\documentclass[a4paper,12pt]{article}

\usepackage{amsmath,amsfonts} 

\begin{document}

% mathematische Funktionen und Bezeichner
\newcommand{\Stab}{\operatorname{Stab}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\Rad}{\operatorname{Rad}}

\newcommand{\I}{\ |\ }
\newcommand{\Gn}{\Gamma_n}
\newcommand{\Dn}{{\Delta_n}}
\newcommand{\w}{\omega}
\newcommand{\Ny}[1]{{\nu^{#1}}}
\newcommand{\zerl}{\models}
\newcommand{\partition}{\vdash}
\newcommand{\verf}{\mathop{|\!\!\!\sim}}
\newcommand{\pfmaxverf}{\mathop{\verf\cdot}\nolimits_{\text{pf}}}

\newcommand{\N}{{\mathbb{N}}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\NS}{\N^\ast}    % NS ~ N^* ~ N-Stern
  
\newtheorem{main}{Main Theorem}
\renewcommand{\themain}{}
\newtheorem{lemma}{Lemma}
\newtheorem{mainlemma}{Main Lemma}
\renewcommand{\themainlemma}{}

\title{On the Automorphism Group \\  of Solomon's Descent Algebra}
\author{Thorsten Bauer\\ 
        \\
        Mathematisches Seminar\\
        Christian--Albrechts--Universit\"at zu Kiel\\
        D-24118 Kiel, Germany}
\date{}

\maketitle

By $\Dn$ we denote the subalgebra of $\Q S_n$ which is known as
Solomon's Descent Algebra \cite{solomon,reutenauer,loewy}.
This article gives a sketch of a proof for the following
theorem about the automorphisms of $\Dn$.
(Details and related developments will be given
in a forthcoming publication.)
\begin{main}
  For the automorphism group of Solomon's Descent Algebra $\Dn$ 
  the following holds:
$$\Aut(\Dn) = 
\begin{cases}
  \text{\rm\ Inn}(\Dn) & \text{\ if $n$ odd,} \\[1ex]
  \text{\rm\ Inn}(\Dn) \times C_2 & \text{\ if $n$ even.}
\end{cases}
$$
\end{main}

The proof of this theorem splits into four main steps:
\begin{itemize}
\item 
  the reduction of the problem to show
  that a stabilizer of a certain set 
  is generated by inner automorphisms,
\item
  the definition of a graph $\Gn$,
\item
  the mentioned stabilizer is determined
  by its action on $\Gn$,
\item
  the action of the stabilizer on $\Gn$
  is induced by conjugation 
  by invertible elements of $\Dn$, i.e. by inner automorphisms,
  and possibly a central involutionary outer automorphism.
\end{itemize}

\section{Notation}

We write $\N$ for the set of positive integers $\{1,2,\ldots\}$,
$\NS$ for the free monoid generated by $\N$
and $\Q\NS$ for the free algebra generated by $\N$
over the field of the rational numbers $\Q$.
The elements of $\NS$ are written as words in the alphabet $\N$,
e.g. $1\, 3\, 2 \in \NS$ is the product of $1$, $3$ and $2$.
If $w=w_1\ldots w_k$ is a word, we set $|w|:=k$, the {\em length} of $w$.

Let $n\in\N$ and $p=p_1\ldots p_k\in\NS$ such that $p_1,\ldots,p_k\in\N$.
The word $p$ is called a {\em partition} of $n$ ($p\partition n$)
if $p_1+\cdots + p_k=n$ and
$p_1 \geq p_2 \geq \dots \geq p_k$.
We write $p(n)$ for the number of all partitions of $n$.
For each letter $c$ we set 
$$a_c(p):=|\{ i \I p_i=c \}|,$$
the number of occurences of the letter $c$ in $p$.
Then we may write 
$$p=n^{a_n(p)} (n-1)^{a_{n-1}(p)} \ldots 1^{a_1(p)}.$$

\section{Reduction of the problem}
\label{SubReduction}

By a theorem of Malcev \cite{malcev}, \cite[11.6]{pierce} we know
that the set $1+\Rad(\Dn)$ of invertible elements
acts transitively by conjugation
on the set of complements of $\Rad(\Dn)$.

By Frattini's lemma 
\cite[3.3]{kurzweil}\footnote{
  Though the lemma is only stated for finite groups,
  it holds for infinte groups, too.
},
for each complement $H$ of $\Rad(\Dn)$ we have:
\begin{equation}
\label{frattini}
\Aut(\Dn) \ =\  \Stab_{\Aut(\Dn)}(H)\ \overline{(1+\Rad(\Dn))},
\end{equation}
where $\overline{(1+\Rad(\Dn))}$ denotes 
the group of automorphisms induced by conjugation by the elements
of $1+\Rad(\Dn)$.

Now we construct a complement $H$ of $\Rad(\Dn)$
to which the above mentioned theorem and lemma will be applied.

