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\title{A new order on integer partitions}
\author{Étienne Tétreault \thanks{\href{mailto: tetreault.etienne@courrier.uqam.ca}{tetreault.etienne@courrier.uqam.ca}.}\addressmark{1}}
\address{\addressmark{1}Département de mathématiques, Université du Québec à Montréal}
\received{\today}
\abstract{Considering Schur positivity of differences of plethysms of homogeneous symmetric functions, we introduce a new relation on integer partitions. This relation is conjectured to be a partial order, with its restriction to one part partitions equivalent to the classical Foulkes conjecture. We establish some of the properties of this relation via the construction of explicit inclusion of modules whose characters correspond to the plethysms considered. We also prove some stability properties for the number of irreducible occurring in these modules as $m$ grows.}
\keywords{Representation theory, plethysm, symmetric group}

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\begin{document}
	\renewcommand{\proofname}{Proof}
	\renewcommand{\refname}{References}
	
\maketitle

It's been more than eighty years since Littlewood \cite{Littlewood} introduced the plethysm operation on symmetric functions (although the name was only introduced in 1950). This binary operation, denoted by $f[g]$, plays a fundamental role in representation theory, and its calculation raises many interesting questions. Indeed, calculating the ``structure'' coefficients of plethysms of the Schur functions is considered by Stanley \cite{Stanley99} as a key problem in algebraic combinatorics, and it also appears at the forefront of current research in Geometric Complexity Theory. A longstanding conjecture, stated by Foulkes \cite{Foulkes} in 1953, simply concerns inequalities between some of these coefficients in simple cases. With the aim of setting up our own context, we cast the statement of his conjecture in terms of coefficients $a_{\nu,\mu}^{\lambda}$ of the Schur expansion of the plethysm
\[
h_{\nu}[h_{\mu}] = \sum_\lambda a_{\nu,\mu}^{\lambda} \, s_\lambda,
\]
of complete homogeneous symmetric functions, which are well known to be positive integers.
Foulkes' conjecture states that for all $n \leq m$, and all partition $\lambda$ of $mn$, we have $a_{(n),(m)}^{\lambda} \leq a_{(m),(n)}^{\lambda}$. This has been proven to hold for $n \leq 5$ \cite{Thrall} \cite{DentSiemons} \cite{McKay} \cite{CIM}, and when $m$ and $n$ are far enough apart \cite{Brion}. But it still remains open in full generality. 

The question may naturally be extended to pairs of integer partitions $\mu$ and $\nu$ as follows.  Consider the binary relation $\nu\trianglelefteq \mu$ on integer partitions, which holds if and only if $h_{\mu}[h_{\nu}] -h_{\nu}[h_{\mu}]$ is Schur positive, or equivalently $a_{\nu,\mu}^{\lambda}\leq a_{\mu,\nu}^{\lambda}$ for all $\lambda$. Clearly ``$\trianglelefteq$'' is reflexive and antisymmetric (if we exclude the partition $(1)$). The conjecture\footnote{This was first observed by F. Bergeron, personal communication.} here is that it is also transitive; hence it gives us a partial order on integer partitions (see Figure~\ref{fig1}). Foulkes conjecture corresponds to the statement that 
  $$ (n)\trianglelefteq (m),\qquad {\mathrm{iff}}\qquad n\leq m,$$
for one part partitions $(m)$ and $(n)$ associated to positive integers $m$ and $n$.

\begin{figure}[!ht]
\begin{center}
%\includegraphics[width=80mm]{MonPoset6.pdf}
\begin{tikzpicture}[line join=bevel,scale=.4]
 \coordinate (P5) at (10bp,790bp);
 \coordinate (P4) at (10bp,730bp);
 \coordinate (P3) at (10bp,650bp);
\coordinate (P2) at (10bp,570bp);
 \coordinate (P33) at (60bp,590bp);
 \coordinate (P42) at (105bp,585bp);
 \coordinate (P51) at (230bp,610bp);
 \coordinate (P511) at (330bp,520bp);
\coordinate (P32) at (55bp,540bp);
 \coordinate (P41) at (175bp,560bp);
 \coordinate (P22) at (10bp,500bp);
 \coordinate (P222)at (50bp,500bp);
 \coordinate (P321) at (195bp,485bp);
 \coordinate (P5111) at (400bp,460bp);
 \coordinate (P411) at (270bp,465bp);
\coordinate (P31)  at (105bp,505bp);
 \coordinate (P21) at (10bp,425bp);
 \coordinate (P221) at (105bp,410bp);
 \coordinate (P4111) at (330bp,400bp);
 \coordinate (P311) at (180bp,385bp);
 \coordinate (P2211) at (10bp,333bp);
 \coordinate (P211) at (105bp,325bp);
 \coordinate (P3111) at (230bp,310bp);
 \coordinate (P2111) at (170bp,260bp);
%
\node[circle,fill=yellow,node font=\small] (mu_5) at (P5) [draw]  
	{$5$};
\node[circle,fill=yellow,node font=\small] (mu_4) at (P4) [draw]  
 	{$4$};
\node[circle,fill=yellow,node font=\small] (mu_3) at (P3) [draw]  
	{$3$};
\node[circle,fill=yellow,node font=\tiny,minimum size=.5cm] (mu_51) at (P51) [draw] 
  	{$51$};
\node[circle,fill=yellow,node font=\tiny,minimum size=.5cm] (mu_33) at (P33) [draw]
  	{$33$};
\node[circle,fill=yellow,node font=\tiny,minimum size=.6cm] (mu_42) at (P42) [draw] 
  	{$42$};
\node[circle,fill=yellow,node font=\small] (mu_2) at (P2) [draw] 
	{$2$};
\node[circle,fill=yellow,node font=\tiny] (mu_32) at (P32) [draw] 
  	{$32$};
\node[circle,fill=yellow,node font=\tiny,minimum size=.6cm] (mu_41) at (P41) [draw] 
  	{$41$};
\node[circle,fill=yellow,node font=\tiny] (mu_511) at (P511) [draw] 
  	{$\scriptstyle 511$};
\node[circle,fill=yellow,node font=\tiny] (mu_5111) at (P5111) [draw] 
  	{$\scriptstyle 5111$};
\node[circle,fill=yellow,node font=\tiny] (mu_411) at (P411) [draw] 
  	{$\scriptstyle 411$};
\node[circle,fill=yellow,node font=\tiny] (mu_321) at (P321) [draw]
  	 {$\scriptstyle 321$};
\node[circle,fill=yellow,node font=\tiny,minimum size=.6cm] (mu_31) at (P31) [draw] 
	{$31$};
\node[circle,fill=yellow,node font=\tiny] (mu_222) at (P222) [draw] 
  	{$\scriptscriptstyle 222$};
\node[circle,fill=yellow,node font=\tiny,minimum size=.6cm] (mu_22) at (P22) [draw] 
	{$22$};
\node[circle,fill=yellow,node font=\tiny,minimum size=.6cm] (mu_21) at (P21) [draw]  
	{$21$};
\node[circle,fill=yellow,node font=\tiny] (mu_221) at (P221) [draw] 
  	{$\scriptstyle 221$};
\node[circle,fill=yellow,node font=\tiny] (mu_311) at (P311) [draw] 
  	{$\scriptstyle 311$};
\node[circle,fill=yellow,node font=\tiny] (mu_4111) at (P4111) [draw] 
	 {$\scriptstyle 4111$};
\node[circle,fill=yellow,node font=\tiny] (mu_2211) at (P2211) [draw] 
  	{$\scriptstyle 2211$};
\node[circle,fill=yellow,node font=\tiny] (mu_211) at (P211) [draw] 
 	{$211$};
\node[circle,fill=yellow,node font=\tiny] (mu_3111) at (P3111) [draw] 
  	{$\scriptstyle 3111$};
 \node[circle,fill=yellow,node font=\tiny] (mu_2111) at (P2111) [draw] 
  	{$\scriptstyle 2111$};
\draw [line width=1pt,black,->] (mu_51) -- (mu_5);		% 5
	\draw [line width=1pt,black,->] (mu_4) -- (mu_5);
\draw [line width=1pt,black,->] (mu_42) -- (mu_4);		% 4
	\draw [line width=1pt,black,->] (mu_41) -- (mu_4);
	\draw [line width=1pt,black,->] (mu_3) -- (mu_4);
\draw [line width=1pt,black,->] (mu_33) -- (mu_3);		% 3
	\draw [line width=.5pt,black,->] (mu_31) -- (mu_3);
	\draw [line width=1pt,black,->] (mu_2) -- (mu_3);
\draw [line width=1pt,black,->] (mu_32) -- (mu_33);		% 33
\draw [line width=1pt,black,->] (mu_32) -- (mu_42);		% 42
	\draw [line width=.5pt,black,->] (mu_31) -- (mu_42);
	\draw [line width=.5pt,black,->] (mu_411) -- (mu_42);
\draw [line width=1pt,black,->] (mu_41) -- (mu_51);		% 51
	\draw [line width=1pt,black,->] (mu_511) -- (mu_51);
\draw [line width=.5pt,black,->] (mu_222) -- (mu_2);		% 2
	\draw [line width=1pt,black,->] (mu_22) -- (mu_2);
\draw [line width=1pt,black,->] (mu_411) -- (mu_511);	% 511
	\draw [line width=1pt,black,->] (mu_5111) -- (mu_511);	% 511
\draw [line width=1pt,black,->] (mu_31) -- (mu_41);		% 41
	\draw [line width=.5pt,black,->] (mu_321) -- (mu_41);
	\draw [line width=.5pt,black,->] (mu_222) -- (mu_41);
	\draw [line width=1pt,black,->] (mu_411) -- (mu_41);
\draw [line width=.5pt,black,->] (mu_321) -- (mu_32);		% 32
	\draw [line width=1pt,black,->] (mu_22) -- (mu_32);
	\draw [line width=.5pt,black,->] (mu_222) -- (mu_32);
	\draw [line width=.5pt,black,->] (mu_311) -- (mu_32);
\draw [line width=.5pt,black,->] (mu_221) -- (mu_22);		% 22
	\draw [line width=1pt,black,->] (mu_21) -- (mu_22);
\draw [line width=.5pt,black,->] (mu_221) -- (mu_222);	% 222
\draw [line width=.5pt,black,->] (mu_3111) -- (mu_321);	% 321
	\draw [line width=1pt,black,->] (mu_221) -- (mu_321);
\draw [line width=1pt,black,->] (mu_4111) -- (mu_5111);	% 5111
\draw [line width=1pt,black,->] (mu_311) -- (mu_411);	% 411
	\draw [line width=1pt,black,->] (mu_4111) -- (mu_411);	% 411
\draw [line width=1pt,black,->] (mu_311) -- (mu_31);		% 31
	\draw [line width=1pt,black,->] (mu_21) -- (mu_31);
	\draw [line width=.5pt,black,->] (mu_221) -- (mu_31);
\draw [line width=1pt,black,->] (mu_3111) -- (mu_4111);	% 4111
\draw [line width=1pt,black,->] (mu_211) -- (mu_21);		% 21
	\draw [line width=1pt,black,->] (mu_2211) -- (mu_21);
\draw [line width=1pt,black,->] (mu_2211) -- (mu_221);	% 221
	\draw [line width=.5pt,black,->] (mu_211) -- (mu_221);
\draw [line width=1pt,black,->] (mu_211) -- (mu_311);	% 311
	\draw [line width=1pt,black,->] (mu_3111) -- (mu_311);
	\draw [line width=.5pt,black,->] (mu_2211) -- (mu_311);
\draw [line width=.5pt,black,->] (mu_2111) -- (mu_2211);	% 2211
\draw [line width=1pt,black,->] (mu_2111) -- (mu_211);	% 211
\draw [line width=1pt,black,->] (mu_2111) -- (mu_3111);	% 3111
%
\end{tikzpicture}
 \end{center}\caption{Poset structure on partitions}\label{fig1}
 \end{figure}

Our aim is to establish some basic results for the relation ``$\trianglelefteq$'' using an approach to plethysm via the representation theory of the symmetric group. Indeed, coefficients of Schur expansion of plethysm of the form $h_{\mu}[h_{\nu}]$ occur as multiplicities of irreducible representations in some $\s_{nm}$-modules, denoted here by $M_{\mu}^{\nu}$. We construct them using similar techniques as \cite{PagetWildon}, although their construction produce modules associated to plethysm of Schur functions.

