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\documentclass[finalversion]{FPSAC2025}
\articlenumber{44}
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\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{coro}{Corollary}
\newtheorem{conj}{Conjecture}

\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Parn}{{\rm Par}_n}
\newcommand{\ParnQ}{{\rm Par}_n^{\mathbb{Q}}}
\newcommand{\LRsat}{{\rm LR}\text{-}{\rm sat}(n)}
\newcommand{\NLsat}{{\rm NL}\text{-}{\rm sat}(n)}
\newcommand{\modtwo}{\hspace{0.2em} ( {\rm mod}\hspace{0.2em} 2)}


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\title[NL saturation]{Proof of the Newell--Littlewood saturation conjecture}

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\author[J. Min]{\texorpdfstring{Jaewon Min\thanks{\href{mailto:jaewonm2@illinois.edu}{jaewonm2@illinois.edu}. Jaewon Min was partially supported by UIUC Campus Research Board grant RB24025.}\addressmark{1}}{Jaewon Min}}

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\address{\addressmark{1}Dept.~of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801}

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\received{November 15, 2024}

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\abstract{By inventing the notion of \emph{honeycombs}, A.~Knutson and T.~Tao proved the saturation conjecture for Littlewood--Richardson coefficients. The Newell--Littlewood numbers are a generalization of the Littlewood--Richardson coefficients. By introducing honeycombs on a M{\"o}bius strip, we prove the saturation conjecture for Newell--Littlewood numbers posed by S.~Gao, G.~Orelowitz and A.~Yong.  
}

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% MJW \resume{\lipsum[2]}

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\keywords{Littlewood--Richardson coefficients, representation theory, Lie groups, saturation, honeycombs, M{\"o}bius strips}

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\begin{document}

\maketitle
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\section{Introduction}\label{sec1}

\subsection{Background}

The irreducible polynomial representations $V_\lambda$ of $\text{GL}_n\C$ are indexed by the set of partitions 
\begin{equation}\label{eqn1.1}
\Parn := \{ \lambda = (\lambda_1 ,\cdots , \lambda_n ) \in \Z^n \mid \lambda_1 \geq \cdots \geq \lambda_n \geq 0  \};
\end{equation}
see, \textit{e.g.},  \cite{fulton1991representation}. For each $\mu, \nu \in \Parn$, 
\begin{equation}\label{eqn1.2}
V_\mu \otimes V_\nu \cong \bigoplus_{\lambda \in {\rm Par}_n} V_\lambda^{\oplus c_{\mu , \nu}^{\lambda}}.
\end{equation}
The tensor product multiplicities $c_{\mu, \nu}^{\lambda}$ are the \textbf{Littlewood--Richardson coefficients}. 

For each $k \in \N:=\{1,2,3,\ldots\}$ and $\lambda \in {\rm Par}_n$, let $k \lambda := (k \lambda_1 , \cdots , k\lambda_n)$. 

\begin{thm}[Saturation of Littlewood--Richardson coefficients {\rm \cite{MR1671451}}]\label{thm1.1}
Let $\lambda, \mu, \nu \in \Parn$. If there exists $k \in \N$ such that $c_{k \mu , k \nu}^{k \lambda} > 0$, then $c_{\mu , \nu}^{\lambda} > 0$.
\end{thm}

A.~Knutson and T.~Tao proved \Cref{thm1.1} using \textit{honeycombs} \cite{MR1671451}. Honeycombs are combinatorial objects used to count Littlewood--Richardson coefficients. This paper concerns a generalization of \Cref{thm1.1} and its proof. 

The significance of the saturation theorem stems from \textit{Horn's conjecture} \cite{MR140521} which gives a recursive description of linear inequalities, called \emph{Horn's inequalities}, on the eigenvalues of $n\times n$ Hermitian matrices $A$, $B$ and $A+B$. \Cref{thm1.1} combined with earlier work of  A.~A.~Klyachko \cite{MR1654578} proved Horn's conjecture; see W.~Fulton's survey \cite{MR1754641}.

\subsection{Main result}\label{subsec1.2}

We generalize \Cref{thm1.1} and its proof to the  \textbf{Newell--Littlewood numbers}, which are defined, using the Littlewood--Richardson coefficients, as
follows:
\begin{equation}\label{eqn1.4}
N_{\lambda, \mu , \nu}:= \sum_{\alpha , \beta , \gamma \in \Parn} c_{\beta , \gamma}^{\lambda} c_{\gamma , \alpha}^{\mu} c_{\alpha , \beta}^{\nu} \quad (\lambda , \mu , \nu \in \Parn).
\end{equation}

For each $\lambda \in \Parn$, let $|\lambda | := \lambda_1 + \cdots + \lambda_n$. If $c_{\mu, \nu}^\lambda \neq 0$, then $|\mu| + |\nu | = |\lambda |$. According to \cite[Lemma 2.2]{MR4302162},
\begin{equation}\label{eqn1.6}
|\mu| + |\nu | = |\lambda | \quad \Rightarrow \quad N_{\lambda , \mu , \nu} = c_{\mu , \nu}^{\lambda}.
\end{equation}
Thus, Newell--Littlewood numbers  generalize  Littlewood--Richardson coefficients.

