\section{Introduction} 

Let  $G$ be a group, and let   $H$ a subgroup. The \textit{subgroup restriction} or \textit{branching} problem investigates how an irreducible representation of $G$ decomposes into irreducible representations  when restricted to $H.$
 In this article, we study this branching for 
$GL(n)\times GL(m) $  viewed as a subgroup of $GL(n  m)$ via
the tensor product of matrices. 

The Kronecker coefficients are the structure constants for the restriction of irreducible
representations of $GL(n m)$ into irreducibles for
$GL(n)\times GL(m)$.
Since irreducible representations of $GL(n)$ are indexed by partitions
of length at most $n$, the Kronecker coefficients are indexed by
triples of partitions of the same weight, with bounded lengths. Let
$g_{\mu,\nu, \lambda}$ be the Kronecker coefficient corresponding to
the triple $\mu,\nu, \lambda.$ 

Symmetric functions appear naturally in the study of the representations of the general linear group: the
Schur function $s_\lambda(x_1, \ldots, x_n)$  is the character of the irreducible representation of $GL(n)$
indexed by $\lambda$. In this setting, the Kronecker coefficients $g_{\mu,\nu,\lambda}$ can be defined by the expression (whose notation is defined in Eq.\eqref{Cauchy}):
\begin{align*}
s_{\lambda}[XY]=s_{\lambda}(x_1y_1, x_1y_2\cdots,x_{n}y_{m})= \sum_{\mu, \nu} g_{\mu,\nu,\lambda} \, s_\mu[X] s_\nu[Y]
\end{align*}
where all partitions appearing in the equation have the same weight, and bounded lengths.

Even the most basic questions about the Kronecker
coefficients remain unsolved eight decades after the pioneering
investigations of Murnaghan and Littlewood~\cite{ Murnaghan:1937,
  Murnaghan:1938, Murnaghan:1955, Littlewood:1956,
  Littlewood:1958}. They are nonnegative integers -- what do they
count? How do we compute them? What can we say about them?

Littlewood--Richardson coefficients provide a benchmark for the sort of answers we would like to obtain.  
They are also the solution of a  closely related group restriction problem: They govern the restriction  from a symmetric group 
to its Young subgroups.

The Littlewood--Richardson coefficients
are known to count several classes
of combinatorial structures, including points in polytopes,
Littlewood-Richardson tableaux, and  hives. No such
combinatorial interpretations are known for the Kronecker coefficients.
It is a fact that the dilated Kronecker coefficients  do not count integer points in dilations of polytopes. 
See Example~(\ref{one-onecubed}) for a simple example.


Identify a triple of partitions of lengths  $\le a, b, c$ (respectively)  with a point in $\mathbb{Q}^{a+b+c}$. 
 The set of triples of partitions whose
corresponding Kronecker coefficient is nonzero is known to have the structure of a finitely
generated semigroup (\cite{Christandl-PhD, Klyachko, Manivel:asymptotics1}). This semigroup
generates a rational polyhedral cone, called the  \emph{Kronecker cone}  and denoted by $PKron_{a,b,c}$.  
Its walls (i.e. facets) are described by a finite set of inequalities.
Can we find these inequalities? Is this cone saturated, or do there exist \emph{holes}, that is,
 points where the Kronecker coefficient is zero,  inside it? If so, where are the holes located? 
 
 The equivalent problem for the similarly defined
\emph{Littlewood-Richardson cone} is a major chapter in algebraic
combinatorics. The faces of the cone are given by the famous Horn
inequalities.  The saturation theorem of Knutson
and Tao says that there are no holes in the Littlewood-Richardson
cone. A survey of these results can be found in~\cite{Fulton}.  Standard techniques in integer programming
allow us to decide if a point belongs to a rational cone in polynomial
time. Therefore, deciding if a Littlewood-Richardson coefficient is
zero or not can also be decided in polynomial time~\cite{Burgisser:Ikenmeyer-SIAM, Mulmuley:GCT3}. 
On the other hand, computing a Littlewood--Richardson coefficient is a  \#P problem \cite{Burgisser:Ikenmeyer-FPSAC}.

In striking contrast with the Littlewood-Richardson coefficients, the
Kronecker coefficients do not satisfy the saturation property
There are holes in the Kronecker cone.
Counterexamples and related conjectures can be found in~\cite{King-Sevilla, BOR-CC, Christandl-PhD}.  Figure~\ref{fig:Holes} illustrates their location in a small, visualizable case. 
\begin{figure}\center

\includegraphics[width=.6\textwidth]{HOLE.png}
\caption{Holes in the Kronecker cone when all three partitions are of length 2. The point at $(i,j,k)$ is black if $g_{(24-i, i) (24-j, j) (24-k, k)}$ is nonzero (assuming, $j\leq i\leq k\leq 24/2$). The points with red crosses, or no dots are 0. Remark that the top face has both zeroes and nonzero values. These are the holes in the polytope.}
\label{fig:Holes}
\end{figure}
The inequalities describing the faces of the Kronecker
cone, the analogues of the Horn inequalities, are known only for some
very particular cases~\cite{Klyachko, Bravyi, Kirillov:saturation,
  Ressayre, Christandl:Harrow:Mitchison}.
  
Our main goal in this work is to understand the \emph{Kronecker function} $\kappa_{m,n,l}$,  a function of the parts of a triple of partitions $(\mu,  \nu, \lambda)$
of lengths bounded by $n, m $ and $l$ respectively, 
\begin{equation}
\kappa_{n,m,l} (\mu,  \nu, \lambda) = \kappa_{n,m,l} (\mu_1, \ldots, \mu_n, \nu_1,\ldots, \nu_m, \lambda_1, \ldots, \lambda_l): = g_{\mu, \nu,\lambda}
\end{equation}
and where $g_{\mu, \nu,\lambda}$ is  the Kronecker coefficient indexed by the partitions  $(\mu,  \nu, \lambda)$. 


We give an elementary proof of the known fact that $\kappa_{n,m,l}$ is a piecewise quasipolynomial (see Theorem~\ref{thm:AlwaysVecPart}). Then we apply the understanding gained by our approach
to the various questions raised in this introduction.


The piecewise quasipolynomiality of the Kronecker function has been
the center of much interest. It has recently been studied by
Mulmuley~\cite{GCT6}, Christandl, Doran, and Walter~\cite{CDW},
Baldoni, Vergne, and Walter~\cite{BaldoniVergneWalter},
Kahle and Micha\l ek~\cite{KahleMichalek}, and Pak and Panova~\cite{Pak:Panova:complexityKC}, and Manivel~\cite{ Manivel:asymptotics1}, among others.  Indeed, this phenomenon is a consequence of the powerful $[Q,R]=0$ theorem of Meinrenken-Sjamaar~\cite{qr0} on the piecewise quasipolynomial
behavior of multiplicity functions. In the case where $\mu$, $\nu$ and
$\lambda$ are partitions of lengths at most $2$, $2$, and $4$,
respectively, the piecewise  quasipolynomial is explicitly computable from formulas given in~\cite{BriandOrellanaRosasquasi}.


\subsection{Article organization and main contributions}
In this work we propose an elementary approach towards the study of the Kronecker function: We deduce the piecewise quasipolynomiality by passing through polytopes in a new way. To set it up, we begin in Section~\ref{sec-Polytopes} with a basic survey on polytopes and quasipolynomials. This is sufficient to understand the mechanics of our strategy. The examples appearing in this section will play a major role in Section~\ref{sec-F22}. 

The key idea is to start from Cauchy's definition of  the Schur function as a  quotient of alternants, and then deduce a relation between Kronecker coefficients and points in a polytope.
To illustrate the process, we start in Section~\ref{sec-F22} with the smallest nontrivial example, that in which two of the partitions have length $\le 2$. We  provide concrete visualizations of the Kronecker functions~$\kappa_{2,2,2}$  and~$\kappa_{2,2,4}$ since the polyhedra involved are of small dimension.  We give an explicit closed form (Theorem~\ref{7termF22Alternant}) for the Kronecker coefficients in terms of coefficients of the vector partition function $F_{2,2}.$  We also  identify a \textit{single} coefficient of $F_{2,2}$ as an upper bound for the Kronecker coefficient (Theorem~\ref{thm:AtomicIsMax}).  Our polyhedral approach plays a crucial role here.

In Section~\ref{sec-Fnm} we consider the general situation.
We prove in Theorem~\ref{thm:AlwaysVecPart} that there exists a nontrivial change of variables which converts the $F_{n,m}$  into a form recognizable as a  vector partition function. This facilitates our analysis since it returns us to the realm of Taylor series from Laurent series.

 We call the coefficients of the vector partition function  $F_{n,m}$ {\em atomic Kronecker coefficients}. They share some properties with the reduced Kronecker coefficients, a family of coefficients lying  between the Littlewood-Richardson coefficients and the Kronecker coefficients that we will introduce later in the article (See Section \ref{RKC}). They  contain enough information to recover the value of any Kronecker coefficient. They may also sometimes coincide with the Kronecker coefficients. In fact, a Kronecker coefficient that  is atomic is  always reduced. However, they are simpler objects. It is immediate from their definition  that the atomic Kronecker coefficients count integer points in polytopes.  The polytopes obtained in this way are particularly elegant because their dimension coincides with the degree of the Kronecker piecewise quasipolynomial (Theorem~\ref{thm:DegreeandPeriod}).

The next two cases after $\kappa_{2,2,4}$,  namely $\kappa_{2,3,6}$ and  $\kappa_{3,3,9}$, involve polyhedra of dimensions 8 and~26,
 bigger than what we can actually visualize. In Sections~\ref{sec-F23} and~\ref{sec-F33}, we analyse the corresponding vector partition  functions $F_{2,3}, F_{3,3}$ from a different perspective.  We investigate what happens on the faces of the associated polyhedral cones $K_{2,3}$ and $K_{3,3}.$
 We show that the coefficients of the vector partition function  behave differently on the different faces. In particular, we show that, whereas for the vector partition function $F_{2,2},$ the coefficients are identically one on {\em both} facets,


  for $F_{2,3}$ and $F_{3,3}$ there is only {\em one} facet on which the atomic Kronecker coefficients are identically one.   

In Section~\ref{Dilation} we study the dilated Kronecker coefficients,~$g_{k\lambda, k\mu,k\nu}$, defined for fixed $\lambda, \mu,$ and $\nu$, and $k \in \mathbb{N}$. We express these as a subseries of vector partition generating functions which implies that these are given by quasipolynomials in~$k$.  Using the vector partition generating function, we describe
in \ref{sec-Diagonal} how to use methods from analytic combinatorics in several variables to determine asymptotic estimates for the values of the coefficients. 

Finally, in Section~\ref{sec-MR-other-families}, using the observation that many families of  coefficients related to the representation theory of the general linear group can be defined by the evaluation of a Schur function on a {\em finite} alphabet, we describe how our approach for analysing the Kronecker coefficients can be applied to  other families, including the Littlewood--Richardson, Heisenberg, and plethystic coefficients.



\section{A Quick Introduction to Polytopes and Quasipolynomials} 
\label{sec-Polytopes}

This section is a primer on polytope point enumeration and
quasipolynomiality. It can be skipped by those familiar with the topic. For more details we recommend one of the following comprehensive introductions:~\cite{DeLoeraHemmeckeKoppe, BeckSinai, BeckSanyal}. The examples we have chosen for this section are directly relevant in our study of the Kronecker coefficients.


\subsection{Polyhedra and polytopes} A \emph{polyhedron} $\mathcal{P}$ is
  the set of solutions of a (finite) system  and  inequalities:
\[
\mathcal{P} = \{ {\mathbf{x}} \in \RR^d : A{\mathbf{x}} \le {\mathbf{b}} \},
\]
for a fixed matrix $A$ and vector ${\mathbf{b}}$, where the ``$\le$" sign is to
be understood componentwise. That is, a polyhedron is the intersection
of finitely many half spaces. A polyhedron is said to be \emph{rational} if both $A$ and ${\mathbf{b}}$ have
integer entries. (If instead, all coefficients are rational, we could
always clear the denominators to make an integral system). A \emph{polytope} is a bounded polyhedron. Equivalently, a
polytope is the convex hull of a finite number of points. A polytope is said to be rational when it has rational
vertices.  Note that any dilation of a polytope contains only a finite
number of integer points. The dimension of a polytope is defined as the dimension of the smallest affine space that contains it, i.e. the affine space spanned by its vertices.
A \emph{$k$-simplex} is a $k$-dimensional polytope which is the convex
hull of $k+1$ vertices. 

\subsection{Quasipolynomials} A function $\phi : \NN \to \QQ$ is a 
 \emph{
    (one-variable) quasipolynomial} if there exist polynomials
  $p_0, p_1, \ldots, p_{k-1}$ in $\QQ[t]$ and a natural number $m>0$,
  a \emph{period\/} of $\phi$, such that
\[
\phi(t) = p_i(t), \text{ for } t \equiv i \mod m.
\]
The polynomials $p_i$ are the \emph{constituents} of $\phi$. The degree of a quasipolynomial is the maximum of the degrees of its components.
  If $m_0$ is the minimum period for $\phi$, then we refer to it as the \emph{minimal
  period} of $\phi$. It is easy to see that $m_0$ divides any possible
period $m$.



\begin{example}\label{ZeroExample} 
Let  $\mathcal{P}$ be the one-dimensional polytope $[0, 1/2]$, and consider its integer dilations $ k \mathcal{P} = [0, k/2]$. This is illustrated in Figure~\ref{fig:zeroexample}.  We are interested in counting the number of integer points in the dilation of $\mathcal{P}$: 
\[
\phi_\mathcal{P}(k) := | \ZZ \cap k \mathcal{P} |
\] For example, $\phi(3)=2$. Alternatively, we want to count the number of nonnegative integer solutions to the inequality $0\le s_1\le k/2$. Then
\[
\phi_\mathcal{P}(k) = \left\lfloor \frac{k}{2} \right\rfloor+1 =
  \begin{cases}
    \frac{k+2}{2} &   \text{ if } k \equiv 0 \mod 2\\
  \frac{k+1}{2}     &  \text{ if } k  \equiv 1 \mod 2   \end{cases}..
\]
is a linear quasipolynomial of period 2. 

\begin{figure}\center
\usetikzlibrary{arrows}
\begin{tikzpicture}
\draw[latex-] (-0.9,0) -- (2.4,0) ;
\draw[-latex] (-0.9,0) -- (2.4,0) ;
\foreach \x in  {-1/2, 0,1/2, 1, 3/2, 2}
\draw[shift={(\x,0)},color=black] (0pt,3pt) -- (0pt,-3pt);
\foreach \x in {0,1,2}
\draw[shift={(\x,0)},color=black] (0pt,0pt) -- (0pt,-3pt) node[below] 
{$\x$};
\draw[*-*] (-0.08,0) -- (0.58,0);
\draw[very thick] (0,0) -- (0.5,0);

\draw[*-*] (-0.08,0.2) -- (1.08,0.2);
\draw[very thick] (0,0.2) -- (1,0.2);

\draw[*-*] (-0.08,0.4) -- (1.58,0.4);
\draw[very thick] (0,0.4) -- (1.5,0.4);

\draw[*-*] (-0.08,0.6) -- (2.08,0.6);
\draw[very thick] (0,0.6) -- (2,0.6);
\end{tikzpicture}

\caption{The one dimensional polytope~$\mathcal{P}=[0, 1/2]$ and its first
  four integer dilations. The volume of the $k$-th dilation is $\left\lfloor \frac{k}{2} \right\rfloor+1$, a quasipolynomial in $k$.}
\label{fig:zeroexample}
\end{figure}
\end{example}


Example~\ref{ZeroExample} illustrates that 
  one-variable quasipolynomials appear when counting integral points in polytopes. The following insight of Ehrhart tells us that this
  is always the case.

\begin{theorem}[Ehrhart~\cite{Ehrhart}] 
\label{thm:Ehrhart} Let $\mathcal{P}$ be a rational
  polytope in $\RR^d$. Let $\phi_\mathcal{P}(k) = | \ZZ^d \cap k \mathcal{P} |$, the number of integer points   in $k \mathcal{P}$. Then $\phi_\mathcal{P}$ is a quasipolynomial in $k$.     Moreover,
  the degree of $\phi_\mathcal{P}$ equals the dimension of $\mathcal{P}$.       All constituents of the quasipolynomial have the volume of $\mathcal{P}$ as their leading coefficient.
  
  The   smallest positive integer $k$ such that $k\mathcal{P}$ has integer   coordinates yields a period.  In particular, if the polytope has integral vertices, then $\phi_\mathcal{P}$ is a polynomial. 
\end{theorem}
  
Not all one-variable quasipolynomials count integer points in the dilations of a polytope. Those that do will be called \emph{Ehrhart functions}. 
The presence of the floor function in Example~\ref{ZeroExample} necessitates working with congruence classes.




\subsection{Partition functions} A \emph{partition of a positive integer} $n$  is a way of writing $n$ as a sum of nonnegative integers, where we 
do not take into consideration the order of the summands. 
Two partitions that differ only in the number of zero summands are considered the same.  We  consider  partitions whose \emph{parts} (nonzero summands) belong to a fixed finite multi-subset of nonnegative integers $S$,  that is, where there could be different copies of the same part.

The \emph{partition function} $p_S : \NN \to \NN$ evaluated at $n$ gives the number of  partitions of $n$ with parts in $S$. The partition function $p_S$ can also be defined by the formal power series:
\[
 \prod_{a_i\in S}\frac{1}{1-t^{a_i}}
= \sum_{n\in\NN} p_S(n) t^{n}.
 \]



\begin{example}~\label{SecondExample} Let
  $S=\{1,1,1\}$. Then $p_S(n)$ is the number of partitions of $n$ with
  three distinct copies of $1$. This is also the number of
  nonnegative integer solutions to the equation $s_0+s_1 \le n$, and
  the number of integer points in the dilations $n \mathcal{P}$ of the
  $2$-simplex with vertices $(0,0), (1,0), (0,1)$. In fact, $p_S$ is given  by
\[
p_S(n)=\frac{(n+1)(n+2)}{2}.
\]
This is a polynomial since all vertices have integer coordinates, as predicted by Ehrhart's Theorem.

\begin{figure}\center
\usetikzlibrary{arrows}
\begin{tikzpicture}[scale=.65]
\draw[step=1.0,draw=gray!60,thin] (-0.5,-0.5) grid (4.5,4.5);
\foreach \i in {0,...,4}
{ \fill[very thick, opacity=0.9-\i/5] (0,0) -- (\i,0) -- (0,\i) -- (0,0);
 \foreach \j in {0,...,\i}
 { \filldraw(\i-\j,\j) circle (2pt);   
 }
}
\end{tikzpicture}
\caption{A visualization of the two dimensional $2$-simplex with vertices $(0,0), (1,0),  (0,1)$  and some integral dilations bounded by vertices $(0,0), (k,0)$ and $(0,k)$ for $k=2, 3, 4, \dots$.}
\label{fig:simplex}
\end{figure}
\end{example}



\begin{example}\label{FirstExample} Let $S=\{1, 1,2\}$. By
  definition, the partition function $p_S(n)$  counts the number of
  nonnegative integer solutions of the linear equation \[ x_1 + x_2 +
    2x_3 =n.  
\]
We consider $x_1$ to be a slack variable and rewrite this as an inequality:
\[
x_2+2 x_3\le n.
\]
Thus, $x_3$ should satisfy $ 0\le x_3 \le \frac{n}{2} $.  After $x_3$
is chosen, the value of $x_2$ can be any value from zero to $n-2
x_3$. We conclude that the partition function is a quadratic
quasipolynomial of period $2$. Indeed,
\[
p_{\{1,1,2\}}(n)= \sum_{x_3=0}^{\frac{n}{2}} (n-2x_3 +1)=
\begin{cases}
\frac{n^2}{4}+ n +1, \,\,\text{ if}   \,\,n \equiv 0 \mod 2 \\
\frac{n^2}{4}+ n +\frac{3}{4}, \,\,\text{ if}   \,\,n \equiv 1 \mod 2 
\end{cases}.
\]
\end{example}




\subsection{Cones and Chambers} 
A \emph{rational polyhedral cone} is the set of all linear
  combinations with nonnegative real coefficients of a finite set of
  generators ${\bf a}_1,  {\bf a}_2, \ldots,  {\bf a}_n$ in $\NN^d$. If the  generators are linearly independent, then we say that the cone  $\mathcal{P}$ is \emph{simplicial}. Let $A$ be a $d\times n$  matrix with column vectors ${\bf a}_1,  {\bf a}_2, \ldots,  {\bf a}_n$. Let $pos(A)$ be the polyhedral cone generated by the columns of $A$. In general,   $pos(A)$ is not a simplicial cone.

Given $\sigma \subseteq \{1,2,\ldots,n\}$, let $A_\sigma$ be 
the submatrix of $A$  consisting of those columns ${\bf a}_i$ with $i \in \sigma$. Let $\ZZ A_\sigma$ be the integral lattice spanned by the columns of $A_\sigma$.  A subset $\sigma$ is a \emph{basis} if $\operatorname{rank}(A)=\operatorname{rank}(A_\sigma)$. 


The \emph{chamber complex} is the polyhedral subdivision of the cone $pos(A)$ which is defined as the common refinement of the simplicial cones $pos(A_\sigma)$, where $\sigma$ runs over all bases. The relation between rational polyhedral cones and polytopes is made explicit in the following theorem.