%The original Foulkes' conjecture can then be interpreted as the existence of some injective homomorphism from $M_{(m)}^{(n)}$ to $M_{(n)}^{(m)}$. One potential candidate for such a homomorphism is the Foulkes-Howe map \cite{Howe}. While it is not always injective (as shown in \cite{MN}), McKay showed \cite{McKay} that it is sufficient to show that it is injective when $m = n+1$.  Naturally, one is led to consider similar injective homomorphisms from $M_{\mu}^{\nu}$ to $M_{\nu}^{\mu}$. Hence we construct a generalization of the Foulkes-Howe map, and use it with other techniques to show that our conjecture holds in many cases.
% In particular, we prove that for all natural numbers $k$ and all partitions $\mu$, we have $(1^k) \trianglelefteq \mu$ and $\mu \trianglelefteq \mu^k$. We also show that the Foulkes' conjecture implies that  $\mu\trianglelefteq (k)$ for all $k \leq n+1$, whenever the greatest part of $\mu$ is $n$. In terms of these modules, we further show some stability properties, namely that the irreducible representation associated to $\lambda+(\tilde{m}n)$ appears at least as many times in $M_{\mu+\tilde{\mu}}^{\nu}$ as the one associated to $\lambda$ does in in $M_{\mu}^{\nu}$.

\acknowledgments{The author would like to thank François Bergeron for being the one who first remarked this potential order relation, for his general support and for his many comments and suggestions for this article.}

\section{Main results}

This extended abstract is a summary of our result in \cite{Montexte}. We skip here some of the proofs, but they can be found in this preprint, along with many examples of our definitions and constructions. For symmetric functions, notations used here are mostly those of \cite{Macdonald}. Other classical combinatorial and representation theory notions are recalled in many texts, for example in \cite{Bergeron}.

We describe in section \ref{construction} the $\s_{nm}$-modules $M_{\mu}^{\nu}$, whose Frobenius transform of its character is the plethysm $h_{\nu}[h_{\mu}]$, using techniques of \cite{PagetWildon}. The fact that $\nu \trianglelefteq \mu$ is then the existence of an injective morphism $M_{\mu}^{\nu} \hookrightarrow M_{\nu}^{\mu}$. In section \ref{morphisms}, we show the following result:

\begin{prop}\label{injmorphism}
If there is injective morphisms $\mathcal{F}_1: M_{\mu}^{\nu_1} \hookrightarrow M_{\nu_1}^{\mu}$ and  $\mathcal{F}_2: M_{\mu}^{\nu_2} \hookrightarrow M_{\nu_2}^{\mu}$, we can construct an injective morphism  $\mathcal{F}_1 \odot \mathcal{F}_2: M_{\mu}^{\nu_1 \uplus \nu_2} \hookrightarrow M_{\nu_1 \uplus \nu_2}^{\mu}$.
\end{prop}

We also construct in this section the generalized Foulkes-Howe map $\mathcal{F}_{\mu,\nu}: M_{\mu}^{\nu} \rightarrow M_{\nu}^{\mu}$. Using this map and proposition \ref{injmorphism}, we prove the following results in section \ref{results1}:

\begin{thm}\label{bigthm}
\begin{enumerate}[label=(\alph*)]
\item For any positive integer $n$ and partition $\mu$, there is an injection 

$M_{\mu}^{1^n} \hookrightarrow M_{1^n}^{\mu}$. 
\item For any positive integer $k$ and partition $\mu$, there is an injection $M_{\mu}^{\mu \uplus 1^k} \hookrightarrow M_{\mu \uplus 1^k}^{\mu}$.
\item For any positive integer $k$ and partition $\mu$, there is an injection $M_{\mu}^{\mu^k} \hookrightarrow M_{\mu^k}^{\mu}$.
\item If the Foulkes' conjecture is true up to $n-1$, then for all partition $\mu$ such that $\mu_1 \leq n$, there is an injection $M_{(n)}^{\mu} \hookrightarrow M_{\mu}^{(n)}$.
\end{enumerate}
\end{thm}

In section \ref{ssh}, we describe semistandard homomorphisms as in \cite{James}, and we show how to use them in our setting. Using these morphisms, we prove the following stability properties in section \ref{results2}:

\begin{thm}\label{thm1}
For any partition $\tilde{\mu}$ of an integer $\tilde{m}$, if $\text{dim} \ \text{Hom}_{\s_{nm}}(S^{\lambda},M_{\mu}^{\nu}) = r$, then we have $\text{dim} \ \text{Hom}_{\s_{n(m+\tilde{m})}}(S^{\lambda+(n\tilde{m})},M_{\mu+\tilde{\mu}}^{\nu}) \geq r$. 
\end{thm}

\begin{thm}\label{thm2}
If the number of parts of $\mu$ is $\tilde{m}$ and $\text{dim} \ \text{Hom}_{\s_{nm}}(S^{\lambda},M_{\mu}^{\nu}) = r$, then we have $\text{dim} \ \text{Hom}_{\s_{(m+2\tilde{m})n}}(S^{\lambda+(2^{\tilde{m}n})},M_{\mu+(2^{\tilde{m}})}^{\nu}) \geq r$.
\end{thm}

%We conclude with some perspectives for future research about this order.

\section{Description of the modules}\label{construction}

\subsection{Representation theory of the symmetric group}

As a means of deriving properties of plethysms, we consider representations of the symmetric group $\s_n$. The necessary link between the two subjects of study is established via the classical Frobenius transform of characters. This is a linear transform which encodes a character as a symmetric function, with the property that irreducible characters are sent to the basis of Schur symmetric function. All the fundamental questions of representation theory of $\s_n$ may be efficiently translated into calculation in the ring $\Lambda$ of symmetric functions. We use a polynomial construction of these modules, as in \cite{Bergeron}.

A \textit{diagram} is simply a finite subset $\bm{d}$ of $\mathbb{N} \times \mathbb{N}$, whose elements are called \textit{cells}. A cell $(i,j)$ is often geometrically represented in $\mathbb{N} \times \mathbb{N}$ by the \textit{box} having vertices $(i,j)$, $(i+1,j)$, $(i,j+1)$ and $(i+1,j+1)$. The $k^{th}$ row (resp. column) of a diagram $\bm{d}$ is the subset of cells such that $j = k-1$ (resp. $i = k-1$). The \textit{row-reading order} of the cells of $\bm{d}$ is the order such that $(i,j) < (i',j')$ if $j < j'$ or if $j=j'$ and $i < i'$.

A diagram is said to be a \textit{Ferrers diagram} (using French convention) if for every $(i,j) \in \bm{d}$, the diagram also contains every $(i',j') \in \mathbb{N} \times \mathbb{N}$ such that $i' \leq i$ and $j' \leq j$. If $\lambda_k$ is the number of cells in the $k^{th}$ row of a Ferrers diagram, and $|\bm{d}| =n$, then $\lambda = (\lambda_1,\lambda_2,..)$ is a decreasing sequence of positive integers who sums to $n$, {\rm i.e.} a \textit{partition} of $n$, denoted $\lambda \vdash n$. This sequence completely determines the diagram, and we write $\lambda$ for both the partition and its associated Ferrers diagram.

Let $\Omega$ be a set. A \textit{tableau} $\tau$ of \textit{shape} $\lambda$ with entries in $\Omega$ is a map $\tau: \lambda \rightarrow \Omega$. It is often displayed as a filing of the boxes of $\lambda$. The $\mathbb{C}$-vector space formally spanned by these tableaux is a $(\s_{\Omega} \times \s_{\lambda})$-bimodule, where $\s_{\Omega}$ acts on variables, {\sl i.e.}  $(\rho \cdot \tau)(i,j) = \rho(\tau(i,j))$ for $\rho \in \s_{\Omega}$; and $ \s_{\lambda}$ acts on places (or cells), {\sl i.e.}  $(\tau \cdot \pi)(i,j) = \tau(\pi(i,j))$ for $\pi \in \s_{\lambda}$. We then consider $R_{\lambda}$ (resp. $C_{\lambda}$), the subgroup of permutations in $\s_{\lambda}$ which only permutes cells lying in a same row (resp. column) in a tableau of shape $\lambda$. If, for $x \in \Omega$, we denote $\gamma_x = |\tau^{-1}(x)|$, the sequence $\gamma = (\gamma_x)_{x \in \Omega}$ is called the \textit{content} of $\tau$.

If $|\Omega| = |\lambda|=n$, we may consider the set of bijective tableaux $t: \lambda \rightarrow \Omega$, usually denoted by lowercase Latin letters. Since, in that case, $\s_{\Omega} \cong \s_{\lambda}$, for our purpose it suffices to consider the action of $\s_n$ on the $\mathbb{C}$-span of these bijective tableaux. Unless specified, we choose $\Omega$ to be the set $\bm{x} = \{x_1,...,x_n\}$, but we can make isomorphic constructions for any set of size $n$.

For each bijective tableau $t$ of shape $\lambda$, we associate the \textit{tableau monomial}, which is $\bm{x}^{t} = \displaystyle \prod_{(i,j) \in [\lambda]} t(i,j)^j$. The $\mathbb{C}$-span of those monomials is called the \textit{permutation module} associated to $\lambda$ and denoted $M^{\lambda}$. In this module, we define the polynomials $\Delta_{t} = \displaystyle \sum_{\pi \in C_{t}} \varepsilon(\pi) \pi \cdot \bm{x}^{t}$. The $\mathbb{C}$-span of those polynomials, called the \textit{Specht module} associated to $\lambda$, is denoted $S^{\lambda}$. We recall some well-known facts about these modules (see Sagan, \cite{Sagan}):

\begin{prop}
\begin{itemize}
\item The set $\{S^{\lambda} \ | \ \lambda \vdash n\}$ is a complete list of non-isomorphic irreducible representations of $\s_{n}$; 
\item The Frobenius transform of the character of $M^{\lambda}$ is $h_{\lambda}$;
\item The Frobenius transform of the character of $S^{\lambda}$ is $s_{\lambda}$.
\end{itemize}
\end{prop}   

\subsection{Wreath product}

For $n,m \in \mathbb{N}$, let $G$ (resp. $H$) be a subgroup of $\s_m$ (resp. $\s_n$). Also, let $f: H \rightarrow \text{Aut}(G^n)$ be the morphism such that for every $h \in H$, $f(h)$ sends $(g_1, ..., g_n) \in G^n$ to $(g_{h^{-1}(1)}, ..., g_{h^{-1}(n)})$. We define the \textit{wreath product} of $G$ with $H$, denoted $G \wr H$, to be the semidirect product $G^n \rtimes_f H$. 

Let $V$ be a $G$-module and $W$ be a $H$-module. The tensor product $V^{\otimes n}$ is naturally a $G^n$-module, and it is also a $H$-module, where $H$ acts by permuting the components of a tensor. The interaction of the two actions correspond to the wreath product, so it is in fact a $G\wr H$-module. Also, there is a canonical surjection $G \wr H \twoheadrightarrow H$, so we can construct the inflated $G \wr H$-module $\infl_{H}^{G \wr H} W$. We then define $V \oslash W$ to be the $G \wr H$-module given by $V^{\otimes n} \otimes \infl_{H}^{G \wr H} W$. When $G= \s_m$ and $H = \s_n$, the operation $\oslash$ mimics the plethysm of symmetric functions, in the following sense:

\begin{prop}
Let $V$ be a $\s_m$-module such that the Frobenius transform of its character is $f$, and let $W$ be a $\s_n$-module such that the Frobenius transform of its character is $g$. Then, the Frobenius transform of the character of $(V \oslash W)\big\uparrow_{\s_m \wr \s_n}^{\s_{nm}}$ is the plethysm $g[f]$.
\end{prop}

So, if we want to study the plethysms $h_{\nu}[h_{\mu}]$, we can do it by studying the modules $(M^{\mu} \oslash M^{\nu­})\big\uparrow_{\s_m \wr \s_n}^{\s_{nm}}$.

