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\title{Combinatorics of Newell--Littlewood Numbers}

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\author[S.~Gao, G.~Orelowitz, N.~Ressayre, and A.~Yong]{Shiliang Gao\thanks{\href{mailto:sgao23@illinois.edu}{sgao23@illinois.edu}. SG was partially supported by an NSF Graduate Research Fellowship.}\addressmark{1}, Gidon Orelowitz\thanks{\href{gidono2@illinois.edu}{gidono2@illinois.edu}. GO was partially supported by NSF RTG grant DMS 1937241.}\addressmark{1}, Nicolas Ressayre\thanks{\href{mailto:ressayre@math.univ-lyon1.fr}{ressayre@math.univ-lyon1.fr}}\addressmark{2}, \\ \and Alexander Yong\thanks{\href{mailto:ayong@illinois.edu}{ayong@illinois.edu}. AY was partially supported by 
NSF RTG grant DMS 1937241, a Simons Collaboration Grant, and the UIUC Center for Advanced Study.}\addressmark{1}}

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\address{\addressmark{1}Dept.~of Mathematics, University of Illinois at Urbana--Champaign, Urbana, IL \\ \addressmark{2}Institut Camille Jordan (ICJ), Universit\'e Claude Bernard Lyon I}

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\abstract{We give an exposition of recent developments in the study of Newell--Littlewood numbers. These
are the tensor product multiplicities of Weyl modules in the stable range. They are also the structure coefficients
of the Koike--Terada basis of the ring of symmetric functions. Two types of combinatorial results are exhibited, those obtained combinatorially starting from the definition of the numbers, and those that also employ geometric and/or representation theoretic methods.}

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\keywords{tensor product multiplicities, eigencones, Newell--Littlewood numbers}

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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above
\section{Introduction}
The \emph{Newell--Littlewood numbers} \cite{Newell, Littlewood} are defined by
\begin{equation}
\label{eqn:Newell--Littlewood}
N_{\mu,\nu,\lambda}=\sum_{\alpha,\beta,\gamma} c_{\alpha,\beta}^{\mu}
c_{\alpha,\gamma}^{\nu}c_{\beta,\gamma}^{\lambda},
\end{equation}
where the indices are partitions 
in 
\[{\sf Par}_n=\{(\lambda_1, \lambda_2, \ldots, \lambda_n)\in {\mathbb Z}_{\geq 0}^n:
\lambda_1\geq \lambda_2\geq \cdots \geq \lambda_n\}.\]
Here, $c^{\mu}_{\alpha,\beta}$ is the  
\emph{Littlewood--Richardson coefficient}; these are of interest in combinatorics, representation theory and algebraic geometry; see, \emph{e.g.}, the 
books \cite{Fulton.Harris, FultonYT, ECII}. This extended abstract mostly summarizes \cite{GOY, GOY2, GORY} but
we also mention related follow-up work.

For an $n$-dimensional vector space $V$ over ${\mathbb C}$ and $\lambda\in {\sf Par}_n$, the \emph{Weyl module} (or \emph{Schur functor}) ${\mathbb S}_{\lambda}(V)$ is an irreducible ${\sf GL}(V)$-module 
(see, \emph{e.g.}, \cite[Lectures~6 and 15]{Fulton.Harris}).
The Littlewood--Richardson coefficients are the tensor product multiplicities
\[{\mathbb S}_{\mu}(V) \otimes {\mathbb S}_{\nu}(V)\cong\bigoplus_{\lambda \in {\sf Par}_n}
{\mathbb S}_{\lambda}(V)^{\oplus c_{\mu,\nu}^{\lambda}}.\]

The Newell--Littlewood numbers arise similarly, where ${\sf GL}(V)$ is replaced
by other classical Lie groups ${\sf G}$. Suppose $W$ 
is a complex vector space, with a fixed nondegenerate symplectic or orthogonal form $\omega$.
%Fix a basis $\{\varepsilon_1,
%\varepsilon_2,\ldots,\varepsilon_{2n+\delta}\}$ such that 
%\[\omega(\varepsilon_k,
%\varepsilon_{2n+1+\delta-k})=\pm \omega(\varepsilon_{2n+1+\delta-k},\varepsilon_k)=1, 
%\text{\ \ if $1\leq i\leq n+\delta$}\] 
%(other pairings are zero).
%where $\dim W = 2n+ \delta$ and $\delta\in \{0,1\}$.  
Let ${\sf G}$
be the subgroup of ${\sf SL}(W)$ preserving $\omega$. Then ${\sf G}={\sf SO}_{2n+1}$
if $\dim W=2n+1$ and $\omega$ is orthogonal. It is ${\sf G}={\sf Sp}_{2n}$ if $\dim W=2n$
and $\omega$ is symplectic. Finally, ${\sf G}={\sf SO}_{2n}$ if $\dim W=2n$ and $\omega$
is orthogonal. These are groups in the $B_n, C_n, D_n$ 
series of the Cartan-Killing classification, respectively.

If $\lambda\in {\sf Par}_n$, H.~Weyl's construction \cite{Weyl} (see also
\cite[Lectures~17 and 19]{Fulton.Harris})
gives
 a ${\sf G}$-module ${\mathbb S}_{[\lambda]}(W)$. These modules are irreducible, except in type $D_n$, where irreducibility holds if $\lambda_n=0$. In the \emph{stable range} $\ell(\mu)+\ell(\nu)\leq n$, 
\begin{equation}
\label{eqn:NLexpansion}
{\mathbb S}_{[\mu]}(W)\otimes  {\mathbb S}_{[\nu]}(W) \cong\bigoplus_{\lambda\in {\sf Par}_n}
{\mathbb S}_{[\lambda]}(W)^{\oplus N_{\mu,\nu,\lambda}};
\end{equation}
this is \cite[Corollary~2.5.3]{Koike}. 
In particular, 
$N_{\mu,\nu,\lambda}$ is independent of ${\sf G}$ \cite[Theorem~2.3.4]{Koike}.

%%Our thesis is that, like the Littlewood--Richardson coefficients, the Newell--Littlewood numbers form a subfamily of  the general multiplicities whose combinatorics deserves separate study. Indeed, we reinforce the parallel with the Littlewood--Richardson coefficients by developing the topic from first principles and symmetric function basics. 