 A function
 $g : \mathbb{N}^n \to \mathbb{Q}$ is a {\em multivariate quasipolynomial} if there exists an $n$--dimensional lattice $\Lambda \subseteq \mathbb{Z}^n$, a set $\{ \mathbf{\lambda}_\mathbf{i} \}$ of coset representatives of $\mathbb{Z}^n/ \Lambda$,
  and polynomials $p_i \in \mathbb{Q}[\mathbf{t}]$ such that $g(\mathbf{t}) = p_i(\mathbf{t})$, for $\mathbf{t} \in \mathbf{\lambda}_\mathbf{i} + \Lambda$.

\begin{theorem}[Blakley~\cite{Blakley}, Sturmfels~\cite{Sturmfels}]
\label{Blakley} 
 Let $A$ be a $d\times n$  matrix with column vectors
  $S=\{{\mathbf{a}}_1, {\mathbf{a}}_2, \ldots, {\mathbf{a}}_n \} \subseteq \NN^d$. There exists a finite
  decomposition of $\ZZ^d \cap pos(A)$ such that $p_A({\mathbf{b}}) $ is a
  multivariable quasipolynomial of degree $n-d$ in each part.
 The number
  $n-d$ is the dimension of the polytope
  $\{ {\mathbf{x}} \, | \, A {\mathbf{x}} = {\mathbf{b}}, \text{ and } x_i \ge 0 \}$.
\end{theorem}



We call the function $p_A({\mathbf{b}}) $ that satisfies the description in Theorem~\ref{Blakley}, a  {\em
piecewise quasipolynomial function.}





\subsection{Vector partitions with restricted parts}
\label{VectorPartition}
 A \emph{vector partition} of ${\mathbf{b}} \in \NN^d$ is a way of decomposing ${\mathbf{b}}$ as a  sum of nonzero vectors in $\NN^d$.  Two vector partitions that  differ only in the order of their nonzero summands are considered the same, and the number of zero vectors in the decomposition is not  relevant.

 We are interested in partitions whose \emph{parts} (nonzero summands) belong to a fixed finite sub-multiset of $ \NN^d$.  The \emph{vector partition function} $p_S : \NN^d \to \NN$ is the function that evaluated at ${\mathbf{b}}$ gives the number of vector partitions of ${\mathbf{b}}$ with parts in $S$.


Given a vector ${\mathbf{x}}=(x_1, x_2, \ldots,
x_n)\in\NN^n$, we define $\mathbf{t}^{{\mathbf{x}}}=t_1^{x_1}t_2^{x_2}\ldots
t_n^{x_n}$. Then, much like the univariate case, the vector partition function  $p_S$ is defined by the formal power series
 \[
 \prod_{{\mathbf{a}}\in S} \frac{1}{1-{\mathbf{t}}^{{\mathbf{a}}}}
= \sum_{{\mathbf{b}} \in\NN^n} p_S({\mathbf{b}}) {\mathbf{t}}^{{\mathbf{b}}}.
 \] 

 A nonnegative integer
 solution ${\mathbf{x}}$ for the system of linear equation $A {\mathbf{x}}= {\mathbf{b}}$, where $A$ is
 the $d\times |S|$ matrix whose columns are the vectors in $S,$  encodes a vector partition of ${\mathbf{b}}$.  In our work, it turns out that 
 our matrices always contains a copy of the $n\times n$ identity matrix~$I_{n}$.

\subsection{A vector partition function}\label{F22}
\begin{example}\label{ThirdExample}
Let $p_S(n,m)$ count the number of vector partitions of ${\mathbf{b}}=(n,m)$ with
parts in $S=\{ (1,0),(0,1),(1,1),(1,2)\}$. Equivalently,  this is the number of nonnegative integer solutions ${\mathbf{x}}$ to the system $A {\mathbf{x}}= \mathbf{b}$, where
\[
A=
\begin{bmatrix}
 1 & 0 &  1  & 1 \\
0 & 1  & 1   &2   \\
\end{bmatrix} \qquad \mathbf{b}=\begin{bmatrix}n\\m\end{bmatrix}
\]
To determine one such partition, it suffices to determine the number
of parts equal to $(1,1)$ and $(1,2)$ in it. The standard basis vectors
serve as slack variables here,  consequently, the multiplicities of $(1,1)$ and $(1,2)$ should fulfill the  inequalities: 
 
\begin{align}\label{polytope22}
  \begin{cases}
 x_3+x_4 \le n\\
 x_3+2x_4 \le m
    \end{cases}

\end{align}
Therefore, the vector partition function $p_S$ counts nonnegative integer points in the polytope defined by the inequalities~\eqref{polytope22}. The three different possibilities are illustrated in Figure~\ref{fig:polytopes}.
\end{example}

\begin{figure}
\begin{minipage}{.32\textwidth}\center
\usetikzlibrary{arrows}
\begin{tikzpicture}[scale=.4]
{ 
\draw[step=1.0,draw=gray!60,thin] (-0.5,-0.5) grid (8.5,8.5);


\draw[very thick] (0,8.2) -- (0,0) -- (8.2,0);

\node at (8.9,0) {$x_3$}; 
\node at (0,8.9) {$x_4$}; 
\fill[opacity=.4] (6,0) -- (0,0) -- (0,3);
\node at (6,-0.5) {$m$}; 
\node at (-0.5,3) {$\frac{m}{2}$}; 
\draw(0,7) -- (7,0);
\draw(0,3) -- (6,0);
\node at (7,-0.5) {$n$}; 
\node at (-0.5,7) {$n$}; 

}\end{tikzpicture}
\end{minipage}
\begin{minipage}{.32\textwidth}\center
\usetikzlibrary{arrows}
\begin{tikzpicture}[scale=.4]
{ 
\draw[step=1.0,draw=gray!60,thin] (-0.5,-0.5) grid (8.5,8.5);

\draw[very thick] (0,8.2) -- (0,0) -- (8.2,0);


\node at (8.9,0) {$x_3$}; 
\node at (0,8.9) {$x_4$}; 
\fill[opacity=.4] (3,0) -- (0,0) -- (0,3);
\node at (8,-0.5) {$m$}; 
\node at (-0.5,4) {$\frac{m}{2}$}; 
\draw(0,4) -- (8,0);
\draw(0,3) -- (3,0);
\node at (3,-0.5) {$n$}; 
\node at (-0.5,3) {$n$}; 
}\end{tikzpicture}
\end{minipage}
\begin{minipage}{.32\textwidth}\center
\usetikzlibrary{arrows}
\begin{tikzpicture}[scale=.4]
{ 
\draw[step=1.0,draw=gray!60,thin] (-0.5,-0.5) grid (8.5,8.5);
\fill[opacity=.4] (6,0) -- (0,0) -- (0,4) -- (4,2);
\draw[very thick] (0,8.2) -- (0,0) -- (8.2,0);

\node at (8.9,0) {$x_3$}; 
\node at (0,8.9) {$x_4$}; 

\node at (8,-0.5) {$m$}; 
\node at (-0.5,4) {$\frac{m}{2}$}; 
\draw(0,4) -- (8,0);
\draw(0,6) -- (6,0);
\node at (6,-0.5) {$n$}; 
\node at (-0.5,6) {$n$}; 

}\end{tikzpicture}
\end{minipage}
\caption{Three possibilities for the polytope defined by  the inequalities in Eqn.~\eqref{polytope22}}
\label{fig:polytopes}
\end{figure}



\begin{enumerate}

\item[(I)]
If $m \le n$ the first equation is redundant. We are counting integer points in the 2-simplex  defined by $x_3 \ge 0$, $x_4 \ge 0$, and $x_3+2 x_4 \le m$.  The number of solutions is given by the quadratic quasipolynomial $p_{1,1,2}(m) $ of Example~\ref{FirstExample}.


\item[(II)] If $n \le \frac{m}{2}$, it is the second equation that is redundant. We are counting integer points in the standard  2-simplex defined by  $x_3 \ge 0$, $x_4 \ge 0$, and $x_3+ x_4 \le n$.

This was solved in Example~\ref{SecondExample}. The number of solutions is given by the quadratic polynomial 
$p_S(n)={\binom{n+2}{2}}$.  


\item[(III)] Finally, if $\frac{m}{2} < n < m,$ both inequalities are relevant. We are counting the number of points in the polytope with vertices 
$(0,0),(n,0),(0,\frac{m}{2}),$ $ (2n-m,m-n)$. We need to multiply by $2$ to get integer vertices, so the Ehrhart theorem says that the resulting quasipolynomial has  period $ 2$ on $m$.  Indeed,  
by fixing $s_1$ and counting the possibilities for $s_0,$ we have 
$p_S(n,m)=\sum_{s_1=0}^{\lfloor\frac{m}{2}\rfloor} (1+\min(m-2s_1, n-s_1))$ and $m-n<\lfloor\frac{m}{2}\rfloor,$ and hence 
\[
p_S(n,m)=
\begin{cases}
nm-\frac{n^2}{2}-\frac{m^2}{4}+\frac{n+m}{2}+1, \,\,\text{ if}   \,\,m \equiv 0 \mod 2 \\
nm-\frac{n^2}{2}-\frac{m^2}{4}+\frac{n+m}{2}+\frac{3}{4}, \,\,\text{ if}   \,\,m \equiv 1 \mod 2 
\end{cases}
\]
\[=nm-\frac{n^2}{2}-\frac{m^2}{4}+\frac{n+m}{2}+\frac{7}{8}+\frac{(-1)^m}{8}.\]

\end{enumerate}

\begin{figure}
\begin{minipage}{.35\textwidth}\center
\usetikzlibrary{arrows}
\begin{tikzpicture}[scale=.4]
{ 
\draw[step=1.0,draw=gray!60,thin] (-0.5,-0.5) grid (8.5,8.5);
\fill[opacity=.4] (8.2,8.2) -- (0,0) -- (4.1,8.2);
\fill[opacity=.6] (8.2,8.2) -- (0,0) -- (8.2,0);
\fill[opacity=.2] (0,8.2) -- (0,0) -- (4.1,8.2);
\draw[very thick] (0,0) -- (8.2,8.2);
\draw[very thick] (0,0) -- (4.1,8.2);
\draw[very thick] (0,8.2) -- (0,0) -- (8.2,0);
\foreach \i in {0,...,8}
 { \foreach \j in {0,...,8}
   { \filldraw(\i,\j) circle (1pt);   
   }
 }
 \node[draw,color=white] at (1,5) {\bf II}; 
\node[draw, color=white] at (5,7) {\bf III}; 
\node[draw, color=white] at (5,2) {\bf I}; 
\node at (8.9,0) {$n$}; 
\node at (0,8.9) {$m$}; 
}\end{tikzpicture}
\end{minipage}
\begin{minipage}{.6\textwidth}
\begin{tabular}{cll
\hline
  Region& &$p_S(n,m)$\\
  \hline
  I& $m\leq n$ & $\frac{m^2}{4}+ m + \frac{7}{8} + \frac{(-1)^m}{8}$\\
  II & $2n\leq m$& $\frac{n^2}{2}+\frac{3n}{2}+1$\\
  III  &$ n\leq m \leq 2n$ & $nm - \frac{n^2}{2} -\frac{m^2}{4} +\frac{n+m}{2} + \frac{7}{8} + \frac{(-1)^m}{8}$ \\%\bottomrule
  \hline
 \end{tabular}
\end{minipage}


\caption{The chambers giving the value of $p_S(n,m)$, the number of vector partitions of $(n,m)$ with parts in $S=\{(1,0), (0,1), (1,1), (1,2) \}$.}
\label{fig:chambers}
\end{figure}






 

\section{A vector partition function related to the Kronecker coefficients.}
\label{sec-F22}

In this section we show how to use Cauchy's definition of the Schur function as a quotient of alternants to deduce  the quasipolynomiality of the Kronecker function.
 Schur polynomials play a central role in the representation theory of   the general linear group. Schur polynomials of $n$ variables are 
  the characters of the irreducible polynomial representations of $GL(n)$.
  
     
   For our purposes Cauchy's definition, as a quotient
  of alternants, works best.  
Suppose $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_n)$ is a partition (or more generally, a vector in $\mathbb{N}^n$). The alternant $a_\lambda(x_1, x_2, \ldots, x_n)$ is defined as the polynomial obtained by antisymmetrizing the monomial $x^{\lambda}$. When $\delta_n=(n-1, n-2, \ldots, 1,0)$,   $a_\delta$ is the  \emph{Vandermonde determinant,} and   
\[
a_{\delta_n}(x_1, x_2, \ldots, x_n)=\prod_{1\le j < k \le n } (x_j-x_k).
\]
Since $a_\lambda(x_1, x_2, \ldots, x_n)$  is a skew-symmetric polynomial, it vanishes unless $\lambda_1, \lambda_2, \ldots, \lambda_n $ are all different.  As a result, there is no loss in assuming that $\lambda_1> \lambda_2> \ldots >  \lambda_n \ge 0$. 
We write $\lambda=\alpha+{\delta_n}$, where $\alpha$ is always a partition, possibly with repeated parts. Then, 
\[
a_\lambda[X]=a_\lambda(x_1, x_2, \ldots, x_n)=\det(x_i^{\lambda_j})_{i,j}.
\]


Let $\alpha$ be a partition of length $\le n$.
The \emph{ Schur polynomial} indexed by  $\alpha$ in the variables $x_1, x_2, \ldots, x_n$ is defined as the quotient of alternants:
\begin{align}\label{Cauchy}
s_\alpha[X]=s_\alpha(x_1, x_2, \ldots, x_n)= \frac{a_{\alpha+{\delta_n}}(x_1, x_2, \ldots, x_n) } {a_{{\delta_n} } (x_1, x_2, \ldots, x_n)}.
\end{align}
As a quotient of anti-symmetric polynomials,  $s_\alpha(x_1, x_2, \ldots, x_n)$ is  symmetric. On the other hand, since  $a_{\alpha+{\delta_n}}$ is divisible by $a_{\delta_n}$,    $s_\alpha(x_1, x_2, \ldots, x_n)$ is a polynomial.


\begin{remark}
Cauchy's definition of the Schur function is a particular case of both the Weyl  character formula  and  the Kostant partition formulas. This fact provides the link between our approach and those
of Christandl, Doran, and Walter~\cite{CDW} and Baldoni, Vergne, and Walter~\cite{BaldoniVergneWalter}.


\end{remark}



Schur polynomials will allows to define the Kronecker coproduct of Schur functions without any reference to representation theory.
Let $\mu$, $\nu$, and $\lambda$ be three partitions of the same weight satisfying that $\ell(\mu) \le n$, $\ell(\nu) \le m$, and $\ell(\lambda) \le  nm$. 
Let $X=\{x_1,\ldots,x_{n}\}$, $Y=\{y_1,\ldots,y_{m}\}$, and set $XY=\{x_1y_1,\ldots,x_{n}y_{m}\}$. 
Define 
\begin{align*}
s_{\lambda}[XY]:=s_{\lambda}(x_1y_1, x_1y_2\cdots,x_{n}y_{m}).
\end{align*}
This is a symmetric function in the $x$'s and the $y$'s separately. Since Schur functions form an integral basis for the algebra of symmetric functions, we can write  \begin{align}\label{comultiplication}
s_{\lambda}[XY]= \sum_{\mu, \nu} g_{\mu,\nu,\lambda} \, s_\mu[X] s_\nu[Y].
\end{align}
The structure coefficients $g_{\mu,\nu,\lambda}$ for this operation (comultiplication)  are the \emph{Kronecker coefficients.} 





Combining Cauchy's definition of a Schur polynomial in Eq.~\eqref{Cauchy} as a quotient of alternants with the comultiplication formula \eqref{comultiplication}, we obtain the identity
\begin{equation}\label{main}
\frac{{a_{\delta_n}[X]}{a_{\delta_m}[Y]}}{a_{\delta_{nm}}[XY]}   \,\,   a_{\lambda+\delta_{nm}}[XY]= \sum_{\mu, \nu} g_{\mu,\nu,\lambda} \,  a_{\mu+\delta_n}[X] a_{\nu+\delta_m}[Y].
\end{equation}

In the preceding formula, the factor $\frac{{a_{\delta_n}[X]}{a_{\delta_m}[Y]}}{a_{\delta_{nm}}[XY]} $ on the left is a rational function. The alternant $a_{\delta_{nm}}[XY]$ is a polynomial  consisting of $(nm)!$ nonzero monomials. On the right side of the equality, we obtain a polynomial, as opposed to a rational function, because the denominator of the rational function on the left divides the alternant $a_{\delta_{nm}}[XY]$. (There are a lot of cancellations.)

To avoid having to consider  the $(nm)!$ monomials in $a_{\delta_{nm}}[XY]$, we  select the smallest of these monomials, and construct a vector partition function which carries a lot of information about the Kronecker coefficients. 

The remainder of this section will be devoted to the analysis of the particular case $n=m=2$ for the Kronecker coefficients.  The  general situation will be studied  in Section~\ref{sec-Fnm}.  


\subsection{The Kronecker coefficients indexed by triples of partitions of lengths $\leq 2,2,4$, respectively. }

Our approach is best illustrated   in the simplest nontrivial case of three partitions with length at most $2, 2, $ and $4$, respectively.
We dedicate the rest of this section to this particular case.
We show how to describe the corresponding vector partition  function as a signed sum of vector partition functions, in such a way that the corresponding polytope is of minimal dimension. It may be
interesting to compare our approach to~\cite{Rosas-PhD} where the Kronecker coefficients for certain shapes are computed from Cauchy's definition of the Schur function.

Let $\lambda$ be a fixed partition of length $\le 4$.
Schur functions are homogeneous polynomials. Without loss of information, we set $X=\{1, x\}$, $Y=\{1,y\}$, $XY=\{1,x,y,xy\}$ in Eq.~\eqref{main}:  
\begin{equation}\label{KroneckerCauchy}
\frac{{a_{\delta_2}[X]}{a_{\delta_2}[Y]}}{a_{\delta_{4}}[XY]} \,\,   a_{\lambda+\delta_{4}}[XY]=
	 \sum_{\mu, \nu}  g_{\mu,\nu,\lambda} \,  x^{\mu_2}y^{\nu_2} + \text{lex. gr.}
\end{equation}
where \lq\lq\text{lex. gr.} "  \ stands for \lq\lq lexicographically greater terms." 

Recall that we are multiplying a rational function by a polynomial. However, the result is polynomial. (This is because $a_{\delta_{4}}[XY]$ divides $a_{\lambda+\delta_{4}}[XY] $.)
What would happen if, in the previous equation, we replaced  the alternant by its leading  monomial   $S( a_{\lambda+\delta_{4}}[XY])$?
We would have a  Laurent series (and not a polynomial).
\begin{equation*} 
\frac{{a_{\delta_2}[X]}{a_{\delta_2}[Y]}}{a_{\delta_{4}}[XY]} \,\,  S( a_{\lambda+\delta_{4}}[XY])=
	 \sum_{\mu, \nu} \tilde g_{\mu,\nu,\lambda} \,  x^{\mu_2}y^{\nu_2} + \text{lex. gr.}
\end{equation*}
This is a well--defined  Laurent series, convergent  under  the assumption that  $0<|xy|< |y|<|x|<1$.  


Let us compute all terms appearing in the previous equation. First, we extract the lexicographically least monomial simply by taking the product of the terms along the main diagonal.  This gives:
\[S( a_{\lambda+\delta_{4}}[XY])=
(xy)^{\lambda_4}\cdot y^{\lambda_3+1} \cdot x^{\lambda_2+2} ={ x^2y}\cdot x^{\lambda_4+\lambda_2} y^{\lambda_4+\lambda_3}\]
while the product-cum-quotient of alternants reduces to 
\[\frac{1}{{ x^2 y}\,\,(1-y/x)(1-xy)(1-x)(1-y)}.\]
Note that we obtain the factor $(1-y/x)$ from $(x-y)$ after factoring out $x$. Also note that our ordering\footnote{When we extract coefficients, we must also respect this ordering for the extractions to be meaningful.} has been chosen with care to cancel the monomial $x^2y.$
Simplifying the resulting expression we obtain a rational function that we will denote by $\bar{F}_{2,2}(x,y)$:
\begin{equation}\label{atomic1stdef}
\bar{F}_{2,2}(x,y)
=\frac{1}{(1-y/x)(1-xy)(1-x)(1-y)}
= \sum_{\mu, \nu}  \tilde g_{\mu,\nu,\lambda} x^{\mu_2-\lambda_2-\lambda_4} y^{\nu_2-\lambda_3-\lambda_4}
\end{equation}


 Restating Eqn.~\eqref{KroneckerCauchy}, we can also use the series $\bar{F}_{2,2}(x,y)$ to compute the actual Kronecker coefficients:
\begin{multline}\label{Kronecker1stdef}
\sum_{\mu, \nu}   g_{\mu,\nu,\lambda} x^{\mu_2} y^{\nu_2}
+ \text{lex. gr.}
=
\dfrac{a_{\lambda+\delta_4}[XY]}{(1-y/x)(1-xy)(1-x)(1-y)x^2y}
=a_{\lambda+\delta_4}[XY]\dfrac{\bar{F}_{2,2}(x,y)}{x^{2}y}.
\end{multline}
\smallskip
\begin{lemma}[A vector partition function]
After the change of basis  $x=s_1$ and $y=s_0s_1,$ the Laurent series $\bar{F}_{2,2}(x,y)$ becomes a vector partition function:
 \begin{align}\label{defF22}
  \nonumber
 F_{2,2}(s_0, s_1)&=\sum \tilde g_{\mu,\nu,\lambda} s_0^{\nu_2-\lambda_3-\lambda_4} s_1^{\mu_2+\nu_2-\lambda_2-\lambda_3-2\lambda_4}\\
 &=\frac{1}{(1-s_0)(1-s_1)(1-s_0s_1)(1-s_0s_1^2)}.
 \end{align}
 \end{lemma}
This change of variables is desirable as it returns us to the realm of Taylor series.  Note that the assumptions $0<|xy|< |y|<|x|<1$, that define the domain of convergence of this series translate to $0 < |s_0|, |s_1|  < 1 $. This is precisely the vector partition function of Example~\ref{F22}.