In \cite{module}, D. Blessenohl and H. Laue 
define elements\footnote{
  In \cite{module} these elements are not indexed
  by the partitions themselves but by listing them
  in the lexicographically decreasing order: $\Ny{(j)}$
  instead of $\Ny{p}$ if $p$ is the $j$-th partition.}
$\Ny{p}$, $p\partition n$,
with the following properties \cite[1.2 Proposition]{module}:
\begin{enumerate}
\renewcommand{\labelenumi}{{\rm (\alph{enumi})}}
\item
  $\Ny{p}$ is an idempotent for each $p\partition n$.
\item
  $\Ny{p}\Ny{r}\ =\ 0$ for all partitions $p,r$ such that $p\neq r$.
\item
  $\sum\limits_{p \partition n} \Ny{p} \ =\  1$.
\item
  $\Dn \ =\ \langle\  \Ny{p} \I p \partition n \ \rangle_\Q
  \oplus\ \Rad(\Dn)$.
\end{enumerate}

The set $H:=\langle\  \Ny{p} \I p \partition n \ \rangle_\Q$
is a subalgebra-complement of $\Rad(\Dn)$ in $\Dn$.
We observe that 
$$H \cong \underbrace{\Q \oplus \cdots \oplus \Q}_{p(n)},$$
and that $\{ \Ny{p} \I p \partition n\}$ is the unique set 
of $p(n)$ mutually orthogonal idempotents $\neq 0$ of $H$.

By Equation (\ref{frattini}) it suffices to show
that $\Stab_{\Aut(\Dn)}(H)$ is generated by inner automorphisms
in the case that $n$ is even
and by inner automorphisms and an involutionary
outer automorphism in the case that $n$ is odd.

\section{Directed graph of partitions}
\label{SubGraph}

Now we define a directed graph $\Gn$,
the nodes of which are the partitions of $n$.
The node $r$ is called connected with $p$ ($r \pfmaxverf p$)
if $|r|=|p|+1$
and there exist letters $c,d\in\N$ such that
$c\neq d$, $a_c(r)=a_c(p)+1$, $a_d(r)=a_d(p)+1$
and $a_{c+d}(r)+1=a_{c+d}(p)$,
i.e. $p$ can be obtained from $r$ by
coalescing two different letters of $r$
and reordering the letters to obtain a partition.

The shape of the graph $\Gamma_7$
is shown in Figure \ref{fig1}.
\begin{figure}
  \begin{center}
    \setlength{\unitlength}{1.111111em}
    \begin{picture}(21.6,33.3)
      \put(8.1,32.8){\makebox(0,0){7}}
      \put(2.7,27.4){\makebox(0,0){6 1}}
      \put(8.1,27.4){\makebox(0,0){5 2}}
      \put(13.5,27.4){\makebox(0,0){4 3}}
      \put(0.0,22.0){\makebox(0,0){5 1 1}}
      \put(5.4,22.0){\makebox(0,0){4 2 1}}
      \put(10.8,22.0){\makebox(0,0){3 3 1}}
      \put(16.2,22.0){\makebox(0,0){3 2 2}}
      \put(2.7,16.6){\makebox(0,0){4 1 1 1}}
      \put(8.1,16.6){\makebox(0,0){3 2 1 1}}
      \put(13.5,16.6){\makebox(0,0){2 2 2 1}}
      \put(5.4,11.2){\makebox(0,0){3 1 1 1 1}}
      \put(10.8,11.2){\makebox(0,0){2 2 1 1 1}}
      \put(8.1,5.8){\makebox(0,0){2 1 1 1 1 1}}
      \put(8.1,0.4){\makebox(0,0){1 1 1 1 1 1 1}}
      \thicklines
      \put(3.2,27.9){\vector(1,1){4.5}} % von 6 1 zu 7
      \put(8.1,27.9){\vector(0,1){4.5}} % von 5 2 zu 7
      \put(13.1,27.9){\vector(-1,1){4.5}} % von 4 3 zu 7
      \put(0.2,22.5){\vector(1,2){2.3}} % von 5 1 1 zu 6 1
      \put(5.2,22.5){\vector(-1,2){2.3}} % von 4 2 1 zu 6 1
      \put(5.6,22.5){\vector(1,2){2.3}} % von 4 2 1 zu 5 2
      \put(15.5,22.5){\vector(-3,2){6.8}} % von 3 2 2 zu 5 2
      \put(6.1,22.5){\vector(3,2){6.8}} % von 4 2 1 zu 4 3
      \put(11.0,22.5){\vector(1,2){2.3}} % von 3 3 1 zu 4 3
      \put(2.5,17.1){\vector(-1,2){2.3}} % von 4 1 1 1 zu 5 1 1
      \put(7.4,17.1){\vector(-3,2){6.8}} % von 3 2 1 1 zu 5 1 1
      \put(7.9,17.1){\vector(-1,2){2.3}} % von 3 2 1 1 zu 4 2 1
      \put(8.3,17.1){\vector(1,2){2.2}} % von 3 2 1 1 zu 3 3 1
      \put(13.7,17.1){\vector(1,2){2.3}} % von 2 2 2 1 zu 3 2 2
      \put(5.2,11.7){\vector(-1,2){2.3}} % von 3 1 1 1 1 zu 4 1 1 1
      \put(10.6,11.7){\vector(-1,2){2.2}} % von 2 2 1 1 1 zu 3 2 1 1
      \put(7.9,6.3){\vector(-1,2){2.3}} % von 2 1 1 1 1 1 zu 3 1 1 1 1
    \end{picture}
  \end{center}
  \caption{The graph $\Gamma_7$}\label{fig1}
\end{figure}