The Schur functions $s_{\lambda}$ form a basis of the ring
$\Lambda$ of symmetric functions. It is the ``universal character'' of ${\mathbb S}_{\lambda}(V)$ for ${\sf GL}$. In a similar fashion, \cite[Section~2]{Koike} establishes universal characters 
of ${\mathbb S}_{[\lambda]}(W)$ for ${\sf Sp}$. This \emph{Koike--Terada basis}
$\{s_{[\lambda]}\}$ of  $\Lambda$ satisfies
\begin{equation}
\label{eqn:polyver}
s_{[\mu]}s_{[\nu]}=\sum_{\lambda}N_{\mu,\nu,\lambda}s_{[\lambda]},
\end{equation}
where $\mu,\nu,\lambda$ are arbitrary partitions. (Now, \cite{Koike} also defines a  basis for ${\sf SO}$ with the same structure coefficients. Which one we use is simply a matter of choice.)
%This basis specializes to the characters for fixed ${\sf Sp}_{2n}$, just as the specialization 
%\begin{equation}
%\label{eqn:thespecialization}
%s_{\lambda}\mapsto s_{\lambda}(x_1,x_2,\ldots,x_n,0,0,\ldots)
%\end{equation}
%does for ${\sf GL}_n$. Their work discusses ``modification rules'' (cf.~\cite{King70, King75}) to \emph{non-positively} compute multiplicities outside the stable range. 

Section~\ref{sec:2} outlines the combinatorially derived results found in \cite{GOY}. Section~\ref{sec:3} 
presents the results from \cite{GORY} which resolve a conjecture from \cite{GOY2} and furthermore
explains the connection to \emph{eigencones}.  A number of problems remain in this subject;
some of these are stated in Section~\ref{sec:4}.

 
\section{Results using purely combinatorial methods}\label{sec:2}

In this section we summarize results that can be obtained directly from (\ref{eqn:Newell--Littlewood}). 

\subsection{Basic observations}

This lemma is stated as \cite[Lemma~2.2]{GOY} without claims of originality by the authors:

\begin{lemma}[Facts about the Newell--Littlewood numbers]\label{prop:easy}
\gap
\begin{itemize}
\item[(I)] $N_{\mu,\nu,\lambda}$ is invariant under any of the $3!$-permutations of 
the indices $(\mu,\nu,\lambda)$.
\item[(II)] $N_{\mu,\nu, \lambda}=c_{\mu,\nu}^{\lambda}$ if $|\mu|+|\nu|=|\lambda|$.
\item[(III)] $N_{\mu,\nu, \lambda}=0$ unless $|\mu|,|\nu|,|\lambda|$ satisfy the triangle inequalities (possibly with equality), \emph{i.e.}, $|\mu|+|\nu|\geq |\lambda|$, 
$|\mu|+|\lambda|\geq |\nu|$, and $|\lambda|+|\nu|\geq |\mu|$.\item[(IV)] $N_{\mu,\nu, \lambda}=0$ if $|\nu\wedge \lambda|+|\mu\wedge\nu|<|\nu|$, where $\nu\wedge\lambda$ is the partition whose $i$-th part  is $\min(\nu_i,\lambda_i)$. 
\item[(V)] $N_{\mu,\nu, \lambda}=0$ unless $|\lambda|+|\mu|+|\nu|\equiv 0 \ (\!\!\!\mod 2)$.
\item[(VI)] $N_{\mu,\nu,\lambda}=N_{\mu',\nu',\lambda'}$ where $\mu'$ is the conjugate partition of $\mu$, \emph{etc.} 
\end{itemize}
\end{lemma} 

This is an observation used in \cite{GOY}:

\begin{proposition}[{\cite[Proposition~2.3]{GOY}}]
\label{prop:reformulate}
$N_{\mu,\nu,\lambda}=\sum_{\alpha\subseteq \mu\wedge \nu} \langle s_{\mu/\alpha}s_{\nu/\alpha}, s_{\lambda}\rangle$. 
\end{proposition}
%%We describe some salient features of $s_{[\mu]}s_{[\nu]}$. 
\subsection{Shape of $s_{[\mu]}s_{[\nu]}$}

Let $\mu \Delta \nu=(\mu\setminus \nu)\cup (\nu\setminus\mu)$ be the symmetric difference of the Young diagrams of $\lambda$ and $\mu$. Define ${\sf Par}$ to be the set of all integer partitions. This theorem is proved in \cite{GOY} using Young tableau combinatorics based on a \emph{demotion} procedure. In \cite{GORY} it is further studied in connection to the Robinson--Schensted--Knuth
correspondence to prove Theorem~\ref{th:GOYconj}.

\begin{theorem}[{\cite[Theorem~3.1]{GOY}}]
\label{thm:interval}
Fix $\mu,\nu\in {\sf Par}$. 
\begin{itemize}
\item[(I)] Let $k\in {\mathbb Z}_{\geq 0}$. There exists $\lambda\in {\sf Par}$ with $|\lambda|=k$ and
$N_{\mu,\nu,\lambda}>0$ if and only if 
\[k\equiv |\mu\Delta\nu| \ ({\mathrm{mod}}\  2)  \text{\ \ and \ $|\mu\Delta\nu|\leq k \leq |\mu|+|\nu|$.}\]
\item[(II)] If $N_{\mu,\nu,\lambda}>0$ with $|\lambda|>|\mu\Delta\nu|$, there
exists $\lambda^{\downarrow\downarrow}$ such that $N_{\mu,\nu,\lambda^{\downarrow\downarrow}}>0$, $\lambda^{\downarrow\downarrow}\subset\lambda$
and $|\lambda^{\downarrow\downarrow}|=|\lambda|-2$.
\item[(III)] If $N_{\mu,\nu,\lambda}>0$ with $|\lambda|<|\mu|+|\nu|$, there exists
$\lambda^{\uparrow\uparrow}$ such that $N_{\mu,\nu,\lambda^{\uparrow\uparrow}}>0$,
$\lambda \subset \lambda^{\uparrow\uparrow}$ and 
$|\lambda^{\uparrow\uparrow}|=|\lambda|+2$.
\end{itemize}
\end{theorem}