\subsection{The modules \texorpdfstring{$M_{\mu}^{\nu}$}{Mmn}}

We now describe a combinatorial description of the modules $(M^{\mu} \oslash M^{\nu­})\big\uparrow_{\s_m \wr \s_n}^{\s_{nm}}$ for $\mu \vdash m$ and $\nu \vdash n$. Let $\bm{x} = \{x_1,...,x_{nm}\}$ and $\mathcal{M}_{\mu}$ be the set of injective tableaux $t: \mu \rightarrow \bm{x}$. Also, denote $\mathcal{N}_{\mu,\nu}$ the set of injective tableaux $\bm{T}: \nu \rightarrow \mathcal{M}_{\mu}$ such that the union of the entries of each entry $t$ of $\bm{T}$ is $\bm{x}$. Thus, an element of $\mathcal{N}_{\mu,\nu}$ is a tableau of shape $\nu$ whose entries are tableaux of shape $\mu$ and such that each element of $\bm{x}$ appears exactly one time. By permuting all the entries at once, the $\mathbb{C}$-span of the elements of $\mathcal{N}_{\mu,\nu}$ is a $\s_{nm}$-module, isomorphic to $\mathbb{C}[\s_{nm}]$. 
For example, if $\mu=(2,1)$ and $\nu=(2,2)$, then an element of $\mathcal{N}_{\mu,\nu}$ is
\[
\bm{T}=
\ytableausetup{notabloids, boxsize=5em}
\begin{ytableau}
\tydu{x_1}{x_5}{x_4} & \tydu{x_6}{x_2}{x_9} \\
\tydu{x_{10}}{x_3}{x_{12}} & \tydu{x_9}{x_{11}}{x_7}
\end{ytableau}.
\]
We would like to have a monomial that is invariant under the row permutations of both $\bm{T}$ and each of its entries. First, if we consider the tableau monomial of an entry $t$ of $\bm{T}$ as defined earlier, it is invariant under row permutations of $t$, but it does not keep track of the entries of $t$ in the first row. This is a problem, because we have to know which variables are in the first row of each entry of $\bm{T}$. We can multiply the tableau monomial by each variable $x_i$ that appears in $t$, so that the $\s_m$-module structure is preserved. We obtain what we call the \textit{higher tableau monomial}, which is
\[
\xi^{t} = \displaystyle \left( \prod_{(i,j) \in \mu} t(i,j) \right) \bm{x}^{t} = \prod_{(i,j) \in \mu} t(i,j)^{j+1}.
\]
In order to use each $\xi^{t}$ to construct a monomial that is invariant under the row permutations of $\bm{T}$, consider the ring $\mathbb{C}\left[ X \ | \ X \ \text{is a monomial of} \ \mathbb{C}[\bm{x}] \ \right]$ ({\rm i.e.} we consider each monomial of $\mathbb{C}[\bm{x}]$ as a variable of this polynomial ring), and denote $*$ its multiplication. In this ring, we can define the tableau monomial of $\bm{T}$, named \textit{plethystic tableau monomial}, to be 
\[
\xi^{*\bm{T}} = \displaystyle \Star{(i,j) \in \nu}{}(\xi^{\bm{T}(i,j)})^{*(j+1)}.
\]
For example, the plethystic tableau monomial of $\bm{T}$ as above is 
\[
\xi^{*\bm{T}} = (x_{10}x_3x_{12}^2)*(x_9x_{11}x_7^2)*(x_1x_5x_4^2)^{*2}*(x_6x_2x_9^2)^{*2}.
\] 
%We say that $\bm{T}$ is in row-reading order if, for $(i,j) \in \nu$ such that $t_{\nu}(i,j) = \ell$, the entry $\bm{T}(i,j)$ is the tableau with values in $\{x_{m(\ell-1)+1},...,x_{m\ell}\}$ in row-reading order. We denote this tableau $\bm{T}_{\mu,\nu}$. 
%For example, the tableau $\bm{T}_{(2,1),(2,2)}$ is
%\[
%\ytableausetup{notabloids, boxsize=5em}
%\begin{ytableau}
%\tydu{x_7}{x_8}{x_9} & \tydu{x_{10}}{x_{11}}{x_{12}} \\
%\tydu{x_1}{x_2}{x_3} & \tydu{x_4}{x_5}{x_6}
%\end{ytableau}.
%\]
Let $M_{\mu}^{\nu}$ be the $\s_{nm}$-module generated by the $\xi^{*\bm{T}}$ for $\bm{T} \in \mathcal{N}_{\mu,\nu}$. %Note that it is isomorphic to $\mathbb{C}[\s_{nm}] \cdot \xi^{*\bm{T}_{\mu,\nu}}$, so it is a cyclic module. 

We have the following result, which is implicit in \cite{PagetWildon}:

\begin{prop}
The representation $M_{\mu}^{\nu}$ is isomorphic to the representation $(M^{\mu} \oslash M^{\nu­})\big\uparrow_{\s_m \wr \s_n}^{\s_{nm}}$.
\end{prop}

So, we can think of an element of $M_{\mu}^{\nu}$ as a monomial with exponents corresponding to $\nu$ whose variables are monomials with exponents corresponding to $\mu$ and such that this ``monomial of monomials'' contains each variable in $\bm{x} = \{x_1,...,x_{nm}\}$ exactly once. Note that $M^{\mu}_{(1)} \cong M^{\mu}$, the usual permutation module.

\section{The morphisms}\label{morphisms}

\subsection{Decomposition via the tensor product}

For two partitions $\nu_1,\nu_2$, we define their union $\nu_1 \uplus \nu_2$ to be the partition that has the parts of $\mu_1$ followed by the parts of $\mu_2$, and then reordered to obtain a partition. For example, $(2,2) \uplus (3,1) = (3,2,2,1)$. Also, if $t_1$ is a tableau of shape $\nu_1$ and $t_2$ is a tableau of shape $\nu_2$, we define their union $t_1 \uplus t_2$ to be the tableau obtained by stacking $t_1$ and $t_2$, and then reordering the rows so that it is a tableau of shape $\nu_1 \uplus \nu_2$.
%For example, we have
%\[
%t_1 = 
%\begin{ytableau}
%x_2 & x_3 \\
%x_1 & x_8
%\end{ytableau}; \qquad t_2 = 
%\begin{ytableau}
%x_7 \\
%x_5 & x_4 & x_6
%\end{ytableau}
%\qquad \implies \qquad
%t_1 \uplus t_2 = 
%\begin{ytableau}
%x_7 \\
%x_2 & x_3\\
%x_1 & x_8\\
%x_5 & x_4 & x_6
%\end{ytableau}.
%\]
We show in \cite{Montexte} the following results:

\begin{prop} \label{Isotens}
Let $\mu$ be a partition of $m$ and $\nu$ be a partition of $n$. Suppose that there is $\nu_1,\nu_2$ such that $\nu = \nu_1 \uplus \nu_2$, with $\nu_1 \vdash n_1$ and $\nu_2 \vdash n_2$. Then, we have an isomorphism $M_{\mu}^{\nu} \cong \left( M_{\mu}^{\nu_1} \otimes M_{\mu}^{\nu_2}\right) \big\uparrow_{\s_{mn_1} \times \s_{mn_2}}^{\s_{mn}}$.
%\begin{proof}
%Consider a tableau $\bm{T} \in \mathcal{N}_{\mu,\nu}$. We can write it as $\bm{T}_1 \uplus \bm{T}_2$, with $\bm{T}_1$ of shape $\nu_1$ and $\bm{T}_2$ of shape $\nu_2$, both having entries that are tableaux of shape $\mu$ and such that the union of the entries of all these tableaux is $\bm{x} = \{x_1,...,x_{nm}\}$. Because the rows are preserved by this split, $\xi^{*\bm{T}_1 \uplus \bm{T}_2} \mapsto \xi^{*\bm{T}_1} \otimes \xi^{*\bm{T}_2}$ is a homomorphism, and it is clearly bijective, as a row permutation in $\bm{T}_1$ or $\bm{T}_2$ corresponds to the same permutation in the corresponding row of $\bm{T}_1 \uplus \bm{T}_2$, and vice versa. So, we have an isomorphism.
%\end{proof}
\end{prop}

 For example, if $\mu=(3,2)$ and $\nu=(2,2)$, then $\nu = (2) \uplus (2)$, and if we consider
\[
\bm{T} = 
\ytableausetup{boxsize=4em}
\begin{ytableau}
\tytd{x_3}{x_{17}}{x_{18}}{x_1}{x_2} & \tytd{x_6}{x_{12}}{x_{10}}{x_{15}}{x_{16}} \\
\tytd{x_4}{x_{19}}{x_{11}}{x_8}{x_{14}} & \tytd{x_{13}}{x_5}{x_9}{x_7}{x_{20}}
\end{ytableau};
\qquad
\bm{T}_1 =
\ytableausetup{boxsize=4em}
\begin{ytableau}
\tytd{x_4}{x_{19}}{x_{11}}{x_8}{x_{14}} & \tytd{x_{13}}{x_5}{x_9}{x_7}{x_{20}}
\end{ytableau} ; \qquad \bm{T}_2 = 
\ytableausetup{boxsize=4em}
\begin{ytableau}
\tytd{x_3}{x_{17}}{x_{18}}{x_1}{x_2} & \tytd{x_6}{x_{12}}{x_{10}}{x_{15}}{x_{16}} 
\end{ytableau}.
\]
then the elements $\xi^{*\bm{T}}$ and $\xi^{*\bm{T}_1} \otimes \xi^{*\bm{T}_2}$ carry the same information.

When we decompose $\mu$ in the same way, we have a weaker result:

\begin{prop}
Let $\mu$ be a partition of $m$ and $\nu$ be a partition of $n$. Suppose that there is $\mu_1,\mu_2$ such that $\mu = \mu_1 \uplus \mu_2$, with $\mu_1 \vdash m_1$ and $\mu_2 \vdash m_2$. Then, we have an injection $\left(M_{\mu_1}^{\nu} \otimes M_{\mu_2}^{\nu} \right) \big\uparrow_{\s_{nm_1} \times \s_{nm_2}}^{\s_{nm}} \hookrightarrow M_{\mu}^{\nu}$.
%\begin{proof}
%An element of $\left(M_{\mu_1}^{\nu} \otimes M_{\mu_2}^{\nu} \right) \big\uparrow_{\s_{nm_1} \times \s_{nm_2}}^{\s_{nm}}$ is of the form $\xi^{*\bm{T}_1} \otimes \xi^{*\bm{T}_2}$, where $\bm{T}_1, \bm{T}_2$ are tableaux of shape $\nu$, the first one having entries that are tableaux of shape $\mu_1$, the second one having entries that are tableaux of shape $\mu_2$, and such that the union of the entries of all these tableaux is $\bm{x}$. If $(i,j)$ is the $k^{th}$ cell of $\nu$ (in row-reading order), denote $\bm{T}_1(i,j) = t_k$ and $\bm{T}_2(i,j) = t_k'$. Then, define $s_k = t_k \uplus t_k'$ and let $\bm{T}$ be the tableau of shape $\nu$ such that $\bm{T}(i,j) = s_k$. Consider the homomorphism that sends $\xi^{*\bm{T}_1} \otimes \xi^{*\bm{T}_2}$ to $\displaystyle \frac{1}{\nu!} \sum_{\sigma \in R_{\bm{T}_1} \times R_{\bm{T}_2}} \sigma \cdot \xi^{*\bm{T}}$. By construction, this morphism is injective.
%\end{proof}
\end{prop}

For example, if $\mu=(3,2)$ and $\nu=(2,2)$ as before, then $\mu = (3) \uplus (2)$, and if we have
\[
\bm{T} = 
\ytableausetup{boxsize=4em}
\begin{ytableau}
\tytd{x_3}{x_{17}}{x_{18}}{x_1}{x_2} & \tytd{x_6}{x_{12}}{x_{10}}{x_{15}}{x_{16}} \\
\tytd{x_4}{x_{19}}{x_{11}}{x_8}{x_{14}} & \tytd{x_{13}}{x_5}{x_9}{x_7}{x_{20}}
\end{ytableau}; \qquad
\bm{T}_1 =
\ytableausetup{boxsize=4em}
\begin{ytableau}
\tyt{x_3}{x_{17}}{x_{18}} & \tyt{x_6}{x_{12}}{x_{10}} \\
\tyt{x_4}{x_{19}}{x_{11}} & \tyt{x_{13}}{x_5}{x_9}
\end{ytableau} ; \qquad \bm{T}_2 = 
\ytableausetup{boxsize=3.5em}
\begin{ytableau}
\tyd{x_1}{x_2} & \tyd{x_{15}}{x_{16}} \\
\tyd{x_8}{x_{14}} & \tyd{x_7}{x_{20}}
\end{ytableau},
\]
then $\xi^{*\bm{T}_1} \otimes \xi^{*\bm{T}_2}$ is sent to $\displaystyle \frac{1}{4} \sum_{\sigma \in R_{\bm{T}_1} \times R_{\bm{T}_2}} \sigma \cdot \xi^{*\bm{T}}$.

By combining these two results, we obtain a proof of \ref{injmorphism} in \cite{Montexte}.

