      \path[fill=white] (\x,3) circle (.2);
      \node at (\x,3) {$z_2$};
      \path[fill=white] (\x,5) circle (.2);
      \node at (\x,5) {$z_1$};
    }

    % GT row 3
    \node at (1,6) {$0$};
    \draw[line width=0.5mm,red,fill=white] (3,6) circle (.35); \node at (3,6) {$c_1$};
    \node at (5,6) {$0$};
    \draw[line width=0.5mm,blue,fill=white] (7,6) circle (.35); \node at (7,6) {$c_2$};
    \node at (9,6) {$0$};
    \node at (11,6) {$0$};
    \draw[line width=0.5mm,darkgreen,fill=white] (13,6) circle (.35); \node at (13,6) {$c_3$};
    \node at (15,6) {$0$};
    % GT row 2
    \node at (1,4) {$0$};
    \node at (3,4) {$0$};
    \draw[line width=0.5mm,darkgreen,fill=white] (5,4) circle (.35); \node at (5,4) {$c_3$};
    \draw[line width=0.5mm,blue,fill=white] (7,4) circle (.35); \node at (7,4) {$c_2$};
    \node at (9,4) {$0$};
    \node at (11,4) {$0$};
    \node at (13,4) {$0$};
    \node at (15,4) {$0$};
    % GT row 1
    \node at (1,2) {$0$};
    \node at (3,2) {$0$};
    \draw[line width=0.5mm,blue,fill=white] (5,2) circle (.35); \node at (5,2) {$c_2$};
    \node at (7,2) {$0$};
    \node at (9,2) {$0$};
    \node at (11,2) {$0$};
    \node at (13,2) {$0$};
    \node at (15,2) {$0$};
    % GT row 0
    \node at (1,0) {$0$};
    \node at (3,0) {$0$};
    \node at (5,0) {$0$};
    \node at (7,0) {$0$};
    \node at (9,0) {$0$};
    \node at (11,0) {$0$};
    \node at (13,0) {$0$};
    \node at (15,0) {$0$};
    % z_1 row
    \draw[line width=0.5mm,red,fill=white] (0,5) circle (.35); \node at (0,5) {$c_1$};
    \draw[line width=0.5mm,red,fill=white] (2,5) circle (.35); \node at (2,5) {$c_1$};
    \node at (4,5) {$0$};
    \draw[line width=0.5mm,darkgreen,fill=white] (6,5) circle (.35); \node at (6,5) {$c_3$};
    \draw[line width=0.5mm,darkgreen,fill=white] (8,5) circle (.35); \node at (8,5) {$c_3$};
    \draw[line width=0.5mm,darkgreen,fill=white] (10,5) circle (.35); \node at (10,5) {$c_3$};
    \draw[line width=0.5mm,darkgreen,fill=white] (12,5) circle (.35); \node at (12,5) {$c_3$};
    \node at (14,5) {$0$};
    \node at (16,5) {$0$};
    % z_2 row
    \draw[line width=0.5mm,darkgreen,fill=white] (0,3) circle (.35); \node at (0,3) {$c_3$};
    \draw[line width=0.5mm,darkgreen,fill=white] (2,3) circle (.35); \node at (2,3) {$c_3$};
    \draw[line width=0.5mm,darkgreen,fill=white] (4,3) circle (.35); \node at (4,3) {$c_3$};
    \draw[line width=0.5mm,blue,fill=white] (6,3) circle (.35); \node at (6,3) {$c_2$};
    \node at (8,3) {$0$};
    \node at (10,3) {$0$};
    \node at (12,3) {$0$};
    \node at (14,3) {$0$};
    \node at (16,3) {$0$};
    % z_3 row
    \draw[line width=0.5mm,blue,fill=white] (0,1) circle (.35); \node at (0,1) {$c_2$};
    \draw[line width=0.5mm,blue,fill=white] (2,1) circle (.35); \node at (2,1) {$c_2$};
    \draw[line width=0.5mm,blue,fill=white] (4,1) circle (.35); \node at (4,1) {$c_2$};
    \node at (6,1) {$0$};
    \node at (8,1) {$0$};
    \node at (10,1) {$0$};
    \node at (12,1) {$0$};
    \node at (14,1) {$0$};
    \node at (16,1) {$0$};
    % The extra labels
    \node at (1.00,6.8) {$ 1$};
    \node at (3.00,6.8) {$ 2$};
    \node at (5.00,6.8) {$ 3$};
    \node at (7.00,6.8) {$ 4$};
    \node at (9.00,6.8) {$ 5$};
    \node at (11.00,6.8) {$ 6$};
    \node at (13.00,6.8) {$ 7$};
    \node at (15.00,6.8) {$ 8$};
    \node at (-.75,1) {$ 3$};
    \node at (-.75,3) {$ 2$};
    \node at (-.75,5) {$ 1$};
\end{tikzpicture}
\]
    We use colors ${\color{red} c_1} > {\color{blue} c_2} >  {\color{darkgreen} c_3}$.
    The Boltzmann weight of this state is $(1+\beta z_1)z_1^3 z_2^2 z^2_3$.
\end{example}