In 2021, S.~Gao, G.~Orelowitz and A.~Yong \cite[Conjecture~5.5, 5.6]{MR4302162} conjectured a generalization of \Cref{thm1.1}.
In \emph{ibid.}, this conjecture was proved for the special cases that $\lambda = \mu = \nu$ \cite[Theorem~4.1]{MR4302162} and for $n=2$ \cite[Theorem~4.1]{MR4302162}. In \cite[Corollary~6.1]{gao2022newelllittlewoodnumbersiiieigencones}, S.~Gao, G.~Orelowitz, N.~Ressayre, and A.~Yong gave a computational proof of the cases when $n \leq 5$.
Our main result is a complete proof of said conjecture from \cite[Conjecture~1.1]{MR4302162}, 
by modifying the proof of \Cref{thm1.1} in \cite{MR1671451}.

\begin{thm}[Newell--Littlewood saturation {\rm \cite[Theorem 1.2]{min2024proofnewelllittlewoodsaturationconjecture}}]\label{thm1.2}
Let  $\lambda,\mu,\nu \in \Parn$ satisfying $|\lambda|+|\mu|+|\nu| \equiv 0 \modtwo$. 
If there exists $k \in \N$ such that $N_{k\lambda,k\mu,k\nu} >0$, then $N_{\lambda,\mu,\nu} >0$.
\end{thm}

This follows from the technical center of this paper, \Cref{thm3.2}, and is introduced at the end. In view of \eqref{eqn1.6}, \Cref{thm1.2} immediately implies the saturation of Littlewood--Richardson coefficients.

We now discuss consequences of proving \Cref{thm1.2}. Analogous to the Horn's inequalities, S.~Gao, 
G.~Orelowitz and A.~Yong 
\cite[Theorem~1.3]{MR4529925} defined \textit{extended Horn inequalities} (which we will not restate here) and proved that they are necessary conditions for $N_{\lambda , \mu , \nu} >0$. Additionally, they conjectured the converse; our paper also confirms this conjecture, due to \cite[Corollary 8.5]{gao2022newelllittlewoodnumbersiiieigencones}.

\begin{coro}{\rm \cite[Conjecture 1.4]{MR4529925}}
If $(\lambda , \mu , \nu) \in (\Parn)^3$ satisfies the extended Horn inequalities and $|\lambda| + |\mu| + |\nu| \equiv 0 \modtwo$, then $N_{\lambda , \mu , \nu} >0$.
\end{coro}

Therefore, the extended Horn inequalities and $|\lambda| + |\mu| + |\nu| \equiv 0 \modtwo$ completely determine the set
\begin{equation}
{\rm NL} := \{ ( \lambda , \mu , \nu) \in (\Parn)^3 \mid N_{\lambda , \mu , \nu} > 0 \}.
\end{equation}

Another application is  to the eigenvalues of a family of complex matrices. Let
\begin{equation}
\ParnQ := \{ \lambda = (\lambda_1 ,\cdots , \lambda_n ) \in \Q^n \mid \lambda_1 \geq \cdots \geq \lambda_n \geq 0  \},
\end{equation}
\begin{equation}\label{eqn1.5}
\NLsat := \{ (\lambda , \mu , \nu) \in (\ParnQ)^3 \mid \exists k >0, \quad N_{k \lambda, k \mu , k \nu} >0 \}.
\end{equation}
In \cite[Proposition 3.1]{gao2022newelllittlewoodnumbersiiieigencones}, S.~Gao, G.~Orelowitz, N.~Ressayre and A.~Yong proved that $\NLsat$ describes an analogue of the Horn problem for matrices in $\mathfrak{sp}_{2n} \C \cap \mathfrak{u}_{2n} \C$. \Cref{thm1.2} shows that $\NLsat$ can be simplified to ${\rm NL}$.

Lastly, \Cref{thm1.2} is related to the conjecture suggested in \cite[Section~7]{MR1671451}. Given a split reductive group $G$ over $\C$, it has a root system and its irreducible representation is indexed by a dominant integral weight $\lambda$. Write the dual weight as $\lambda^*$ and the tensor product multiplicities by $c_{\mu, \nu}^{\lambda}(G)$.