%\begin{proof}[Proof of proposition \ref{injmorphism}]
%For $\nu = \nu_1 \uplus \nu_2$, suppose that we have injections $\mathcal{F}_1: M_{\mu}^{\nu_1} \hookrightarrow M_{\nu_1}^{\mu}$ and $\mathcal{F}_2: M_{\mu}^{\nu_2} \hookrightarrow M_{\nu_2}^{\mu}$. Because induction and tensor product are functorial constructions, we can construct the morphism $\mathcal{F} = \left(\mathcal{F}_1 \otimes \mathcal{F}_2\right) \big\uparrow_{\s_{mn_1} \times \s_{mn_2}}^{\s_{mn}}$ such that we have the following composition:
%\[
%M_{\mu}^{\nu} \tilde{\rightarrow} \left( M_{\mu}^{\nu_1} \otimes M_{\mu}^{\nu_2}\right) \big\uparrow_{\s_{mn_1} \times \s_{mn_2}}^{\s_{mn}} \xrightarrow{\mathcal{F}} \left( M^{\mu}_{\nu_1} \otimes M^{\mu}_{\nu_2}\right) \big\uparrow_{\s_{mn_1} \times \s_{mn_2}}^{\s_{mn}} \hookrightarrow M_{\nu}^{\mu}.
%\]
%This chain of morphisms from $M_{\mu}^{\nu}$ to $M_{\nu}^{\mu}$ is denoted $\mathcal{F}_1 \odot \mathcal{F}_2$. It remains to show that this morphism is injective. But the tensor product and the induction are exact functors in a category of representations of a finite group. It means that the injectivity of $\mathcal{F}_1$ and $\mathcal{F}_2$ implies the injectivity of $\mathcal{F}$, hence the injectivity of $\mathcal{F}_1 \odot \mathcal{F}_2$, which concludes the proof.
%\end{proof}

\subsection{Decomposition using permutation modules}\label{permmod}

For a partition $\mu$, let $\mu^n$ be $\underbrace{\mu \uplus ... \uplus \mu}_{n \ \text{times}}$, and consider the permutation module $M^{\mu^n}$. By proposition \ref{Isotens} (for $\nu=(1)$), this module is isomorphic to the module $(M^{\mu})^{\otimes n}\big\uparrow_{(\s_{n})^m}^{\s_{nm}}$. So, we can write an element of $M^{\mu^n}$ as $\xi^{t_1} \otimes ... \otimes \xi^{t_n}$, where each $t_i$ is a tableau of shape $\mu$ and each variable of $\bm{x} = \{x_1,..,x_{nm}\}$ is used exactly once. Let $\bm{T} \in \mathcal{N}_{\mu,\nu}$ be such that if $(i,j)$ is its $\ell^{th}$ cell in row-reading order, then, $\bm{T}(i,j) = t_{\ell}$. We then define the projection
\[
\begin{array}{ccll}
\phi: & M^{\mu^n} & \xtwoheadrightarrow & M_{\mu}^{\nu} \\
         & \xi^{t_1} \otimes ... \otimes \xi^{t_n} & \longmapsto           & \xi^{*\bm{T}}
\end{array}.
\]
Reciprocally, for $\xi^{*\bm{T}} \in M_{\mu}^{\nu}$, if we denote $\bm{T}(i,j) = t_{\ell}$ when $(i,j)$ is the $\ell^{th}$ cell of $\bm{T}$ in row-reading order, there is an injective homomorphism
\[
\begin{array}{crlc}
\widetilde{\phi}: & M_{\mu}^{\nu} & \hooklongrightarrow & M^{\mu^n} \\
                    & \xi^{*\bm{T}}	& \longmapsto	         & \displaystyle \frac{1}{\nu!} \sum_{\sigma \in R_{\nu}} \xi^{t_{\sigma(1)}} \otimes ... \otimes \xi^{t_{\sigma(n)}}.
\end{array},
\]
We can easily see that $\phi \circ \widetilde{\phi} = \text{Id}$. 
For example, let $\mu=(3,1)$ and $\nu=(2,1)$. An element of $\mathcal{N}_{\mu,\nu}$ is
\[
\bm{T} = 
\ytableausetup{boxsize=4em, aligntableaux = center}
\begin{ytableau}
\tytu{*(green)x_8}{*(green)x_{10}}{*(green)x_3}{*(blue)x_{11}} \\
\tytu{*(magenta)x_5}{*(magenta)x_6}{*(magenta)x_2}{*(red)x_7} & \tytu{*(orange)x_4}{*(orange)x_9}{*(orange)x_1}{*(teal)x_{12}}
\end{ytableau}.
\]
The corresponding element of $M_{\mu}^{\nu}$ is $\xi^{*\bm{T}} = (x_{\textcolor{magenta}5}x_{\textcolor{magenta}6}x_{\textcolor{magenta}2}x_{\textcolor{red}7}^{\textcolor{red}2})*(x_{\textcolor{orange}4}x_{\textcolor{orange}9}x_{\textcolor{orange}1}x_{\textcolor{teal}{12}}^{\textcolor{teal}2})*(x_{\textcolor{green}8}x_{\textcolor{green}{10}}x_{\textcolor{green}3}x_{\textcolor{blue}{11}}^{\textcolor{blue}2})^{*2}$.  We then have:
%Using the isomorphism between $(M^{\mu})^{\otimes n}\big\uparrow_{(\s_{n})^m}^{\s_{nm}}$ and $M^{\mu^n}$, it is equivalent to the following element of $M^{(3,3,3,1,1,1)}$:
%
%\begin{align*}
% \frac{1}{2} \left(  x_{\textcolor{magenta}5}x_{\textcolor{magenta}6}x_{\textcolor{magenta}2}x_{\textcolor{orange}4}^{\textcolor{orange}2}x_{\textcolor{orange}9}^{\textcolor{orange}2}x_{\textcolor{orange}1}^{\textcolor{orange}2}x_{\textcolor{green}8}^{\textcolor{green}3}x_{\textcolor{green}{10}}^{\textcolor{green}3}x_{\textcolor{green}3}^{\textcolor{green}3}x_{\textcolor{red}7}^{\textcolor{red}4}x_{\textcolor{teal}{12}}^{\textcolor{teal}5}x_{\textcolor{blue}{11}}^{\textcolor{blue}6} + x_{\textcolor{orange}4}x_{\textcolor{orange}9}x_{\textcolor{orange}1}x_{\textcolor{magenta}5}^{\textcolor{magenta}2}x_{\textcolor{magenta}6}^{\textcolor{magenta}2}x_{\textcolor{magenta}2}^{\textcolor{magenta}2}x_{\textcolor{green}8}^{\textcolor{green}3}x_{\textcolor{green}{10}}^{\textcolor{green}3}x_{\textcolor{green}3}^{\textcolor{green}3}x_{\textcolor{teal}{12}}^{\textcolor{teal}4}x_{\textcolor{red}7}^{\textcolor{red}5}x_{\textcolor{blue}{11}}^{\textcolor{blue}6}   \right) 
%\end{align*}.

%This polynomial is projected on $M_{\mu}^{\nu}$ as follows:

\begin{align*}
\widetilde{\phi}(\xi^{*\bm{T}}) 	&= \frac{1}{2} \left(  x_{\textcolor{magenta}5}x_{\textcolor{magenta}6}x_{\textcolor{magenta}2}x_{\textcolor{red}7}^{\textcolor{red}2} \otimes x_{\textcolor{orange}4}x_{\textcolor{orange}9}x_{\textcolor{orange}1}x_{\textcolor{teal}{12}}^{\textcolor{teal}2} \otimes x_{\textcolor{green}8}x_{\textcolor{green}{10}}x_{\textcolor{green}3}x_{\textcolor{blue}{11}}^{\textcolor{blue}2} + x_{\textcolor{orange}4}x_{\textcolor{orange}9}x_{\textcolor{orange}1}x_{\textcolor{teal}{12}}^{\textcolor{teal}2} \otimes x_{\textcolor{magenta}5}x_{\textcolor{magenta}6}x_{\textcolor{magenta}2}x_{\textcolor{red}7}^{\textcolor{red}2} \otimes x_{\textcolor{green}8}x_{\textcolor{green}{10}}x_{\textcolor{green}3}x_{\textcolor{blue}{11}}^{\textcolor{blue}2} \right) \\
\phi(\widetilde{\phi}(\xi^{\bm{T}})) 	&= \frac{1}{2} \left(  (x_{\textcolor{magenta}5}x_{\textcolor{magenta}6}x_{\textcolor{magenta}2}x_{\textcolor{red}7}^{\textcolor{red}2})*(x_{\textcolor{orange}4}x_{\textcolor{orange}9}x_{\textcolor{orange}1}x_{\textcolor{teal}{12}}^{\textcolor{teal}2})*(x_{\textcolor{green}8}x_{\textcolor{green}{10}}x_{\textcolor{green}3}x_{\textcolor{blue}{11}}^{\textcolor{blue}2})^{*2} \right. \\
					&\qquad \left. + (x_{\textcolor{orange}4}x_{\textcolor{orange}9}x_{\textcolor{orange}1}x_{\textcolor{teal}{12}}^{\textcolor{teal}2})*(x_{\textcolor{magenta}5}x_{\textcolor{magenta}6}x_{\textcolor{magenta}2}x_{\textcolor{red}7}^{\textcolor{red}2})*(x_{\textcolor{green}8}x_{\textcolor{green}{10}}x_{\textcolor{green}3}x_{\textcolor{blue}{11}}^{\textcolor{blue}2})^{*2} \right),
\end{align*}

which is exacty $\xi^{\bm{T}}$. These morphisms are essential in our setting, as we use them in sections \ref{GFHM} and \ref{ssh}.

\subsection{Generalized Foulkes-Howe map}\label{GFHM}

We have already seen that the fact that $\nu \trianglelefteq \mu$ is equivalent to the existence of an injective homomorphism $M_{\mu}^{\nu} \hookrightarrow M_{\nu}^{\mu}$. We describe in the previous section a morphism from $M_{\mu}^{\nu}$ to $M^{\mu^n}$, and another from $M^{\nu^m}$ to $M_{\nu}^{\mu}$. The only missing part is a morphism from $M^{\mu^n}$ to $M^{\nu^m}$.