\subsection{Newell--Littlewood polytopes} 
We now turn to ``polytopal'' aspects of the Newell--Littlewood numbers.
Fix $\lambda,\mu,\nu\in {\sf Par}_n$. Let $\alpha_{i}^j, \beta_i^j, \gamma_{i}^j \in {\mathbb R}$
for $1\leq i,j\leq n$ and consider the linear constraints:

\begin{enumerate}
\item \emph{Non-negativity}: For all $1\leq i,j\leq n$, $\alpha_{i}^j,\beta_i^j,\gamma_i^j\geq 0$.
\item \emph{Shape constraints}: For all $k$,
\begin{enumerate}
\item
$\sum_j \alpha_k^j + \sum_i \beta_i^k = \mu_k$,
\item
$\sum_j \gamma_k^j + \sum_i \alpha_i^k = \nu_k$,
\item
$\sum_j \beta_k^j + \sum_i \gamma_i^k = \lambda_k$.
\end{enumerate}
\item \emph{Tableau/semistandardness constraints}: For all $k,l$:
\begin{enumerate}
\item
$\sum_{j} \alpha_{k+1}^j + \sum_{i\leq l} \beta_i^{k+1} \leq \sum_{j} \alpha_k^j + \sum_{i<l} \beta_i^k$,
\item
$\sum_{j} \gamma_{k+1}^j + \sum_{i\leq l} \alpha_i^{k+1} \leq \sum_{j} \gamma_k^j + \sum_{i<l} \alpha_i^k$,
\item
$\sum_{j} \beta_{k+1}^j + \sum_{i\leq l} \gamma_i^{k+1} \leq \sum_{j} \beta_k^j + \sum_{i<l} \gamma_i^k$.
\end{enumerate}
\item \emph{Ballot constraints}: For all $k,l$:
\begin{enumerate}
\item
$\sum_{i<k} \alpha_{l}^i \geq \sum_{i\leq k} \alpha_{l+1}^{i}$,
\item
$\sum_{i<k} \beta_{l}^i \geq \sum_{i\leq k} \beta_{l+1}^{i}$,
\item
$\sum_{i<k} \gamma_{l}^i \geq \sum_{i\leq k} \gamma_{l+1}^{i}$.
\end{enumerate}
\end{enumerate}

\begin{definition}[{\cite[Section~5]{GOY}}]
The \emph{Newell--Littlewood polytope} is
\[{\mathcal P}_{\mu,\nu,\lambda}=\{(\alpha_i^j,\beta_i^j,\gamma_i^j)\in {\mathbb R}^{3n^2}: \text{(1)-(4) hold}\}\subset {\mathbb R}^{3n^2}.\]
\end{definition}

\begin{theorem}[{\cite[Theorem~5.1]{GOY}}]
\label{thm:thepolytope}
$N_{\mu,\nu,\lambda}=\#({\mathcal P}_{\mu,\nu,\lambda}\cap {\mathbb Z}^{3n^2})$.
\end{theorem}

\begin{example}
We illustrate the correspondence asserted by Theorem \ref{thm:thepolytope}. Let $\mu = (2)$, $\nu = (2,1)$, and $\lambda = (2,1)$.  Write $\alpha_i^j, \beta_i^j$ and $\gamma_i^j$ in terms of matrices $[\alpha],[\beta]$ and $[\gamma]$ so that $[\alpha]_{i,j} = \alpha_i^j, [\beta]_{i,j}=\beta_i^j$ and $[\gamma]_{i,j} = \gamma_i^j$. Then ${\mathcal P}_{\mu,\nu,\lambda}\cap \mathbb{Z}^{12}$ would be the two triples
\[\big([\alpha],[\beta],[\gamma]\big) = \left(\begin{pmatrix}
  0 & 1\\
  0 & 0
\end{pmatrix}, \begin{pmatrix}
  1 & 0\\
  0 & 0
\end{pmatrix}, \begin{pmatrix}
  1 & 1\\
  0 & 0
\end{pmatrix}
\right)\text{ or }
\left(\begin{pmatrix}
  1 & 0\\
  0 & 0
\end{pmatrix}, \begin{pmatrix}
  1 & 0\\
  0 & 0
\end{pmatrix}, \begin{pmatrix}
  1 & 0\\
  0 & 1
\end{pmatrix}
\right)
\]
%If we order the values of $(\alpha_i^j)_{i,j}$ by $(\alpha_1^1,\alpha_1^2,\dots, \alpha_1^n, \dots, \alpha_n^1,\dots, \alpha_n^n)$, and similarly order $(\beta_i^j)_{i,j}$ and $(\gamma_i^j)_{i,j}$ then $\{((0,1,0,0),(1,1,0,0),(1,1,0,1)), ((0,1,0,0),(1,0,0,1),(2,0,0,1))\} = 
 implying $N_{\mu,\nu,\lambda}=2$.  The first triple corresponds to the triple of LR tableaux contributing, respectively, to $c^{\mu}_{\alpha,\beta}, c^{\nu}_{\gamma,\alpha}$ and $c^{\lambda}_{\beta,\gamma}$ where $\alpha = (1), \beta = (1), \gamma = (2)$:
\[\tableau{\ & 1}\,,\qquad
\tableau{\ & \  \\
1}\,,\qquad 
\tableau{\ & 1\\
1}\,.\]
Similarly, the second triple corresponds to these LR tableaux
\[\tableau{\ & 1}\,,\qquad
\tableau{\ & 1 \\
\ }\,,\qquad  
\tableau{\ & 1\\
2}\,.\]
which contribute, respectively, to $c^{\mu}_{\alpha,\beta}, c^{\nu}_{\gamma,\alpha}$ and $c^{\lambda}_{\beta,\gamma}$ with $\alpha = (1), \beta = (1), \gamma = (1,1)$. \qed
\end{example}

That $N_{\lambda,\mu,\nu}$ counts lattice points in a polytope can be proved using A.~Berenstein--A.~Zelevinsky~\cite[Section~2.2]{BZ} on more general tensor product multiplicities, together with \cite[Corollary~2.5.3]{Koike}. The proof of Theorem~\ref{thm:thepolytope} in \cite{GOY} uses
a self-contained approach, similar to one in the preprint version of \cite{Mulmuley} for the
Littlewood--Richardson coefficients.

\subsection{Streched Newell--Littlewood numbers}
A conjecture of W.~Fulton (proved in \cite{KTW}) states that 
\[ c_{\mu,\nu}^{\lambda}=1\implies c_{k\mu,k\nu}^{k\lambda}=1, \text{ for all } k\geq 1.\] 

\begin{example}[{\cite[Example~5.24]{GOY}}]
One checks that
\[N_{(1,1),(1,1),(1,1)}=(c_{(1),(1)}^{(1)})^3=1 \text{\ but 
$N_{(2,2),(2,2),(2,2)}=(c_{(1,1),(1,1)}^{(1,1)})^3+(c_{(2),(2)}^{(2)})^3=2$.}\]
Therefore, the analogue of Fulton's conjecture for $N_{\nu,\mu,\lambda}$ does not hold.
\qed
\end{example}

Define a function 
\[{\mathfrak c}_{\mu,\nu}^{\lambda} \colon {\mathbb Z}_{\geq 1}\to {\mathbb N} \text{ \ 
by $k\mapsto c_{k\mu,k\nu}^{k\lambda}$}.\] 
R.~C.~King--C.~Tollu--F.~Toumazet \cite{King} conjecture that this function is interpolated
by a polynomial with nonnegative rational coefficients. The polynomiality property was
proved by H.~Derksen--J.~Weyman \cite{DW:poly}. Consequently,  the polynomial
${\mathfrak c}_{\mu,\nu}^{\lambda}$ is called the \emph{Littlewood--Richardson polynomial}.
The positivity conjecture is still open.