The following observation is useful in the present discussion. We respect our coefficient ordering by noting that when we say the coefficient of $x^i y^j$ in $\bar{F}_{2,2}(x,y)$, denoted $[x^iy^j]\bar{F}_{2,2}(x,y)$ we mean $[y^j][x^i]\bar{F}_{2,2}(x,y)$ since our series expansion prioritizes $x$. The order is interchangeable in extractions of $F_{2,2}(s_0,s_1)$, since it is a finite product of Taylor (geometric) series. In our analysis of the coefficients, we choose between the original series $\bar{F}_{2,2}(x,y)$ in the variables $x,y,$ and the vector partition function $F_{2,2}(s_0,s_1)$ depending upon which is more convenient.  More precisely, we use Eqn.~\eqref{defF22} to apply techniques from  the theory of vector partition functions and polyhedral geometry.  However Eqn.~\eqref{Kronecker1stdef} is the more natural choice to analyse the alternant $a_{\lambda+\delta_4}[XY].$ 
 \begin{proposition}\label{barF22}  The coefficient of $x^i y^j$ in $\bar{F}_{2,2}(x,y)$, which is also the coefficient of $s_0^j s_1^{i+j}$ in the Taylor expansion of $F_{2,2}(s_0,s_1)$,  is nonzero if and only if $j\geq 0$ and $i+j\geq 0.$ 
 \end{proposition}


  It follows immediately from this or from Eqn.~\eqref{atomic1stdef} and Section~\ref{F22}, that 

 
 \begin{proposition}\label{ineqatomiccone22} 
 The atomic Kronecker coefficient  $\tilde g_{\mu, \nu, \lambda}$ is nonzero if and  only if 
\begin{align}\label{Bravyi}
\begin{cases}
&  \lambda_2+\lambda_3+2\lambda_4 \le \mu_2 + \nu_2 \qquad\text{(First Bravyi Inequality)}\\
& \lambda_3+\lambda_4 \le \nu_2.  \qquad  \qquad\phantom{firstx }\text{(Second Bravyi Inequality)}
\end{cases}
\end{align}
In this case, 
\[\tilde g_{\mu, \nu, \lambda}= p_S(\nu_2-(\lambda_3+\lambda_4),  \mu_2+\nu_2-
(\lambda_2+\lambda_4 )-(\lambda_3+\lambda_4)),\] a  quasipolynomial of degree 2.
\end{proposition}

\begin{corollary} The value of the atomic Kronecker coefficient  depends only on the values of the two linear forms $\nu_2-(\lambda_3+\lambda_4)$ and $  \mu_2+\nu_2-
(\lambda_2+\lambda_4 )-(\lambda_3+\lambda_4)$. In particular, when either one is equal to zero, the corresponding atomic Kronecker coefficient is equal to 1. 



\end{corollary}


The rest of Section~\ref{sec-F22} will study the Kronecker coefficients indexed by partitions of lengths bounded by $2,2$, and $4$ in a unified way. We analyse the relationship between the atomic Kronecker coefficients,~$\tilde g_{\mu, \nu, \lambda}$  and the actual Kronecker coefficients. 
 The strategy is elementary: we compute using both Eqns.~\eqref{Kronecker1stdef}  and \eqref{defF22}.
 
  It is important to keep in mind that for the rest of Section~\ref{sec-F22},  we always refer to the situation where $\mu$ and $\nu$ have length bounded by 2, and   the length of $\lambda$ is less or equal to 4.
 
We postpone the introduction of the analogue of  vector partition functions $F_{2,2}$ and  $\bar F_{2,2}$ for the general situation  until  Section \ref{sec-Fnm}.





\subsection{From $\bar F_{2,2}$ to an exact expression for Kronecker coefficients.}
The rational series $F_{2,2}$ is not directly the generating series for the Kronecker coefficients because we truncated some polynomials in its construction. The main result of this section, Theorem~\ref{thm:MAIN1}, is an exact formula for the Kronecker coefficients in the $n=m=2$ case. 

We ask ourselves: What do the terms in $a_{\lambda+\delta_{nm}}[XY]-S(a_{\lambda+\delta_{nm}}[XY])$ look like?  The difference is described in the precise affine combinations of the parts of $\lambda$ appearing in the left-hand side of Eq.~\eqref{main}. In the case $n=m=2,$ we identify these affine combinations. Since $\ell(\lambda)\le 4,$ the number of terms (here, $4!=24$) in the  expansion of the alternant $a_{\lambda+\delta_4}[XY]$ is fixed. However, it turns out that only \textit{seven} terms contribute to the Kronecker coefficient.


The following theorem explicitly identifies which terms of the alternant contribute to the Kronecker coefficient. The polynomial in Theorem~\ref{7termF22Alternant} is minimal in the following sense:  Example~\ref{Ex7terms} below exhibits a combination in which all seven terms contribute nontrivially to the Kronecker coefficient, with NO  cancellation between any pairs of terms. 
 
 
 
 \begin{theorem}\label{7termF22Alternant}\label{thm:MAIN1}
Assume $\ell(\lambda)\leq 4,$ and $\ell(\mu), \ell(\nu)\leq 2.$ Also assume $\mu_2\geq \nu_2.$ Then the  Kronecker coefficient $g_{\mu, \nu, \lambda}$ is equal to each of the following:
\begin{enumerate}
\item the coefficient of $x^{\mu_2} y^{\nu_2}$ in 
 \[ P_\lambda(x,y)\bar{F}_{2,2}(x,y)\]
 where $P_\lambda(x,y)$ is the polynomial consisting of the following seven terms:
 \begin{multline}\label{explicitF22extraction}
 y^{\lambda_3+\lambda_4}  (x^{\lambda_2+\lambda_4} 
 -x^{\lambda_2+\lambda_3+1}-x^{\lambda_1+\lambda_4+1}
 +x^{\lambda_1+\lambda_3+2  })\\
 +y^{\lambda_2+\lambda_4+1}(-x^{\lambda_3+\lambda_4-1}+x^{\lambda_2+\lambda_3+1}
 +x^{\lambda_1+\lambda_4+1})
 \end{multline}
 
A monomial $y^b x^a$ in $P_\lambda$ makes a nonzero contribution to $g_{\mu,\nu,\lambda}$ if and only if 
 $ b\leq \nu_2 \text{ and } b+a\leq \mu_2+\nu_2$. 
 
 \item the following 7-term linear combination of Ehrhart functions $p_S(n,m)$:
 %
 \begin{multline}\label{Exact}g_{\mu,\nu,\lambda}=
 p_S(\nu_2-(\lambda_3+\lambda_4), \nu_2-(\lambda_3+\lambda_4)+ \mu_2-(\lambda_2+\lambda_4))\\
 %
 -p_S(\nu_2-(\lambda_3+\lambda_4), \nu_2-(\lambda_3+\lambda_4)+ \mu_2-(\lambda_2+\lambda_3+1))\\
 %
 -p_S(\nu_2-(\lambda_3+\lambda_4), \nu_2-(\lambda_3+\lambda_4)+ \mu_2-(\lambda_1+\lambda_4+1))\\
 %
+ p_S(\nu_2-(\lambda_3+\lambda_4), \nu_2-(\lambda_3+\lambda_4)+ \mu_2-(\lambda_1+\lambda_3+2))\\
-p_S(\nu_2-(\lambda_2+\lambda_4+1), \nu_2-(\lambda_2+\lambda_4+1)+\mu_2-(\lambda_3+\lambda_4-1))\\
+p_S(\nu_2-(\lambda_2+\lambda_4+1), \nu_2-(\lambda_2+\lambda_4+1)+\mu_2-(\lambda_2+\lambda_3+1))\\
+p_S(\nu_2-(\lambda_2+\lambda_4+1), \nu_2-(\lambda_2+\lambda_4+1)+\mu_2-(\lambda_1+\lambda_4+1))
 %
 \end{multline}
 %
 \end{enumerate}
   \end{theorem}
 

   


\begin{example}\label{Ex7terms} (\textbf{Minimality of the polynomial in Theorem~\ref{7termF22Alternant}.}) Let $\lambda=(12,7,4,1), \mu=\nu=(12,12).$ 
From Theorem~\ref{7termF22Alternant}, the Kronecker coefficient 
$g_{\mu, \nu, \lambda}$ is the coefficient of $x^{12}y^{12}$ in the product $P_\lambda\bar{F}_{2,2}(x,y),$  where 
\[P_{\lambda}=y^5(x^8-x^{12}-x^{14}+x^{18}) +y^9(-x^{4}+x^{12}+x^{14}).\]
Using Eq.~(\ref{Exact}), we have 
\begin{multline*}
g_{\mu,\nu,\lambda}=p_S(7,11)-p_S(7,7)-p_S(7,5)+p_S(7,1)-p_S(3,11)+p_S(3,3)+p_S(3,1)\\
=32-20-12+2-10+6+2=0.
\end{multline*}

This example is noteworthy because the Kronecker coefficient vanishes, but there is no cancellation between pairs of the seven coefficients above. By definition, the atomic coefficient is the contribution from the first monomial $y^5x^8$ in the expansion of $P_\lambda$ above, (it is also the lexicographically least monomial),  hence~$\tilde{g}_{\mu, \nu, \lambda}=p_S(7,11)=32.$

\end{example}  
 
 
 \begin{figure}\center
\includegraphics[width=.6\textwidth]{FIGURE.pdf}
 \begin{comment}
 \begin{tikzpicture}
\begin{axis}[grid=both,
 axis lines=middle,
ticklabel style={fill=blue!5!white!5},
 x tick label style={rotate=45, anchor=east, align=center},
 xmin=0,xmax=10.3,
 xtick={-1,1,2,4,5,6,7,9,10},
 xticklabels={,$\lambda_3+\lambda_4-1$ ,$\lambda_2+\lambda_4$ ,$\lambda_2+\lambda_3+1$, {\color{gray}$n/2+1$}  ,$\lambda_1+\lambda_4+1$, $\lambda_1+\lambda_3+2$,, {\color{gray}$n$},},
 ymin=0,ymax=10.3,
 ytick={-1,1,2,4,5, 6,7,9,10},yticklabels={,$\lambda_3+\lambda_4$,$\lambda_2+\lambda_4+1$ , $\lambda_2+\lambda_3+2$,{\color{gray}$n/2+2$}, $\lambda_1+\lambda_4+2$ , $\lambda_1+\lambda_3+3$,,{\color{gray}$n$},},
]

 \addplot[holdot] coordinates{(1,1)(2,2)(4,4)(4,6)};
 \addplot[boldot] coordinates{(1,2)(2,1)(4,1)(4,2)(6,1)(6,2)(7,1)};

\addplot[soldot] coordinates{(2,1)(4,2)(1,4)(1,6)(6,2)(7,1)};
 
 \addplot[moldot] coordinates{(1,2)(4,1)(2,6)(2,4)(1,7)(6,1)};

 \addplot[color=red]coordinates{(10,0)(0,10)};
\end{axis}
 \node at (5.3,-1.5) {exponent of $x$};
 \node[rotate=90] at (-2.7,2.3) {exponent of $y$};


\end{tikzpicture}
\end{comment}

\caption{The expansion of  $x^{-2}y^{-1}\cdot  a_{\lambda+\delta_4}$.}
\label{fig:alternant_lambda_4}
\end{figure}


 \begin{proof}[Proof of Theorem~\ref{7termF22Alternant}] 
 Eqn.~(\ref{Exact}) follows from the polynomial $P_\lambda$ because of the following observation:
 If $x^ay^b$ is a monomial in $P_\lambda$ making a nonzero contribution to $g_{\mu,\nu,\lambda}$, that value is 
 
 \[\pm[s_0^{\nu_2-b}s_1^{\mu_2+\nu_2-a-b}]F_{2,2}=\pm p_S(\nu_2-b,\mu_2+\nu_2-a-b).\] This follows from the substitution $x\mapsto s_1$ and $y\mapsto s_0s_1$, Proposition~\ref{barF22} and Section~\ref{F22}. 
 
 Now we establish Eqn.~(\ref{explicitF22extraction}), the first equality of the theorem.
  Proposition~\ref{barF22} says that   the Kronecker coefficient $g_{\mu,\nu,\lambda}$ is obtained by extracting the coefficient of $x^{\mu_2} y^{\nu_2}$ in the product of  $\bar{F}_{2,2}$ with  the expansion of  $(x^{-2}y^{-1})$ times the alternant $a_{\lambda+\delta_4}.$ 
   
Let $\lambda$ be a fixed partition of $n.$ 
We represent the 24 monomial terms of  $x^{-2}y^{-1}a_{\lambda+\delta_4}$ using Cartesian coordinates, as  in Figure~\ref{fig:alternant_lambda_4}. The lattice point $(a,b)$ (corresponding  to the monomial $x^ay^b$) 
has been marked with a plus sign or a minus sign to indicate the coefficient of $x^ay^b;$  the nonzero coefficients in the alternant are always $\pm 1.$ The horizontal (resp. vertical) axis  shows the possible exponents of $x$ (resp. $y$).

The first observation is the ordering 
 \begin{multline*}\lambda_3+\lambda_4-1<\lambda_2+\lambda_4<\min(\lambda_2+\lambda_3+1, \lambda_1+\lambda_4+1)\\
 \leq \max(\lambda_2+\lambda_3+1, \lambda_1+\lambda_4+1)<\lambda_1+\lambda_3+2.
 \end{multline*}
 
 The second observation is that 
 \[\min(\lambda_2+\lambda_3, \lambda_1+\lambda_4)
 \leq \frac{n}{2}\le\max(\lambda_2+\lambda_3, \lambda_1+\lambda_4),\]
  since $(\lambda_2+\lambda_3)+(\lambda_1+\lambda_4)=n.$
  (In particular the inequalities are strict unless 
  $\lambda_2+\lambda_3= \lambda_1+\lambda_4.$)
 
Thus there are two possible total orders.
Figure~\ref{fig:alternant_lambda_4} shows the case
$\lambda_2+\lambda_3<\lambda_1+\lambda_4;$ but the argument is identical for the other case, since it corresponds to switching the appropriate pair of rows and columns.  In particular, note the important fact that the lattice point weights  corresponding to the lines $y=\lambda_2+\lambda_3+2, y=\lambda_1+\lambda_4+2$ are identical, and likewise for the lines 
$x=\lambda_2+\lambda_3+1, x=\lambda_1+\lambda_4+1.$
 
 Proposition~\ref{barF22} says that the  only monomials  of $x^{-2}y^{-1}a_{\lambda+\delta_4}$ making a nonzero contribution to the computation of $g_{\mu,\nu,\lambda}$ are those $x^ay^b$ satisfying
 $(n/2)\ge \nu_2\geq b$ and $n\ge \mu_2+\nu_2\geq a+b.$  The second inequality immediately implies that only lattice points below the line $x+y=n$ can contribute (and hence these are the only ones shown in Figure~\ref{fig:alternant_lambda_4} . In the case depicted in Figure~\ref{fig:alternant_lambda_4}, the first inequality implies in addition that only points below the line $y=\lambda_1+\lambda_4+2$ can contribute. 
 
 This argument eliminates all monomials in $x^{-2}y^{-1}a_{\lambda+\delta_4}$ except for the nine points appearing in the  region bounded by the dimension of the coordinate axes and the lines $y=\frac{n}{2}+2,$ $x+y=n$ in Figure~{\ref{fig:alternant_lambda_4}}. 
 An additional argument will now show that indeed the contributions of  monomials $x^{\lambda_3+\lambda_4-1}y^{\lambda_2+\lambda_3+2}$ and
 $x^{\lambda_2+\lambda_4}y^{\lambda_2+\lambda_3+2}$  (along the line $y=\lambda_2+\lambda_3+2$) are equal but of opposite signs. 

The coefficient of $x^{\mu_2}y^{\nu_2}$ in the product of each monomial $x^ay^b$ in the above polynomial with $\bar{F}_{2,2}(x,y)$ is equal to the coefficient of 
 $x^{\mu_2-a}y^{\nu_2-b}$ in $\bar{F}_{2,2}(x,y).$   
 From Section~\ref{F22} and Proposition~\ref{barF22}, this coefficient equals $p_S(n_b, m_{b,a})$ where $n_b=\nu_2-b, m_{b,a}=(\nu_2-b )+(\mu_2-a ).$  Note that $a<b$ implies $\mu_2-a>\mu_2-b \geq (\nu_2-b),$ and hence $m_{b,a} > 2 n_b.$  It follows from (II) in Figure~\ref{fig:chambers} that $p_S(n_b, m_{b,a})=\binom{n_b+2}{2}$ is independent of $a,$ the exponent of $x.$
 
 
 Now apply these observations to the two monomials corresponding to $b=\lambda_2+\lambda_3+2$ and $a_i\in\{\lambda_3+\lambda_4-1, 
 \lambda_2+\lambda_4\}, i=1,2.$  Since  $a_i<b$ and 
  $\mu_2\geq \nu_2,$ we have $\mu_2-a_i\geq \nu_2-a_i>\nu_2-b.$    A necessary condition for either monomial to make a nonzero contribution is for the exponent $\nu_2-b$ of $y$  to be nonnegative. In the present situation, this forces the exponent $\mu_2-a_i$ of $x$ to be nonnegative as well. Hence either both monomials $x^{\mu_2-a_i}y^{\nu_2-b}, i=1,2,$ contribute to the Kronecker coefficient, or neither does.  Since $a_i<b$, from the preceding paragraph, the contributions are independent of the $a_i$ and \textit{equal}  (to $\binom{\nu_2-b+2}{2}$), and since the monomials come with opposite sign, their combined contribution is zero.
 