Let $\xi^{t_1} \otimes ... \otimes \xi^{t_n}$ be an element of $(M^{\mu})^{\otimes n}\big\uparrow_{(\s_{n})^m}^{\s_{nm}}$, which is isomorphic to $M^{\mu^n}$. Denote $x_{i,j}$ the $j^{th}$ entry (in row-reading order) of $t_i$. By construction, the set of all $x_{i,j}$ is equal to $\bm{x}$, so it is only a relabelling of the variables. Then, for $1 \leq j \leq m$, consider the tableau $s_j: \nu \rightarrow \{x_{1,j}, ..., x_{m,j}\}$ in row-reading order. We define the following morphism:
\[
\begin{array}{ccll}
\Psi: & M^{\mu^n} & \longrightarrow & M^{\nu^m} \\
         & \xi^{t_1} \otimes ... \otimes \xi^{t_n} & \longmapsto   & \displaystyle \frac{1}{\mu!} \sum_{\sigma \in R_{t_1} \times ... \times R_{t_n}} \sigma \cdot \xi^{s_1} \otimes ... \otimes \xi^{s_m}
\end{array}.
\] 
%The sum is there to ensure that this morphism is well defined for any choice of $t_i$ such that its higher tableau monomial is $\xi^{t_i}$. 
For example, if $\mu = (2,2)$ and $\nu = (2,1)$, take the monomial $x_{\textcolor{magenta}{1}}x_{\textcolor{magenta}{2}} x_{\textcolor{orange}{5}}^{\textcolor{orange}{2}}x_{\textcolor{orange}{6}}^{\textcolor{orange}{2}}x_{\textcolor{green}{9}}^{\textcolor{green}{3}}x_{\textcolor{green}{10}}^{\textcolor{green}{3}}x_{\textcolor{red}{3}}^{\textcolor{red}{4}}x_{\textcolor{red}{4}}^{\textcolor{red}{4}}x_{\textcolor{teal}{7}}^{\textcolor{teal}{5}}x_{\textcolor{teal}{8}}^{\textcolor{teal}{5}} x_{\textcolor{blue}{11}}^{\textcolor{blue}{6}}x_{\textcolor{blue}{12}}^{\textcolor{blue}{6}}$ of $M^{\mu^n}$, which corresponds to the tensor product $x_{\textcolor{magenta}{1}}x_{\textcolor{magenta}{2}}x_{\textcolor{red}{3}}^{\textcolor{red}{2}}x_{\textcolor{red}{4}}^{\textcolor{red}{2}} \otimes x_{\textcolor{orange}{5}}x_{\textcolor{orange}{6}}x_{\textcolor{teal}{7}}^{\textcolor{teal}{2}}x_{\textcolor{teal}{8}}^{\textcolor{teal}{2}} \otimes x_{\textcolor{green}{9}}x_{\textcolor{green}{10}}x_{\textcolor{blue}{11}}^{\textcolor{blue}{2}}x_{\textcolor{blue}{12}}^{\textcolor{blue}{2}}$. It can be written as $\xi^{t_1} \otimes \xi^{t_2} \otimes \xi^{t_3}$, with
\[
t_1 =
\begin{ytableau}
*(red)x_3 & *(red)x_4 \\
*(magenta)x_1 & *(magenta)x_2
\end{ytableau},
\  t_2=
\begin{ytableau}
*(teal)x_7 & *(teal)x_8 \\
*(orange)x_5 & *(orange)x_6
\end{ytableau},
\  t_3=
\begin{ytableau}
*(blue)x_{11} & *(blue)x_{12} \\
*(green)x_9 & *(green)x_{10}
\end{ytableau}.
\]
We then construct the following tableaux:
\[
s_1 =
\begin{ytableau}
*(green)x_9  \\
*(magenta)x_1 & *(orange)x_5
\end{ytableau} ,
\  s_2=
\begin{ytableau}
*(green)x_{10}  \\
*(magenta)x_2 & *(orange)x_6
\end{ytableau}, 
\ s_3=
\begin{ytableau}
*(blue)x_{11}  \\
*(red)x_3 & *(teal)x_7
\end{ytableau},
\ s_4=
\begin{ytableau}
*(blue)x_{12}  \\
*(red)x_4 & *(teal)x_8
\end{ytableau},
\]
%so that $\xi^{s_1} \otimes \xi^{s_2} \otimes \xi^{s_3} \otimes \xi^{s_4} =  x_{\textcolor{magenta}{1}}x_{\textcolor{orange}{5}}x_{\textcolor{green}{9}}^{\textcolor{green}{2}} \otimes x_{\textcolor{magenta}{2}}x_{\textcolor{orange}{6}}x_{\textcolor{green}{10}}^{\textcolor{green}{2}} \otimes x_{\textcolor{red}{3}}x_{\textcolor{teal}{7}}x_{\textcolor{blue}{11}}^{\textcolor{blue}{2}} \otimes x_{\textcolor{red}{4}}x_{\textcolor{teal}{8}}x_{\textcolor{blue}{12}}^{\textcolor{blue}{2}}$.
 We can conclude that the image of the monomial by $\Psi$ is
\[
\displaystyle \frac{1}{4} \sum_{\sigma \in R_{t_1} \times R_{t_2} \times R_{t_3}} \sigma \cdot x_{\textcolor{magenta}{1}}x_{\textcolor{orange}{5}}x_{\textcolor{magenta}{2}}^{\textcolor{magenta}{2}}x_{\textcolor{orange}{6}}^{\textcolor{orange}{2}} x_{\textcolor{red}{3}}^{\textcolor{red}{3}}x_{\textcolor{teal}{7}}^{\textcolor{teal}{3}}x_{\textcolor{red}{4}}^{\textcolor{red}{4}}x_{\textcolor{teal}{8}}^{\textcolor{teal}{4}} x_{\textcolor{green}{9}}^{\textcolor{green}{5}}x_{\textcolor{green}{10}}^{\textcolor{green}{6}}x_{\textcolor{blue}{11}}^{\textcolor{blue}{7}} x_{\textcolor{blue}{12}}^{\textcolor{blue}{8}}.
\]
Now consider the composition of homomorphisms $\phi \circ \Psi \circ \widetilde{\phi}: M_{\mu}^{\nu} \rightarrow M_{\nu}^{\mu}$. When $\mu=(m)$, $\nu=(n)$, this map is the \textit{Foulkes-Howe map} states in the language of symmetric groups. This map was developed as a tool to prove the Foulkes' conjecture. For this reason, we will call this homomorphism the \textit{generalized Foulkes-Howe map}, and we denote it $\mathcal{F}_{\mu,\nu}$. We use this map in the next section. 
%We can have a better understanding of this composition by using the following diagram:
%
%\begin{center}
%\begin{tikzcd}
%M_{\mu}^{\nu} 
%	\arrow[r, dashed, "\mathcal{F}_{\mu,\nu}"] 
%	\arrow[d, "\widetilde{\phi}"]
%& M_{\nu}^{\mu} \\
%M^{\mu^n}
%	\arrow[r, "\Psi"]
%& M^{\nu^m}
%	\arrow[u, "\phi"] \\
%\end{tikzcd}
%\end{center}


%We have the following result for the (regular) Foulkes-Howe map, proven by McKay \cite{McKay}:
%
%\begin{thm}[Propagation theorem]
%If $\mathcal{F}_{(m),(n)}$ is injective, then the map $\mathcal{F}_{(m),(n+1)}$ is also injective.
%\end{thm}
%
%Moreover, we know that for $m \leq 5$, the map $\mathcal{F}_{(m),(m+1)}$ is injective. Using the theorem inductively, $\mathcal{F}_{(m), (n)}$ is injective for $n \geq m+1$. The case $m=n$ is trivial (although the map $\mathcal{F}_{(m),(m)}$ is not always injective), so the Foulkes' conjecture holds for $m \leq 5$. In the future, we hope to find propagation theorems for other (conjectured) chains in the relation $\trianglelefteq$, because in can help to show that this relation is transitive. 

\section{First results}\label{results1}

We can use the tools we just developed to prove theorem \ref{bigthm}. Full proofs are given in \cite{Montexte}. Part $(a)$ is the following proposition:

\begin{prop}
For any positive integer $n$ and partition $\mu$, there is an injection $M_{\mu}^{1^n} \hookrightarrow M_{1^n}^{\mu}$. 
%\begin{proof}
%
%We use the generalized Foulkes-Howe map, so we have the composition
%\begin{center}
%\begin{tikzcd}
%\mathcal{F}_{\mu,1^n}: M_{\mu}^{1^n} 
%	\arrow[r, hookrightarrow, "\widetilde{\phi}"] 
%& M^{\mu^n} 
%	\arrow[r, "\Psi"] 
%& M^{(1^n)^m} 
%	\arrow[r, twoheadrightarrow, "\phi"]
%& M_{1^n}^{\mu} 
%\end{tikzcd}
%\end{center}
%Let $\bm{T} \in \mathcal{N}_{\mu,1^n}$, and denote $\bm{T}(0,i-1) = t_i$. As $R_{1^n} = \{\text{Id}\}$, the injection $\widetilde{\phi}$ sends $\xi^{*\bm{T}}$ to $\xi^{t_1} \otimes ... \otimes \xi^{t_n}$, so we can see that it is in fact a bijection. Let $s_j: 1^n \rightarrow \{x_1,...,x_{nm} \}$ be the injective tableau such that $s_j(0,k-1)$ is the $j^{th}$ entry of $t_k$ (in row-reading order). Then, $\Psi$ maps $\xi^{t_1} \otimes ... \otimes \xi^{t_n}$ to $\displaystyle \frac{1}{\mu!} \sum_{\sigma \in R_{t_1} \times ... \times R_{t_n}} \sigma \cdot \xi^{s_1} \otimes ... \otimes \xi^{s_m}$. Finally, $\phi$ maps this element to $\displaystyle \frac{1}{\mu!} \sum_{\sigma \in R_{t_1} \times ... \times R_{t_n}} \sigma \cdot \xi^{*\bm{T}}$, where $\bm{T}: \mu \rightarrow \{s_1,..,s_m\}$ is in row-reading order. 
%
%We have to show that this map is injective. Note that as the entries of $\bm{T} \in \mathcal{N}_{1^n,\mu}$ don't have row symmetries, we have that if there is $\bm{T'}$ such that $\xi^{*\bm{T}} = \xi^{*\bm{T'}}$, then the permutation $\sigma \in \s_{nm}$ such that $\bm{T} = \sigma \cdot \bm{T'}$ is an element of $R_{t_1} \times ... \times R_{t_n}$. So, the element $\displaystyle \frac{1}{\mu!} \sum_{\sigma \in R_{t_1} \times ... \times R_{t_n}} \sigma \cdot \xi^{*\bm{T}}$ is completely determined by $\bm{T}$ and the subgroup $R_{t_1} \times ... \times R_{t_n}$. It follows that we can find the unique $\xi^{t_1} \otimes ... \otimes \xi^{t_n}$ that has this image under $\phi \circ \Psi$, and because $\widetilde{\phi}$ is an isomorphism, we have that $\mathcal{F}_{\mu,1^n}$ is injective.
%\end{proof}
\end{prop}
%
For example, if $\mu=(2,1)$ and $n=2$, we have that (using the generalized Foulkes-Howe map) 

\[
\mathcal{F}_{\mu,1^n} \left( (x_{\textcolor{magenta}{2}}x_{\textcolor{magenta}{3}}x_{\textcolor{red}{4}}^{\textcolor{red}{2}})*(x_{\textcolor{orange}{5}}x_{\textcolor{orange}{6}}x_{\textcolor{teal}{1}}^{\textcolor{teal}{2}})^{*2} \right) = \displaystyle \frac{1}{4} \sum_{\sigma \in \s_{{\textcolor{magenta}{\{2,3\}}}} \times \s_{{\textcolor{orange}{\{5,6\}}}}} \sigma \cdot (x_{\textcolor{magenta}{2}}x_{\textcolor{orange}{5}}^{\textcolor{orange}{2}})*(x_{\textcolor{magenta}{3}}x_{\textcolor{orange}{6}}^{\textcolor{orange}{2}})*(x_{\textcolor{red}{4}}x_{\textcolor{teal}{1}}^{\textcolor{teal}{2}})^{*2},
\]

and we can see that the image completely determine the element we chose from $M_{\mu}^{1^n}$.

As a corollary, we obtain part $(b)$ of theorem \ref{bigthm}:

\begin{cor}
For any positive integer $k$ and partition $\mu$, there is an injection $M_{\mu}^{\mu \uplus 1^k} \hookrightarrow M_{\mu \uplus 1^k}^{\mu}$.
%\begin{proof}
%We have $\mu1^k = \mu \uplus 1^k$, and we have the bijective morphism $\text{Id}: M_{\mu}^{\mu} \tilde{\rightarrow} M_{\mu}^{\mu}$ and the injective morphism $\mathcal{F}_{\mu,1^k}: M_{\mu}^{1^k} \hookrightarrow M_{1^k}^{\mu}$ given by the preceding proposition. So, we can construct the injective morphism $\text{Id} \odot \mathcal{F}_{\mu,1^k}: M_{\mu}^{\mu1^k} \hookrightarrow M_{\mu1^k}^{\mu}$. 
%\end{proof}
\end{cor} 

%For example, we have for $\mu=(2)$ and $k=2$ that using this morphism,
%\[
%(x_{\textcolor{magenta}{2}}x_{\textcolor{magenta}{3}})*(x_{\textcolor{red}{6}}x_{\textcolor{red}{7}})*(x_{\textcolor{orange}{1}}x_{\textcolor{orange}{4}})^{*2}*(x_{\textcolor{teal}{5}}x_{\textcolor{teal}{8}})^{*3} \quad  \mapsto \quad \displaystyle \sum_{\sigma \in \s_{\textcolor{orange}{\{1,4\}}} \times \s_{\textcolor{teal}{\{5,8\}}}} \sigma \cdot (x_{\textcolor{magenta}{2}}x_{\textcolor{magenta}{3}}x_{\textcolor{orange}{1}}^{\textcolor{orange}{2}}x_{\textcolor{teal}{5}}^{\textcolor{teal}{3}})*(x_{\textcolor{red}{6}}x_{\textcolor{red}{7}}x_{\textcolor{orange}{4}}^{\textcolor{orange}{2}}x_{\textcolor{teal}{8}}^{\textcolor{teal}{3}}),
%\]
%and we can see that there is no other monomial of $M_{(2)}^{(2,1,1)}$ that has this image.