\begin{definition}[{\cite[Section~5.4]{GOY}}]
The \emph{Newell--Littlewood function} is
${\mathfrak N}_{\mu,\nu,\lambda} \colon {\mathbb Z}_{\geq 1}\to {\mathbb N}$ by $k\mapsto N_{k\mu,k\nu,k\lambda}$.
\end{definition}

${\mathfrak N}_{\mu,\nu,\lambda}(k)$ cannot always be interpolated by a single
polynomial:

\begin{theorem}[{\cite[Theorem~5.25]{GOY}}]
\label{prop:floorex}
There exist $\lambda,\mu,\nu$ such that ${\mathfrak N}_{\mu, \nu,\lambda}(k)\not\in {\mathbb R}[k]$.
\end{theorem}
\begin{proof}
One argues that ${\mathfrak N}_{(1,1),(1,1),(1,1)}(k)=\left\lceil{\frac{k+1}{2}}\right\rceil$,
which is clearly non-polynomial.
\end{proof}
Recent work of R. C. King \cite{StrechedNL} extensively studies ${\mathfrak N}_{\mu,\nu,\lambda}(k)$. One of his results is
%Theorem~\ref{thm:thepolytope} implies that ${\mathfrak N}_{\mu, \nu,\lambda}(k)$ is an Ehrhart quasi-polynomial. That is
%\[{\mathfrak N}_{\mu, \nu,\lambda}(k) = \sum_{i=0}^{N} c_{i}(k)k^i\]
%for some $N\in \mathbb{Z}_{>0}$,
%where each $c_i(k)$ is a periodic function with integral period. 

\begin{theorem}[{\cite[Corollary~2.2]{StrechedNL}}]
	For any $\lambda,\mu,\nu\in \Par_n$, there exist $P_e(k),P_o(k)\in \mathbb{Q}[k]$ such that
	\[{\mathfrak N}_{\mu, \nu,\lambda}(k) = 
	 \begin{cases}
	 P_e(k) & \text{for }k\text{ even,} \\
	 P_o(k) & \text{for }k\text{ odd.}
	 \end{cases}
	 \]
	
	%%The Newell--Littlewood function ${\mathfrak N}_{\mu,\nu,\lambda}(k)$ is an Ehrhart quasi-polynomial in $k$ with minimum quasi-period at most $2$.
\end{theorem}

\subsection{Multiplicity-freeness}
\begin{definition}[{\cite[Section~6]{GOY}}]
A pair $(\mu,\nu)\in {\sf Par}^2$ is \emph{NL-multiplicity-free} if (\ref{eqn:polyver}) contains no multiplicity, 
\emph{i.e.}, each $N_{\mu,\nu,\lambda}\in\{0,1\}$ for all $\lambda\in {\sf Par}$.
\end{definition}

\begin{theorem}[{\cite[Theorem~6.1]{GOY}}]
\label{thm:mfrep}
A pair $(\mu,\nu)\in {\sf Par}^2$ is NL-multiplicity-free if and only if
\begin{itemize}
\item[(I)] $\mu$ or $\nu$ is either a single box or $\emptyset$;
\item[(II)] $\mu$ is a single row and $\nu$ is a rectangle (or vice versa); or
\item[(III)] $\mu$ is a single column and $\nu$ is a rectangle (or vice versa).
\end{itemize}
\end{theorem}

Theorem~\ref{thm:mfrep} is an analogue of a theorem of \cite{Stembridge} for Schur functions.



\subsection{Version of T.~Lam--A.~Postnikov--P.~Pylyavskyy's theorems}

If $\alpha,\beta\in {\sf Par}$ then $\alpha\vee\beta\in {\sf Par}$ has parts
$\max(\alpha_i,\beta_i)$ (where one postpends $0$'s to $\alpha$ or $\beta$
as necessary). Given two skew shapes $\nu/\alpha$ and $\mu/\beta$, let 
\[(\nu/\alpha)\wedge(\mu/\beta):=(\nu \wedge \mu) /(\alpha \wedge \beta) \text{ \ 
and $(\nu/\alpha)\vee (\mu/\beta) := (\nu \vee \mu) / (\alpha \vee \beta)$.}\] 
Set
${\rm sort}_{1}(\nu,\mu) := (\rho_{1},\rho_{3},\rho_{5},\ldots)\text{\ and ${\rm sort}_{2}(\nu,\mu) := (\rho_{2},\rho_{4},\rho_{6},\ldots)$}$, 
where $(\rho_{1},\rho_{2},\ldots):=\nu\cup \mu$. In what follows, $\frac{\nu+\mu}{2}$ means coordinate-wise
addition and division. Moreover, $\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ are taken coordinate-wise.

Define $f\in \Lambda$ to be \emph{Schur nonnegative} if $f=\sum_{\lambda} a_{\lambda} s_{\lambda}$
with $a_{\lambda}\geq 0$ for all $\lambda \in {\sf Par}$.

\begin{theorem}[\cite{AJM}]\label{thm:sschurmj}
Let $\nu/\alpha$ and $\mu/\beta$ be skew shapes. These are Schur nonnegative:
\begin{enumerate}
    \item $s_{(\nu/\alpha)\wedge(\mu/\beta)}s_{(\nu/\alpha)\vee (\mu/\beta)} - s_{\nu/\alpha}s_{\mu/\beta}$
    \item $s_{\lfloor \frac{\nu + \mu}{2} \rfloor / \lfloor \frac{\alpha + \beta}{2}\rfloor}s_{\lceil \frac{\nu + \mu}{2} \rceil / \lceil \frac{\alpha + \beta}{2}\rceil} - s_{\nu/\alpha}s_{\mu/\beta}$
    \item $s_{\textup{sort}_{1}(\nu,\mu)/\textup{sort}_{1}(\alpha,\beta)}s_{\textup{sort}_{2}(\nu,\mu)/\textup{sort}_{2}(\alpha,\beta)} - s_{\nu/\alpha}s_{\mu/\beta}$
\end{enumerate}
\end{theorem}