\begin{thm}{\rm\cite[Theorem 1.1]{MR2415306}}\label{thm1.3}
Let $G$ be a split reductive group over $\C$. Then there exists $k_G \in \N$ with following property: if $\lambda , \mu , \nu$ are dominant integral weights such that $\lambda^* + \mu + \nu$ is in the root lattice,
\begin{equation}
\exists k \in \N \text{ such that } c_{k\mu , k\nu}^{k \lambda}(G) >0 \quad \Rightarrow \quad c_{k_G \mu , k_G \nu}^{k_G \lambda}(G) >0.
\end{equation}
\end{thm}

\begin{conj}{\rm\cite[Conjecture 1.4]{MR2253157}}\label{con1.1}
If the root system of $G$ is simply laced, then $k_G$ can be chosen as $1$.
\end{conj}

In particular, we are interested in the cases when $G = {\rm SO}_{2n+1}\C, \hspace{0.2em} {\rm Sp}_{2n} \C, \hspace{0.2em} {\rm SO}_{2n} \C$. In \cite[Theorem 1.1]{MR2415306}, M.~Kapovich and J.~J.~Millson proved that $k_G = 4$. Additionally, P.~Belkale and S.~Kumar \cite[Theorem 6, 7]{MR2580675} proved that $k_G=2$ if $G$ is ${\rm SO}_{2n+1}\C$ or ${\rm Sp}_{2n} \C$. S.~V.~Sam \cite[Theorem 1.1]{MR2855088} proved that $k_G=2$ when $G = {\rm SO}_{2n+1}\C, \hspace{0.2em}{\rm Sp}_{2n} \C,\hspace{0.2em}{\rm SO}_{2n} \C$, by using quiver representations, extending the proof of \Cref{thm1.1} given by H.~Derksen and  J.~Weyman~\cite{MR1758750}.

The possibility that $k_G=1$ when $G = {\rm SO}_{2n}  \C$ remains open. For recent work concerning  ${\rm SO}_{2n}  \C$ and ${\rm Spin}_{2n}  \C$, see, \textit{e.g.}, \cite{MR2508902, MR4261948}.

Let $G = {\rm SO}_{2n+1}\C,\hspace{0.2em} {\rm Sp}_{2n} \C,\hspace{0.2em}{\rm SO}_{2n} \C$. For the classical Lie groups, irreducible representations are indexed by the set of partitions $\Parn$; see, \textit{e.g.}, \cite{fulton1991representation, MR885807}. $l(\lambda)$ denotes the number of non-zero components of $\lambda = (\lambda_1 , \cdots , \lambda_n)$. According to \cite[Theorem 3.1]{MR991410}, 
\begin{equation}\label{eqn33}
l(\mu)+l(\nu) \leq n \quad \Rightarrow \quad N_{\lambda, \mu , \nu} = c_{\mu , \nu}^{\lambda}(G).
\end{equation}
The condition \eqref{eqn33} imposed on $\mu , \nu \in \Parn$ is called the \textit{stable range}. The next result is an immediate consequence of \Cref{thm1.2}:

\begin{coro}
Let $G = {\rm SO}_{2n+1}\C, \hspace{0.2em}{\rm Sp}_{2n} \C,\hspace{0.2em}{\rm SO}_{2n} \C$. Suppose $\lambda, \mu , \nu \in \Parn$ and $l(\mu)+l(\nu) \leq n$. If there exists $k \in \N$ such that $c_{k\mu, k \nu}^{k \lambda}(G) >0$, then $c_{\mu , \nu}^\lambda(G) >0$. 
\end{coro}

Thus, $k_G$ from \Cref{con1.1} may be taken as $1$ for $G = {\rm SO}_{2n+1}\C, \hspace{0.2em}{\rm Sp}_{2n} \C,\hspace{0.2em}{\rm SO}_{2n} \C$ if $(\mu , \nu)$ is in the stable range. 


\section{Saturation of Littlewood--Richardson coeffcients}

In this paper, $B$ is fixed to be the two-dimensional real vector space
\begin{equation}\label{eqn2.1}
B:=\{(x,y,z) \in \mathbb{R}^{3} \mid x+y+z = 0\}.
\end{equation}
Let the \textbf{lattice points} of $B$ be
\begin{equation}\label{eqn2.2}
B_{\mathbb Z}:= \{(x,y,z) \in \mathbb{Z}^{3} \mid x+y+z = 0\}.
\end{equation}

A.~Knutson and T.~Tao constructed a directed graph $\Delta_n$; see \cite[Figure 3]{MR1671451}. Write $V_{\Delta_n}$ as the set of vertices of $\Delta_n$. If $h: V_{\Delta_n} \rightarrow B$ satisfies conditions to be a \textit{configuration}, $h$ is called a \textbf{honeycomb}; see \cite[Section 2.1]{MR1671451} for details. Also, see \cite[Section 2.2]{MR1671451} to check how they read off three partitions $\lambda , \mu , \nu \in \Parn$ from a honeycomb $h$, which is \textit{boundary condition} of $h$ denoted by $\partial (h)$.