Using the proposition \ref{injmorphism}, we also prove in \cite{Montexte} the two last parts of theorem \ref{bigthm}:

\begin{prop}
For any positive integer $k$ and partition $\mu$, there is an injection $M_{\mu}^{\mu^k} \hookrightarrow M_{\mu^k}^{\mu}$.
%\begin{proof}
%We will prove it by induction, If $k=1$, we can use the identity map. Then, suppose that there is an injective morphism $\mathcal{F}: M_{\mu}^{\mu^k} \hookrightarrow M_{\mu^k}^{\mu}$, and consider the case $k+1$. Then, we have $\mu^{k+1} = \mu^k \uplus \mu$, so using the identity map on $M_{\mu}^{\mu}$, we can construct the morphism $\mathcal{F} \odot \text{Id}: M_{\mu}^{\mu^{k+1}} \hookrightarrow M_{\mu^{k+1}}^{\mu}$, which is also injective.
%\end{proof}
\end{prop}

\begin{prop}
If the Foulkes' conjecture is true up to $n-1$, then for all partition $\mu$ such that $\mu_1 \leq n$, there is an injection $M_{(n)}^{\mu} \hookrightarrow M_{\mu}^{(n)}$.
%\begin{proof}
%Suppose that $\mu = (\mu_1,...,\mu_k)$, so that $\mu = \mu_1  \uplus ... \uplus \mu_k$. If $\mu_{\ell}$ is the largest part such that $\mu_{\ell} = n$, denote $\mathcal{F}_{\ell+i}$ (for $1 \leq i \leq n-\ell$) the injective morphism from $M_{(\mu_{\ell+i})}^{(n)}$ to $M_{(n)}^{(\mu_{\ell+i})}$, which exists as we supposed the Foulkes' conjecture holds in all those cases. Also, let $\text{Id}$ be the identity map on $M_{(n)}^{(n)}$. Then, the morphism $\underbrace{\text{Id} \odot ... \odot \text{Id}}_{\ell \ \text{times}} \odot \mathcal{F}_{\ell+1} \odot ... \odot \mathcal{F}_{n}$, from $M_{\mu}^{(n)}$ to $M_{(n)}^{\mu}$, is composed of injective morphisms, so it is an injection.
%\end{proof} 
\end{prop} 

\section{Decomposition via semistandard homomorphisms}\label{ssh}

\subsection{Semistandard homomorphisms}

We use here the description of \cite{James} for a basis of the homomorphisms from $S^{\lambda}$ to a permutation module $M^{\gamma}$, and mostly use its notations.

Let $\lambda \vdash n$ and $\tau: \lambda \rightarrow \mathbb{N}^*$ be a tableau of content $\gamma$. We denote $T(\lambda,\gamma)$ the set of such tableaux. Remark that up to a relabelling of the entries, we can always choose $\gamma$ to be in decreasing order, so that it corresponds to a partition. Also, using the identification $\s_{\lambda} \cong \s_n$, the $\s_n$-module $\mathbb{C}[T(\lambda,\gamma)]$ is isomorphic to $M^{\gamma}$, where we now consider this module to be generated by the monomials $\xi^t$ (instead of the $\bm{x}^t$, as in the usual definition). In effect, for any bijective tableau $t: \gamma \rightarrow \bm{x}$, the map $f_t: \mathbb{C}[T(\lambda,\gamma)] \rightarrow M^{\gamma}$ such that $f_t(\tau) = \displaystyle \prod_{(i,j) \in \lambda} t(i,j)^{\tau(i,j)}$ is an isomorphism.
%For example, if $\lambda=(3,2)$ and $\gamma=(2,2,1)$  we have
%\[
%\tau = 
%\begin{ytableau}
%*(red) 2	&*(red)2 \\
%*(magenta)1	&*(magenta)1	&*(orange) 3
%\end{ytableau};
%\qquad
%t_{\lambda} = 
%\begin{ytableau}
%*(red) x_4	&*(red)x_5 \\
%*(magenta)x_1	&*(magenta)x_2	&*(orange) x_3
%\end{ytableau};
%\qquad
%f_{t_{\lambda}}(\tau) = x_{\textcolor{magenta}1}x_{\textcolor{magenta}2}x_{\textcolor{red}4}^{\textcolor{red}2}x_{\textcolor{red}5}^{\textcolor{red}2}x_{\textcolor{orange}3}^{\textcolor{orange}3}.
%\]
%Note that the tableau $\tau_{\lambda, \gamma}$ (of shape $\lambda$ and content $\gamma$ in row-reading order) is such that $f_{t_{\lambda}}(\tau_{\lambda, \gamma}) = \xi^{t_{\gamma}}$. 

Denote by $r_{\tau}$ the number of $\sigma \in R_{\tau}$ such that $\tau \cdot \sigma = \tau$. For every $\tau \in T(\lambda,\gamma)$, we define a homomorphism $\widehat{\Theta}_{\tau}$ by setting
\[
\begin{array}{crll}
\widehat{\Theta}_{\tau}: & M^{\lambda} & \longrightarrow & M^{\gamma} \\
				&  \xi^t	    & \mapsto		 & \displaystyle \frac{1}{r_{\tau}} \sum_{\sigma \in R_{\lambda}} f_t(\tau \cdot \sigma)
\end{array}
\]
%For example, using the same $\lambda, \gamma, t_{\lambda}$ and $\tau$ as in the previous example, we have
%\[
%\begin{array}{crll}
%\widehat{\Theta}_{\tau}:	&M^{\lambda} & \longrightarrow & \mathbb{C}[T(\lambda,\gamma)]  \cong  M^{\gamma} \\
%					& x_1x_2x_3x_4^2x_5^2	      & \mapsto	     & 
%\ f_{t_{\lambda}}\left(
%\ytableausetup{notabloids}
% \begin{ytableau}
%*(red) 2	&*(red)2 \\
%*(magenta)1	&*(magenta)1	&*(orange) 3
%\end{ytableau}
%+
%\begin{ytableau}
%*(red) 2	&*(red)2 \\
%*(magenta)1	&*(orange)3 &*(magenta)1
%\end{ytableau}
%+
%\begin{ytableau}
%*(red) 2	&*(red)2 \\
%*(orange)3 &*(magenta)1	&*(magenta)1
%\end{ytableau}
%\right) \\
%&& = &
%x_{\textcolor{magenta}1}x_{\textcolor{magenta}2}x_{\textcolor{red}4}^{\textcolor{red}2}x_{\textcolor{red}5}^{\textcolor{red}2}x_{\textcolor{orange}3}^{\textcolor{orange}3} + x_{\textcolor{magenta}1}x_{\textcolor{magenta}3}x_{\textcolor{red}4}^{\textcolor{red}2}x_{\textcolor{red}5}^{\textcolor{red}2}x_{\textcolor{orange}2}^{\textcolor{orange}3}+x_{\textcolor{magenta}2}x_{\textcolor{magenta}3}x_{\textcolor{red}4}^{\textcolor{red}2}x_{\textcolor{red}5}^{\textcolor{red}2}x_{\textcolor{orange}1}^{\textcolor{orange}3} \\ 
%
%\end{array}
%\]
If $i$ is the inclusion $i: S^{\lambda} \hookrightarrow M^{\lambda}$, we define $\Theta_{\tau}: S^{\lambda} \rightarrow M^{\gamma}$ to be the morphism $\widehat{\Theta}_{\tau} \circ i$. Note that $\Theta_{\tau} = 0$ if and only if $\tau$ has a column containing two equal entries. 

We say that a tableau $\tau$ with entries in $\mathbb{N}$ is \textit{semistandard} if it is weakly increasing in the rows (from left to right) and strictly increasing in the columns (from bottom to top). When $\tau \in T(\lambda,\gamma)$ is semistandard, we say that $\Theta_{\tau}$ is a \textit{semistandard homomorphism}. These homomorphisms have the nice following property:

\begin{prop}
The set $\{ \Theta_{\tau} \ | \ \tau \in T(\lambda,\gamma) \ \text{semistandard} \}$ is a basis for $\HOM_{\s_r}(S^{\lambda},M^{\gamma})$.
\end{prop}

\subsection{Application for the modules \texorpdfstring{$M_{\mu}^{\nu}$}{Mmn}}

We want to use the semistandard homomorphisms to study the modules $M_{\mu}^{\nu}$. First, combining the semistandard homomorphisms and the morphisms of section \ref{permmod}, we obtain the following setting:

\begin{center}
\begin{tikzcd}
S^{\lambda}
	\arrow[r]
	\arrow[rr, bend left, "\Theta_{\tau}"] 
& M_{\mu}^{\nu}
	\arrow[r, hook, "\widetilde{\phi}"]
	\arrow[rr, bend right, "\phi \circ \widetilde{\phi} = \text{Id}"]
&M^{\mu^n}
	\arrow[r, two heads, "\phi"]
&M_{\mu}^{\nu}
\end{tikzcd}
\end{center}

So for $\lambda$ a partition of $nm$ and $\tau$ a semistandard tableau of shape $\lambda$ and content $\mu^n$, if we define $\overline{\Theta}_{\tau} = \phi \circ \Theta_{\tau}$, we have that $\overline{\Theta}_{\tau}$ is a generating set of $\HOM_{\s_{nm}}(S^{\lambda}, M_{\mu}^{\nu})$. Hence, we can have a combinatorial description of the decomposition of the modules $M_{\mu}^{\nu}$. However, as $\phi$ is a projection, this set is far from being linearly independent, and this tool is generally not efficient to calculate this decomposition, although it can be a good tool to make proofs about these modules.

\section{Stability properties}\label{results2}

We use semistandard homomorphisms to prove that some sequences of plethysm coefficients are increasing. To do so, we generalize the arguments of de Boeck in \cite{deBoeck}. We only put here the lemmas needed to show theorems \ref{thm1} and \ref{thm2}. The proofs of these lemmas and theorems are in \cite{Montexte}. 

 For two tableaux $\tau_1,\tau_2$, we define the \textit{join} of the two tableaux, denote $\tau_1 \vee \tau_2$, to be the tableau such that the row $i$, when read from left to right, consists of the row $i$ of $\tau_1$ followed by the row $i$ of $\tau_2$.
%For example:
%\[
%\tau_1 = \begin{ytableau} 2 & 3 \\ 1 & 1 & 2 \\ 0 & 0 & 0 \end{ytableau}; \qquad \tau_2 = \begin{ytableau} 2 & 3 \\ 0 & 1 \end{ytableau}; \qquad \tau_1 \vee \tau_2 = \begin{ytableau} 2 & 3 \\ 1 & 1 & 2 & 2 & 3 \\ 0 & 0 & 0 & 0 & 1 \end{ytableau}.
%\] 
If $\tau_1$ is of shape $\mu_1$ and $\tau_2$ is of shape $\mu_2$, then $\tau_1 \vee \tau_2$ is of shape $\mu_1+\mu_2$, where we use the componentwise addition of integer vectors.

We use in this first lemma the tableau $\tau_{(nm'),\mu'^n}$ for $\mu' \vdash m'$, which is the only semistandard tableau of shape $(nm')$ and content $\mu'^n$. 
%For example, if $m'=4$, $n=3$ and $\mu'=(3,1)$, we have
%\[
%\tau_{(nm'),\mu'^n} = \begin{ytableau}1 & 1 & 1 & 2 & 2 & 2 & 3 & 3 & 3 & 4 & 5 & 6 \end{ytableau}
%\]
For any $\mu \vdash m$ and $\nu \vdash n$, we have the following lemma:

\begin{lem}
Let $\tau \in T(\lambda,\mu^n)$ such that $\overline{\Theta}_{\tau}: S^{\lambda} \rightarrow M_{\mu}^{\nu}$ is non-zero. Then, for any partition $\tilde{\mu}$ of an integer $\tilde{m}$, the tableau $\tilde{\tau} = \tau \vee \tau_{(n\tilde{m}),\tilde{\mu}^n}$ is such that $\overline{\Theta}_{\tilde{\tau}}: S^{\lambda + (n\tilde{m})} \rightarrow M_{\mu + \tilde{\mu}}^{\nu}$ is also non-zero.
%\begin{proof}[Proof (sketch)]
%As the morphism is non-zero and $S^{\lambda}$ is a cyclic module, we have that for the row-reading order tableau $t_{\lambda}$, $\overline{\Theta}_{\tau}(\Delta_{t_{\lambda}}) \neq 0$. Choose a monomial appearing in $\overline{\Theta}_{\tau}(\Delta_{t_{\lambda}})$ with a non-zero coefficient, and let $\bm{T} \in \mathcal{N}_{\mu,\nu}$ be such that $\xi^{*\bm{T}}$ is equal to this monomial. As $\overline{\Theta}_{\tau} = \phi \circ \Theta_{\tau}$ and $\phi$ is surjective, there exists $\rho \in T(\lambda, \mu^n)$ such that $\phi(f_{t_{\lambda}}(\rho)) = \xi^{*\bm{T}}$. 
%We have
%\[
%\overline{\Theta}_{\tau}(\Delta_{t_{\lambda}}) = \displaystyle \frac{1}{r_{\tau}} \sum_{(\sigma, \pi) \in R_{\lambda} \times C_{\lambda}} \varepsilon(\pi) \pi \sigma \cdot \phi(f_{t_{\lambda}}(\tau)).
%\]
%Denote $B_{t_{\lambda}}(\tau,\bm{T}) = \{ (\sigma, \pi) \in R_{\lambda} \times C_{\lambda} \ | \ \phi(f_{t_{\lambda}}(\tau \cdot \sigma\pi)) = \xi^{*\bm{T}} \}$. Then, the coefficient of $\xi^{*\bm{T}}$ is
%%\[
%%\overline{\Theta}_{\tau}(\Delta_{t_{\lambda}}) = \displaystyle \frac{1}{r_{\tau}} \left( \sum_{(\sigma, \pi) \in B_{t_{\lambda}}(\tau,\bm{T})} \varepsilon(\pi) \pi \sigma \cdot \phi(f_{t_{\lambda}}(\tau)) + \sum_{\substack{(\sigma, \pi) \in R_{\lambda} \times C_{\lambda}\\ (\sigma,\pi) \not\in B_{t_{\lambda}}(\tau,\bm{T})}} \varepsilon(\pi) \pi \sigma \cdot \phi(f_{t_{\lambda}}(\tau)) \right).
%%\]
%\[
%\displaystyle \frac{1}{r_{\tau}} \sum_{(\sigma,\pi) \in B_{t_{\lambda}}(\tau,\bm{T})} \varepsilon(\pi) \quad \neq 0.
%\]
%Now consider the morphism $\overline{\Theta}_{\tilde{\tau}}$. To do so, first define
%\[
%A_k = \{i \in \{1,...,n\tilde{m}\} \ | \tau_{(n\tilde{m}),\tilde{\mu}^n}(i+1,0) \equiv k \ (\text{mod} \ n)\}.
%\]
%Then, let $\tilde{\bm{x}}$ be the set $\{x_{nm+1},x_{nm+2},...,x_{n(m+\tilde{m})}\}$, and partition this set in subsets $\tilde{\bm{x}}^{(k)}$ such that $\tilde{\bm{x}}^{(k)} = \{x_{nm+i} \ | \ i \in A_k \}$. If $t_{\nu}(i,j)=k$, consider the tableau $\tilde{\bm{T}} \in \mathcal{N}_{\mu+\tilde{\mu},\nu}$ such that $\tilde{\bm{T}}(i,j)  = \bm{T}(i,j) \vee t_{k}$, where $t_k$ is the tableau of shape $\tilde{\mu}$ with entries in $\tilde{\bm{x}}^{(k)}$ in row-reading order. Also, consider the bijective tableau $s: (n\tilde{m}) \rightarrow \tilde{\bm{x}}$ in row-reading order and $\tilde{t} = t_{\lambda} \vee s$. The tableaux $\tilde{\bm{T}}$ and $\tilde{t}$ are constructed so that $\phi(f_{\tilde{t}}(\rho \vee \tau_{(n\tilde{m}),\tilde{\mu}^n})) = \xi^{*\tilde{\bm{T}}}$. 
%
%As shown with more details in \cite{Montexte}, the construction of $\tilde{\tau}$ implies that the coefficient of $\xi^{*\tilde{\bm{T}}}$ is 
%%To prove the lemma, it is enough to show that the coefficient of $\xi^{*\tilde{\bm{T}}}$ in $\overline{\Theta}_{\tilde{\tau}}(\Delta_{\tilde{t}})$ is also non-zero. Using the same technique, this coefficient is
%%\[
%%\displaystyle \frac{1}{r_{\tilde{\tau}}} \sum_{(\tilde{\sigma},\tilde{\pi}) \in B_{\tilde{t}}(\tilde{\tau},\tilde{\bm{T}})} \varepsilon(\tilde{\pi}).
%%\]
%%%We can see that $C_{\lambda+(n\tilde{m})} = C_{\lambda}$, because it only permutes the ``$\bm{T}$ part'' of $\tilde{\bm{T}}$, and the ``$\tau$ part'' of $\tilde{\tau}$.  As $(\tilde{\sigma},\pi) \in B_{\tilde{t}}(\tilde{\tau},\tilde{\bm{T}}) \subseteq R_{\lambda+(n\tilde{m})} \times C_{\lambda}$ is such that $\phi(f_{\tilde{\tau}}(\tilde{\tau} \cdot \tilde{\sigma}\pi)) = \xi^{*\tilde{\bm{T}}}$, by definition, this implies that $\tilde{\tau} \cdot \tilde{\sigma}\pi = (\tau \cdot \sigma\pi) \vee (\tau_{(n\tilde{m}),\tilde{\mu}^n} \cdot \omega)$ for $(\sigma, \pi) \in B_{t_{\lambda}}(\tau,\bm{T})$ and $\omega \in R_{(n\tilde{m})}$; else, $\pi$ would permute some variables that are not in the ``$\tau$ part'' of $\tilde{\tau}$, so the image under $\phi$ could not be $\xi^{*\tilde{\bm{T}}}$. Moreover, as $\xi^{*\tilde{\bm{T}}}$ is fixed, there is only one choice for the entries of $\tau_{(n\tilde{m}),\tilde{\mu}^n} \cdot \omega$, so there is always exactly $\tilde{\mu}^n!$ possible permutations in $R_{(n\tilde{m})}$ to do that. 
%%As shown with more details in \cite{Montexte}, the construction of $\tilde{\tau}$ implies that the coefficient of $\xi^{*\tilde{\bm{T}}}$ is 
%\[
%\displaystyle \frac{\tilde{\mu}^n!}{r_{\tilde{\tau}}} \sum_{(\sigma,\tau) \in B_{t_{\lambda}}(\tau,\bm{T})} \varepsilon(\pi),
%\]
%which is a non-zero multiple of the coefficient of $\xi^{*\bm{T}}$, so it is also non-zero.
%\end{proof}
\end{lem}
%
This lemma is the main ingredient to prove theorem \ref{thm1}.
%
%%\begin{proof}[Proof of theorem \ref{thm1}]
%%As the tableaux of the form $\tau \vee \tau_{(n\tilde{m}),\tilde{\mu}^n}$ are not semistandard, the associated morphisms may be linearly dependants, so we have to prove that they are not. Let $\{\overline{\Theta}_{\tau_1}, ..., \overline{\Theta}_{\tau_r}\}$ be a basis of $\text{Hom}_{\s_{nm}}(S^{\lambda},M_{\mu}^{\nu})$. Denote $\tilde{\tau}_i = \tau_i \vee \tau_{(n\tilde{m}),\tilde{\mu}^n}$, and suppose that there exists $\alpha_1,...,\alpha_r$ such that $\alpha_1\overline{\Theta}_{\tilde{\tau}_1} + ... + \alpha_r\overline{\Theta}_{\tilde{\tau}_r} = 0$. It follows that 
%%\[
%%\displaystyle \sum_{i=1}^r \alpha_i \overline{\Theta}_{\tilde{\tau}_i}(\Delta_{t_{\lambda+(n\tilde{m})}}) = 0,
%%\]
%%so any term of this sum vanishes; in particular, for any $\bm{T} \in \mathcal{N}_{\mu, \nu}$, the coefficient of $\xi^{*\tilde{\bm{T}}}$ is $0$. But by the preceding lemma, it means that the coefficient of $\xi^{* \bm{T}}$ times the non-zero constant $\frac{\tilde{\mu}!^n}{r_{\tilde{\tau}}}$ is zero, which means that this coefficient must be zero. As this is true for any monomial of $M_{\mu}^{\nu}$, we have $\alpha_1\overline{\Theta}_{\tau_1} + ... + \alpha_r\overline{\Theta}_{\tau_r} = 0$. But this means that $\alpha_1 = ... =  \alpha_r = 0$, because this set is a basis. Hence, $\{\overline{\Theta}_{\tilde{\tau}_1}, ...,\overline{\Theta}_{\tilde{\tau}_r}\}$ must also be linearly independent.
%%\end{proof}

We also generalize another stability property, originally due to Dent \cite{Dent}. For a tableau $\tau'$ of shape $(2^k)$ and another tableau $\tau$, we have that $\tau' \vee \tau$ corresponds to adding two columns of length $k$ at the left of $\tau$. Moreover, we write $\tau_{(2^k),(2^k)}$ for the only semistandard tableau of shape and content $(2^k)$.
%For example, if $k=6$, then
%\[
%\tau_{(2^k),(2^k)} = \begin{ytableau} 6 & 6 \\ 5 & 5 \\ 4 & 4 \\ 3 & 3 \\ 2 & 2 \\ 1 & 1 \end{ytableau}
%\]
Using this, we prove the following lemma:

\begin{lem}
Let $\tau \in T(\lambda,\mu^n)$ such that $\overline{\Theta}_{\tau}: S^{\lambda} \rightarrow M_{\mu}^{\nu}$ is non-zero. Then, if the number of parts of $\mu$ is $\tilde{m}$, the tableau $\tilde{\tau} = \tau_{(2^{\tilde{m}n}),(2^{\tilde{m}n})} \vee \tau$ is such that $\overline{\Theta}_{\tilde{\tau}}: S^{\lambda + (2^{\tilde{m}n})} \rightarrow M_{\mu + (2^{\tilde{m}})}^{\nu}$ is also non-zero.
%\begin{proof}[Proof (Sketch)]
%Consider the set of variables $\tilde{\bm{x}} = \{x_{nm+1},...,x_{n(m+2\tilde{m})}\}$ and the bijective tableau $t: (2^{n\tilde{m}}) \rightarrow \tilde{\bm{x}}$ in row-reading order. For $\tilde{t} = t \vee t_{\lambda}$, which is a bijective tableau of shape $\lambda + (2^{\tilde{m}n})$ with entries in $\bm{x} \cup \tilde{\bm{x}}$, consider 
%\[
%\overline{\Theta}_{\tilde{\tau}}(\Delta_{\tilde{t}}) = \displaystyle \frac{1}{r_{\tilde{\tau}}} \sum_{(\tilde{\sigma},\tilde{\pi}) \in R_{\tilde{t}} \times C_{\tilde{t}}} \varepsilon(\tilde{\pi}) \tilde{\pi} \tilde{\sigma} \cdot \phi(f_{\tilde{t}}(\tilde{\tau})).
%\] 
%We claim (and prove in \cite{Montexte}) that $\tilde{\sigma} \in R_{\tilde{t}}$ is such that $\displaystyle \sum_{\tilde{\pi} \in C_{\tilde{t}}} \varepsilon(\tilde{\pi}) \tilde{\pi} \tilde{\sigma} \cdot f_{\tilde{t}}(\tilde{\tau}) \ \neq 0$ if and only if there is $\sigma \in R_{\lambda}$ such that $\tilde{\tau} = \tau_{(2^{\tilde{m}n}),(2^{\tilde{m}n})} \vee (\tau \cdot \sigma)$ and $\displaystyle \sum_{\pi \in C_{\lambda}} \varepsilon(\pi) \pi \sigma \cdot f_{t_{\lambda}}(\tau) \ \neq 0$. 
%%The \textit{if} part is by construction of $\tilde{\tau}$. For the \textit{only if} part, we know that this sum is zero if and only if $\tilde{\tau} \cdot \tilde{\sigma}$ has two equal entries in the same column. For row $j > \lambda_1'$, there is only the entries $j$. For row $j=\lambda_1'$, we know that as $\tau$ is semistandard, this row contains only values that are greater or equal to $j$. So, if $\tilde{\sigma}$ permutes the ``$\tau_{(2^{\tilde{m}n}),(2^{\tilde{m}n})}$ part'', it would be by entries greater or equal to $j$, but these values are already in these columns, above. So, we would have $\displaystyle \sum_{\tilde{\pi} \in C_{\tilde{t}}} \varepsilon(\tilde{\pi}) \tilde{\pi} \tilde{\sigma} \cdot f_{t_{\lambda}}(\tilde{\tau}) \ = 0$. Proceeding with the same argument down to the first row, we have that the ``$\tau_{(2^{\tilde{m}n}),(2^{\tilde{m}n})}$ part'' is preserved by $\tilde{\sigma}$, which prove the claim.
%We also have that $C_{\tilde{t}} \cong (\s_{\tilde{m}n})^2 \times C_{\lambda}$. This allows us to write
%\[
%\overline{\Theta}_{\tilde{\tau}}(\Delta_{\tilde{t}}) = \displaystyle \frac{1}{r_{\tilde{\tau}}} \sum_{(\sigma,\pi,\omega) \in R_{\lambda} \times C_{\lambda} \times (\s_{\tilde{m}n})^2} \varepsilon(\pi) \varepsilon(\omega) \ \phi(f_{\tilde{t}}((\tau_{(2^{\tilde{m}n}),(2^{\tilde{m}n}))} \cdot \omega) \vee (\tau \cdot \sigma\pi))).
%\] 
%
%Now let's take $\tau$ such that $\overline{\Theta}_{\tau} \neq 0$. We know that there exists $\bm{T} \in \mathcal{N}_{\mu,\nu}$ such that the coefficient of $\xi^{*\bm{T}}$ in $\overline{\Theta}_{\tau}(\Delta_{t_{\lambda}})$ is non-zero. Let $\rho \in T(\lambda, \mu^n)$ such that $\phi(f_{t_{\lambda}}(\rho)) = \xi^{*\bm{T}}$. We know from the previous lemma that this coefficient is 
%\[
%\displaystyle \frac{1}{r_{\tau}} \sum_{(\sigma,\pi) \in B_{t_{\lambda}}(\tau, \bm{T})} \varepsilon(\pi).
%\]
%Now consider the partition of $\tilde{\bm{x}}$ into $n$ subsets 
%\[
%\tilde{\bm{x}}^{(k)} = \{x_{nm+2k-1},x_{nm+2k},..., x_{nm+2(n-1)\tilde{m}+2k-1}, x_{nm+2(n-1)\tilde{m}+2k}\}.
%\]
%Let $t_k: (2^{\tilde{m}}) \rightarrow \tilde{\bm{x}}^{(k)}$ be in row-reading order, and denote $\tilde{\bm{T}} \in \mathcal{N}_{\mu+(2^{\tilde{m}}),\nu}$ to be the tableau such that if $t_{\nu}(i,j) = k$, we have $\tilde{\bm{T}}(i,j) = t_k \vee \bm{T}(i,j)$. This tableau is such that $\phi(f_{\tilde{t}}(\tau_{(2^{\tilde{m}n}),(2^{\tilde{m}n})} \vee \rho)) = \xi^{*\tilde{\bm{T}}}$. 
%%The coefficient of $\xi^{*\tilde{\bm{T}}}$ in $\overline{\Theta}_{\tilde{\tau}}(\Delta_{\tilde{t}})$ is
%%\[
%%\displaystyle \frac{1}{r_{\tilde{\tau}}} \sum_{(\tilde{\sigma},\tilde{\pi}) \in B_{\tilde{t}}(\tilde{\tau},\tilde{\bm{T}})} \varepsilon(\tilde{\pi}).
%%\]
%Using the claim, we show in \cite{Montexte} that $\omega$ must be an even permutation uniquely determined by $(\sigma,\tau)$, so we have
%%By the claim, we can restrict ourselves to $(\sigma,\pi) \in B_{t_{\lambda}}(\tau,\bm{T})$ and $\omega \in \s_{\tilde{m}n}^2$ such that $(\tilde{\sigma},\tilde{\pi}) = (\sigma, \pi\omega)$. Remark that if $\phi(f_{\tilde{t}}(\tilde{\tau})) = \phi(f_{\tilde{t}}(\tilde{\tau}'))$, then there must be $\upsilon \in \s_{\tilde{m}n}$ such that $\upsilon \cdot \tilde{\tau} = \tilde{\tau}'$ (we use here the left action, {\rm i.e.} it permutes the values). So, if the right action of $(\sigma,\pi,\omega) \in B_{t_{\lambda}}(\tau,\bm{T}) \times (\s_{\tilde{m}n})^2$ is the same as the left action of a certain $\upsilon$, it must be that $\upsilon \cdot \tau = \tau \cdot \sigma\pi$ and $\upsilon \cdot \tau_{(2^{\tilde{m}n}),(2^{\tilde{m}n})} = \tau_{(2^{\tilde{m}n}),(2^{\tilde{m}n})} \cdot \omega$. This implies that $\omega$ is completely determined by $(\sigma,\pi)$. Moreover, $\omega$ must be an even permutation, as it is a composition of permutations that exchange the two values in a row with the two values of another row. So, we have:
%\[
%\displaystyle \frac{1}{r_{\tilde{\tau}}} \sum_{(\tilde{\sigma},\tilde{\pi}) \in B_{\tilde{t}}(\tilde{\tau},\tilde{\bm{T}})} \varepsilon(\tilde{\pi}) = \frac{1}{r_{\tilde{\tau}}} \sum_{(\sigma,\pi) \in B_{t_{\lambda}}(\tau, \bm{T})} \varepsilon(\pi)\varepsilon(\omega) = \frac{1}{r_{\tilde{\tau}}} \sum_{(\sigma,\pi) \in B_{t_{\lambda}}(\tau, \bm{T})} \varepsilon(\pi),
%\]
%which is a non-zero multiple of the coefficient of $\xi^{*\bm{T}}$, so it is non-zero.
%\end{proof}
\end{lem}