Let $\states_{\lambda,w}$ denote the same model as $\overline{\states}_{\lambda,w}$ with additional colored configurations $\tt{b}'_1$, defined as $\tt{b}_1$ except for ${\color{red} c_j} < {\color{blue} c_i}$, whenever the colors ${\color{red} c_j} < {\color{blue} c_i}$ do not cross in $\overline{\states}_{\lambda,w}$; \textit{i.e.}, whenever $(i, j)$ is not an inversion of $w w_0$.
In this case, we can remove configurations $\tt{b}^{\dagger}$ and $\tt{b}_1$ for colors ${\color{red} c_i} > {\color{blue} c_j}$ from the model without changing the possible states.
We extend the definition of the above $R$-matrix, except we require that the bottom left two configurations only appear when colors ${\color{red} c_i}$ and ${\color{blue} c_j}$ cross, and we add the following additional two configurations when they do not cross:
\begin{equation}
\label{eq:noncrossing_weights}
\begin{array}{c@{\hspace{80pt}}c}
\begin{tikzpicture}[scale=0.7]
\draw (0,0) to [out = 0, in = 180] (2,2);
\draw (0,2) to [out = 0, in = 180] (2,0);
\draw[line width=0.5mm, blue, fill=white] (0,0) circle (.35);
\draw[line width=0.5mm, red, fill=white] (0,2) circle (.35);
\draw[line width=0.5mm, red, fill=white] (2,2) circle (.35);
\draw[line width=0.5mm, blue, fill=white] (2,0) circle (.35);
\node at (0,0) {$c_j$};
\node at (0,2) {$c_i$};
\node at (2,2) {$c_i$};
\node at (2,0) {$c_j$};
\path[fill=white] (1,1) circle (.3);
\node at (1,1) {$z_i,z_j$};
\end{tikzpicture}
&
\begin{tikzpicture}[scale=0.7]
\draw (0,0) to [out = 0, in = 180] (2,2);
\draw (0,2) to [out = 0, in = 180] (2,0);
\draw[line width=0.5mm, red, fill=white] (0,0) circle (.35);
\draw[line width=0.5mm, blue, fill=white] (0,2) circle (.35);
\draw[line width=0.5mm, blue, fill=white] (2,2) circle (.35);
\draw[line width=0.5mm, red, fill=white] (2,0) circle (.35);
\node at (0,0) {$c_i$};
\node at (0,2) {$c_j$};
\node at (2,2) {$c_j$};
\node at (2,0) {$c_i$};
\path[fill=white] (1,1) circle (.3);
\node at (1,1) {$z_i,z_j$};
\end{tikzpicture} \\
(1 + \beta z_i) z_j & (1 + \beta z_j) z_ i
\end{array}
\end{equation}
(We have simply interchanged the bottom left two Boltzmann weights from the $R$-matrix in \cref{fig:colored_R_matrix}.)
Indeed, this allows us to show the model is integrable.

\begin{proposition}
\label{prop:Lascoux_YBE}
Consider the modified $L$-matrix and $R$-matrix given above for $\states_{\lambda,w}$.
Then \cref{prop:YBE} holds for this (colored) model $\states_{\lambda,w}$.
\end{proposition}

\begin{theorem}
\label{thm:5vertex_Lascoux}
We have
$
L_{w\lambda}(\zz; \beta) = Z(\states_{\lambda,w}; \zz; \beta).
$
\end{theorem}

This can be proven using the train argument or a direct combinatorial argument by replacing first touching corners with crossings and \cref{thm:5vertex_atoms}.