\begin{thm}[\rm {\cite[Theorem 4]{MR1671451}}]\label{thm2.1}
Let $\lambda,\mu,\nu \in \Parn$. Then $c_{\mu , \nu}^{\lambda}$ counts the
number of honeycombs $h$ satisfying:
\begin{itemize}
\item $\partial (h) = (\lambda , \mu , \nu)$, and
\item $\forall v \in V_{\Delta_n}$, $h(v) \in B_\Z$.
\end{itemize}
\end{thm}

\begin{thm}[\rm {\cite[Theorem 2]{MR1671451}}]\label{thm2.2}
For any honeycomb $h$ with $\partial (h) \in \Z^{3n}$, there exists a honeycomb $g$ such that:
\begin{itemize}
\item $\partial (g) = \partial (h)$, and
\item $\forall v \in V_{\Delta_n}$, $g(v) \in B_\Z$.
\end{itemize}
\end{thm}

See \cite[Section 5]{MR1671451} to check the construction of honeycomb $g$ in \Cref{thm2.2}, named as \textit{largest lift}. As a corollary, we have \Cref{thm1.1}.

\begin{proof}[Proof of \Cref{thm1.1}]
Suppose $\lambda, \mu, \nu \in \Parn$ and $k \in \N$ such that $c_{k \mu, k \nu}^{k \lambda}>0$. By \Cref{thm2.1}, there exists a honeycomb  $h$ such that
\begin{equation}\label{eqn2.9}
\partial (h) = (k \lambda , k \mu , k \nu).
\end{equation}
Since $k>0$, $\frac{1}{k} h $ is also a honeycomb; for instance, see \cite[Lemma 2.1]{min2024proofnewelllittlewoodsaturationconjecture}. Then
\begin{equation}\label{eqn2.10}
\partial \left(\frac{1}{k} h \right) =(\lambda , \mu , \nu).
\end{equation}
Apply \Cref{thm2.2} to find a honeycomb $g$ such that $\partial (g) = \partial (\frac{1}{k}h)$ and $g(v) \in B_\Z$ for all $v \in V_{\Delta_n}$. By \Cref{thm2.1} once more, $c_{\mu, \nu}^{\lambda}>0$.
\end{proof}

\section{Graphs embedded in M{\"o}bius strips}

Define a directed graph $\widetilde{\Gamma}_n$ of which vertices and edges are
\begin{align}
V_{\widetilde{\Gamma}_n} &= \{\widetilde{A}_{i,j} \mid i, j \in \Z, 0\leq i \leq n \} \cup \{\widetilde{B}_{i,j} \mid i, j \in \Z, 0\leq i \leq n\}, \\
E_{\widetilde{\Gamma}_n} &= \{ (\widetilde{A}_{i,j}, \widetilde{B}_{i,j}) \mid i, j \in \Z ,0 \leq i \leq n\} \\
& \cup \{(\widetilde{A}_{i,j}, \widetilde{B}_{i-1,j}) \mid i,j \in \Z,  1 \leq i \leq n \} \\
& \cup \{ (\widetilde{A}_{i,j}, \widetilde{B}_{i-1,j-1}) \mid i, j \in \Z , 1 \leq  i \leq n\}.
\end{align}
Here, we denote a directed edge from $U$ to $W$ as $(U,W)$. As in \Cref{fig5},  $\widetilde{\Gamma}_{n}$ is an infinite strip composed of $(n-1)$-number of layers of hexagons. There are vertices connected to exactly one edge in \Cref{fig5}, namely $\widetilde{A}_{0,j}, \widetilde{B}_{n,j}$ for $j \in \Z$. Such vertices of $\widetilde{\Gamma}_{n}$ are the \textbf{boundary vertices} in $\widetilde{\Gamma}_n$.

\begin{figure}
\centering
\includegraphics[width=\textwidth]{figure_5b_MobTinCov.png}
\caption{The directed graph $\widetilde{\Gamma}_{5}$.}
\label{fig5}
\end{figure}

We now define a graph $\Gamma_{n}$, which will be a ``quotient graph'' of 
$\widetilde{\Gamma}_n$. Intuitively, ``slice'' $\widetilde{\Gamma}_n$ into pieces by using trapezoids as in \Cref{fig41}. We want to identify all trapezoids as one, which corresponds to the quotient graph $\Gamma_n$. For instance, four bold vertices of $\widetilde{\Gamma}_5$ in \Cref{fig41} are identified as a vertex of $\Gamma_5$. 

\begin{figure}
\centering
\includegraphics[width=\textwidth]{figure_41_equiv1.png}
	\caption{Identify $\widetilde{\Gamma}_{5}$ to have $\Gamma_5$.}
\label{fig41}
\end{figure}

To be precise, identify the vertices of $\widetilde{\Gamma}_{n}$ using the equivalence relation $\sim$ defined by
\begin{equation}\label{eqn3.3}
\widetilde{A}_{i,j} \sim \widetilde{B}_{-i+n,-i+j+2n}, \quad \widetilde{B}_{i,j} \sim \widetilde{A}_{-i+n,-i+j+2n}. \quad (i,j \in \Z, 0 \leq i \leq n).
\end{equation}