We refer the reader to \cite[Definition~2.1.1]{Koike} for a definition of $s_{[\lambda]}\in \Lambda$ as a determinant
in terms of complete homogeneous symmetric functions.
\begin{definition}[{\cite[Section~7.3]{GOY}}]
$f\in \Lambda$ is \emph{Koike--Terada nonnegative} if $f=\sum_{\lambda} b_{\lambda}s_{[\lambda]}$
has $b_{\lambda}\geq 0$ for every $\lambda \in {\sf Par}$. 
\end{definition}
Combining Theorem~\ref{thm:sschurmj} with
Proposition~\ref{prop:reformulate} implies:

\begin{theorem}[{\cite[Theorem~7.4]{GOY}}]
\label{thm:meetjoin}
The following are  Koike--Terada nonnegative:
\begin{enumerate}
    \item ${s}_{[\nu\wedge \mu]}  
    {s}_{[\nu \vee\mu]} -{s}_{[\nu]}   {s}_{[\mu]} $
    \item ${s}_{[\lfloor \frac{\nu + \mu}{2} \rfloor]}    {s}_{[\lceil \frac{\nu + \mu}{2} \rceil]} -{s}_{[\nu]}  {s}_{[\mu]} $
    \item ${s}_{[\textup{sort}_{1}(\nu,\mu)]}    {s}_{[\textup{sort}_{2}(\nu,\mu)]}  - {s}_{[\nu]}   {s}_{[\mu]} $
\end{enumerate}
\end{theorem}


\section{Nonvanishing results using geometric methods; connection to eigencones}\label{sec:3}

We now turn to the results of \cite{GORY}, whose proofs rely on a mix of geometry and combinatorics. Fix $n\in \naturals$. Let
$
\nls(n)=\{(\lambda,\mu,\nu)\in(\Par_n)^3\,:\, N_{\lambda,\mu,\nu}> 0\}.
$
Indeed, $\nls$ is a finitely generated semigroup \cite[Section~5.2]{GOY}. An
approximation of it is the saturated semigroup:
$$
\nlss(n)=\{(\lambda,\mu,\nu)\in (\Par_n^\mathbb{Q})^3\,:\, \text{there exists } t>0 \quad N_{t\lambda,t\mu,t\nu}\neq 0\},
$$
where
$\Par_n^\mathbb{Q}=\{(\lambda_1,\ldots,\lambda_n)\in\mathbb{Q}^n\,:\,\lambda_1\geq\ldots\geq\lambda_n\geq
0\}$. 
By Lemma~\ref{prop:easy}(II),
a subproblem asks to study  
\[\lrss(n)=\{(\lambda,\mu,\nu)\in (\Par_n^\mathbb{Q})^3\,:\, \text{there exists } t>0 \quad c_{t\lambda,t\mu}^{t\nu}> 0\}.
\]
In fact, A.~Klyachko \cite{Klyachko} characterized $\lrss(n)$.
%The solution to that case is 1990's combined breakthrough work of
%A.~Klyachko \cite{Klyachko} and A.~Knutson-T.~Tao \cite{KT:saturation}. 
For $I=\{i_1< \cdots<i_d\}\subseteq {\mathbb Z}_{>0}$,
set 
\[\tau(I):=(i_d-d\geq \cdots\geq i_2-2 \geq i_1-1)\in {\sf Par}_d.\]

\begin{theorem}[\cite{Klyachko}]\label{thm:classicalHorn} % \cite{KT:saturation})
$(\lambda,\mu,\nu)\in \lrss(n)$ if and only if $|\lambda|=|\mu|+|\nu|$, and
for every $d<n$, and every triple of subsets $I,J,K\subseteq [n]$ of cardinality $d$ such that $c_{\tau(I),
\tau(J)}^{\tau(K)}> 0$,
\[
\sum_{k\in K}\lambda_k\leq
\sum_{i\in
I}\mu_i+\sum_{j\in J}\nu_j.\]
\end{theorem}

For our next result, we need a definition. Let $\lambda^{(1)},\dots,\lambda^{(s)}\in \Par_n$ for $s\geq 3$. 
Think of the indices $1,\dots,s$ as elements of $\integers/s\integers$. 
The \emph{multiple Newell--Littlewood number} \cite{GORY} is
\[N_{\lambda_1,\dots,\lambda_s}=\sum_{(\alpha_1,\dots,\alpha_s)\in(\Par_n)^{s}}\prod_{i\in\integers/s\integers}
c_{\alpha_i\,\alpha_{i+1}}^{\lambda^{(i)}}. \]
When $s=3$, we recover (\ref{eqn:Newell--Littlewood}). We have a representation-theoretic interpretation of these
numbers. The second author is pursuing a study of these numbers defined for any graph $G=(V,E)$ (the multiple Newell--Littlewood numbers being those for a cycle). 
%To $A=\{i_1<\cdots<i_r\}\subset[n]$, associate the partition
%\begin{equation}
%\label{taudef}
%\tau(A)=(i_r-r\geq\cdots\geq i_1-1).
%\end{equation}


In \cite[Conjecture~1.4]{GOY2}, three of the authors
conjectured a description of the semigroup $\nls(n)$. That assertion
subsumes Conjecture~\ref{conj:saturation} and a description of $\nlss$ using \emph{extended Horn inequalities} \cite[Definition~1.2]{GOY2}. In \cite{GORY} one finds a resolution of the latter part of the conjecture, giving a second description of $\nlss(n)$; this is Theorem~\ref{th:GOYconj}.
%Using the known descriptions of $\spss(n)$ \cite{BK,GITEigen,algo}
%we deduce from Theorem~\ref{th:cones=} a \emph{minimal}
%list of inequalities defining $\nlss(n)$:

For $A = \{i_1 < \ldots < i_r\}\subseteq [n]$, let $\lambda_A$ be
the partition using the only parts indexed by $A$; namely,
$\lambda_A=(\lambda_{i_1},\dots,\lambda_{i_r})$. Let $|\lambda_A|=\sum_{i\in A}\lambda_i$. 

\begin{theorem}[{\cite[Theorem~1.5]{GORY}}]
  \label{th:GOYconj}
 $(\lambda,\mu,\nu)\in
\nlss(n)$ if and only if
\[
0\leq |\lambda_A|-|\lambda_{A'}|+|\mu_B|-|\mu_{B'}|+|\nu_C|-|\nu_{C'}|\]
for any subsets $A,A',B,B',C,C'\subset [n]$ such that
\begin{enumerate}
\item $A \cap A'= B \cap B' = C \cap C' = \emptyset$;
\item $|A|+|A'|=|B|+|B'|=|C|+|C'|=|A'|+|B'|+|C'|$;
\item $N_{\tau(A'), \tau(B), \tau(C'), \tau(A), \tau(B'), \tau(C)}\neq 0$. 
\end{enumerate}
\end{theorem}

\begin{example} For $n=2$, the table below gives the Horn inequalities (together with $|\nu|=|\lambda|+|\mu|$) and the Extended Horn inequalities:

\begin{center}
\begin{tabular}{c|c}
Horn inequalities & Extended Horn/Klyachko inequalities\\
\hline \hline
$\nu_1\leq \lambda_1+\mu_1$ & $\nu_1\leq \lambda_1+\mu_1$, $\lambda_1\leq \mu_1+\nu_1$, $\mu_1\leq \nu_1+\lambda_1$\\
\hline
$\nu_2\leq \lambda_1+\mu_2$,  & $\nu_2\leq \lambda_1+\mu_2$, $\lambda_2\leq \mu_1+\nu_2$, $\mu_2\leq \nu_1+\lambda_2$,\\
$\nu_2\leq \lambda_2+\mu_1$  & $\nu_2\leq \lambda_2+\mu_1$, $\lambda_2\leq \mu_2+\nu_1$, $\mu_2\leq \nu_2+\lambda_1$\\
\hline
$|\nu|= |\lambda|+|\mu|$,  & $|\nu|\leq |\lambda|+|\mu|$, $|\lambda|\leq |\mu|+|\nu|$, $|\mu|\leq |\nu|+|\lambda|$\\
\hline
\ & $\lambda_1+\mu_2\leq \lambda_2+\mu_1 + |\nu|$, $\mu_1+\nu_2\leq \mu_2+\nu_1 + |\lambda|$\\
\ & $\nu_1+\lambda_2\leq \nu_2+\lambda_1 + |\mu|$, $\lambda_1+\nu_2\leq \lambda_2+\nu_1 + |\mu|$\\
\ &  $\mu_1+\lambda_2\leq \mu_2+\lambda_1 + |\nu|$,  $\nu_1+\mu_2\leq \nu_2+\mu_1 + |\lambda|$
\end{tabular}
\end{center}
In this case, both lists are minimal, but this is not true for larger $n$.
\end{example}

We now derive minimal inequalities.
\begin{definition}
\label{lem:tau02A}
For $A,A'\subset [n]$, write $A=\{\alpha_1<\cdots<\alpha_a\}$ and
  $A'=\{\alpha'_1<\cdots<\alpha'_{a'}\}$.
Define $\tau^0(A,A'),\tau^2(A,A') \in \Par_{a+a'}$ as follows:
$$
\begin{array}{ll}
\tau^2(A,A')_k=a+|A'\cap [\alpha_k,n]|& \text{for all } k=1,\dots,a;\\
  \tau^2(A,A')_{l+a}=|A\cap [\alpha'_{a'+1-l}]| & \text{for all } l=1,\dotsc,a';\\
\tau^0(A,A')_k=n-a-a'+|[\alpha_k,n] - (A\cup A')| &\text{for all } k=1,\dotsc,a;\\
  \tau^0(A,A')_{l+a}=|[\alpha'_{a'+1-l}]- (A\cup A')|&\text{for all } l=1,\dotsc,a'.
\end{array}
$$
\end{definition}

For a partition $\lambda = (\lambda_1,\ldots,\lambda_k) \subseteq (a^b)$, \emph{i.e.}, the rectangle with $a$ columns and $b$ rows. Define $\lambda^\vee$ \emph{with respect to $(a^b)$} to be the partition $(a-\lambda_b,a-\lambda_{b-1},\ldots,a-\lambda_1)$ where we set $\lambda_i = 0$ for $i>k$. We will denote this by $\lambda^{\vee[a^b]}$. 
This is an NL-generalization of Theorem~\ref{thm:classicalHorn}.


\begin{theorem}[{\cite[Theorem~1.2]{GORY}}]
  \label{th:minlistineq}
$(\lambda,\mu,\nu)\in
\nlss(n)$ if and only if
\[0\leq |\lambda_A|-|\lambda_{A'}|+|\mu_B|-|\mu_{B'}|+|\nu_C|-|\nu_{C'}|\]
for any subsets $A,A',B,B',C,C'\subset [n]$ such that
\begin{enumerate} 
\item
$A \cap A'= B \cap B' = C \cap C' = \emptyset$;
\item \label{cond:sumr} 
$|A|+|A'|=|B|+|B'|=|C|+|C'|=|A'|+|B'|+|C'|=:r$;
\item \label{cond:c1}
$c_{\tau^0(A,A')^{\vee[(2n-2r)^r]}\;
    \tau^0(B,B')^{\vee[(2n-2r)^r]}}^{\tau^0(C,C')}=
c_{\tau^2(A,A')^{\vee[r^r]}\; \tau^2(B,B')^{\vee[r^r]}}^{\tau^2(C,C')}=1$.
\end{enumerate}
Moreover, this list of inequalities is irredundant.
\end{theorem}





%%The \emph{Horn inequalities} (\ref{eq:ineq}) were introduced in A.~Horn's 1962 paper \cite{Horn}. The inequalities
%%have a pre-history \cite{Fulton:survey,Bhatia}.


We now state a result that factors the
NL-coefficients on the boundary of $\nlss(n)$. It is an analogous to 
\cite[Theorem~7.4]{DW:comb} and \cite[Theorem~1.4]{KTT:factorLR} 
for $c_{\lambda,\mu}^{\nu}$. 
Let $\lambda\in\Par_n$ and $A,A'\subset[n]$.
Write $A'=\{i'_1<\cdots<i'_s\}$ and $A=\{i_1<\cdots<i_t\}$ and set
\[
\lambda_{A,A'}=(\lambda_{i'_1},\dots,\lambda_{i'_s},-\lambda_{i_t},\dots,-\lambda_{i_1})
\text{\ \ and \ $
\lambda^{A,A'}=\lambda_{[n]-(A\cup A')}$, \emph{etc}.}\]

\begin{theorem}[{\cite[Theorem~1.3]{GORY}}]
\label{th:reduction}
  Let $A,A',B,B',C,C'\subset [n]$ satisfy conditions 1, 2, and 3 from Theorem~\ref{th:minlistineq}.
For $(\lambda,\mu,\nu)\in(\Par_n)^3$ such that
$0=|\lambda_A|-|\lambda_{A'}|+|\mu_B|-|\mu_{B'}|+|\nu_C|-|\nu_{C'}|$,
\[N_{\lambda,\mu,\nu}=c_{\lambda_{A,A'},\mu_{B,B'}}^{\nu_{C,C'}^*} N_{\lambda^{A,A'},\mu^{B,B'},\nu^{C,C'}}.\]
\end{theorem}



Famously, in \cite{Klyachko} one finds a relation between $\lrss(n)$ and the \emph{Hermitian eigencone}. Let $\mathcal{H}(n,\mathbb{C})$ be the set of $n\times n$ complex Hermitian matrices. For $M\in \mathcal{H}(n,\mathbb{C})$, let $\lambda(M) \in \{(\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n):\lambda_i\in \mathbb{R}\}$ be its eigenvalues in weakly decreasing order. 