This lemma is also the main ingredient to prove theorem \ref{thm2}, by a similar proof than the one of \ref{thm1}.

%\section{Conclusion}
%
%In the future, we hope to find more cases where there is an injection $M_{\mu}^{\nu} \hookrightarrow M_{\nu}^{\mu}$, i.e. more cases where $\nu \trianglelefteq \mu$. We also hope to find additional clues that this is an order relation, like propagation theorems in the style of the one of McKay. It would also be great to find a way to use efficiently the semistandard homomorphisms. 
%
%In general, this conjecture deserved to be studied more. In addition to being a nice generalization of the Foulkes' conjecture, it would be a great step toward the understanding of the plethysm coefficients. 

%\bibliographystyle{plain}
%\bibliography{02Tetreault}
\printbibliography

%\appendix
%
%\pagebreak
%
%\section{Representation theory of the symmetric group}
%
%As a means of deriving properties of plethysms, we consider representations of the symmetric group $\s_n$. The necessary link between the two subjects of study is established via the classical Frobenius transform of characters. This is a linear transform which encodes a character as a symmetric function, with the property that irreducible characters are sent to the basis of Schur symmetric function. All the fundamental questions of representation theory of $\s_n$ may be efficiently translated into calculation in the ring $\Lambda$ of symmetric functions. We use a polynomial construction of these modules, as in \cite{Bergeron}.
%
%A \textit{diagram} is simply a finite subset $\bm{d}$ of $\mathbb{N} \times \mathbb{N}$, whose elements are called \textit{cells}. A cell $(i,j)$ is often geometrically represented in $\mathbb{N} \times \mathbb{N}$ by the \textit{box} having vertices $(i,j)$, $(i+1,j)$, $(i,j+1)$ and $(i+1,j+1)$. The $k^{th}$ row (resp. column) of a diagram $\bm{d}$ is the subset of cells such that $i = k-1$ (resp. $j = k-1$). We order the cells of $\bm{d}$ in reverse lexicographical order, so that $(i',j') < (i,j)$ if and only if $j' < j$ or $j'=j$ and $i' < i$. 
%
%A \textit{tableau} $\tau$ of \textit{shape} $\bm{d}$ with entries in a set $\Omega$ is a map $\tau: \bm{d} \rightarrow \Omega$. It is often displayed as a filing of $\bm{d}$, with the value $\tau(i,j)$ appearing in the box corresponding to the cell $(i,j)$. Since we think of this in terms of Cartesian coordinates, our diagrams and tableaux are drawn using the French convention.
%
%The $\mathbb{C}$-vector space formally spanned by these tableaux is a $(\s_{\Omega} \times \s_{\bm{d}})$-bimodule, where $\s_{\Omega}$ acts on values, {\sl i.e.}  $(\rho \cdot \tau)(i,j) = \rho(\tau(i,j))$ for $\rho \in \s_{\Omega}$; and $ \s_{\bm{d}}$ acts on places (or cells), {\sl i.e.}  $(\tau \cdot \pi)(i,j) = \tau(\pi(i,j))$ for $\pi \in \s_{\bm{d}}$. We then consider $R_{\tau}$ (resp. $C_{\tau}$), the subgroup of permutations in $\s_{\bm{d}}$ which only permutes cells lying in a same row (resp. column) in $\tau$. For $a \in \Omega$, if we denote $\gamma_a = |\tau^{-1}(a)|$ (i.e. the number of occurrences of $a$ in $\tau$), then the sequence $\gamma = (\gamma_a)_{a \in \Omega}$ is called the \textit{content} of $\tau$.
%
%The \textit{row-reading word} of a tableau $\tau$ is the word on $\Omega$ obtained by reading the entries of the tableau from left to right in each row, starting with the bottom one. If $\Omega$ is an ordered set and the reading word of $\tau$ doesn't have inversions, we say that it is in \textit{row-reading order}. If we fix $\bm{d}$ and $\gamma$, the row-reading order tableau is unique, and we denote it $\tau_{\bm{d},\gamma}$.
%
%A diagram $\bm{d}$ is said to be a \textit{Ferrers diagram} if for every $(i,j) \in \bm{d}$, the diagram also contains every $(i',j') \in \mathbb{N} \times \mathbb{N}$ such that $i' \leq i$ and $j' \leq j$. If $\lambda_i$ is the number of cells in the $i^{th}$ row of a Ferrers diagram, then $\lambda = (\lambda_1,...,\lambda_k)$ is a decreasing sequence of positive integers, {\rm i.e.} a \textit{partition}, and this sequence completely determines the diagram. We denote by $\lambda$ both the partition and its associated Ferrers diagram.
%
%When $|\lambda| = |\Omega| =r$, we may consider the set of bijective (i.e. of content $(1,...,1)$ ) tableaux $t: \lambda \rightarrow \Omega$, usually denoted by lowercase Latin letters. Since, in that case, $\s_{\Omega} \cong \s_{[\lambda]}$, for our purpose it suffices to consider the action of $\s_r$ on the $\mathbb{C}$-span of these bijective tableaux. Unless specified, we choose $\Omega$ to be the set $\bm{x} = \{x_1,...,x_r\}$, but we can make isomorphic constructions for any set of size $r$. The row-reading tableau is then denoted $t_{\lambda}$, $R_{t_{\lambda}}$ is denoted $R_{\lambda}$ and $C_{t_{\lambda}}$ is denoted $C_{\lambda}$. There is a bijection between the $\mathbb{C}$-span of bijective tableaux of shape $\lambda$ and $\s_r$, by associating a tableau $t$ to the permutation $\pi$ such that $t = \pi \cdot t_{\lambda}$. So, as a $\s_r$-module, it is isomorphic to the left regular representation $\mathbb{C}\s_r$.
%
%For each bijective tableau $t$ of shape $\lambda$, we associate the \textit{tableau monomial}, which is $\bm{x}^{t} = \displaystyle \prod_{(i,j) \in [\lambda]} t(i,j)^j$. If we consider the polynomial ring $\mathbb{C}[\bm{x}]$ as a $\s_r$-module, where $\pi \in \s_r$ permutes the variables, then we have that $\pi \cdot \bm{x}^{t} = \bm{x}^{\pi \cdot t}$, so $t \mapsto \bm{x}^{t}$ is a $\s_r$-module homomorphism. The image of this homomorphism is the following $\s_r$-submodule of $\mathbb{C}[\bm{x}]$:
%\[
%M^{\lambda} = \mathbb{C}[\bm{x}^{t} \ | \ t \ \text{bijective tableau of shape} \ \lambda].
%\]
%It is called the \textit{permutation module} associated to $\lambda$. It is equal to $\mathbb{C}\s_r\bm{x}^{t_{\lambda}} = \mathbb{C}[\pi \cdot \bm{x}^{t_{\lambda}} \ | \ \pi \in \s_r]$, so it is a cyclic module. 
%
%Also, for a bijective tableau of shape $\lambda$, we define the following polynomial:
%\[
%\Delta_{t} = \displaystyle \sum_{\pi \in C_{t}} \varepsilon(\pi) \pi \cdot \bm{x}^{t}.
%\]
%This polynomial also has the property that $\pi \cdot \Delta_{t} = \Delta_{\pi \cdot t}$ for all $\pi \in \s_r$ and for all bijective tableaux $t$, so $t \mapsto \Delta_{t}$ is also a $\s_r$-module homomorphism, whose image is the following submodule of $M^{\lambda}$:
%\[ 
%S^{\lambda} = \mathbb{C}[\Delta_{t} \ | \ t \ \text{bijective tableau of shape} \ \lambda].
%\]
%It is called the Specht module associated to $\lambda$. We can show that we have $\Delta_{\tau} = \displaystyle \prod_{(x,y)} (x - y)$, where indices run of pairs $(x,y)$ of entries of $\tau$ such that $x$ occurs in the same column as $y$, but in a lower row.
%
%We recall some well-known facts about these modules (see Sagan, \cite{Sagan}):
%
%\begin{prop}
%\begin{itemize}
%\item The set $\{S^{\lambda} \ | \ \lambda \vdash r\}$ is a complete list of non-isomorphic irreducible representations of $\s_{r}$; 
%\item The Frobenius transform of the character of $M^{\lambda}$ is $h_{\lambda}$;
%\item The Frobenius transform of the character of $S^{\lambda}$ is $s_{\lambda}$.
%\end{itemize}
%\end{prop}   

\end{document}