The vertices of $\Gamma_n$ are representatives of the equivalence classes $[\widetilde{P}]$ for each $\widetilde{P} \in V_{\widetilde{\Gamma}_n}$; we have the quotient map induced by the equivalence relation: 
\begin{equation}
	p_v : V_{\widetilde{\Gamma}_{n}} \rightarrow V_{\Gamma_{n}}, \quad \widetilde{P} \mapsto [\widetilde{P}].
\end{equation}

\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{figure_6b_MobTin.png}
\caption{The graph $\Gamma_{5}$.}
\label{fig6}
\end{figure}

Next, we define an equivalence relation $\equiv$ on the edges in $\widetilde{\Gamma}_n$. Write a directed edge $\widetilde{e} = ({\sf tail}(\widetilde{e}), {\sf head}(\widetilde{e}))$. For each $\widetilde{e} = (\widetilde{A},\widetilde{B})$ and $\tilde{e}' = (\widetilde{A}' , \widetilde{B}')$, set
\begin{equation}
	\widetilde{e} \equiv \tilde{e}' \iff \widetilde{A} \sim \widetilde{A}' , \widetilde{B} \sim \widetilde{B}' \text{ or } \widetilde{A} \sim \widetilde{B}' , \widetilde{B} \sim \widetilde{A}'.
\end{equation}
The edges of $\Gamma_n$ are representatives of equivalence classes $[\widetilde{e}]$ for each $\widetilde{e} \in E_{\widetilde{\Gamma}_n}$. Here, $[\widetilde{e}]$ is a \emph{non-directed} edge connecting $p_v({\sf tail}(\widetilde{e}))$ and $p_v ({\sf head}(\widetilde{e}))$. We denote a non-directed edge $e = \{ A, B \}$ if $e$ connects vertices $A$ and $B$. The quotient map is defined by
\begin{equation}
	p_e : E_{\widetilde{\Gamma}_n} \rightarrow E_{\Gamma_n}, \quad \widetilde{e} \mapsto [\widetilde{e}].
\end{equation}

From \eqref{eqn3.3}, $\widetilde{A}_{i,j} \sim \widetilde{A}_{i,j+3n}$ and $\widetilde{B}_{i,j} \sim \widetilde{B}_{i,j+3n}$ for all indices. Therefore, there are $3n(n+1)$-many equivalence classes in $V_{\widetilde{\Gamma}_n}$, each represented by $\widetilde{A}_{i,j}$ for $0 \leq i \leq n, 1 \leq j \leq 3n$. Set $A_{i,j} := p_v (\widetilde{A}_{i,j})$ for $0\leq i \leq n, 1 \leq j \leq 3n$.

In summary, $\Gamma_n$ is a finite graph embedded in a M{\"o}bius strip. For instance, consider $\Gamma_5$ in \Cref{fig6}. Each of the vertices $A_{1,1},A_{2,1},A_{3,1},A_{4,1}, A_{5,1}$ are connected to $A_{5,10},A_{4,9},A_{3,8},A_{2,7},A_{1,6}$, respectively.

\section{Covering space of M{\"o}bius strips}

Fix $\delta \in \mathbb{N}$. For each $k \in \mathbb{Z}$, define subsets of $B$
\begin{align}\label{eqn3.15}
D_{\delta}^{(2k)}& := \{ (x,y,z) \in B \mid (k-1)\delta \leq x \leq k \delta, \hspace{1em} (k-1)\delta \leq y \leq k\delta \}, \\
D_{\delta}^{(2k+1)}&:= \{ (x,y,z) \in B \mid (k-1)\delta \leq x \leq k \delta, \hspace{1em}  k\delta \leq y \leq (k+1)\delta \}, \\
\widetilde{B}_{\delta}&:=\bigcup_{k \in \mathbb{Z}} D_{\delta}^{(k)}.
\end{align}
$\widetilde{B}_{\delta}$ is depicted in \Cref{fig7}, as an infinite zigzag strip. Here, $D_{\delta}^{(k)}$ is a rhombus. In \Cref{fig7}, there are six rhombi, which are $D_{\delta}^{(0)}, D_{\delta}^{(-1)} , \cdots , \cdots D_{\delta}^{(-5)} $, from left to right.

\begin{figure}
\centering
\includegraphics[width=\textwidth]{figure_7_BDCov.png}
\caption{The infinite strip $\widetilde{B}_{\delta}$ contained in $B$.}
\label{fig7}
\end{figure}

We want to define a quotient space $B_\delta$ of $\widetilde{B}_\delta$. Intuitively, we ``slice'' $\widetilde{B}_\delta$ into pieces and identify them into one to construct $B_\delta$. See \Cref{fig43}, where the four bold points are identified as one element in $B_\delta$.

\begin{figure}
\centering
\includegraphics[width=\textwidth]{figure_43_equiv2.png}
\caption{The M{\"o}bius strip $B_\delta$ and its covering space $\widetilde{B}_\delta$.}
\label{fig43}
\end{figure}

To write a formal definition, define an equivalence relation on $B$, namely
\begin{equation}\label{eqn3.5}
(x,y,z) \sim (y-2\delta, x-\delta, z+3\delta).
\end{equation}
Denote the quotient map by $q : B \rightarrow B/ \sim$. Define $B_{\delta} := q( \widetilde{B}_{\delta} )$. $\widetilde{B}_\delta$ is an infinite strip whereas $B_\delta$ is a M{\"o}bius strip; see \Cref{fig43}.