\begin{theorem}[\cite{Klyachko}]\label{LRHermitian}
	Let $(\lambda,\mu,\nu)\in(\Par^{\mathbb{Q}}_n)^3$. Then $(\lambda,\mu,\nu)\in \lrss(n)$ if and only if there exists $M_1,M_2,M_3\in \mathcal{H}(n,\mathbb{C})$ such that $(\lambda,\mu,\nu) = (\lambda(M_1),\lambda(M_2),\lambda(M_3))$ and $M_1+M_2=M_3$.
\end{theorem}

%Let $V={\mathbb C}^{2n}$
%with the standard basis $\{\vec e_1,\vec e_2,\ldots,\vec e_{2n}\}$. Define
%a skew-symmetric bilinear form $\omega(\bullet,\bullet): V\times V\to {\mathbb C}$ using the block matrix 
%$\Omega:=\left[\begin{matrix} 0 & I_n \\ -I_n & 0 \end{matrix}\right]$. By definition, the \emph{symplectic group} 
%$G=\Sp(2n,\CC)$ is the group of
%automorphisms of $V$ that preserve this bilinear form.
%
% Given a $n\times n$ matrix $A=(A_{ij})_{1\leq i,j\leq n}$ let
%  $A^T$ be its transpose. The Lie
%  algebra $\sp(2n,\CC)$ is the set of matrices $M\in {\sf Mat}_{2n\times 2n}(\CC)$
%  such that $M^T \Omega+\Omega M=0$; namely,
%  \begin{equation}
%    \label{eq:sp2n}
%    \sp(2n,\CC)=
%    \left\{
%    \begin{pmatrix}
%      A&B\\
%C&-A^T
%    \end{pmatrix}
%:
%\begin{array}{l}
%A,B,C \mbox{ of size }n\times n,\\
%B^T=B
%\mbox{ and } C^T=C
%\end{array}
%\right\}
%  \end{equation}
%which has complex dimension $2n^2+n$.
% Thus, \eqref{eq:sp2n} gives
%\begin{equation}
%\sp(2n,\CC)\cap \mathcal{H}(2n,\mathbb{C})=
%\left\{
%    \begin{pmatrix}
%      A&B\\
% \overline B^T& -A^T
%    \end{pmatrix} : \overline A^T=A
%\mbox{ and } B^T=B
%\right\}.
%\label{eq:sp2nu}
%\end{equation}
%For any $M\in \sp(2n,\CC)\cap \mathcal{H}(2n,\mathbb{C})$, $M$ is diagonalizable. Moreover, ordering the eigenvalues in weakly decreasing order, we have
%\[\lambda(M) \in \{(\lambda_1\geq \lambda_2\geq \ldots \geq \lambda_n\geq -\lambda_n\geq \ldots \geq -\lambda_1): \lambda_i\in \mathbb{R}\}.\]
%For $\lambda = (\lambda_1,\ldots,\lambda_n) \in \Par_n^\mathbb{R}$, set $\widehat \lambda = (\lambda_1,\ldots,\lambda_n,-\lambda_n,\ldots,-\lambda_1)$.

For $\lambda\in {\sf Par}_n$,  let $\widehat\lambda:=(\lambda_1,\ldots,\lambda_n,-\lambda_n,\ldots,-\lambda_1)$.
 This is an analogue of Theorem~\ref{LRHermitian}:

\begin{theorem}[{\cite[Proposition~3.1]{GORY}}]
	$(\lambda,\mu,\nu)\in \nlss(n)$ if and only if there exists 	
\begin{equation}\label{eqn:Nov12abc}	
	M_1,M_2,M_3\in \left\{
    \begin{pmatrix}
      A&B\\
 \overline B^T& -A^T
    \end{pmatrix} : \overline A^T=A
\mbox{ and } B^T=B
\right\}
\end{equation}
	 such that $M_1+M_2=M_3$ and
	$(\widehat\lambda,\widehat\mu,\widehat\nu) = (\lambda(M_1),\lambda(M_2),\lambda(M_3))$. 
	\end{theorem}
The above results of \cite{GORY} are proved using \cite[Theorem~1.1]{GORY} which shows $\nlss(n)$ 
is the truncation of the \emph{saturated tensor cone} for ${\sf Sp}_{2m}$ whenever $m\geq n\geq 1$.
The latter object was studied in \cite{BK} and minimal inequalities as well as an eigencone description were given. The set of matrices in (\ref{eqn:Nov12abc}) is $\sp(2n,\CC)\cap \mathcal{H}(2n,\mathbb{C})$ as used in \cite{BK}. The result \cite[Theorem~1.1]{GORY} is trivial if $m\geq 2n$ by 
(\ref{eqn:NLexpansion}); the content is the case $m<2n$. The proof uses the third authors' work on ``GIT-semigroups'' and a dose of Schubert calculus.

%Now,
%\begin{equation}
 % \label{eq:dimSchub}
 % \codim(\overline{BF_I})= |\tau(I^0)^{\vee[(2n-2r)^r]}| + 1/2(|\tau(I^2)^{\vee[r^r]}| + |I\cap [n]|).
%\end{equation}
%Moreover,
%\begin{equation}
 % \label{eq:dimGr}
  %\dim(\Gr_\omega(r,2n))= r(2n-2r)+\frac{r(r+1)}{2}.
  %\end{equation}

Finally, the second author has proved a characterization of $N_{\lambda,\mu,\nu}>0$ in terms of 
``short exact cycles'' of abelian $p$-groups. This is analogous to the short exact sequences characterization 
for $c_{\lambda,\mu}^{\nu}>0$ due to T.~Klein \cite{Klein}. Additionally, the second author has proved a characterization of non-zeroness of multiple Newell--Littlewood numbers in terms of ``long exact cycles'' of abelian $p$-groups.  Details will appear elsewhere.

\section{Some open problems}\label{sec:4}

\subsection{Saturation}

Knutson--Tao's  \emph{Saturation Theorem} \cite{KT:saturation} states
$c^{\nu}_{\lambda,\mu}>0$ if and only if $c^{t\nu}_{t\lambda,t\mu}>0\ \text{for some }t\in \mathbb{Z}_{>0}$.
% (see, \emph{e.g.}, Fulton's survey \cite{Fulton:survey}). 

\begin{conjecture}[{NL-Saturation \cite[Conjecture~5.5]{GOY}}]
  \label{conj:saturation}
Let $(\lambda,\mu,\nu)\in(\Par_n)^3$. Then 
$N_{\lambda,\mu,\nu}\neq 0$ if and only if $|\lambda|+|\mu|+|\nu|$ is
even and there exists
$t>0$ such that $N_{t\lambda,t\mu,t\nu}\neq 0$.
\end{conjecture}
In \cite{GORY}, Theorem~\ref{th:minlistineq} is used to prove 
Conjecture~\ref{conj:saturation} for  $n\leq 5$, by computer-aided calculation of Hilbert bases. 
In earlier work, \cite[Corollary~5.16]{GOY} proved the $n=2$ case, by combinatorial
reasoning. In addition, we have:
\begin{theorem}[{\cite[Theorem~5.7]{GOY}}]
	Conjecture~\ref{conj:saturation} holds if $\lambda,\mu$, or $\nu$ is a row or a column.
\end{theorem}

Conjecture~\ref{conj:saturation} generalizes \cite[Corollary~4.5]{GOY} which is the case $\lambda=\mu=\nu$.
That result addresses a matter in H.~Hahn's notion of \emph{detection} which is motivated by R.~Langlands'
\emph{beyond endoscopy proposal} towards his functoriality conjecture; see \cite{Hahn}.