Note that we could use the train proof and then use the combinatorial proof to show
Equation~\eqref{eq:Lascoux_into_atoms} as a consequence (thus yielding an alternative proof of~\cite[Theorem~5.1]{Monical16}).

\begin{example}
We consider replacing the $\tt{b}^{\dagger}$ corresponding to ${\color{red}c_1}$ and ${\color{darkgreen}c_3}$ in $\states_{\emptyset,1}$ for $m = n = 3$ with colors ${\color{red}c_1} > {\color{blue}c_2} > {\color{darkgreen}c_3}$.
This introduces a double crossing of the colors ${\color{blue}c_2}$ and ${\color{darkgreen}c_3}$.
We can then resolve this to a valid state in $\states_{\emptyset,w_0}$ by
\[
  \begin{tikzpicture}[scale=0.80, font=\small,baseline=0]
    % Draw the grid and weights
    \foreach \y in {1,3,5}
        \draw (0,\y)--(6,\y);
    \foreach \x in {1,3,5}
      \draw (\x,0)--(\x,6);
    \draw[line width=0.5mm,darkgreen] (5,6) -- (5,5) -- (1,5) -- (1,1) -- (0,1);
    \draw[line width=0.5mm,blue] (3,6) -- (3,3) -- (0,3);
    \draw[line width=0.5mm,red] (1,6) -- (1,5) -- (0,5);
    \foreach \x in {0,2,4,6}
    {
      \draw[fill=white] (\x,5) circle (.35);
      \draw[fill=white] (\x,3) circle (.35);
      \draw[fill=white] (\x,1) circle (.35);
    }
    \foreach \x in {1,3,5}
    {
      \draw[fill=white] (\x,0) circle (.35);
      \foreach \y in {1,2,3} {
        \draw[fill=white] (\x,2*\y) circle (.35);
        \path[fill=white] (\x,7-2*\y) circle (.2);
        \node at (\x,7-2*\y) {$z_{\y}$};
      }
    }

    % GT row 3
    \draw[line width=0.5mm,red,fill=white] (1,6) circle (.35); \node at (1,6) {$c_1$};
    \draw[line width=0.5mm,blue,fill=white] (3,6) circle (.35); \node at (3,6) {$c_2$};
    \draw[line width=0.5mm,darkgreen,fill=white] (5,6) circle (.35); \node at (5,6) {$c_3$};
    % GT row 2
    \draw[line width=0.5mm,darkgreen,fill=white] (1,4) circle (.35); \node at (1,4) {$c_3$};
    \draw[line width=0.5mm,blue,fill=white] (3,4) circle (.35); \node at (3,4) {$c_2$};
    \node at (5,4) {$0$};
    % GT row 1
    \draw[line width=0.5mm,darkgreen,fill=white] (1,2) circle (.35);, \node at (1,2) {$c_3$};
    \node at (3,2) {$0$};
    \node at (5,2) {$0$};
    % GT row 0
    \node at (1,0) {$0$};
    \node at (3,0) {$0$};
    \node at (5,0) {$0$};
    % z_1 row
    \draw[line width=0.5mm,red,fill=white] (0,5) circle (.35); \node at (0,5) {$c_1$};
    \draw[line width=0.5mm,darkgreen,fill=white] (2,5) circle (.35); \node at (2,5) {$c_3$};
    \draw[line width=0.5mm,darkgreen,fill=white] (4,5) circle (.35); \node at (4,5) {$c_3$};
    \node at (6,5) {$0$};
    % z_2 row
    \draw[line width=0.5mm,blue,fill=white] (0,3) circle (.35); \node at (0,3) {$c_2$};
    \draw[line width=0.5mm,blue,fill=white] (2,3) circle (.35); \node at (2,3) {$c_2$};
    \node at (4,3) {$0$};
    \node at (6,3) {$0$};
    % z_3 row
    \draw[line width=0.5mm,darkgreen,fill=white] (0,1) circle (.35); \node at (0,1) {$c_3$};
    \node at (2,1) {$0$};
    \node at (4,1) {$0$};
    \node at (6,1) {$0$};
    % The extra labels
    \node at (1.00,6.8) {$ 1$};
    \node at (3.00,6.8) {$ 2$};
    \node at (5.00,6.8) {$ 3$};
    \node at (-.75,1) {$ 3$};
    \node at (-.75,3) {$ 2$};
    \node at (-.75,5) {$ 1$};
    