By the equivalence relation on $B$, $D_{\delta}^{(k)}$ is identified to $D_{\delta}^{(k-3)}$ for all $k \in \mathbb{Z}$. For instance, $D_{\delta}^{(0)}$ and $D_{\delta}^{(-3)}$, $D_{\delta}^{(-1)}$ and $D_{\delta}^{(-4)}$, $D_{\delta}^{(-2)}$ and $D_{\delta}^{(-5)}$ are identified by the map $q$ in \Cref{fig7}.

\section{M{\"o}bius honeycombs}

\begin{figure}
\centering
\includegraphics[width=\textwidth]{figure_9_MConfig.png}
\caption{Image of $\widetilde{h}$ contained in $\widetilde{B}_{\delta}$ when $n=5$.}
\label{fig9}
\end{figure}

Define a direction map $d: E_{\widetilde{\Gamma}_n} \rightarrow B$ by mapping
\begin{equation}
(\widetilde{A}_{i,j}, \widetilde{B}_{i-1,j-1}) \mapsto (0,-1,1), \quad
(\widetilde{A}_{i,j}, \widetilde{B}_{i-1,j}) \mapsto (1,0,-1), \quad
(\widetilde{A}_{i,j}, \widetilde{B}_{i,j})  \mapsto (-1,1,0).   
\end{equation}
As in \Cref{fig5}, $d$ maps each southeast edges to $(0,-1,1)$, southwest edges to $(1,0,-1)$, and north edges to $(-1,1,0)$. Define a function $\widetilde{h} : V_{\widetilde{\Gamma}_{n}} \rightarrow B$ satisfying
\begin{equation}\label{eqn3.9}
\widetilde{h}({\sf head}(\widetilde{e}))-\widetilde{h}({\sf tail}(\widetilde{e})) \in \{ a \cdot v \in B \mid a \geq 0, v= d(\widetilde{e})\}, \quad \widetilde{e} \in E_{\widetilde{\Gamma}_n}.
\end{equation}

Consider $\widetilde{A}_{0,1}, \widetilde{A}_{0,2} ,\cdots , \widetilde{A}_{0,3n}$, which are representatives of equivalence classes of boundary vertices. For instance, in \Cref{fig5}, these vertices are on the lowest level, from right to left. Add conditions on $\widetilde{h}$ so that it satisfies
\begin{subequations}\label{eqn3.7}
\begin{align}
\widetilde{h} (\widetilde{A}_{0,j}) & \in \{(-2 \delta , 2 \delta - \xi , \xi) \mid 4\delta \leq \xi \leq 5\delta \}, \quad (1 \leq j \leq n) \\
\widetilde{h} (\widetilde{A}_{0,j}) & \in \{(- \delta ,  \delta - \xi , \xi) \mid 2\delta \leq \xi \leq 3\delta \}, \quad (n+1 \leq j \leq 2n) \\
\widetilde{h} (\widetilde{A}_{0,j}) & \in \{(0 ,  - \xi , \xi) \mid 0 \leq \xi \leq \delta \}, \quad (2n+1 \leq j \leq 3n).
\end{align}
\end{subequations}

When $n=5$, for each $1 \leq j \leq 5$, $\widetilde{A}_{0,j}$ should be mapped to the line segment connecting $(-2\delta, -3\delta, 5\delta)$ and $(-2\delta, -2\delta, 4\delta)$, which is in the boundary of $D_\delta^{(-4)}$; see \Cref{fig7}. The cases of $6 \leq j \leq 10$ and $11\leq j \leq 15$ can be interpreted in similar fashion. 