\subsection{Analogue of M.~Kleber's conjecture} 

Fix a rectangle $R=a\times b$ and consider all products 
$s_{\lambda}s_{\lambda^{\vee[R]}}$. M.~Kleber \cite[Section~3]{Kleber} conjectured that these products, ranging over unordered pairs
$(\lambda,\lambda^{\vee[R]})$ are linearly independent in $\Lambda$. 

\begin{problem}[{\cite[Problem 7.2]{GOY}}]\label{prob:kleber}
Are the products ${s}_{[\lambda]}  {s}_{[\lambda^{\vee[R]}]}$, indexed over unordered pairs of partitions 
$(\lambda,\lambda^{\vee[R]})$ contained in $R$, linearly independent in $\Lambda$? 
\end{problem}

By Lemma~\ref{prop:easy}(II), M.~Kleber's conjecture implies a ``yes''
answer to Problem~\ref{prob:kleber}.

\subsection{A unimodality conjecture}
There seems to be another ``structural'' aspect of (\ref{eqn:polyver}). Define 
\[h^{\mu,\nu}_{t}=\sum_{\lambda: |\lambda|=|\mu\Delta\nu|+2t} N_{\mu,\nu,\lambda}.\]

A sequence $(a_k)_{k=0}^{N}$ is \emph{unimodal} if  
there exists $0\leq m\leq N$ such that 
\[0\leq a_0\leq a_1\leq \ldots\leq  a_m \geq a_{m+1}
\geq \ldots a_{N-1}\geq a_N.\]
\begin{conjecture}[{\cite[Conjecture~3.7]{GOY}}]
\label{conj:log}
The sequence $\{h^{\mu,\nu}_{t}\}_{t=0}^{|\mu\wedge \nu|}$ is unimodal.
\end{conjecture}
Conjecture~\ref{conj:log} is true for all $s_{[\mu]}s_{[\nu]}$ where
$0\leq |\mu|,|\nu|\leq 7$, and many larger cases \cite{GOY}. Theorem~\ref{thm:interval} (II) and (III) suggest a proof approach
for Conjecture~\ref{conj:log}: construct chains in Young's poset, each element $\lambda$ appearing $N_{\mu,\nu,\lambda}$-many times, ``centered'' at $m$.

%\begin{example}
%Continuing the previous example, $\{h^{\mu,\nu}_{t}\}_{t=0}^{3}=2,5,4$. Here $m=1$
%and we are suggesting that the following chains demonstrate the unimodality:
%\begin{align*}
%(1,1)\subset & (2,2)  \subset (4,2)\\
%(2)\subset & (2,1,1)  \subset (4,1,1)\\
%\ & (3,1)  \\
%\ & (3,1) \subset (3,2,1)\\
%\ & (4)  \subset (5,1)
%\end{align*}
%There is choice in the chains; in the first and third chains we could 
%interchange the roles of $(2,2)$ and $(3,1)$.\qed
%\end{example}



\subsection{The associativity relation}
Since $N_{\mu,\nu,\lambda}$ are structure constants for the Koike--Terada basis, the associativity relation
$({s}_{[\mu]}   {s}_{[\nu]} )  {s}_{[\lambda]} ={s}_{[\mu]}   ({s}_{[\nu]}   {s}_{[\lambda]} )$, 
implies for any $\mu,\nu,\lambda,\tau\in {\sf Par}$ that:
\begin{equation}
\label{eqn:assoc}
\sum_{\theta} N_{\mu,\nu,\theta}N_{\theta,\lambda,\tau}=\sum_{\theta} N_{\nu,\lambda,\theta}N_{\mu,\theta,\tau}.
\end{equation}

\begin{problem}[{\cite[Problem~7.1]{GOY}}]
Give a bijective proof of (\ref{eqn:assoc}) using the definition (\ref{eqn:Newell--Littlewood}).
\end{problem}

Let ${\overline c}_{\lambda,\mu,\nu}$ be the structure constants for a ring $R$ with basis $\{[\lambda]:\lambda\in \Par_n\}$. Define ${\overline N}_{\mu,\nu,\lambda}$ with these coefficients, as in 
(\ref{eqn:Newell--Littlewood}). The ${\overline N}_{\mu,\nu,\lambda}$ are structure constants for an associative,
commutative ring if ${\overline c}_{\lambda,\mu}^{\nu}=\alpha c_{\lambda,\mu}^{\nu}$ for a scalar $\alpha$.
\emph{What are other examples?}




%\acknowledgements{}

%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
%\bibliography{NL-FPSAC2}{}
%\bibliographystyle{plain}

\printbibliography

\end{document}

%Fix $m\geq n \in \NN$ and consider the symplectic Lie
%algebra $\sp(2m,\CC)$. 
%By postpending $0$'s, $\Par_n$ embeds into $\Par_m$. 
%Newell--Littlewood numbers are tensor
%product multiplicities for $\sp(2m,\CC)$ in the \emph{stable range} \cite[Corollary~2.5.3]{Koike}:
%\begin{equation}
%\text{ for all } (\lambda,\mu,\nu)\in(\Par_n)^3\quad \mbox{if $m\geq 2n$ then }
%\mult^m_{\lambda,\mu,\nu}=N_{\lambda,\mu,\nu}.
%\label{eq:N=m}
%\end{equation}
%Now, \eqref{eq:N=m} immediately implies
%\begin{equation}
%\label{eqn:firstobs}
%\nlss(n)={\sf SatTensor}(\sp(m))\cap(\Par_n^\QQ)^3, \text{\ for any $m\geq 2n$.}
%\end{equation}
%Our first result says the relationship of $\nlss$ to ${\sf SatTensor}(\sp)$ is \emph{independent} of the stable range. 
%\begin{theorem}
%  \label{th:cones=}
%For any $m\geq n\geq 1$,
%$$
%\nlss(n)={\sf SatTensor}(\sp(m))\cap(\Par_n^\QQ)^3.
%$$
%\end{theorem}
%


\end{document}