 The remaining seven nonzero lattice points (boxed in Figure~\ref{fig:alternant_lambda_4}) are precisely those corresponding to  the monomials in the polynomial $P_\lambda.$ \end{proof}



 
 \begin{remark}
Calculations of $\kappa_{2,2,4}$ were previously explicitly worked out in~\cite{BriandOrellanaRosasquasi} using an identity describing the Kronecker coefficient as a linear combination of \textit{reduced} Kronecker coefficients (See section \ref{RKC})\cite[Theorem 4]{BriandOrellanaRosasquasi}. Their approach differs from ours, but does permit determination that the number of chambers in the corresponding chamber complex is~$74$. This approach can also be compared with  \cite{Rosas-PhD} where a combinatorial interpretation for $\kappa_{2,2,4}$ was found as the difference of the number of integer points in two rectangles (mod 2).  
 \end{remark}
   
   

  \subsection{Examples}\label{SheilaExamples}
We illustrate these results with some examples. Since we invoke the computation of the coefficients $p_S(n,m)$ of $s_0^ns_1^m$ in $F_{2,2}(s_0, s_1)$ from Section~\ref{F22} extensively, we record the values of the following special coefficients:
\[p_S(n,0)=1=p_S(0,m);\quad p_S(n,1)=2 \text{ for }n>0;\quad p_S(1,m)=3 \text{ for }m\geq 2.\]
 

\begin{example} Let $\lambda=(6,5,4,1), \mu=\nu=(9,7).$ 
Note that $\lambda_3+\lambda_4=5, \mu_2+\nu_2=14,$ 
and thus, by checking the inequalities \eqref{1stalternantInequality} and \eqref{2ndalternantInequality},
 we see that  only three of the seven terms from the polynomial $P_\lambda$  contribute to $g_{\mu,\nu,\lambda}, $ specifically those in the expansion of
\[y^{5}(x^{6}-x^{8}) +y^{7}(-x^{4} ).\]
We obtain the value
\[g_{\mu,\nu,\lambda}
 =p_S(2,3)-p_S(2,1)-p_S(0,3)=2.\]
\end{example}

\begin{example}\label{ex:Atomic}
Our format is well suited to compute dilated Kronecker coefficients\footnote{They are also known as stretched Kronecker coefficients},  as we shall see in greater detail in Section~\ref{Dilation}. Assume $k$ is a positive integer, and let $k\lambda=(6k,3k,2k), k\mu=(7k,4k)$ and $k\nu=(8k,3k)$. These are the dilations of the triple $((6,3,2), (7,4), (8,3))$.  The Kronecker coefficients are all atomic and given by the quasipolynomial $p_S(k, 2k) = (k+1)(k+2)/2$,  which counts integer points in dilations of a standard 2-simplex. This is precisely the polytope dilation in Figure~\ref{fig:simplex}
\end{example}



 \subsection{Inequalities that determine when a Kronecker coefficient is atomic}
 
A careful analysis of the order relations between the exponents appearing in Eqn.~\eqref{explicitF22extraction}  yields the following result.
 
 \begin{corollary}\label{atomicisKron}The atomic coefficient equals the Kronecker coefficient if 
 \begin{enumerate}
 \item $\lambda_3+\lambda_4\le \nu_2\leq \lambda_2+\lambda_4$ 
 and
 \item $(\lambda_3+\lambda_4)+(\lambda_2+\lambda_4) \leq 
 \mu_2+\nu_2\le 
 (\lambda_3+\lambda_4)+\min(\lambda_2+\lambda_3, \lambda_1+\lambda_4).$ 
 \end{enumerate}
 When $\lambda_2+\lambda_3\le \lambda_1+\lambda_4,$ these inequalities can be combined into a more pleasing form to give a sufficient condition for equality of atomic and Kronecker coefficients:
 \[\lambda_4\le \mu_2+\nu_2-(\lambda_2+\lambda_3+\lambda_4)\le \lambda_3
 \le \nu_2-\lambda_4\le \lambda_2.\]
 
 \end{corollary}
 \begin{proof} 
 We now examine the exponents in Eqn.~\eqref{explicitF22extraction} more carefully.  
 %
 The two exponents of $y$ in Eqn.~\eqref{explicitF22extraction} are ordered as follows:
 \begin{equation}\label{yInequality} \lambda_3+\lambda_4<\lambda_2+\lambda_4+1.\end{equation}
The five exponents of $x$ are ordered thus:
 %
 \begin{multline}\label{xInequality}\lambda_3+\lambda_4-1<\lambda_2+\lambda_4<\min(\lambda_2+\lambda_3+1, \lambda_1+\lambda_4+1)\\
 \leq \max(\lambda_2+\lambda_3+1, \lambda_1+\lambda_4+1)<\lambda_1+\lambda_3+2.
 \end{multline}
Compare with~\cite{Rosas-PhD}.  In turn, by considering the total degree sequences of the monomials, which constitute a set of size six, we have the following two chains of inequalities:
 \begin{multline} \label{1stalternantInequality}
 (\lambda_3+\lambda_4)+(\lambda_2+\lambda_4)
 <(\lambda_3+\lambda_4) +\min(\lambda_2+\lambda_3+1, \lambda_1+\lambda_4+1)\\
 \leq (\lambda_3+\lambda_4) +\max(\lambda_2+\lambda_3+1, \lambda_1+\lambda_4+1)\\
 < (\lambda_3+\lambda_4) +(\lambda_1+\lambda_3+2)
 \end{multline}
 \vskip-.2in
 and
 \vskip-.3in
 \begin{multline} \label{2ndalternantInequality}
 (\lambda_2+\lambda_4+1)+(\lambda_3+\lambda_4-1)\\
 <(\lambda_2+\lambda_4+1) +\min(\lambda_2+\lambda_3+1, \lambda_1+\lambda_4+1)\\
 \leq (\lambda_2+\lambda_4+1) +\max(\lambda_2+\lambda_3+1, \lambda_1+\lambda_4+1).
 \end{multline}

 A monomial $y^b x^a$ will make a nonzero contribution to 
 $g_{\mu,\nu,\lambda}$ if and only if 
 \[ b\leq \nu_2 \text{ and } b+a\leq \mu_2+\nu_2.\]
 Hence the subset of 7 monomials in Eqn.~\eqref{explicitF22extraction}  contributing to the Kronecker coefficient $g_{\mu, \nu, \lambda}$ is  determined by where the number $\mu_2+\nu_2$ falls in the consecutive  intervals determined by each of the inequalities Eqn.~\eqref{1stalternantInequality} and \eqref{2ndalternantInequality}, and also where $\nu_2$ falls in Eqn.~\eqref{yInequality}.
 

 
  The atomic coefficient comes from the least monomial,  which is the first one in the first line of ~\eqref{explicitF22extraction}.  The first condition in the above statement,  in view of the inequality \eqref{2ndalternantInequality},   eliminates the possibility of any contribution to the Kronecker coefficient from the monomials in  the second  line of \eqref{explicitF22extraction}.
 
 The second condition then eliminates all but the first monomial, 
  because  only monomials with total degree not exceeding 
 \[(\lambda_3+\lambda_4)+\min (\lambda_2+\lambda_3, \lambda_1+\lambda_4)\] 
 can contribute.  
 Hence we are left with only the atomic coefficient. \end{proof} 
 
 From this we can easily deduce some conditions on the parts which ensure that the atomic Kronecker coefficients are an upper bound for the Kronecker coefficients.   In fact 
 Theorem~\ref{thm:AtomicIsMax} below states that NO restrictions on the parts of $\lambda$ are needed.  The proof is quite technical, and appears in the Appendix.
 
 \begin{theorem}~\label{thm:AtomicIsMax}  The atomic coefficient is an upper bound for the Kronecker coefficient.
 
  \end{theorem}
  



 
 
 
 \subsection{Bravyi's  vanishing conditions for the Kronecker coefficients.}
 A theorem of Murnaghan implies that if $\mu_2+\nu_2<\lambda_2+\lambda_3+\lambda_4,$ then the Kronecker coefficient vanishes (see Eqn.~\eqref{MurnaghanIn}). The following stronger result (due to Bravyi) follows from our methods.  

\begin{proposition}[Bravyi~\cite{Kirillov:saturation, Bravyi}]
\label{nonzeroKron}  

Assume $\ell(\lambda)\leq 4, \ell(\mu), \ell(\nu)\leq 2.$  The Kronecker coefficient is zero if $\nu_2<\lambda_3+\lambda_4$ or $\mu_2+\nu_2<\lambda_2+\lambda_3+2\lambda_4 .$  Equivalently, if the Kronecker coefficient $g_{\mu, \nu,\lambda}$  is nonzero then Bravyi's inequalities~\eqref{Bravyi} are satisfied. \end{proposition}
\begin{proof} We assume without loss of generality that $\mu_2\geq \nu_2.$
We will use the polynomial $P_\lambda$ of Theorem~\ref{7termF22Alternant}.

First suppose $\nu_2<\lambda_3+\lambda_4$.  Then we have 
$\nu_2<\lambda_3+\lambda_4\le \lambda_2+\lambda_4.$  Examining the polynomial 
$P_\lambda$ in~\eqref{explicitF22extraction}, we see that none of the monomials $y^bx^a$ makes a contribution since the condition $b\leq \nu_2$ is violated for both exponents $b$ of $y$ in $P_\lambda.$

Now suppose $\mu_2+\nu_2<\lambda_2+\lambda_3+2\lambda_4.$  Observe that 
$\lambda_2+\lambda_3+2\lambda_4$ is precisely the sum of the exponents for the first monomials $y^bx^a$ in each of the first two lines of the polynomial in~\eqref{explicitF22extraction}. Hence the  condition $b+a\leq \mu_2+\nu_2$ is violated for these two monomials.   But the sum of exponents $b+a$ for each of the other monomials in~\eqref{explicitF22extraction} is strictly greater than the sum for the first monomial in each line, so the condition is violated for all the monomials in $P_\lambda.$ \end{proof}

\subsection{A closed formula for the reduced Kronecker coefficients.}


Long ago, Murnaghan observed that the sequences of Kronecker coefficients obtained by increasing the value of $k$ in
$ g(\lambda+(k), \mu+(k), \nu+(k))$
 always  stabilize. 
   Their stable value is called the \emph{reduced Kronecker coefficient} and denoted by $
 \bar g_{\bar \lambda, \bar  \mu, \bar  \nu}
$, where, given a partition $\lambda$, we denote by $\bar \lambda$ the partition obtained from $\lambda$ deleting its first part. 

The reduced Kronecker coefficients are conjectured to verify the saturation hypothesis~\cite{Klyachko, Kirillov:saturation}. They share with the atomic
Kronecker coefficients the property that~\cite{BOR-JA} that the value of
 Kronecker coefficients can be computed as an alternated sum of reduced Kronecker coefficients. We will have more to say about the reduced Kronecker coefficients  in Section \ref{RKC}.

Murnaghan also discovered  a necessary condition for the Kronecker coefficient $g_{\lambda, \mu, \nu}$ to be nonzero. The following inequality has to hold:
\begin{align}\label{MurnaghanIn}
|\bar \lambda| \le |\bar \mu| + |\bar \nu|.
\end{align}




 
 






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 \begin{proposition}\label{Stability2parts}  Let $\lambda_2 \ge \mu_2\geq \nu_2.$ Assume $\lambda$ has at most two parts. Then $g_{\mu,\nu,\lambda}$ is independent of $\lambda_1$ as soon as $\lambda_1\geq \mu_2+\nu_2.$
Moreover, the stable value is $p_S(\nu_2, \nu_2+\mu_2-\lambda_2)-p_S(\nu_2, \nu_2+\mu_2-\lambda_2-1).$ Explicitly,  let $\ell = \nu_2+\mu_2-\lambda_2$, then,
 
  \begin{align}\label{reduced224}
  \bar g_{(\lambda_2),(\mu_2),(\nu_2)}&= \begin{cases}
\frac{\ell}{2} + \frac{ 3 + (-1)^{\ell}}{4} =  \left \lfloor \frac{\ell}{2} \right \rfloor +1 &\text{ if } \mu_2+\nu_2 \ge  \lambda_2\\
0 &\text{ if } \mu_2+\nu_2< \lambda_2
\end{cases}
 \end{align} 
 \begin{proof} We have $\lambda_3=\lambda_4=0.$
 For any statement $S$, we write $\delta(S) $ to mean 1 if $S$ is true and 0 otherwise.
 %
 From Eqn.~(\ref{Exact}) of Theorem~\ref{7termF22Alternant}, the  Kronecker coefficient is
 \begin{multline*
 p_S(\nu_2, \nu_2+ \mu_2-\lambda_2)
 -p_S(\nu_2, \nu_2+ \mu_2-(\lambda_2+1))\\
 %
 -p_S(\nu_2, \nu_2+ \mu_2-(\lambda_1+1))\cdot {\scriptstyle\mathbf{\delta(\mu_2+\nu_2\geq \lambda_1+1)}}
+  p_S(\nu_2, \nu_2+ \mu_2-(\lambda_1+2))\cdot {\scriptstyle\mathbf{\delta(\mu_2+\nu_2\geq \lambda_1+2)}}\\
-p_S(\nu_2-(\lambda_2+1), \nu_2-(\lambda_2+1)+\mu_2+1)
+p_S(\nu_2-(\lambda_2+1), \nu_2-(\lambda_2+1)+\mu_2-(\lambda_2+1))\\
+0\cdot p_S(\nu_2-(\lambda_2+1), \nu_2-(\lambda_2+1)+\mu_2-(\lambda_1+1))
 %
 \end{multline*}


 The zero coefficient in the last line is explained by the fact that $ (\lambda_1+\lambda_2)/2\geq \mu_2\geq \nu_2, $ and thus we always have $\mu_2+\nu_2\leq \lambda_1+\lambda_2.$  
 
The  hypothesis that  $\lambda_1\geq \mu_2+\nu_2$  eliminates the two terms with  $\lambda_1$ in their arguments,   establishing a stable value.  The reduced  Kronecker coefficient is thus given by 
 %
 \begin{multline}\label{PropReduced}
 \mathbf{p_S}(\nu_2, \nu_2+ \mu_2-\lambda_2)
 -\mathbf{p_S}(\nu_2, \nu_2+ \mu_2-(\lambda_2+1))\\
 %
-\mathbf{p_S}(\nu_2-(\lambda_2+1), \nu_2+\mu_2-\lambda_2)
+\mathbf{p_S}(\nu_2-(\lambda_2+1), \nu_2+\mu_2-2(\lambda_2+1))
 %
 \end{multline}
 
Of these four terms, since $\lambda_2\ge\mu_2\ge\nu_2,$ the third and  fourth terms are immediately eliminated because the first argument is negative: 
 $\nu_2-\lambda_2-1\le -1.$   
 
 If $\mu_2+\nu_2< \lambda_2,$  both first and second terms are identically zero.
 
 If $\mu_2+\nu_2= \lambda_2,$ only the first term appears, but it must be 1 from the boundary values $p_S(n,0)=1=p_S(0,m)$  recorded in Section~\ref{SheilaExamples}.
 
If $\mu_2+\nu_2> \lambda_2,$  the first two Ehrhart functions both appear.  Since  $\lambda_2-\mu_2\ge 0$ implies $\nu_2\ge\ell=\nu_2+\mu_2-\lambda_2,$ 
 both  are computed using the quasipolynomial corresponding to Region I in Figure~\ref{fig:chambers}, and consequently  the reduced Kronecker coefficient is 
\[\mathbf{p_S}(\nu_2,\ell)-\mathbf{p_S}(\nu_2,\ell-1)
=\frac{\ell}{2} + \frac{ 3 + (-1)^{\ell}}{4}.\] 
The proof is now complete.
 \end{proof}
 \end{proposition}
 Using the symmetry of the Kronecker coefficients with respect to the three partitions, we immediately have:

\begin{figure}\center
\begin{minipage}[t]{.6\textwidth}
 \begin{tikzpicture}
\node[inner sep=0pt] (picture1) at (0,0)
    {\includegraphics[width=\textwidth]{cone.pdf}};%
    \node[label=left:{$(0,0,0)$}] at (-3.4,-.9) {$\bullet$};
    
    
\end{tikzpicture}
\end{minipage}
\hfill
\begin{minipage}[b]{.38\textwidth}
\begin{tikzpicture}[scale=.8]
 \foreach \n in {1,2,3}{
        \node at ({\n*360/3+45}:4cm) (n\n) {};
        \draw[color=white] (0,0)--(n\n);
        
    }
    \fill[opacity=.4] (0,0) -- (45:4cm) -- (165:4cm);
  \fill[opacity=.2, color=- gray] (0,0) -- (45:4cm) -- (285:4cm);
     \fill[opacity=.6, color=blue!20!gray] (0,0) -- (285:4cm) -- (165:4cm);
  \draw[color=white] (n1) -- (n2) -- (n3) -- (n1);
  \node at (105:1cm) {$III$};
  \node at (-15:1cm) {$II$}; 
  \node at (225:1cm) {$I$};
\end{tikzpicture}
\vspace{2cm}
\mbox{}
\end{minipage}

\caption{The chamber complex for the reduced Kronecker coefficients indexed by three one--row shapes. The reduced coefficients are zero outside  the tetrahedra,
one on all the boundaries, and grow linearly as we move towards the center of one of the chambers. The interior of the cone is divided into three chambers.}
\label{fig:RKCchamber}
\end{figure}

 \begin{corollary} Fix $\lambda_2, \mu_2, $ and $\nu_2$. Let $a=\max(\lambda_2, \mu_2, \nu_2)\geq b\geq c=\min(\lambda_2, \mu_2, \nu_2)$ be a total ordering of $\lambda_2, \mu_2, \nu_2.$ 
 Set  $\ell =b+c-a.$  Then
   \begin{align}
   \bar g_{(\lambda_2),(\mu_2),(\nu_2)}&=    \bar g_{(a),(b),(c)}
   =
   \begin{cases}
\frac{\ell}{2} + \frac{ 3 + (-1)^{\ell}}{4} =  \left \lfloor \frac{\ell}{2} \right \rfloor +1 &\text{ if } \ell \ge  0\\
0 &\text{ if } \ell < 0.
\end{cases}
 \end{align} 
The chamber complex for this quasipolynomial is illustrated  in Figure~\ref{fig:chambers}. The walls are the hyperplanes $I) : \mu_2+\nu_2=\lambda_2$,
$II ) : \mu_2+\lambda_2=\nu_2$, $III ) : \lambda_2+\nu_2=\mu_2.$ The  reduced Kronecker coefficient  indexed by points on any of these walls always has value  equal to one.
 \end{corollary}
 
We have obtained the counting function for the number of integer points in the  one-dimensional polytope of Figure~\ref{fig:zeroexample}, which was  studied in 
Example~\ref{ZeroExample}.

\begin{remark}
Both Kirillov and   Klyachko have conjectured  that the reduced Kronecker coefficients satisfy the saturation hypothesis (See~\cite{Klyachko}, Conjecture 6.2.4, and  \cite{Kirillov:saturation}, Conjecture 2.33). (Note that Kirillov calls the reduced Kronecker coefficients the extended Littlewood--Richardson coefficients.) 
 We have a showed a stronger result. The quasipolynomial $G_{(\lambda_2,\mu_2,\nu_2)}(k)$ defined by  
$G_{(\lambda_2,\mu_2,\nu_2)}(k):=\bar g_{k\lambda_2,k\mu_2,k\nu_2}$
is an Ehrhart function as it counts integer points in polytopes.
\end{remark}




\subsection{The relative positions of the cones associated to the Kronecker, the reduced Kronecker and the Littlewood--Richardson coefficients}\label{RKC}


Murnaghan discovered an unexpected relationship between the Kronecker and the Littlewood--Richardson coefficients: In the particular case where Eq~\eqref{MurnaghanIn} is an equality, and the first parts of the partitions are ``big enough", the Kronecker coefficient $g_{\lambda, \mu, \nu}$ coincides with the Littlewood-Richardson
coefficient $c^{\bar \lambda}_{\bar \mu, \bar \nu}$.  Equivalently, Murnaghan's result can be expressed in terms of the reduced Kronecker coefficients,
\[
 \bar g_{\bar \lambda, \bar  \mu, \bar  \nu}= c_{\bar \lambda, \bar  \mu, \bar  \nu}
\]
For this reason the reduced Kronecker coefficients are sometimes called extended Littlewood--Richardson coefficients.

The Littlewood--Richardson coefficients are both particular instances of Kronecker and reduced Kronecker coefficients.  We call the cone defined by the non-zero atomic Kronecker coefficients the {\em atomic cone}; see Proposition \ref{ineqatomiccone22}.
We ask, what is the relation between the positions of these different families of coefficients (the Kronecker, the reduced Kronecker, and the Littlewood--Richardson coefficients) with respect to the atomic cone?  
 
The atomic Kronecker coefficients attain their minimum values (one) at its boundary:
$
\nu_2-\lambda_3-\lambda_4=0$ or $
\mu_2+\nu_2-\lambda_2-\lambda_3-2\lambda_4=0
$ (compare with inequality~\eqref{Bravyi}).  
When we dilate 
  the  three indexing partitions,  the values of the atomic Kronecker coefficients inside the cone are always strictly increasing. 
Moreover, on the face defined by 
$
\mu_2+\nu_2-\lambda_2-\lambda_3-2\lambda_4=0
$, the atomic Kronecker coefficients coincide with the Kronecker coefficient. 


 \begin{theorem}\label{LRfaceofKron}
 The Littlewood--Richardson cone coincides with  the intersection of the  face   of the Kronecker cone defined by the first Bravyi inequality:
 \[
  \lambda_2+\lambda_3+2\lambda_4 = \mu_2 + \nu_2  
 \]
 with the hyperplane $\lambda_4=0$.
 \end{theorem}

\begin{proof}
Let $c_{\alpha, \beta}^\gamma=\bar g_{\alpha, \beta}^\gamma=g_{\lambda,\mu,\nu}$ with $\alpha=\bar \mu$,  $\beta=\bar \nu$  and $\gamma=\bar \lambda$. 
 We want to
see where $(\lambda,\mu,\nu)$ sits in relation with the Kronecker cone.


Suppose that  $c_{\alpha, \beta}^\gamma>0$.
Since $\alpha$ and $\beta$ have just one part, 
Pieri's rule tells us that the length of $\gamma$ can be at most two, and hence 
$\lambda_4=0$, and $ \lambda_2+\lambda_3 = \mu_2 + \nu_2  $.
On the other hand, using the inequalities described in Corollary~\ref{atomicisKron} we easily see that all atomic Kronecker coefficients in
this wall are indeed Kronecker coefficients.