The last condition on $\widetilde{h}$ is
\begin{equation}\label{eqn3.8}
\widetilde{P}_1 \sim \widetilde{P}_2 \hspace{0.5em} \in V_{\widetilde{\Gamma}_n} \quad \Rightarrow \quad \widetilde{h}(\widetilde{P}_1) \sim \widetilde{h}(\widetilde{P}_2) \hspace{0.5em} \in B.
\end{equation}

For fixed $\delta \in \mathbb{N}$, $\widetilde{h} : V_{\widetilde{\Gamma}_{n}} \rightarrow B$ is a \textbf{M{\"o}bius honeycomb} of size $\delta$ if $\widetilde{h}$ satisfies \eqref{eqn3.9}, \eqref{eqn3.7} and \eqref{eqn3.8}. In \eqref{eqn3.7}, write $\xi_i$ as the $z$-coordinate of $\widetilde{h}(\widetilde{A}_{0,i})$. Define the \textbf{boundary condition} of $\widetilde{h}$ as $(\xi_1 , \cdots, \xi_{3n})$ and denote it as $\partial (\widetilde{h})$. See \Cref{fig9} and \ref{fig44} for illustrations of M{\"o}bius honeycombs.

\begin{figure}
\centering
\includegraphics[width = \textwidth]{figure_44_equiv3.png}
\caption{Images of $\widetilde{h}$ repeated due to identification.}
\label{fig44}
\end{figure}

\section{Saturation of Newell--Littlewood numbers}

Our goal is to generalize \Cref{thm2.1} and \Cref{thm2.2} from Littlewood--Richardson coefficients to Newell--Littlewood numbers. As a result, we have \Cref{thm3.1} and \Cref{thm3.2}, leading to \Cref{thm1.2}.

\begin{thm}[\rm {\cite[Theorem 3.1]{min2024proofnewelllittlewoodsaturationconjecture}}]\label{thm3.1}
Let $\lambda,\mu,\nu \in \text{\emph{Par}}_{n}$ and $\delta \in \N$ such that $\delta \geq \lambda_1, \mu_1, \nu_1$. Then $N_{\lambda,\mu,\nu}$ counts the number of M{\"o}bius honeycombs $\widetilde{h}$ of size $\delta$ satisfying:
\begin{itemize}
\item $\partial (\widetilde{h}) = (\lambda_{1}+4\delta,\cdots, \lambda_{n}+4\delta, \mu_{1}+2\delta, \cdots, \mu_{n}+2\delta, \nu_{1}, \cdots, \nu_{n})$, and
\item $\forall \widetilde{W} \in V_{\widetilde{\Gamma}_n}$, $\widetilde{h} ( \widetilde{W} ) \in B_{\Z}$.
\end{itemize}
\end{thm}

\begin{proof}
By \Cref{thm2.1}, $c_{\beta ,\gamma}^{\lambda}c_{\gamma , \alpha}^{\mu} c_{\alpha, \beta}^{\nu}$ is the number of ordered triples $(h_{\lambda}, h_{\mu}, h_{\nu})$ of honeycombs satisfying:
\begin{itemize}
\item $\partial (h_{\lambda}) = (\lambda , \beta,\gamma)$, $\partial (h_{\mu}) = (\mu , \gamma, \alpha)$, $\partial (h_{\nu}) = (\nu, \alpha, \beta)$, and
\item $\forall v \in V_{\Delta_n}$, $h_\lambda (v) , h_\mu (v) , h_\nu (v) \in B_\Z$.
\end{itemize}
If $c_{\beta ,\gamma}^{\lambda}c_{\gamma , \alpha}^{\mu} c_{\alpha, \beta}^{\nu} \neq 0$, then $\delta \geq \alpha_1, \beta_1, \gamma_1$ follows from $\delta \geq \lambda_1, \mu_1, \nu_1$. As a result,
\begin{equation}\label{eqn3.31}
\forall v \in V_{\Delta_n}, \quad h_\lambda (v) , h_\mu (v) , h_\nu (v) \in D_\delta^{(0)}.
\end{equation}

We have infinite copies of three different types of rhombi depicted in \Cref{fig12}. Each type of rhombi is arranged in $B$ as follows.
\begin{itemize}
\item $h_\lambda$ rhombus: $\cdots , D_{\delta}^{(-4)}, D_{\delta}^{(-1)}, D_{\delta}^{(2)}, D_{\delta}^{(5)}, \cdots$
\item $h_\mu$ rhombus: $\cdots , D_{\delta}^{(-5)}, D_{\delta}^{(-2)}, D_{\delta}^{(1)}, D_{\delta}^{(4)}, \cdots$
\item $h_\nu$ rhombus: $\cdots , D_{\delta}^{(-6)}, D_{\delta}^{(-3)}, D_{\delta}^{(0)}, D_{\delta}^{(3)}, \cdots$
\end{itemize}

Gluing pieces along the line segments $\alpha^{*}, \beta^{*}$ and $\gamma^{*}$, we have $\widetilde{h}$ satisfying the given conditions. Therefore, the number of $\widetilde{h}$ satisfying the given conditions is greater than or equal to $N_{\lambda,\mu,\nu}$.