    \draw[->] (7,3) -- (8.5,3);
    
    \begin{scope}[xshift=10cm]
    % Draw the grid and weights
    \foreach \y in {1,3,5}
        \draw (0,\y)--(6,\y);
    \foreach \x in {1,3,5}
      \draw (\x,0)--(\x,6);
    \draw[line width=0.5mm,darkgreen] (5,6) -- (5,5) -- (3,5) -- (3,3) -- (1,3) -- (1,1) -- (0,1);
    \draw[line width=0.5mm,blue] (3,6) -- (3,5) -- (1,5) -- (1,3) -- (0,3);
    \draw[line width=0.5mm,red] (1,6) -- (1,5) -- (0,5);
    \foreach \x in {0,2,4,6}
    {
      \draw[fill=white] (\x,5) circle (.35);
      \draw[fill=white] (\x,3) circle (.35);
      \draw[fill=white] (\x,1) circle (.35);
    }
    \foreach \x in {1,3,5}
    {
      \draw[fill=white] (\x,0) circle (.35);
      \foreach \y in {1,2,3} {
        \draw[fill=white] (\x,2*\y) circle (.35);
        \path[fill=white] (\x,7-2*\y) circle (.2);
        \node at (\x,7-2*\y) {$z_{\y}$};
      }
    }

    % GT row 3
    \draw[line width=0.5mm,red,fill=white] (1,6) circle (.35); \node at (1,6) {$c_1$};
    \draw[line width=0.5mm,blue,fill=white] (3,6) circle (.35); \node at (3,6) {$c_2$};
    \draw[line width=0.5mm,darkgreen,fill=white] (5,6) circle (.35); \node at (5,6) {$c_3$};
    % GT row 2
    \draw[line width=0.5mm,blue,fill=white] (1,4) circle (.35); \node at (1,4) {$c_2$};
    \draw[line width=0.5mm,darkgreen,fill=white] (3,4) circle (.35); \node at (3,4) {$c_3$};
    \node at (5,4) {$0$};
    % GT row 1
    \draw[line width=0.5mm,darkgreen,fill=white] (1,2) circle (.35);, \node at (1,2) {$c_3$};
    \node at (3,2) {$0$};
    \node at (5,2) {$0$};
    % GT row 0
    \node at (1,0) {$0$};
    \node at (3,0) {$0$};
    \node at (5,0) {$0$};
    % z_1 row
    \draw[line width=0.5mm,red,fill=white] (0,5) circle (.35); \node at (0,5) {$c_1$};
    \draw[line width=0.5mm,blue,fill=white] (2,5) circle (.35); \node at (2,5) {$c_2$};
    \draw[line width=0.5mm,darkgreen,fill=white] (4,5) circle (.35); \node at (4,5) {$c_3$};
    \node at (6,5) {$0$};
    % z_2 row
    \draw[line width=0.5mm,blue,fill=white] (0,3) circle (.35); \node at (0,3) {$c_2$};
    \draw[line width=0.5mm,darkgreen,fill=white] (2,3) circle (.35); \node at (2,3) {$c_3$};
    \node at (4,3) {$0$};
    \node at (6,3) {$0$};
    % z_3 row
    \draw[line width=0.5mm,darkgreen,fill=white] (0,1) circle (.35); \node at (0,1) {$c_3$};
    \node at (2,1) {$0$};
    \node at (4,1) {$0$};
    \node at (6,1) {$0$};
    % The extra labels
    \node at (1.00,6.8) {$ 1$};
    \node at (3.00,6.8) {$ 2$};
    \node at (5.00,6.8) {$ 3$};
    \node at (-.75,1) {$ 3$};
    \node at (-.75,3) {$ 2$};
    \node at (-.75,5) {$ 1$};
  \end{scope}
  \end{tikzpicture}
\]
\end{example}

Finally, we construct another variation on the model $\overline{\states}_{\lambda,w}$ where instead of adding $\tt{b}'_1$ for certain colors (and removing the corresponding $\tt{b}_1$ and $\tt{b}^{\dagger}$), we replace $\tt{b}_1$ with $\tt{b}'_1$ for all colors.
Let $\states'_{\lambda,w}$ denote this modified model.
We also use the following $R$-matrix given by \cref{fig:colored_R_matrix} except we \emph{replace} the two bottom left configurations by Equation~\eqref{eq:noncrossing_weights}.
This satisfies the Yang--Baxter equation (the proof is the same as \cref{prop:YBE}):

\begin{proposition}
\label{prop:Lascoux_prime_YBE}
Consider the modified $L$-matrix and $R$-matrix given above for $\states'_{\lambda,w}$.
Then \cref{prop:YBE} holds for this model $\states'_{\lambda,w}$.
\end{proposition}

\begin{theorem}
\label{thm:modified2}
We have
$
L_{w\lambda}(\zz; \beta) = Z(\states'_{\lambda,w}; \zz; \beta).
$
\end{theorem}

\begin{proposition}
There exists a weight-preserving bijection $\xi \colon \states'_{\lambda,w} \to \states_{\lambda,w}$.
\end{proposition}

\begin{proof}[Proof sketch]
Construct a bijection: For each pair of crossing colors, replace the northeast $\tt{b}'_1$ with $\tt{b}^{\dagger}$ and the other $\tt{b}'_1$ and $\tt{b}^{\dagger}$ with $\tt{b}_1$.
\end{proof}