Let $N$ be any number such that the sequences $(N-\lambda_2,\lambda_2,\lambda_3),(N-\mu_2,\mu_2),(N-\nu_2,\nu_2)$ are partitions.
It remains to see whether  the Kronecker coefficient
$g_{(N-\lambda_2,\lambda_2,\lambda_3),(N-\mu_2,\mu_2),(N-\nu_2,\nu_2)}$ is reduced. 
For this, we will use the bound described in Theorem 1.5 of~\cite{BOR-JA}. It says that such a  Kronecker coefficient is stable as soon
as $N\ge\lambda_2+\mu_2+\nu_2+\left\lfloor \frac{\lambda_3}{2} \right\rfloor=2\lambda_2+\left\lfloor \frac{\lambda_3}{2} \right\rfloor$. But since $(N-\lambda_2,\lambda_2,\lambda_3)$
has to be a partition, the smallest possible value for $N$ will be that one that corresponds to partition $( \lambda_2,\lambda_2,\lambda_3)$. That is,  $2\lambda_2+\lambda_3 \ge 2\lambda_2+\left\lfloor \frac{\lambda_3}{2} \right\rfloor.$ 

   \end{proof}
 

Bravyi derived a a third inequality that the Kronecker coefficients must satisfy in order to be positive~\cite[p.6]{Kirillov:saturation, Bravyi}),  namely 
 \begin{equation}\label{3rdBravyi} |\mu_2-\nu_2|\leq \min(\lambda_1-\lambda_3,  \lambda_2-\lambda_4).\qquad\text{{\em (Third Bravyi Inequality})} \end{equation}
 
Define the  {\em Bravyi cone} to be the polyhedral  cone obtained by  taking the intersection of the atomic cone with the region defined by the third Bravyi inequality.
All nonzero Kronecker coefficients lie within the Bravyi cone. We can ask where the Littlewood--Richardson (LR) coefficients lie inside the cone.
More precisely, 
we are asking for those nonzero Littlewood--Richardson coefficients (inside the Kronecker cone) coming from the  identities:
\[
 g_{\lambda,\mu,\nu} = \bar g_{\bar \lambda,\bar \mu,\bar \nu} =  c_{ \bar \mu,\bar  \nu}^{\bar  \lambda}.
\] 
 Now the Littlewood-Richardson coefficient $c^{(\lambda_2, \lambda_3)}_{(\mu_2), (\nu_2)}$ is nonzero iff the skew-shapes $(\lambda_2, \lambda_3)/(\mu_2)$ and $(\lambda_2, \lambda_3)/ (\nu_2)$ are horizontal strips, or equivalently iff 
 $$ \lambda_2\geq \mu_2\geq \lambda_3 \text{ and } 
  \lambda_2\geq \nu_2\geq \lambda_3,$$ 
 which in turn is equivalent to saying $\mu_2$ and $ \nu_2$ lie in the interval $[\lambda_3, \lambda_2].$ Hence, when the LR coefficient is nonzero,   we must have (since $\lambda_1\geq \lambda_2\geq \lambda_3\geq \lambda_4$),
 $$|\mu_2-\nu_2|\leq \lambda_2-\lambda_3\leq \min(\lambda_1-\lambda_3,  \lambda_2-\lambda_4),$$ 
 and this is the third Bravyi inequality.

\begin{remark}
Note that we have the following implications.  If a Kronecker coefficient is atomic then it is  reduced, the reason being that   the value of $\tilde g$ does not depend on the first part of $\lambda$.
On the other hand, if a Kronecker coefficient satisfies the equality of Murnaghan's condition then it is reduced.
This is Theorem~\ref{LRfaceofKron}.
\end{remark}

Finally, consider next what happens at the intersection of  the hyperplanes defined by the various Bravyi inequalities.
\begin{align*}
\begin{cases}
&\lambda_2+\lambda_3+2\lambda_4 = \mu_2+\nu_2,  \qquad \text{(first Bravyi)}\\
&\lambda_3+\lambda_4 = \nu_2, \qquad \qquad \qquad \,\,\, \,\, \text{(second Bravyi)}\\
&\mu_2 - \nu_2 = \lambda_2 - \lambda_4, \qquad \qquad  \,\,\,\,  \text{(third Bravyi)}\\
\end{cases}
\end{align*}
The first and third Bravyi inequalities give  $\mu_2 = \lambda_2+\lambda_4$ while 
the second and third give  $\lambda_2+\lambda_3=\mu_2$.
Therefore, they intersect at the plane with parametric equations
$
\lambda_2=t, \lambda_3=s, \lambda_4=s, \mu_2=s+t, \nu_2=2s
$

A triple in this family of partitions satisfies Murnaghan's condition iff $s=0$. That is, we are 
looking at the coefficients $g_{(*,t),(*,t),(*)}=1$. This corresponds to tensoring with the trivial representation.
 
 \begin{proposition}  Let $\lambda=(u, t,s,s), \mu=(u+s, t+s), \nu=(u+t, 2s)$ where $u\geq t\geq s.$   Then for all $k\ge 1,$ 
 $$g_{k\mu,k\nu,k\lambda}=1=\tilde g_{k\mu,k\nu,k\lambda}.$$ 
 That is, the \textit{dilated} Kronecker coefficient 
 equals 1, as does the atomic dilated Kronecker coefficient.
 
 
 \end{proposition}
 
 
 
  \begin{proof}  Note that we are computing the  Kronecker coefficient for the \textit{dilated} triple $(k\lambda,k\mu, k\nu).$  We will use the 7-term polynomial in Theorem~\ref{7termF22Alternant} to determine which  coefficients of $F_{2,2}$ contribute to $g_{k\mu,k\nu,k\lambda}.$
   We check that the inequalities of Corollary~\ref{atomicisKron} are satisfied:  
    $k(\lambda_3+\lambda_4)=2ks=k\nu_2<k(\lambda_2+\lambda_4)+1=k(t+s)+1\text{ and}$
  
  $k(\lambda_3+\lambda_4)+k(\lambda_2+\lambda_4)=k(3s+t) \leq 
 k(\mu_2+\nu_2)=k(3s+t)\le 
 k(\lambda_3+\lambda_4)+k\min(\lambda_2+\lambda_3, \lambda_1+\lambda_4)=k(3s+t).$
 


 
 Hence the Kronecker and atomic coefficients coincide, and they both equal 
 the coefficient of 
 $x^{k(t+s)}y^{2ks} $ in 
 $x^{k(t+s)} y^{2ks} \bar{F}_{2,2},$ which is  just $p_S(0,0)=1,$  as claimed.

  \end{proof}
  
 A triple of partitions $(\lambda, \mu, \nu)$ of the same weight is stable if $g_{(k\lambda, k\mu, k\nu)}$ equals 1 for all $k.$  We have shown that all triples of partitions lying at the intersection of the three walls (i.e. facets) defined by the inequalities of  Bravyi  are stable. 
Stable triples are relevant because the sequences $g_{\alpha+n\lambda, \beta+n\mu, \gamma+\nu}$ stabilize for $n$ sufficiently large. Stembridge asks for the location of all stable triples~\cite{Stembridge}. It remains to decide whether there are more of them.
  \begin{corollary}
  All triples of partitions  at the intersection of the three facets of the Bravyi cone 
 
   are stable. That is, if we fix $u\geq t\geq s$, the sequence of Kronecker coefficients
 \[
 g(\lambda+k(u,t,s,s), \mu+k(u+s,t+s), \nu+k(u+t,2s))
 \]
is eventually constant.
  \end{corollary}


\section{The vector partition function $F_{n,m}$}
\label{sec-Fnm}
\label{nmKronecker}
Having completed our analysis of the $n=m=2$ case, we examine the extent to which these results generalize. First we show that we can make a variable substitution to convert
$\frac{{a_{\delta_n}[X]}{a_{\delta_m}[Y]}}{a_{\delta_{nm}}[XY]}$ into a vector partition function. This is the result of Theorem~\ref{thm:MAIN2}.

Let us go back to  beginning, that is, back to Eq.\eqref{Cauchy}. 
Let $S(p)$ denote the smallest monomial in $p$ with respect to the lexicographic order. By truncating the alternants in Eq.\eqref{Cauchy}, we obtain a new family of coefficients $\tilde g_{\mu,\nu,\lambda} $ closely related to the Kronecker coefficients:
\begin{align}\label{tildeKronecker}
\frac{{a_{\delta_n}[X]}{a_{\delta_m}[Y]}}{a_{\delta_{nm}}[XY]} S( a_{\lambda+\delta_{nm}}[XY])=  \sum_{\mu, \nu} \tilde g_{\mu,\nu,\lambda} \,  S(a_{\mu+\delta_n}[X] ) S( a_{\nu+\delta_m}[Y]) +\text{ lex. g. t.}.
\end{align}
We proceed by first expanding all the Vandermonde determinants involved as a product of linear binomial factors. We want to factor the binomial terms so that we obtain a product of  terms of the form $1$ minus a Laurent monomial. We  can do this in such a way that the resulting Laurent series converges in a nonzero domain if we follow the lexicographic ordering, and always factor the smallest monomial in each binomial.

We consider the special alphabets $X=\{1,x_1,x_2,\ldots ,x_{n-1}\},\quad$ $Y=\{1,y_1,y_2,\ldots, y_{m-1}\}, n,m\geq 2.$ Then $XY=\{1, x_i, y_j, x_iy_j:
1\leq i\leq n-1, 1\leq j\leq m-1\}.$
The set $XY$ is ordered as follows:
\begin{equation}\label{ordering}
1>x_i>x_{i+1}, y_j>y_{j+1}, x_i>y_j, x_i y_j>x_ky_\ell \text{ if }i< k,\text{ or } i=k \text{ and } j<l.
\end{equation}
The following claim is clear.
\begin{lemma}The smallest term  in $S( a_{\lambda+\delta_{nm}}[XY])$, with respect to the lexicographic ordering, is the product of the monomials in the main diagonal of the  matrix of the alternant $a_{\lambda+\delta_{nm}}[XY].$
\end{lemma}

 Similarly, to compute the smallest term, with respect to the lex ordering, in each of the two remaining alternants $S( a_{\mu+\delta_{n}}[X])$ and $S( a_{\nu+\delta_{m}}[Y])$, we take the product of the monomials in the main diagonal of the corresponding  matrices.
We  obtain a Laurent series
\begin{align*}\label{Laurentseries}
 \sum_{\mu, \nu} g_{\mu,\nu,\lambda} \,  {\bf x}^{\,\,\,l_1(\mu,\nu,\lambda)} {\bf y}^{\,\,\,l_2(\mu,\nu,\lambda)} 
\end{align*}
where $l_1(\mu,\nu,\lambda)$ and $l_2(\mu,\nu,\lambda)$ are linear combinations of the parts of $\mu, \nu$ and $\lambda$. It is a product of binomial terms of the form 1 minus a Laurent monomial. Finally, we perform a change of basis which we describe in detail below, to ensure that we get a convergent Taylor series expansion.

For example, for $n=m=3$, the substitution is
$x_{1} = s_{1} t_{1}, $ 
$ x_{2} = s_{1} s_{2} t_{1}^{2}, $
$ y_{1} = s_{0} s_{1} s_{2} t_{1}^{2}, $ and 
$ y_{2} = s_{0} s_{1} s_{2} t_{1}^{3}$
More precisely, in order to guarantee convergence of our series, we assume in Eqn.~\eqref{ordering} that 
$1>|x_1|>|x_2|>..>|x_{n-1}|>|y_1|>|y_2|>...>|y_{m-1}|
 >|x_1y_1|>|x_1y_2|>...>|x_1y_{m-1}|\\
\ldots 
  >|x_{n-1}y_1|>|x_{n-1}y_2|>...>|x_{n-1}y_{m-1}|
$.

We define the rational function $G_{n,m}$  by
$$G_{n,m}=\frac{{a_{\delta_n}[X]}{a_{\delta_m}[Y]}}{a_{\delta_{nm}}[XY]}.$$
Observe that we have the Vandermonde expansion
$$a_{\delta_n}[X]=\prod_{i=1}^{n-1}(1-x_i)\prod_{1\le i <j\le n-1} (x_i-x_j),$$
and similarly for the second alternant $a_{\delta_m}[Y].$
For the alternant in the denominator we have 
\[a_{\delta_{nm}}[XY
=a_{\delta_n}[X]\cdot a_{\delta_m}[Y]\cdot A \cdot B\cdot C\cdot D\cdot E\cdot F,\]
where 
$$A=\prod_{j=1}^{m-1}\prod_{i=1}^{n-1}(x_i-y_j),  \,\,
B=\prod_{i=1}^{n-1}\prod_{j=1}^{m-1}(1-x_iy_j)$$
and
$C= \prod_{i=1}^{n-1}\prod_{j=1}^{m-1}(x_i-x_i y_j)\cdot 
\prod_{j=1}^{m-1}\prod_{i=1}^{n-1}(y_j-x_i y_j)$
\[=\left(\prod_{j=1}^{m-1}(1-y_j)\right)^{n-1}(\prod_{i=1}^{n-1} x_i^{m-1}) 
\cdot \left(\prod_{i=1}^{n-1}(1-x_i )\right)^{m-1} (\prod_{j=1}^{m-1}y_j^{n-1}),
\]
\[D=\prod_{\stackrel {k=1}{k\neq i}}^{n-1}\prod_{i=1}^{n-1}\prod_{j=1}^{m-1} (x_k-x_iy_j)
\cdot \prod_{\stackrel {k=1}{k\neq j}}^{m-1}\prod_{j=1}^{m-1}\prod_{i=1}^{n-1} (y_k-x_iy_j),\]
\[E=\prod_{j\neq \ell=1}^{m-1}\prod_{1\le i<k\le n-1} (x_i y_j -x_k y_\ell),\]
\begin{multline*}
F= \prod_{j=1}^{m-1}\prod_{1\le i<k\le n-1} (x_i y_j -x_k y_j)
\cdot \prod_{i=1}^{n-1}\prod_{1\le j<\ell\le m-1} (x_i y_j -x_i y_\ell)\\
=\prod_{j=1}^{m-1}y_j^{n-1\choose 2}\prod_{1\le i<k\le n-1} (x_i  -x_k )^{m-1}
\cdot \prod_{i=1}^{n-1}x_i^{m-1\choose 2}\prod_{1\le j<\ell\le m-1} ( y_j -y_\ell)^{n-1
.\end{multline*}

It follows that the quotient of alternants $G_{n,m}$ simplifies to $\dfrac{1}{ABCDEF}.$

\smallskip

Note that each factor in $A, C, D, E, F$ can be rewritten in the form $(1-M)$ where $M$ is a \textit{Laurent} monomial in the $x_i$ and the $y_j.$ The factors of $B$ are already in this form.
For instance, in $E$ we can rewrite each factor as 
$$x_iy_j-x_ky_\ell=x_iy_j(1-x_ky_\ell x_i^{-1} y_j^{-1}).$$

Thus, the following definition for $\bar{F}_{n,m}(X,Y)$  makes sense.
\begin{definition}[$\bar{F}_{n,m}(X,Y)$ ]\label{defineFnm}

There are positive integers $a_i, b_j$ such that in 
the product 
 \[G_{n,m}  \prod_{i=1}^{n-1}x_i^{a_i} \prod_{i=j}^{m-1}y_j^{b_j}, \]
all  factors are of the form $(1-M)^{-1}$ where $M$ is a Laurent monomial in the $x_i$ and the $y_j.$ We define $\bar{F}_{n,m}(X,Y)$ to be this product, i.e. we have 
\begin{equation}\label{Laurentseries}
G_{n,m}\prod_{i=1}^{n-1}x_i^{a_i} \prod_{i=j}^{m-1}y_j^{b_j}
=\bar{F}_{n,m}.
\end{equation}
\end{definition}

 We now show that there is a different set of $(n+m-2)$ variables $s_i, i=0, \ldots n-1, t_j, j=1,\ldots, m-2,$ such that by effecting a judicious (and non-obvious)  change of variables, $\bar{F}_{n,m}(X,Y)$ becomes a product of factors of the form $(1-M)^{-1}$ where each Laurent monomial $M$ in $X,Y$ is a  monomial with nonnegative exponents in the new variables $S,T.$  In other words, $F_{n,m}(S,T)$ is a vector partition function in the new variables.

 
We claim that the quotients of consecutive terms in the 
sequence \eqref{ordering} become monomials (and not Laurent monomials), after setting, for each  $1 \le i \le n-1$ and $1 \le j \le m-1$,
 \begin{align} \label{changeofvars}
 \begin{cases}
&x_{i}=s_1 s_2.. s_{i} (t_1 t_2 ... t_{m-2})^{i}\\
&y_{j}= (s_0 s_1 ... s_{n-1})  (t_1 t_2 ... t_{m-2})^{n-1} t_1 t_2...t_{j-1} \\
\end{cases}.
         \end{align}
 Note when $n=m=2$, there are no  $t_i$ variables, and we recover the substitution for $F_{2,2}$ in Section~\ref{F22}.
 We have 
\begin{multline}\label{basisvectorsFnm}
\frac{x_{i+1}}{ x_i }= s_{i+1} t_1 t_2 ... t_{m-2}, \quad
 \frac{y_{j+1}}{ y_j} = t_j, \ 1\le j  \le m-2,\\  
 \frac{y_1}{x_{n-1}} = s_0,  \quad \frac{x_1 y_1  }{ y_{m-1}} = s_1,\ 
  \frac{x_i y_1  }{  x_{i-1}y_{m-1}} = s_i ,\   2\le i\le n-1.
 \end{multline}
This establishes our claim.  Hence we have proved: 
\begin{theorem} 
\label{changeofvariables}\label{thm:AlwaysVecPart}\label{thm:MAIN2}
Let $F_{n,m}(S,T)$ be the series obtained after performing the previous substitutions in the series~\eqref{Laurentseries}; its domain of convergence is $\{|s_i|<1, |t_j|<1: 0\le i\le n-1, 1\le j\le m-2\}.$  Then $F_{n,m}$ is a vector partition function.
\end{theorem}
From the preceding discussion we can also conclude:
\begin{corollary}\label{matrixFnm}  Let $A_{n,m}$ be the matrix associated to the vector partition function $F_{n,m}$, as in Section~\ref{VectorPartition}.   Then 
\begin{enumerate}
\item the largest entry is $2n-1;$
\item the number of columns  is $\binom{nm}{2}-\binom{n}{2}-\binom{m}{2}$;
\item the number of rows is $m+n-2;$
\item all the basis vectors appear in the columns of $A_{n,m};$
\item the rank of the matrix is $m+n-2.$
\end{enumerate}
\end{corollary}

\begin{proof} 
The largest entry is obtained by examining the largest possible exponent  of the variables $s_i$ or $t_j$ in the monomials $M$ occurring in the factors $(1-M)$ of $F_{n,m}.$ We have, from the product $B$ of the preceding proof, for $ i\le n-2, j\le m-1,$ the monomial 
\[ x_i y_j=(s_1\ldots s_i)(t_1t_2\ldots t_{m-2})^i\cdot (s_0s_1\ldots s_{n-1})
(t_1t_2\ldots t_{m-2})^{n-1}(t_1t_2\ldots t_{j-1})\]
and clearly the largest exponent here occurs for each of $t_1,\ldots , t_{j-1},$ and it equals $i+(n-1)+1=n+i\le 2n-1.$  The maximum exponent $2n-1$ occurs in the monomial $x_{n-1} y_j.$
Examining the products other than $B,$ we see that all other monomials involve dividing by $x_i$ or $y_j$ or both, so it is clear that they cannot yield a larger exponent.

The number of columns in the matrix $A_{n,m}$ equals the number  of linear factors in $a_{\delta_{nm}}[XY]$ minus the number of linear factors in $a_{\delta_n}[X]$ minus the number of linear 
factors in $a_{\delta_m}[Y];$ since these are all Vandermonde determinants, the second result follows.  For the third  result, observe that the number of rows is simply the number of variables in the set $\{s_i, 0\le i\le n-1, t_j, 1\le j\le m-2\}.$ 

For the last two statements, observe that Eqn.\eqref{basisvectorsFnm} in the preceding proof establishes that all the basis vectors appear as columns of the matrix  $A_{n,m}$, since all the variables $s_i$ and $t_j$ occur as quotients when converting the factors of the Vandermonde in the products $A$-$F$ into the form $(1-M)$ in Eq.~\eqref{basisvectorsFnm}. Hence the rank is the number of rows of the matrix.  \end{proof}

\subsection{The degree of the Kronecker piecewise quasipolynomial}
\label{sec:DegreeandPeriod}

We can now immediately obtain information about the 
degree  of the Kronecker quasipolynomial $\kappa_{n,m,nm}$. Let $X$ be an alphabet of size $n$, and $Y$ an alphabet of size $m$, and let 
\begin{align*}
d&=\frac{n^2m^2}{2}-\frac{n^2}{2}-\frac{m^2}{2}-\frac{nm}{2}-\frac{n}{2}-\frac{m}{2}+2
\end{align*}

\begin{theorem}\label{thm:DegreeandPeriod}
The Kronecker function $\kappa_{n,m,nm}$ is always described by a piecewise quasipolynomial of degree $\le d$. 
\end{theorem}
\begin{proof} The degree of the Kronecker quasipolynomial $\kappa_{n,m,nm}$ is bounded by 
the dimension of the null space of $A_{n,m}.$ 
It is thus equal to the number of columns minus the rank.  By Corollary~\ref{matrixFnm}, this is just $d.$ 

Since Kronecker coefficients are linear combinations of different shifts of this vector partition function, these bounds apply in general.
\end{proof}

The degree of  $\kappa_{n,m,nm}$ has  been obtained by  Baldoni, Vergne, and Walter~\cite{BaldoniVergneWalter, VergneWalter} using the language of moment maps.

In addition to being completely elementary, another advantage of our approach is that the dimension of the polyhedral cones
 involved in the calculation are the minimal possible ones, as they
coincide with the degree of the quasipolynomial.

Still, the dimensions of the polyhedral cones involved grow fast.  For instance, from Theorem~\ref{thm:DegreeandPeriod}, for $n=2, m=3$ we have dimension 8, and for $n=m=3$  the dimension is 26.  In the next examples, we will briefly  explore $F_{2,3},$ and $F_{3,3}.$ In particular, we will look at the faces of the polyhedral cones of the atomic Kronecker coefficients.