Conversely, by slicing $\widetilde{B}_\delta$ into $D_{\delta}^{(k)}$, we can reverse the process above, proving the other side of the inequality.
\end{proof}

\begin{figure}
\centering
\includegraphics[width=\textwidth]{figure_99.png}
\caption{Image of $h_{\lambda},h_{\mu},h_{\nu}$ contained in $D_{\delta}^{(0)}$ when $n=5$.}
\label{fig12}
\end{figure}

In short, $c_{\beta , \gamma}^\lambda c_{\gamma , \alpha}^\nu c_{\alpha, \beta}^\nu$ leads to gluing three honeycombs, which is a M{\"o}bius strip. Here, the boundary of M{\"o}bius strip is chosen as in \Cref{fig44} so that gluing process can be reversed.

For instance, let $n=3$ and $\lambda = \mu = \nu = (3,2,1)$. Since $\lambda_1 = \mu_1 = \nu_1 =3$, take $\delta =3$. In \Cref{fig57}, the number of M{\"o}bius honeycombs satisfying the conditions is $20$. Therefore, $N_{\lambda , \mu , \nu} = 20$. 

\begin{figure}
\centering
\includegraphics[width = \textwidth]{figure_57_NL.png}
\caption{$n=3$, $\delta = 3$, $\lambda = \mu = \nu = (3,2,1)$. Then $N_{\lambda, \mu , \nu} = 20$.}
\label{fig57}
\end{figure}

\begin{thm}[\rm {\cite[Theorem 3.2]{min2024proofnewelllittlewoodsaturationconjecture}}]\label{thm3.2}
Let $\delta \in \mathbb{N}$ and $\widetilde{h}$ be a M{\"o}bius honeycomb of size $\delta$ such that $\partial (\widetilde{h}) = (\xi_{1}, \cdots, \xi_{3n})$ in $\mathbb{Z}^{3n}$ and $\sum_{1\leq j \leq 3n}\xi_{j} \equiv 0 \modtwo $. Then there exists a M{\"o}bius honeycomb $\widetilde{g}$ of size $\delta$ with:
\begin{itemize}
\item $\partial (\widetilde{g}) = \partial (\widetilde{h})$, and
\item $\forall \widetilde{W} \in V_{\widetilde{\Gamma}_n}$, $\widetilde{g} ( \widetilde{W} ) \in B_{\Z}$.
\end{itemize}
\end{thm}

For the existence of $\widetilde{g}$ in \Cref{thm3.2}, see \cite[Section 4]{min2024proofnewelllittlewoodsaturationconjecture} for the construction of \textit{largest lifts} of M{\"o}bius honeycombs and \cite[Section 5]{min2024proofnewelllittlewoodsaturationconjecture} for readjustment. As a corollary, we have \Cref{thm1.2}.

\begin{proof}[Proof of \Cref{thm1.2}]
Suppose that $N_{k\lambda , k\mu, k\nu} >0$. Choose $\delta \in \N$ such that $\delta \geq \lambda_1,\mu_1, \nu_1$. By \Cref{thm3.1}, there exists a M{\"o}bius honeycomb $\widetilde{h}$ of size $k\delta$ satisfying
\begin{equation}
\partial (\widetilde{h}) = (k\lambda_{1}+4k\delta,\cdots, k\lambda_{n}+4k\delta, k\mu_{1}+2k\delta, \cdots, k\mu_{n}+2k\delta, k\nu_{1}, \cdots, k\nu_{n}).
\end{equation}

Due to \cite[Lemma A.7]{min2024proofnewelllittlewoodsaturationconjecture}, $\frac{1}{k} \widetilde{h}$ is a M{\"o}bius honeycomb of size $\delta$ and
\begin{equation}
\partial \left( \frac{1}{k}\widetilde{h} \right) = (\lambda_{1}+4\delta,\cdots, \lambda_{n}+4\delta, \mu_{1}+2\delta, \cdots, \mu_{n}+2\delta, \nu_{1}, \cdots, \nu_{n}).
\end{equation}
In particular, $\partial \left( \frac{1}{k}\widetilde{h} \right) \in \Z^{3n}$ and the sum of its components is $|\lambda| + |\mu| + |\nu| + 6n\delta$, which is an even integer due to the condition $|\lambda| + |\mu| + |\nu| \equiv 0 \modtwo$. Apply \Cref{thm3.2} to find a M{\"o}bius honeycomb $\widetilde{g}$ of size $\delta$ such that
\begin{equation}
\partial \left( \frac{1}{k}\widetilde{h}  \right) = \partial (\widetilde{g}) \hspace{0.5em} \text{and} \hspace{0.5em} \forall \widetilde{W} \in V_{\widetilde{\Gamma}_n}, \hspace{0.5em} \widetilde{g} (\widetilde{W}) \in B_\Z.
\end{equation}

Due to the existence of $\widetilde{g}$, $N_{\lambda , \mu , \nu} >0$ follows from \Cref{thm3.1}.
\end{proof}

\acknowledgements{
We thank Shiliang Gao and Alexander Yong for helpful remarks on this preprint. We thank Jiyang Gao for pointing out the exceptional case of ``white triangle of size $0.5$''.
}

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