%===============================================================================
\section{Lascoux atoms to K-key and set-valued skyline tableaux}
\label{sec:proofofconjecture}

In this section, we sketch the proof of \cref{conj:svt_Lascoux}.

First, recall that we can equate $\lambda$ with a $\{0,1\}$-sequence of length $m$ by considering the Young diagram inside of an $n \times (m - n)$ rectangle and starting at the bottom left, each up step we write a $1$ and each right step we write a $0$.
For example, with $\lambda = 522100$ (so $n = 6$) with $m = 14$, the corresponding $\{0,1\}$-sequence is $11010110001000$.
Next, by forgetting about the color (\textit{i.e.}, replacing every color with a $1$), we obtain the model of Motegi and Sakai~\cite{MS13,MS14}.
Furthermore, every admissible state corresponds to a GT pattern $(\lambda^{(i)})_{i=0}^n$ by letting $\lambda^{(i)}$ be the $(n-i)$-th row of vertical edges (with the top (half) edges being the $0$-th row of the model) to be the $\{0,1\}$-sequence of a partition~\cite[Sec.~3]{MS14}.
We denote this bijection $\fP$ from $\states_{\lambda}$ to GT patterns with top row $\lambda$.
However, the bijection $\fP$ is not weight preserving, but instead we also have to twist by $w_0$, \textit{i.e.}, map $z_i \mapsto z_{n+1-i}$.

We refine the states of $\overline{\states}_{\lambda,w}$ to allow markings at the configurations $\tt{a}_2$.
More precisely, a \defn{marked state} is a pair $(S, M)$ with $S \in \overline{\states}_{\lambda,w}$ such that $M$ is some subset of all configurations of $\tt{a}_2$.
We note that $\fP$ naturally extends to a bijection between marked states $\bigsqcup_{w \in \sym{n}} \overline{\states}_{\lambda,w}$ and marked GT patterns with top row $\lambda$ as each configuration $\tt{a}_2$ in a state $S$ corresponds to a position where a marking is possible in $\fP(S)$.
As before, the weight gets twisted by $w_0$.
Thus, for any state $S \in \overline{\states}_{\lambda,w}$, we have
\begin{equation}
\label{eq:marked_weight}
\wt(S) = \sum_{(S, M)} \beta^{|M|} \wt(S, M) = \sum_{(\fP(S), M)} \beta^{|M|} w_0 \wt(\fP(S), M),
\end{equation}
where the first sum is over all possible markings of $S$ and the second sum is over all possible markings of $\fP(S)$.
We also can do the same refinement for $\states_{\lambda,w}$. The following two theorems are proved by noting that markings correspond to adding larger entries in a marked GT pattern.

\begin{theorem}
\cref{conj:svt_Lascoux} is true, that is to say
\[
\overline{L}_{w\lambda}(\zz; \beta) = \sum_{\substack{T \in \svt^n(\lambda) \\ K(T) = K_{w\lambda}}} \wt_{\beta}(T) = \sum_{S \in \skyline(w\lambda)} \wt_{\beta}(S),
\qquad
L_{w\lambda}(\zz; \beta) = \sum_{\substack{T \in \svt^n(\lambda) \\ K(T) \leq K_{w\lambda}}} \wt_{\beta}(T).
\]
\end{theorem}


%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
VB would like to thank Ben Brubaker, Daniel Bump, and Henrik Gustafsson for useful discussion. 
TS would like to thank Kohei Motegi and Kazumitsu Sakai for invaluable discussions and explanations on their papers, in particular the weights of the five-vertex model that we use here.
TS would also like to thank Cara Monical, Tomoo Matsumura, Oliver Pechenik, and Shogo Sugimoto for useful discussions.
KW would like to thank Ben Brubaker for useful discussions and inspiration.
This paper benefited from computations using \textsc{SageMath}~\cite{sage}. We thank the anonymous referees for useful comments. 
}



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