\subsection{The vector partition function $F_{2,3}$}
\label{sec-F23}

The domain of convergence of the vector partition function $F_{2,3}$ is $|x_1y_2|<|x_1y_1|<|y_2|<|y_1|<|x_1|<1.$  $F_{2,3}$ counts  nonnegative integer solutions to $A_{2,3}{\bf x}={\bf n}$, with $A_{2,3}$ equal to 
\[
\left(\begin{array}{rrrrrrrrrrrr}
1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1  \\
0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 2 & 2  \\
0 & 0 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 3 
\end{array}\right)
\]

We need the following computations for partitions $\lambda,\mu, \nu$ of lengths bounded by 6,2 and 3 respectively.
\begin{eqnarray}&S(a_{\lambda+\delta_6}[XY])&=
(x_1y_2)^{\lambda_6}(x_1y_1)^{\lambda_5+1} y_2^{\lambda_4+2} y_1^{\lambda_3+3}x_1^{\lambda_2+4} \cdot 1^{\lambda_1+5}\\
&&=\mathrm{(x_1^5y_1^4y_2^2)}\quad  x_1^{\lambda_6+\lambda_5+\lambda_2} y_1^{\lambda_3+\lambda_5} y_2^{\lambda_4+\lambda_6},\\
&S(a_{\nu+\delta_3}[Y])&=y_2^{\nu_3} y_1^{\nu_2}\mathrm{y_1},\\
&S(a_{\mu+\delta_2}[X])&= x_1^{\mu_2}.
\end{eqnarray}

As in Section~\ref{sec-F22} (see the definition following Eqn.~\eqref{atomic1stdef}),
we denote by $\bar{F}_{2,3}(x,y)$ the series obtained from $F_{2,3}(x,y)$ by factoring out monomials from each binomial factor so that every factor is of the form $(1-M).$ One then checks that the relation $F_{2,3}=\frac{ \bar{F}_{2,3}}{\mathrm{x_1^{5} y_1^3y_2^2}}$ holds.
Hence Eqn.~\eqref{main} becomes
\[ \frac{ \bar{F}_{2,3}}{(x_1^5y_1^3y_2^2)}\ 
a_{\lambda+\delta_6}[XY]=\sum_{\mu,\nu} g_{\mu,\nu,\lambda} \mathrm{ y_1} x_1^{\mu_2}y_2^{\nu_3} y_1^{\nu_2} +\text{lex. gr.}\]
or
\begin{equation}\label{F23main}  \bar{F}_{2,3}\ 
a_{\lambda+\delta_6}[XY]=\mathrm{x_1^{5} y_1^4y_2^2}\sum_{\mu,\nu} g_{\mu,\nu,\lambda}  x_1^{\mu_2}y_2^{\nu_3} y_1^{\nu_2} +\text{lex. gr.}\end{equation}

Truncate the alternant $a_{\lambda+\delta_6}$ to the lexicographically least monomial, using the expression above.  The defining equation for the atomic coefficients, Eqn.~\eqref{tildeKronecker}, now gives 
\[  \bar{F}_{2,3}\ x_1^{\lambda_6+\lambda_5+\lambda_2} y_1^{\lambda_3+\lambda_5} y_2^{\lambda_4+\lambda_6}=\sum_{\mu,\nu} \tilde g_{\mu,\nu,\lambda}  x_1^{\mu_2}y_2^{\nu_3} y_1^{\nu_2} ,\]
where $\tilde g$ is the atomic Kronecker coefficient as before.  Note that the initial affine combinations of the parts of $\lambda,\mu,\nu$ fortuitously reduce to \textit{linear} combinations.


To make $\bar{F}_{2,3}$ into the vector partition function whose matrix $A_{2,3}$ is given above, we need the substitution of 
Theorem~\ref{thm:AlwaysVecPart}, namely $x_1=s_1t_1, y_1=s_0s_1t_1, y_2=s_0s_1t_1^2.$   This gives 
\[F_{2,3}(s_0,s_1,t_1)
=\sum_{\mu, \nu} \tilde g_{\mu,\nu,\lambda}  M(\mu,\nu,\lambda)\]
where the monomial $M(\mu,\nu,\lambda)$  is given by 
\[t_1^{\mu_2+\nu_2+2\nu_3-(3\lambda_6+\lambda_5+2\lambda_4+2\lambda_3+\lambda_2)}
s_1^{\mu_2+\nu_2+\nu_3-(2\lambda_6+\lambda_5+\lambda_4+2\lambda_3 +\lambda_2 )} 
s_0^{\nu_2+\nu_3-(\lambda_6+\lambda_5+\lambda_4+\lambda_3)}.\]
We conclude that the atomic coefficient is nonzero if and only if the exponents appearing in $M(\mu,\nu,\lambda)$ are all nonnegative.

The dimension of the solution space is rather large. The polytopes involved have dimension $8,$ making them very hard to visualize.
However, some interesting phenomena can be observed by  looking at the restriction of this system of equations to the positive orthant.
 Recall that we are looking for nonnegative solutions to $A_{2,3}{\bf x}={\bf n}$. Let ${\bf n}=(n_1, n_2, n_3)$.

If $n_3=0$ , since we are only considering nonnegative linear combinations of the columns of the matrix, none of the columns other than the first two can appear. We obtain the restricted matrix $A_3=\left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right)$, and $p_{A_3}(n_1,n_2)=1$ is a constant polynomial.  Here we use the notation of Theorem~\ref{Blakley} for the quasipolynomial $p_A({\mathbf{b}}) $ associated to the polytope defined by the  solution space of the matrix equation $A{\bf x}={{\mathbf{b}}}.$ 

On the other hand, if $n_2=0$, we can discard any column where the second entry is not zero. In this case the restricted matrix 
is $A_2=\left(\begin{array}{rrr} 1 & 0 &1 \\ 0 & 1 &1 \end{array}\right)$, and $p_{A_2}(n_1,n_3)$ is a linear piecewise polynomial: We need to solve  the system of inequalities $x_3 \le n_1, x_3 \le n_3$.  Hence $p_{A_2}(n_1,n_3)=1+\min(n_1, n_3).$

Finally, if we set $n_1=0$, the restricted matrix is then  $A_1=\left(\begin{array}{rrrrr}1 & 0 & 1 &1 &1 \\ 0& 1 & 1 & 1 & 2 \end{array}\right)$, and $p_{A_1}(n_2,n_3)$ is a piecewise cubic  quasipolynomial.

Note that  the atomic Kronecker coefficients are identically one {\em only} on the facet defined by  $n_3=0 $. Contrast this with the situation for $F_{2,2}$, where the coefficients are identically one on {\em both} facets: from Figure~\ref{fig:chambers}, we see that 
  $p_{A_{2,2}}(n,m)=1$ if $n=0$ or $m=0.$ 




\subsection{The vector partition function $F_{3,3}$}
\label{sec-F33}
The domain of convergence of the vector partition function $F_{3,3}$ is $|x_2y_2|<|x_2y_1|<|x_1y_2|<|x_1y_1|<|y_2|<|y_1|<|x_2|<|x_1|<1.$  $F_{3,3}$ counts  nonnegative integer solutions to $A_{3,3}{\bf x}={\bf n}$, with
{\tiny
\[
\mkern-10mu 
A_{3,3}=
\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 \\
0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 \\
1 & 1 & 0 & 1 & 1 & 2 & 0 & 1 & 1 & 2 & 1 & 2 & 2 & 3 & 0 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 3 & 3 & 3 & 4 & 3 & 4 & 4 & 5
\end{array}\right).
\]}


The dimension of the solution space is now much larger than the previous case of $F_{2,3}.$ The polytopes involved have dimension $26,$ again rendering visualisation impossible. 
However, a similar phenomenon can be observed by  looking at the restriction of this system of equations to the positive orthant. 


Let
${\bf n}=(n_1,n_2,n_3,n_4)$. Set $n_1=n_2=n_3=0$. That is, we are counting solutions of the system on those integer points belonging to the $n_4$ axis. Since we are only considering
 nonnegative linear combinations of the columns (vector partitions), all variables corresponding to
 a column where one of the first three rows has a nonzero entry should be equal to zero.  We end up counting partitions of $n_4$ with two copies of $1$, Example~\ref{SecondExample}. 
 
 Similar arguments allows us to deduce that:
 
 \begin{enumerate}
 \item 
On the axes $n_1, n_2$ and $n_3$ we are counting partitions with just one part equal to $1$. Therefore, $p_A=1$ a constant 
polynomial.

On the other hand, on the $n_4$ axis,  we are counting partitions with just two different copies of $1$ . Therefore, as we just showed, we obtain the
quadratic polynomial of Example~\ref{SecondExample}. 
\item 
On the $2$-faces,   generated by either axes $n_1$ and $n_2$, or  by  $n_1$ and $n_3$,  or  by $n_2$ and $n_3$, the restricted matrix is $\left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right)$. Therefore, $p_A=1$ is  a constant  polynomial.

On the other hand, on the $2$-face generated by axes $n_1$ and $n_4$, the restricted matrix is $\left(\begin{array}{rrrr} 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \end{array}\right)$. 
Again, we get a piecewise polynomial, this time of degree $2.$ 

Finally, on the $2$-faces generated by $n_2$ and $n_4$, and by  $n_3$ and $n_4$ the restricted matrix is 
$\left(\begin{array}{rrrrrrrrrr} 
 0 & 0 & 1 & 1 & 1 & 1 \\ 
 1 & 1 & 0 &1& 1 & 2 \end{array}\right)$. 
This time we obtain a piecewise quasipolynomial of degree 4.

\item  In any subspace of  $n_4=0 $, there is a unique solution: $p_A=1$ is  a constant  polynomial.


\item If  $n_3=0$  (which coincides with the case $n_2=0$), the resulting quasipolynomial has degree is 7.  The restricted matrix is 
\[
\left(\begin{array}{rrrrrrrrrr} 
 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 
 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1\\
 1 & 1 & 0 & 1 & 1 & 2 & 0 & 1 & 1 & 2 \end{array}\right).
 \]

\item If  $n_1=0$, the resulting quasipolynomial has degree 11. The restricted matrix is
\[
\left(\begin{array}{rrrrrrrrrrrrrr} 
0 & 0& 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1&1&1 \\ 
0 & 0& 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1&1&1 \\
1 & 1& 0 & 1 & 1 & 2 & 0 & 1 & 1 & 2 & 1 & 2&2&3 \end{array}\right).
\]

\end{enumerate}


This situation is reminiscent of the results obtained for $F_{2,3}$.  There is only  one {\em facet} where the atomic Kronecker coefficients are identically one, namely   the facet defined by setting the last coordinate $n_4$ to be equal to zero. 

It only remains to figure out the equations of the facets in terms of the parts of  the three indexing partitions. 

As in the preceding section, 
we need the following computations for partitions $\lambda,\mu, \nu$ of lengths bounded by 9,3 and 3 respectively.
{\small
\begin{eqnarray*}&S(a_{\lambda+\delta_9}[XY])&=
(x_2y_2)^{\lambda_9}(x_2y_1)^{\lambda_8+1} (x_1y_2)^{\lambda_7+2} (x_1y_1)^{\lambda_6+3}  y_2^{\lambda_5+4} y_1^{\lambda_4+5} x_2^{\lambda_3+6} x_1^{\lambda_2+7} \\
&&=\mathrm{(x_1^{12}x_2^{7}y_1^9y_2^6)}\quad  x_2^{\lambda_3+\lambda_8+\lambda_9}
x_1^{\lambda_2+\lambda_6+\lambda_7}
 y_2^{\lambda_5+\lambda_7+\lambda_9} y_1^{\lambda_4+\lambda_6+\lambda_8},\\
&S(a_{\nu+\delta_3}[Y])&=y_2^{\nu_3} y_1^{\mu_2}\quad \mathrm{y_1},\\
&S(a_{\mu+\delta_3}[X])&= x_2^{\mu_3}x_1^{\mu_2}\quad \mathrm{x_1}.
\end{eqnarray*}
}


We denote by $\bar{F}_{3,3}(x,y)$ the series obtained from $F_{3,3}(x,y)$ by factoring out monomials from each binomial factor so that every factor is of the form $(1-M).$ One then checks that the relation $F_{3,3}=\frac{ \bar{F}_{3,3}}{\mathrm{x_1^{11} x_2^7y_1^8y_2^6}}$ holds. Compare with  (Eqn.~\eqref{atomic1stdef}.)
Hence Eqn.~\eqref{main} becomes
\[ \frac{ \bar{F}_{3,3}}{x_1^{11} x_2^7y_1^8y_2^6}\ 
a_{\lambda+\delta_9}[XY]=\sum_{\mu,\nu} g_{\mu,\nu,\lambda} \mathrm{x_1y_1} x_2^{\mu_3}x_1^{\mu_2}y_2^{\nu_3} y_1^{\nu_2} +\text{lex. gr.}\]
Then,
\begin{equation}\label{F33main}  \bar{F}_{3,3}\ 
a_{\lambda+\delta_9}[XY]=\mathrm{x_1^{12} x_2^7y_1^9y_2^6}\sum_{\mu,\nu} g_{\mu,\nu,\lambda}  x_2^{\mu_3}x_1^{\mu_2}y_2^{\nu_3} y_1^{\nu_2} +\text{lex. gr.} \end{equation}

Truncating the alternant on the left using the preceding computation, Eqn.~\eqref{tildeKronecker}   now gives the series for the atomic Kronecker coefficients (by definition).  Note (again) that the initial affine combinations of the parts of $\lambda,\mu,\nu$ fortuitously reduce to linear combinations.
\[\bar{F}_{3,3}\ x_2^{\lambda_3+\lambda_8+\lambda_9}
x_1^{\lambda_2+\lambda_6+\lambda_7}
 y_2^{\lambda_5+\lambda_7+\lambda_9} y_1^{\lambda_4+\lambda_6+\lambda_8}
=\sum_{\mu,\nu} \tilde g_{\mu,\nu,\lambda}  x_2^{\mu_3}x_1^{\mu_2}y_2^{\nu_3} y_1^{\nu_2}.\]


To make $\bar{F}_{3,3}$ into the vector partition function $F_{3,3}$ whose matrix $A_{3,3}$ is given above, we need the substitution of 
Theorem~\ref{thm:AlwaysVecPart}, namely $x_1=s_1t_1, x_2=s_1s_2t_1^2, y_1=s_0 s_1 s_2 t_1^2, y_2=s_0s_1s_2t_1^3.$

This makes 
\[x_2^{\mu_3}x_1^{\mu_2}y_2^{\nu_3} y_1^{\nu_2}
=s_0^{\nu_2+\nu_3}s_1^{\mu_2+\mu_3+\nu_2+\nu_3} s_2^{\mu_3+\nu_2+\nu_3} t_1^{\mu_2+2\mu_3+2\nu_2+3\nu_3}\]
and
\[ x_2^{\lambda_3+\lambda_8+\lambda_9}
x_1^{\lambda_2+\lambda_6+\lambda_7}
 y_2^{\lambda_5+\lambda_7+\lambda_9} y_1^{\lambda_4+\lambda_6+\lambda_8}\\
 =s_0^a s_1^bs_2^c t_1^d,
 \]
 where \begin{align*} 
& a=\lambda_{\geq 4},\ b=\lambda_{\geq 2}+\lambda_{\geq 7},\ c=\lambda_{\geq 3}+\lambda_{\geq 8},\\
&d=\lambda_2+2\lambda_3+2\lambda_4+3\lambda_5
+3\lambda_6+4\lambda_7+4\lambda_8+5\lambda_9;
\end{align*}
here we have written $\lambda_{\geq i} $ to mean the sum of all parts $\lambda_i+\lambda_{i+1}+\ldots$

\begin{proposition}\label{ineqatomiccone}
We conclude that the atomic coefficient $\tilde g_{\mu, \nu, \lambda} $ is nonzero if and only if 
all of the following hold:
\begin{align*}
\begin{cases}
&\nu_2+\nu_3-a\geq 0,\\
&\mu_2+\mu_3+\nu_2+\nu_3-b\geq 0,\\
&\mu_3+\nu_2+\nu_3-c\geq 0,\\
&\mu_2+2\mu_3+2\nu_2+3\nu_3-d\geq 0.\\
\end{cases}
\end{align*}
\end{proposition}
 





\section{Computing Dilated Kronecker coefficients} 
\label{Dilation}

\subsection{Dilated Kronecker Coefficients}
 Fix $\mu,\nu$ and $\lambda$. The family of Kronecker coefficients given by the image of the function
 $k\mapsto g_{k\mu,k\nu,k\lambda}$, for $k=1, 2, \dots$ is a set of  \emph{dilated Kronecker coefficients},  has been the center of a lot of attention.  When the lengths of the partitions $\mu,\nu,\lambda$ are bounded by 2,2, and 4, we can compute them using Theorem~\ref{thm:MAIN1},  and we can also write them as subseries of $F_{n,m}$ in a way that directly connects to Ehrhart functions. 

\subsection{Examples using Theorem~\ref{thm:MAIN1}}
 Theorem~\ref{thm:MAIN1} gives  exact formulas for $g_{k\mu,k\nu,k\lambda}$ when $\ell(\mu), \ell(\nu) \leq 2, \ell(\lambda)\le 4$ and $\mu_2\geq \nu_2.$

 \begin{example}[The Kronecker function is not an Ehrhart function]\label{one-onecubed}
 Consider the dilated Kronecker coefficient $g_{(k,k), (k,k), (k,k)},$ for any positive integer $k$. 
 
By direct computation we see that only the first four monomials of $P_\lambda$ in Theorem~\ref{7termF22Alternant} contribute to the Kronecker coefficient:
 %

 \begin{align}
 g_{(k,k), (k,k), (k,k)} &=[x^ky^k](x^k-2x^{k+1}+x^{k+2})\bar{F}_{2,2}(x,y)\\
 &=[s_0^k s_1^k] (1-2s_1+s_1^2)F_{2,2}(s_0, s_1) \label{eq:gkkk}\\
 &=p_S(k,k)-2p_S(k,k-1)+p_S(k,k-2).
 \end{align}
Using the formula of Section~\ref{F22} for $p_S(n,m)$, we obtain 
 \begin{align}\label{even_odd}
 g_{(k,k), (k,k), (k,k)}=\begin{cases} 1, &k \text{ even}\\
                                                       0, &k \text{ odd}.
                                                       \end{cases}
\end{align}

The sequence $ g_{(k,k), (k,k), (k,k)}$, for $k\geq 0$ illustrates that the Kronecker coefficients cannot  possibly count points in the dilations of a polytope because such sequences are necessarily weakly increasing. That is, the Kronecker function is not an Ehrhart function.  See~\cite{King-Sevilla, BOR-CC} for related results and conjectures. 

\begin{remark}[The holes of the Kronecker cone]
Example~\ref{one-onecubed} also illustrates the origin of the holes in the Kronecker cone that we saw in Figure~\ref{fig:Holes}.  Note that the holes are all in
the face of the Kronecker cone defined by equations $\mu_1=\mu_2$, $\nu_1=\nu_2$, $\lambda_1=\lambda_2$, $\lambda_3=\lambda_4=0$.
It  is  always  the case that the zeroes of the Kronecker cone are on its walls (facets) \cite{Manivel:asymptotics1}.

 This can also be seen for the example in Figure~\ref{fig:Holes} where the holes are all inside the face defined by $\lambda_1=\lambda_2$.


\end{remark}
\end{example}
 

\begin{example}  Consider an example of Baldoni and Vergne~\cite[Section 5.1.1]{BaldoniVergneWalter}, whose methods are quite different from ours. Let  $\lambda=(132,38,19,11), \mu=(110,90),\nu=(120,80).$ We will compute an expression for the dilated Kronecker coefficient 
$g_{k\mu,k\nu,k\lambda}.$

We have $k(\mu_2+\nu_2)=170k,$ $k(\lambda_3+\lambda_4)=30k, 
k(\lambda_2+\lambda_4)=49k, k\min(\lambda_2+\lambda_3, \lambda_1+\lambda_4)=k\min(57, 143)=57k.$ Eqn.~\eqref{explicitF22extraction} of Theorem~\ref{7termF22Alternant} becomes
\begin{equation*
 y^{30k}  (x^{49k} 
 -x^{57k+1}-0\cdot x^{143k+1}
 +0\cdot x^{151k+2  })
 +y^{49k+1}(-x^{30k-1}+x^{57k+1}
 +0\cdot x^{143k+1}).
 \end{equation*}
From  Proposition~\ref{barF22} and Section~\ref{F22},
\[
g_{k\mu,\,k\nu,\,k\lambda}=p_S(50k, 91k)-p_S(50k,83k-1)
-p_S(31k-1, 91k-2)+p_S(31k-1,64k-2).\]
From Section~\ref{F22}, the last two terms cancel each other because both correspond to Region II in Figure 4,
and hence depend only on the first argument $n$ of $p_S(n,m).$  The two remaining terms  correspond to Region III.  The reader can check that using the formula for Region III  gives 
\[g_{k\mu,k\nu,k\lambda}=52k^2+\frac{25}{2}k+\frac{3}{4}+\frac{(-1)^k}{4},\]
in agreement with the result in \cite[Section 5.1.1]{BaldoniVergneWalter}.
\end{example}


\subsection{A generating function for Kronecker coefficient dilations, an approach from analytical combinatorics.}
\label{sec-Diagonal}
The coefficient extraction that we use to write Kronecker coefficients can be done in bulk at the generating function level to encode dilated Kronecker products. Consider the formal series 
\[
\Phi_{\mu,\nu,\lambda}(z):=\sum g_{k\mu,k\nu,k\lambda}  z^k.\]
We can express $\Phi_{\mu,\nu,\lambda}(z)$ as a \emph{diagonal} of a rational function. A diagonal is a subseries defined as follows. Let $\mathbf{x}=x_1, \dots, x_d$. Given a multivariate Taylor series, 
\[A(\mathbf{x})= \sum_{(i_1,\dots,i_d)\in\mathbb{N}^d} a(i_1, i_2,
\dots, i_d) \, x_1^{i_1} x_2^{i_2}\dots x_r^{i_d}\] the \emph{central diagonal}
of this series, denoted  $\operatorname{Diag}_{\mathbf{x};z} A(\mathbf{x})$ is
the univariate series $\sum a(k,k,\dots, k)\, z^k$. More generally, we define the
diagonal along the ray ${\mathbf{r}}=(r_1, r_2, \dots, r_d)$ to be the univariate series:
\[\operatorname{Diag}^{\mathbf{r}}_{{\mathbf{x}};z} A(\mathbf{x}) :=\sum a(kr_1,kr_2,\dots, kr_d) z^k.\] If we apply
$\operatorname{Diag}$ to a rational function, or more generally any function, it is a shorthand for the application of the operator to the series development around the origin with variables considered in order $x_1, x_2, \dots, x_d$.

For example, from Eqn.~\eqref{eq:gkkk} we have
\begin{equation}
\sum_k g_{(k,k), (k,k), (k,k)}\, z^k 
= \Delta \frac{1-2s_1+s_1^2}{(1-s_0)(1-s_1)(1-s_0s_1)(1-s_0s_1^2)}.
\end{equation}

Dilations of the atomic coefficients can be expressed as diagonals of rational functions,  given Eqn.~\eqref{defF22} and Proposition~\ref{ineqatomiccone22}:
\begin{eqnarray*}
\sum \tilde g_{k\mu,k\nu,k\lambda} \,z^k 
&=&
\sum
[s_0^{k(\nu_2-\lambda_3-\lambda_4)}s_1^{k(\mu_2+\nu_2-\lambda_2-\lambda_3-2\lambda_4)}]
  \,\frac{1}{(1-s_0)(1-s_0s_1)(1-s_0s_1^2)(1-s_1)} z^k.\\
&=& \Delta^{(\nu_2-\lambda_3-\lambda_4, \mu_2+\nu_2-\lambda_2-\lambda_3-2\lambda_4)}_{\mathbf{s};z}   F_{2,2}(s_0, s_1).
\end{eqnarray*}

Furthermore, by factoring out the lexicographically least monomial in the expression in Theorem~\ref{thm:MAIN1} we can write the generating function for dilations of arbitrary Kronecker coefficients as diagonals of rational functions.

Generating functions which can be expressed as diagonals
of rational functions are well studied in their own right, particularly from a point of view of asymptotics and computation.  Recently, Bostan et
al.~\cite{BoLaSa17} showed an equivalence between generating functions of binomial sums, and diagonals of rational functions. Thus, Kronecker coefficients are multiple binomial sums.\footnote{This is the class  of  multivariate  sequences that contains the binomial coefficient sequence and that is closed under pointwise addition, pointwise multiplication, linear change of variables and
partial summation.}

There are explicit expressions for the coefficient asymptotics of rational function diagonals. We point the reader to two books which cover the topic~\cite{PeWi13}[Example 10.2.5] and \cite{Mish19}[Chapter 9]. The methods can be used to determ dominant asymptotics of the dilated Kronecker function as $n$ tends to infinity. The first step is to rewrite the quotient $F_{n,m}$ as a sum of rational functions $F_{n,m}=\sum G_i/H_i$ such that the
point $(1,1, \dots,1)$ is a transversal intersection for each
$H_i$. Finding such a form is done via a systematic computation.  We have done this in the case of $F_{2,2}$, and $F_{2,3}$. The computations  for $F_{3,3}$ have recently been completed by Stefan Trandafir. This is the subject of ongoing work. The asymptotics make clear the chambers of quasipolynomiality, and the technique gives computational tests for when the degree of the quasipolynomial will drop. It is directly related to the degree to which the numerator vanishes at the critical point, which in this case is the all ones vector.
Manivel~\cite[Remark 1]{Manivel:asymptotics1} remarked:
\begin{quote}\sl
One interesting implication of the quasipolynomiality property is that, knowing the Kronecker coefficients asymptotically, in fact we know them completely.
\end{quote}

It is an active area of research to develop computer algebra methods to make such diagonal computations effective. There are numerous strategies possible.  For example, there is a method using systems of linear differential equations, made effective in several implementations~\cite{Chyz00, Kout13}. In each case, the algorithm execution is limited by heavy intermediary Gr\"obner basis computations. The two-variable case, $F_{2,2}$, is easily manageable, but the computations are not currently possible for larger $n, m$. 
\section{Final comments}

\subsection{Vector partition functions and other families of coefficients}
\label{sec-MR-other-families}
Most interesting families of  coefficients related to the representation theory of the general linear group can be defined by the evaluation of a Schur function on a {\em finite} alphabet:
\begin{align*}
&&s_\lambda[X+Y] =\sum_{\mu,\nu} c_{\mu,,\nu}^{\lambda} s_{\mu}[X] s_{\nu}[Y], \qquad &&\text{Littlewood Richardson coefficients}\\
&&s_\lambda[XY] =\sum_{\mu,\nu} g_{\mu,\nu,\lambda} s_{\mu}[X] s_{\nu}[Y],  \qquad&&\text{Kronecker coefficients}\\
&&s_\lambda[XY+X+Y] =\sum_{\mu,\nu} h_{\mu,\nu}^{\lambda} s_{\mu}[X] s_{\nu}[Y],  \qquad&&\text{Heisenberg coefficients}\\
&&s_\lambda[s_\theta][X+Y] =\sum_{\mu,\nu} a_{\mu,\nu}^{\lambda} s_{\mu}[X] s_{\nu}[Y],  \qquad&&\text{plethysm coefficients}
\end{align*}
and so on. They all share the common property of being defined by the evaluation of a  Schur function on an alphabet $P(X,Y)$, where  $P(X,Y)$ 
is a polynomial function of two alphabets $X$ and $Y$ with    integer coefficients.

\medskip

The approach presented in this work for the Kronecker coefficients will apply to the study of all such families of coefficients. Therefore, the Littlewood-Richardson,
Heisenberg, and plethysm functions  are all given by quasipolynomials.
 Let us briefly discuss the well-known simple case of the Littlewood--Richardson coefficients. 

\medskip
\medskip
\medskip\medskip


\subsection{The Littlewood--Richardson coefficients and the resultant} 
In this section we explore a beautiful and unexpected relation between the resultant, the alternant $a_{\delta_{n+m}}[X+Y]$,
and the Littlewood-Richardson coefficients.


Let $P$   be a monic polynomial.
 Vieta's formula   expresses the coefficients of  polynomial $P$ as $\pm$ the elementary symmetric
functions evaluated at  its roots. Therefore, we can think of the alphabet $X$ (of size $n=\deg(P)$)
as listing the roots of $P=\prod_{i=1}^n(x-x_i)$.

In this setting, the alternant $a_{\delta_n}[X]$ is just the square of the discriminant of $P$. Similarly, if $Q$ is a 
monic polynomial of degree $m$, then $a_{\delta_m}[Y]$ is the square of the discriminant of $Q$.


Note that for the Littlewood-Richardson coefficients,  we have
\[
\frac{a_{\delta_{|X|}}[X] a_{\delta_{|Y|}}[Y]   }
{a_{\delta_{n+m}}[X+Y]}=\frac{1}{\prod_{j, i} (x_j-y_i)}=\frac{1}{Res(P,Q)}
\]
the product running over all $j=1,\ldots,n$ and $i=1,\ldots,m.$
Elegantly, the function appearing in the denominator of this expression is the resultant of the monic polynomials $P$ and $Q$.

 For fixed $n$ and $m$, the 
  matrix $A_{n,m}$ associated to the vector partition function is an $(n+m)\times(nm) $ matrix with entries equal to $\pm 1$ or zero. For each pair $j,i$ there
 is a column in this matrix, corresponding to the factor $(1-x_j^{-1}y_i)$, 

 with the entry in the top $n$ rows  (indexing the $x_j$)  equal $-1$, and  the entry in the bottom $m$ rows (indexing the $y_i$),  equal to $1$;  the remaining entries are zero. 
 A couple of examples will illustrate this construction.
 \begin{align*}
 A_{2,2}=
 \begin{pmatrix}
-1&-1&\phantom{-}0&\phantom{-}0\phantom{-}\\
\phantom{-}0&\phantom{-}0&-1&-1\phantom{-}\\
\phantom{-}1&\phantom{-}0&\phantom{-}1&\phantom{-}0\phantom{-}\\
\phantom{-}0&\phantom{-}1&\phantom{-}0&\phantom{-}1\phantom{-}\\
 \end{pmatrix}
  \end{align*}
   \begin{align*}
  A_{2,3}=
 \begin{pmatrix}
-1&-1&-1&\phantom{-}0 & \phantom{-} 0&\phantom{-}  0  &  \\
\phantom{-}0 & \phantom{-} 0&\phantom{-}  0 & -1&-1 &-1 &\\
\phantom{-}1&\phantom{-}0&\phantom{-}0 &\phantom{-}1&\phantom{-}0&\phantom{-}0 & \\
\phantom{-}0&\phantom{-}1&\phantom{-}0&\phantom{-}0&\phantom{-}1&\phantom{-}0&\\
\phantom{-}0&\phantom{-}0&\phantom{-}1&\phantom{-}0&\phantom{-}0&\phantom{-}1&
 \end{pmatrix}
 \end{align*}
 
 

 
 Define the stretched Littlewood-Richardson function $P_{\mu,\nu}^{\lambda}(k) := c_{k\mu,k\nu}^{k\lambda}$. The same argument used to show that the Kronecker function is a quasipolynomial implies that the Littlewood-Richardson function is also a quasipolynomial. However, a stronger statement holds.
  
A matrix is  { \em totally unimodular} if its square submatrices all have determinant $\pm 1$ or 0. Equivalently, all nonzero minors are $\pm 1.$   It is straightforward to see that 
$A_{n,m}$ is totally unimodular.  For this we could either use the characterization totally unimodular matrices of  Ghouila-Houri \cite{GH}, or give an inductive argument. 

The total unimodularity of $A_{n,m}$ immediately implies the well-known fact that the Littleword--Richardson function is a polynomial (and not just a quasipolynomial). 
See \cite{Sturmfels, DLS}. 



 \begin{corollary} Let $n$ and $m$ be two fixed integers. Let    $\nu \vdash n$ $\mu \vdash m$, and  $\lambda \vdash n+m$  be partitions.
 
Then the degree of
$P_{\nu,\mu}^{\lambda}$  is always less than or equal to the co-rank of $  A_{\ell(\nu), \ell(\mu)}$:
\[
\deg(P_{\nu,\mu}^{\lambda}) \le (\ell(\nu)-1)(\ell(\mu)-1).
\]
Moreover, this bound is sharp.
 \end{corollary}
 
 \begin{proof} It is easy to see that these matrices always have rank $n+m-1$. The matrix  structure is as follows.  The $nm$ columns are partitioned left to right into $n$ blocks each of size $m.$ The bottom $m$ rows of each block constitute a copy of the $m$ by $m$ identity matrix,    contributing $m$ pivots.

 Since every column has exactly two nonzero entries, a $+1$ and a $-1,$ the sum of the rows is the zero vector.  In the top $n$ rows, each column has exactly one nonzero entry, equal to $(-1)$. Each of rows $2, \dots, n$ contributes one pivot, since it consists of all zeros except for one block of $m$ columns filled with consecutive $(-1)$s.    Hence the co-rank of the matrix is  $nm-(m+n-1)=(n-1)(m-1)$.
 \end{proof}
 
 


 


 


 \subsection{Other families of coefficients: The plethystic and  Heisenberg coefficients.} A similar argument will show that
 the  plethystic and Heisenberg coefficients can be computed as an alternating sum of vector partition functions. The implications
 of this fact will be the theme of another work. The quasipolynomial corresponding to some families of plethysms has been studied by  Kahle and Micha\l ek in
~\cite{KahleMichalek}, and from a different point of view by Colmenarejo~\cite{Colmenarejo-ple}.

\subsection{Are the atomic Kronecker coefficients an upper bound for the Kronecker coefficients?}
Theorem~\ref{thm:AtomicIsMax} suggests the following, using an analogous definition of atomic for other dimensions:
\begin{conjecture} For any dimension, the atomic Kronecker coefficient is an upper bound for the Kronecker coefficient:
$\tilde  g_{\mu,\nu,\lambda}\geq g_{\mu,\nu,\lambda}.$
\end{conjecture} 

If the asymptotics of coefficients of $F_{n,m}$ are known, such a bound would be useful for determining when a coefficient is nonzero. This result could be very useful to find triples 
in the Kronecker polyhedron, such that $g_{\mu,\nu,\lambda}=0$. This would have  importance in Geometric Complexity Theory. Manivel~\cite[Remark 2]{Manivel:asymptotics1} noticed that although the dilated Kronecker coefficient
$g(k\mu, k\nu, k\lambda)$ is a quasipolynomial function of $k\geq 0$,  its highest order term is really polynomial. More precisely, the period at the level of the coefficients increases for the terms of lower degrees. Can we use results on diagonals to prove explicit results of this nature?

\subsection{Towards a combinatorial interpretation}
Let us return to one of our motivating questions: Can we develop a combinatorial interpretation of Kronecker coefficients from this work? A first step could be to determine  the conditions on $\lambda, \mu$ and $\nu$ under which the Kronecker coefficient is an Ehrhart polynomial. A second approach could be to apply the work of Garrabrant and Pak~\cite{GaPa16} wherein they interpret of diagonals of (combinatorial) rational functons using tiling systems. Their constraints are met here, and would connect a new object to Kronecker coefficients. 

Some other
 families of Kronecker coefficients  are known to count integer partitions satisfying some restrictions.  Manivel~\cite{Manivel:rectangular} encountered them looking at rectangular Kronecker coefficients,  A. Garsia, N. Wallach, G. Xin and M. Zabrocki~\cite{Garsia:Wallach:Xin:Zabrocki} found that only some elegant families of  partitions appeared in some particular Kronecker products, (like in $s_{d,d}*s_{d,d}$), I. Pak, G. Panova, and E. Vallejo~\cite{PakPanovaVallejo:TensorSquares} used  partition functions in their attack on the Saxl conjecture, and Pak and Panova~\cite{PakPanova:qBinomial} have expressed some families (again of rectangular Kronecker coefficients) as the difference of two partition functions.  Colmenarejo and Rosas~\cite{Colmenarejo-Rosas, Colmenarejo} determined some other families of Kronecker coefficients given by vector partition functions and plane partitions.

\subsection{Comparison with other approaches}
\label{Comparisons}

Both Christandl, Doran, and Walter~\cite{CDW} and Baldoni, Vergne, and Walter~\cite{BaldoniVergneWalter}
 describe and implement algorithms to compute the Kronecker coefficients. They work in dimensions that are the sum of the lengths of the partitions -- and hence are much greater than what we see here.

 
More recently, Igor Pak and Greta Panova used an elegant approach to obtain a bound for the calculation of the Kronecker coefficients, see the proof of Lemma 5.4 in~\cite{Pak:Panova:complexityKC}. They extracted the Kronecker coefficients from the  generalized Cauchy identity (see Ex I.7.10 in~\cite{Macdonald} or  Ex 7.78 in~\cite{Stanley}):
\[
\sum_{\lambda, \mu, \nu}g_{\lambda,\mu, \nu }  s_{\lambda}[X] s_{\mu}[Y] s_{\nu}[Z] = \prod_{i,j,k} \frac{1}{1-x_iy_jz_k}
\]
They then showed that this could be expressed as a signed sum of three dimensional contingency arrays with certain margins.



\section{Appendix: The atomic Kronecker coefficient is an upper bound for the Kronecker coefficient for the case $2-2-4$}

Recall that in Theorem \ref{thm:AtomicIsMax} we claimed that for   a triple of partitions $\lambda, \mu, \nu$ of the same integer, such that $\ell(\lambda)\leq 4, \ell(\mu), \ell(\nu)\leq 2,$ 
and $\mu_2\geq \nu_2 ,$ the atomic Kronecker coefficient $\tilde g_{\mu,\nu,\lambda}$ is always greater than or equal to the actual Kronecker coefficient $g_{\mu,\nu,\lambda}.$
In this appendix we use our polyhedral geometry approach to prove this result.  Theorem~\ref{7termF22Alternant} and its applications showed how the Kronecker coefficient is completely determined by the functions $p_S(n,m).$  Our proof, depending heavily on the fact that the $p_S(n,m)$  are Ehrhart functions, consists of a careful analysis of the contributions of each term in the polynomial 
$P_\lambda$ in the proof of Theorem~\ref{7termF22Alternant}.

Our arguments will reveal  a remarkable relationship between the seven monomials in 
$P_\lambda.$    For brevity we will label the exponents of 
$y$ and $x$ appearing in Eqn.~\ref{explicitF22extraction} as follows:
\[b=\lambda_3+\lambda_4, a_0=\lambda_2+\lambda_4, 
a_1=\lambda_2+\lambda_3+1, a_2=\lambda_1+\lambda_4+1, 
a_3=\lambda_1+\lambda_3+2.\]
Combining Eqns.~\eqref{yInequality},~\eqref{xInequality},  we have the inequalities
\begin{equation}\label{expIneq}b\leq a_0<\{a_1, a_2\}<a_3.\end{equation}
The polynomial $P_\lambda$ is then
\[P_\lambda=y^b(x^{a_0}-x^{a_1}-x^{a_2}+x^{a_3})
+y^{a_0+1}(-x^{b-1}+x^{a_1}+x^{a_2}).\]
Recall that the first monomial, $y^b x^{a_0},$ is the one that determines the atomic Kronecker coefficient.
We will call this the atomic monomial.
The dependency digraph of Figure~\ref{fig:DepGraph} for the signed monomials 
in $P_\lambda$   is a consequence of Theorem~\ref{7termF22Alternant} and the inequalities~\eqref{expIneq},~\eqref{1stalternantInequality},~\eqref{2ndalternantInequality}. If  $M_1,$ $M_2$ are signed monomials,  a directed edge from node  $M_1$ to node $ M_2$ in the digraph signifies that if $M_1$ makes a nonzero contribution to the Kronecker coefficient (as described by Theorem~\ref{7termF22Alternant}), then so must the monomial $M_2$.
\begin{comment}
\begin{align}\label{DepGraph}
&y^{a_0+1} x^{a_1}\rightarrow -y^{a_0+1} x^{b-1} & 
  &y^{a_0+2} x^{a_2}\rightarrow -y^{a_0+1} x^{b-1} \\
 & &-y^{a_0+1} x^{b-1}\rightarrow y^b x^{a_0} &\\
& -y^b x^{a_1}\rightarrow y^b x^{a_0} & 
& -y^b x^{a_2}\rightarrow y^b x^{a_0}\\
&+y^bx^{a_3}\rightarrow  -y^b x^{a_1} & 
&+y^bx^{a_3}\rightarrow  -y^b x^{a_2}\\
&y^{a_0+1} x^{a_1}\rightarrow -y^{b} x^{a_1} &
  & y^{a_0+2} x^{a_2}\rightarrow -y^{b} x^{a_2}
\end{align}
\end{comment}

\begin{figure}
\begin{center}
\tikzstyle{line} = [draw, -latex']
\begin{tikzpicture}[scale=1.5]
  
  \node (M6) at (-1,6) {$\scriptstyle +y^{a_0+1} x^{a_1}$};
  \node (M7) at (1,6) {$\scriptstyle +y^{a_0+1} x^{a_2}$};
  \node (M5) at (0,5) {$\scriptstyle -y^{a_0+1} x^{b-1}$};
  \node (M1) at (0,4) {$\scriptstyle \mathbf{+y^b x^{a_0}}$};
  \node (M2) at (-1,3) {$\scriptstyle -y^b x^{a_1}$};
  \node (M3) at (1,3) {$\scriptstyle -y^b x^{a_2}$};
  \node (M4) at (0,2) {$\scriptstyle  +y^b x^{a_3}$};
  
  \path [line, blue] (M4) -- (M3); \path [line] (M3) -- (M1);
  \path [line, red] (M4) -- (M2);  \path [line] (M2) -- (M1);
     \path [line]  (M5) -- (M1); 
     \path [line, red] (M6) -- (M5) ;
    \path [line, blue]  (M7) -- (M5);
     \path [line, blue] (M6) -- (M2); 
     \path [line, red]  (M7) -- (M3); 
 
\end{tikzpicture}
\end{center}
\caption{\small Dependency digraph for the monomials in $P_\lambda$  (the atomic monomial is in bold).  The blue and red arrows correspond to the two scenarios described in the proof of Theorem~\ref{thm:AtomicIsMax}.}
\label{fig:DepGraph}
\end{figure}




 We will examine the contribution to $g_{\mu,\nu,\lambda}$ of each of the three non-atomic monomials  in $P_\lambda$ with positive coefficient.  Recalling Eqn.~(\ref{Exact}) from Theorem~\ref{thm:MAIN1}, this in turn will necessarily entail a detailed analysis of the vector partition function 
$p_S(n,m)$ of Section~\ref{F22}.    The final result exhibits the following surprising  phenomenon in the monomials of $P_\lambda.$  We will show that in fact, every non-atomic monomial with positive coefficient can be matched with a monomial with negative coefficient to yield a net nonpositive value (see the coloured arrows in Figure~\ref{fig:DepGraph}). For clarity of  exposition, the technical lemmas have been relegated to the end of the section.


\begin{proof}[Proof of Theorem \ref{thm:AtomicIsMax}]
 The atomic Kronecker coefficient is determined by only  the first monomial
$y^b x^{a_0}.$  In order to prove that the result of the corresponding  coefficient extraction from $F_{2,2}$ is never less 
than the actual Kronecker coefficient, it suffices to show that the contribution of the  three remaining (non-atomic) positively signed  monomials, viz. 
\[+y^{a_0+1} x^{a_1}, +y^{a_0+1} x^{a_2}, +y^bx^{a_3}\]
 is offset by that of the three negative ones,
 \[-y^{b} x^{a_1}, -y^{b} x^{a_2}, -y^{a_0+1} x^{b-1}.\]
More precisely, we say that a positive monomial $+M_1$ is \textit{offset} by a negative monomial $-M_2$ if the contribution of 
$M_1-M_2$ to the Kronecker coefficient is \textit{nonpositive}. 

Lemmas~\ref{pSboundBinomCoeff} to~\ref{lastmonomial} which follow will establish that one of the following two scenarios,  corresponding respectively to the blue arrows and the red arrows in Figure~\ref{fig:DepGraph}, \textit{must} occur.  

The contribution of 
\begin{enumerate}  
\item    $+y^{a_0+1} x^{a_1}$  is offset by $-y^{b} x^{a_1}$ AND
\item $+y^{a_0+1} x^{a_2}$ is offset by $-y^{a_0+1} x^{b-1}$ AND 
\item    $+y^b x^{a_3}$ is offset by $-y^b x^{a_2};$ 
\end{enumerate}
OR the contribution of 
\begin{enumerate}  
\item  $+y^{a_0+1} x^{a_1}$  is offset by $-y^{a_0+1} x^{b-1}$ AND 
\item    $+y^{a_0+1} x^{a_2}$ is offset by$-y^{b} x^{a_2}$  AND 
\item    $+y^b x^{a_3}$ is offset by $-y^b x^{a_1}.$ 
\end{enumerate}

The above two scenarios show that, apart from the monomial $y^bx^{a_0},$  whenever there is a contribution from a positively signed monomial in $P_\lambda$ to the Kronecker coefficient, there is an offsetting negatively signed monomial which also contributes, resulting in a net \textit{nonpositive} contribution.

This completes the proof that the monomial $y^bx^{a_0}$ gives the maximal contribution to the Kronecker coefficient, i.e. that $\tilde g_{\mu,\nu,\lambda}$ is an upper bound. \end{proof}

We now prove the technical lemmas we need on the monotonic behaviour of the  function $p_S(n,m)$.   For brevity, 
throughout these arguments, we will write $c(m)$ for the expression $\frac{7}{8}+\frac{(-1)^m}{8}.$  Note that $c(m)\leq 1$ for all $m.$  


\begin{lemma}\label{pSboundBinomCoeff} The partition function $p_S(n,m)$ satisfies 
\[p_S(n,m)\le p_S(n,m')=\binom{n+2}{2} \text{ whenever }m'\geq 2n.\]
\end{lemma}
\begin{proof} We have three cases.

\noindent
\textbf{Case 1:} Suppose $n\in[0, \frac{m}{2}].$ Then we claim that $p_S(n,m)=\binom{n+2}{2}=p_S(n, m')$ for all $m'\ge 2n.$ 
This is just a consequence of the definition.

\noindent
\textbf{Case 2:}
 Suppose $n\in(\frac{m}{2},m).$  We must show that $p_S(n,m)\le
p_S(n,m')$ for all $m'\ge 2n.$ 

  From Figure~\ref{fig:chambers}, when $\frac{m}{2}\leq n<m,$ $p_S(n,m)$ is given by the formula for Region III, while $p_S(n,m')$ is given by the binomial coefficient $\binom{n+2}{2}.$  Inspecting the third figure in Figure~\ref{fig:polytopes}, and using the fact that the $p_S(n,m)$ count lattice points in the appropriate regions, it is immediate that the difference $p_S(n,m')-p_S(n,m)$ is nonnegative for $n$ in this interval and $m'\geq 2n.$
  
\begin{comment}
We compute the difference $p_S(n,m')-p_S(n,m)$ as a function $f(n)$ of $n$ for $n$ in the interval $(\frac{m}{2},m).$ 
This is 
\begin{align*} &f(n)=\frac{n^2}{2}+\frac{3n}{2}+1-nm+\frac{n^2}{2}+\frac{m^2}{4}-\frac{n+m}{2}-\frac{7}{8}-\frac{(-1)^m}{8}\\
&=n^2+n+1-c(m)+\frac{m^2}{4}-\frac{m}{2}-\frac{mn}{4},\quad \text{where } 1-c(m)=\frac{1-(-1)^m}{8}=0\text{ or } \frac{1}{4}.
\end{align*}
Hence the derivative $f'(n)=2n+1-m>0$ in the domain of $n,$ 
so $f(n)$ is an increasing function of $n$ in this interval.
But $f(\frac{m}{2})=1-c(m)$ which is nonnegative, so $f(n)$ is nonnegative for all $n$ in the interval $(\frac{m}{2},m).$ 
\end{comment}

\noindent
\textbf{Case 3:}
Suppose $n\ge m\ge 0.$  We must show that $p_S(n,m)\le \binom{n+2}{2}=p_S(n,m')$ for all $m'\ge 2n.$ Again this is immediate by the same geometric argument, inspecting the first and third figures in Figure~\ref{fig:polytopes}.

\end{proof}

\begin{comment}
  We compute
\begin{equation*} p_S(n,m) =\frac{m^2}{4} +m +c(m)\leq \frac{n^2}{4}+n +1<\frac{n^2}{2}+\frac{3n}{2} +1=\frac{(n+2)(n+1)}{2}.
\end{equation*}
\end{comment}

\begin{lemma}\label{pSRegionIII2nd} Suppose  $\frac{m}{2}<\frac{M}{2}<n<m<M.$ Then $p_S(n,M)-p_S(n,m)\geq 0.$
\end{lemma}
\begin{proof} Both partition functions are computed according to the formula for Region III in Figure~\ref{fig:chambers}, and hence count lattice points in a convex polytope (see Figure~\ref{fig:polytopes}).  They are therefore increasing functions in each argument.
\end{proof}
\begin{comment}
  We compute:
\begin{align*} &p_S(n,M)-p_S(n,m)=n(M-m)-\frac{1}{4}(M^2-m^2) +\frac{1}{2}(M-m)+c(M)-c(m)\\
&=(M-m)(n-\frac{M+m}{4}+\frac{1}{2}) +c(M)-c(m),
\end{align*}
and this is easily seen to be nonnegative since $c(M)-c(n)=0, \pm 1/4,$ and $n-\frac{M+m}{4}=\frac{(2n-M)+(2n-m)}{4}>0.$ 
\end{comment}



\begin{lemma}\label{pSRegionIII}  Fix $k\geq 0.$ Then  $p_S(n, n+k),$ 
for $0\leq k\leq n,$ is an increasing function of $n.$ 
\end{lemma}
\begin{proof}  From Section~\ref{F22}, we see that the conditions on    $k,n,$ imply that $p_S(n, n+k)$ corresponds to Region III in Figure~\ref{fig:chambers}.
As before, since the function $p_S(n, n+k)$ counts lattice points in a convex polytope (see the third figure in Figure~\ref{fig:polytopes}), it is an increasing function of $n.$ \end{proof}

\begin{comment}
Hence we have 
\begin{multline}\label{pSRegionIIIformula}p_S(n, n+k)
=n(n+k) -\frac{n^2}{2}-\frac{(n+k)^2}{4}+\frac{2n+k}{2} +c(n+k)\\
=\frac{n^2}{4}+n(\frac{k}{2}+1) -\frac{k^2}{4}+\frac{k}{2}+c(n+k),\end{multline}
where we have written $c(m)=\frac{7+(-1)^{m}}{8}.$
Let $n_2>n_1.$ Then 
\begin{align*}&p_S(n_2, n_2+k)-p_S(n_1, n_1+k)\\
&=\frac{n_2^2-n_1^2}{4}+(n_2-n_1)(\frac{k}{2}+1)+c(n_2+k)-c(n_1+k)\\
&=(n_2-n_1)(\frac{n_2+n_1}{4}+\frac{k}{2}+1) +(-1)^k\frac{(-1)^{n_2}-(-1)^{n_1}}{8},
\end{align*}
and this is clearly positive if $n_2>n_1.$\end{proof}
\end{comment}





Consider first the monomial $+y^{a_0+1}x^{a_i}, i=1,2.$ 
Note the crucial fact that from the dependency relations, if either of these monomials contributes a nonzero coefficient, so does the preceding \textit{negative} monomial $-y^{a_0+1}x^{b-1}.$
\begin{lemma} Let $i=1,2.$ Then the net contribution of the monomials 
$y^{a_0+1}(-x^{b-1}+x^{a_i})$ to $g_{\mu,\nu,\lambda}$ is negative or zero. 
\begin{proof}
The value  contributed to $g_{\mu,\nu,\lambda}$ 
by the monomial $+y^{a_0+1}x^{a_i}, i=1,2,$ is the coefficient 
$[x^{\mu_2} y^{\nu_2}]$ in the product $+y^{a_0+1}x^{a_i}\bar{F}_{2,2},$ which in turn is given by the vector partition function
\begin{equation}\label{pS1}p_S(\nu_2-(a_0+1), \nu_2-(a_0+1)+(\mu_2-a_i)).\end{equation}
On the other hand, the contribution from the negative monomial 
$-y^{a_0+1}x^{b-1}$ was shown in the proof of Theorem~\ref{7termF22Alternant} to be coming from Region II in Figure~\ref{fig:chambers}.  It therefore contributes the value
\begin{equation}\label{pSneg2}-p_S(\nu_2-(a_0+1), \nu_2-(a_0+1)+(\mu_2-b+1))=-{\nu_2-(a_0+1)\choose 2}.\end{equation}

But now Lemma~\ref{pSboundBinomCoeff} says the net contribution 
of these two monomials is negative or zero, 
as claimed.
\end{proof}
\end{lemma}

However, this is of course not sufficient to establish our theorem,  because \textit{both} positive monomials  $+y^{a_0+1}x^{a_i}, i=1,2$ can make a nonzero contribution. 
Appealing to the dependency relations, we see that a positive contribution from $+y^{a_0+1}x^{a_i}$ forces a 
 negative contribution from the monomial 
$-y^b x^{a_i},$ for each $i=1,2.$ 

\begin{lemma}\label{Case1A}  If $\mu_2-a_i\le 0,$ then the net contribution of $+y^{a_0+1}x^{a_i}$ and $-y^b x^{a_i}$ is negative or zero. 
\end{lemma}
\begin{proof} The contribution of $+y^{a_0+1}x^{a_i}$ is given by 
the vector partition function Eqn.~\eqref{pS1}, while that of $-y^b x^{a_i}$ is given by 
\begin{equation}\label{pSneg1}-p_S(\nu_2-b, \nu_2-b+(\mu_2-a_i)).\end{equation}
Because $\mu_2-a_i\le 0,$ in each case we have a vector partition function of the form $p_S(n,m)$ where $m<n$.  Hence each vector partition function corresponds to Region I in Figure~\ref{fig:chambers}.  But that function is clearly an increasing function of its second argument, $m.$  Also, we know from the inequalities~\eqref{expIneq} 
above that $\nu_2-(a_0+1)+(\mu_2-a_i)<\nu_2-b+(\mu_2-a_i).$  Hence the claim follows.
\end{proof}

\begin{lemma}\label{Case1Bi}  If $\mu_2-a_i> 0,$ then the net contribution of $+y^{a_0+1}x^{a_i}$ and $-y^b x^{a_i}$ is negative or zero.
\end{lemma}
\begin{proof} We must again carefully examine the respective contributions of these two monomials, which are 
\begin{equation}p_S(\nu_2-(a_0+1), \nu_2-(a_0+1)+(\mu_2-a_i)).\end{equation
and 
\begin{equation}-p_S(\nu_2-b, \nu_2-b+(\mu_2-a_i).\end{equation}
Each function above is of the form $p_S(n,m)$ where $n<m,$ so it is evaluated according to the formula for Region II or Region III in Figure~\ref{fig:chambers}. 
We know $\nu_2-(a_0+1)<\nu_2-b.$
We have three cases to consider:

\noindent
\textbf{Case 1:} Assume $\nu_2-(a_0+1)<\nu_2-b<\mu_2-a_i.$
Then each vector partition function above corresponds to Region II in 
Figure~\ref{fig:chambers}, given by a binomial coefficient so the net contribution is a difference of two binomial coefficients
${\nu_2-(a_0+1)+2\choose 2}-{\nu_2-b+2\choose 2},$
and this is clearly negative in view of the inequality~\eqref{expIneq}.

\noindent
\textbf{Case 2:} Assume $0\leq \nu_2-(a_0+1)< \mu_2-a_i
<\nu_2-b.$  Set $\mu_2-a_i=k, n_1=\nu_2-(a_0+1), n_2=\nu_2-b.$  
Thus we have $0\le n_1<k<n_2.$   In particular, $n_1-n_2\leq -2.$

Since $2n_1<n_1+k,$ we know that $p_S(n_1, n_1+k),$ which is the value of the contribution from the monomial $y^{a_0+1}x^{a_i}$, 
is specified by Region II in Figure~\ref{fig:chambers}, and is therefore given by the binomial coefficient ${n_1+2\choose 2}.$ 

Since $n_2\in (\frac{n_2+k}{2}, n_2+k),$  we conclude similarly that 
the contribution from the monomial $y^{b}x^{a_i}$ is given by computing $p_S(n_2, n_2+k)$ using the formula for Region III.

Hence, using the expression for $p_S(n,n+k)$ for Region III in Figure~\ref{fig:chambers}, the net contribution of $-y^b x^{a_i}+y^{a_0+1} x^{a_i}$ is given by 
\begin{align*} &p_S(n_1, n_1+k)-p_S(n_2, n_2+k)\\
&=\frac{n_1^2+3n_1+2}{2} -\left(\frac{n_2^2}{4}+n_2(\frac{k}{2}+1) -\frac{k^2}{4}+\frac{k}{2}+c(n_2+k) \right)
\end{align*}
Consider the function 
$f(k)=\frac{n_1^2+3n_1+2}{2} -\left(\frac{n_2^2}{4}+n_2(\frac{k}{2}+1) -\frac{k^2}{4}+\frac{k}{2} \right),$ 
 a polynomial in $k.$ 

It is easy to check that $f'(k)=\frac{1}{2}(k-1-n_2)\leq -1 \text{ when } k<n_2,$ 
and hence this is a decreasing function of $k$ with maximum value $f(n_1)$ in the interval $[n_1, n_2-1].$ But 
\begin{align*} &f(n_1)=\frac{n_1^2}{2}-\frac{n_2^2}{4}+\frac{3n_1}{2}-\frac{n_1n_2}{2}-n_2+1+\frac{n_1^2}{4}-\frac{n_1}{2}\\
&=\frac{n_1^2-n_2^2}{4}+(n_1-n_2) +\frac{n_1^2}{2}-\frac{n_1n_2}{2}+1\\
&=(n_1-n_2)[\frac{n_1+n_2}{4}+1+\frac{n_1}{2}]+1
\end{align*}
Since $n_1-n_2\leq -2,$ and the expression in square brackets is at least $\frac{5}{4},$ we see that $f(k)\leq f(n_1+1)<f(n_1)<-\frac{3}{2}.$ 
To find the net contribution of the two monomials, we need to add the value of $c(n_2+k).$ But this is at most 1. It follows that the net contribution is negative.


\noindent
\textbf{Case 3:}   Assume $0 < \mu_2-a_i\leq \nu_2-(a_0+1)
<\nu_2-b.$
Again set $\mu_2-a_i=k, n_1=\nu_2-(a_0+1), n_2=\nu_2-b.$  
The contribution of the monomial $+y^{a_0+1}x^{a_i}$ is $p_S(n_1, n_1+k)$ while that of the monomial $-y^bx^{a_i}$ is 
$-p_S(n_2, n_2+k).$  The inequalities imply that the function $p_S$ corresponds to Region III in Figure~\ref{fig:chambers} in both cases.  Hence Lemma~\ref{pSRegionIII} applies (because $0< k\leq n_1<n_2$), showing that 
the net contribution, $p_S(n_1, n_1+k)-p_S(n_2, n_2+k),$ is indeed negative or zero.
\end{proof}

It remains to consider what happens when the last monomial with positive coefficient in the first line of $P_\lambda,$ $y^b x^{a_3},$ contributes to the Kronecker coefficient.  From the dependency relations, we know that then all the monomials  $y^b x^{a_i}$ must contribute nonzero terms as well, and possibly also one or both monomials 
$y^{a_0+1}x^{a_i}$.   In the latter  case there is also necessarily a negative contribution from $-y^{a_0+1}x^{b}.$

\begin{lemma}\label{lastmonomial}
For each of $i=1,2,$ the net contribution of the two monomials $-y^bx^{a_i}+y^bx^{a_3}$ is always negative or zero. 
\end{lemma}

\begin{proof}  Set $n=\nu_2-b, m_i=n+(\mu_2-a_i), i=1,2,3.$ The contribution of $y^bx^{a_3}$ is $p_S(n, m_3)$, and that of 
$-y^bx^{a_i}, i=1,2,$ is $-p_S(n, m_i).$  Note that $m_3<m_i, i=1,2,$ in view of ~\eqref{expIneq}.

We will examine the behaviour of the function $p_S(n,m)$ according to where $n$ falls in each of the intervals below.  Although there are two categories:

\begin{center} $0<\frac{m_3}{2}<m_3\leq \frac{m_i}{2}<m_i,$
or
$0<\frac{m_3}{2}< \frac{m_i}{2}<m_3<m_i,$\end{center}
  both can be treated by the same arguments, because the same difference of vector partition functions $p_S(n,m)$ comes into play in each case.

\noindent
\textbf{Case 1:} 
If $n\leq \frac{m_3}{2},$ then in either category, both $p_S(n, m_3)$ and $p_S(n, m_i)$ are computed by the formula for Region II in Figure~\ref{fig:chambers}, and hence both equal the binomial coefficient ${n+2\choose 2}.$ The net contribution of $-y^bx^{a_i}+y^bx^{a_3}$ here is zero.

\noindent
\textbf{Case 2:} If $n>m_i,$ then in either category, both $p_S(n, m_3)$ and $p_S(n, m_i)$ are computed by the formula for Region I in Figure~\ref{fig:chambers}. But the quasipolynomial for Region I is clearly an increasing function of the second argument of $p_S,$ and hence, (since $m_3<m_i$), $-p_S(n, m_i)+p_S(n, m_3)$ is negative or zero. 

\noindent
\textbf{Case 3:} Suppose $\frac{m_i}{2}\leq n\leq m_3. $ Then both functions $p_S$ correspond to Region III, 
and Lemma~\ref{pSRegionIII2nd} applies directly to show that $-p_S(n, m_i)+p_S(n, m_3)$ is negative or zero.

\noindent
\textbf{Case 4:} Suppose $m_3\leq n\leq  \frac{m_i}{2}. $ Then 
the monomial $-y^bx^{a_i}$ contributes $p_S(n, m_i)$ which is now a binomial coefficient since $n\leq \frac{m_i}{2}.$  By Lemma~\ref{pSboundBinomCoeff}, the net contribution here is negative or zero. 

\noindent
\textbf{Case 5:}  Suppose 
$0<\frac{m_3}{2}<m_3\leq \frac{m_i}{2}<n<m_i.$

We need to examine the difference 
\[p_S(n,m_3)-p_S(n,m_i),\] where the first function corresponds to Region I and the second to region III.
We will consider the function 
$f(n)=p_S(n,m_3)-p_S(n,m_i)$ on the interval $[m_i/2, m_i].$ 
We have 
\[f(n)=\frac{m_3^2}{4}+m_3+c(m_3) 
-(nm_i-\frac{n^2}{2}-\frac{m_i^2}{4}+\frac{n+m_i}{2}+c(m_i)).\]
One checks that $f'(n)=-(m_i-n+1/2)\le -\frac{3}{2},$ and hence the function is decreasing with maximum at $\frac{m_i}{2}.$
This value is checked to be 
\[f(\frac{m_i}{2})= \frac{1}{4}(m_3^2-\frac{m_i^2}{2}) + (m_3-\frac{3m_i}{4}) +c(m_3)-c(m_i).\]
But $c(m_3)-c(m_i)\leq \frac{1}{4}$ as before, and we have $m_3<m_i.$  Hence $f(\frac{m_i}{2})<-\frac{1}{2}$ is negative, 
and so is $f(n).$

\noindent
\textbf{Case 6:}  Suppose 
$0<\frac{m_3}{2}< \frac{m_i}{2}<m_3<n<m_i.$ 
Exactly the same argument applies to this case, since we still have 
$m_3<n<m_i,$ which was the only inequality we used in the preceding argument.   This completes the proof of the lemma.
\end{proof}

 

\section*{Acknowledgements}
This research was initiated at the Banff International Research Station in the Women in Algebraic Combinatorics II workshop.  It was further supported by the  National Science Foundation under Grant No. DMS-1440140 while the  first and third authors were in residence at the Mathematical Sciences Research
Institute in Berkeley, California, during the summer of 2017. The authors express their gratitude for the support of these two sources, without which this work would not have been possible. The research of MM is partially supported by National Science and Engineering Research Council Discovery Grant RGPIN-04157. During the completion of this work, she was hosted by Institut Denis Poisson (Tours, France) and LaBRI (Bordeaux, France).  The research of MR is partially supported by MTM2016-75024-P, FEDER, and the Junta de Andalucia under grants P12-FQM-2696 and FQM-333.

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