Obviously $\Gn$ has two (three resp.) connected subgraphs
in the case that $n$ is odd (even resp.).
More precisely, the partition $1\ldots 1$ if $n$ is odd
and $1\ldots 1$ and $2\ldots 2$ if $n$ is even
are not connected with any other node.
We observe further that for each $k<n$
the subgraph of $\Gn$ induced by 
all partitions which include the letter $k$
is isomorphic to $\Gamma_{n-k}$.
It is not at all trivial to see
that, if $\varphi$ is an automorphism of $\Gn$,
for each $k\in\N$ this subgraph 
is invariant under $\varphi$.
We get by an inductive argument
the following lemma about $\Gn$:
\begin{lemma}
\label{LmAutom}
For the automorphism group of $\Gn$ holds:
$$\Aut(\Gn) = 
\begin{cases}
  \{ \id \} & \text{\ if $n$ odd,} \\[1ex]
  \{\id,\tau\} & \text{\ if $n$ even,}
\end{cases}
$$
where $\tau$ is the automorphism of $\Gn$
that exchanges the nodes $1\ldots 1$  and $2\ldots 2$
and fixes the other nodes.
\end{lemma}

\section{The action of $\Stab_{\Aut(\Dn)}(H)$ on $\Gn$}
\label{SubAction}

Now let $\varphi$ be an automorphism of $\Dn$
such that $H^\varphi \subseteq H$.

Since the elements $\Ny{p}$, $p\partition n$,
are mutually orthogonal idempotents,
$\varphi$ acts on the set
$\{\Ny{p} \I p\partition n \}$.
Therefore $\varphi$ induces an action on
the set of partitions of $n$.

It turns out that the action of $\varphi$ 
on the set of partitions of $n$
is compatible with the relation $\pfmaxverf$,
i.e. for all partitions $r,p$ of $n$
we have:
$$ r \pfmaxverf p \ \Longleftrightarrow\  r^\varphi \pfmaxverf p^\varphi.$$
Therefore $\varphi$ induces an automorphism of $\Gn$.

By Lemma \ref{LmAutom} we see that the spaces
$\Ny{p}\Dn\Ny{r}$ are  $\varphi$-invariant
for all partitions $p,r$ such that $r\pfmaxverf p$.

By \cite[2.2 Corollary]{loewy}
\footnote{
  In \cite{loewy}, the spaces $\w_p\Lambda^r$ are treated.
  But these are isomorphic to $\Ny{p}\Dn\Ny{r}$.}
we get then
\begin{lemma}
Let $p,r$ partitions of $n$ such that $r\pfmaxverf p$.
Then $\Ny{p}\Dn\Ny{r}$ is a one-dimensional
$\varphi$-invariant space,
and $\varphi|_{\Ny{p}\Dn\Ny{r}}$ has an eigenvalue $\neq 0$.
\end{lemma}

By \cite[2.4 Corollary]{loewy}
we can deduce 
$$\Rad(\Dn) \ =\  \langle\  \Ny{p}\Dn\Ny{r} \I r \pfmaxverf p \
\rangle_{(+,\cdot)}$$
from which 
we see that the action of $\varphi$ on $\Rad(\Dn)$ 
is determined by the action on 
the subspaces $\Ny{p}\Dn\Ny{r}$, $r\pfmaxverf p$.

Now we define a certain subgraph $\Gn^*$ of $\Gn$,
the nodes of which are the partitions of $n$, too.
At first, we set
$$P_n^*:=\{ p\in P_n \I p=p_1 \ldots p_k, p_1 \neq p_k\},$$
i.e. the set of all partitions of $n$ that have at least
two different letters.
For each partition $p=p_1\ldots p_k \in P_n^*$
let
$$\iota(p):=\min\{i \I i\in\{1,\ldots,k\}, p_i\neq p_1\}$$
and
$$\zeta(p):=(p_1+p_{\iota(p)}) p_2\ldots p_{\iota(p)-1} p_{\iota(p)+1}\ldots p_k,$$
i.e. we form the sum of the two largest different letters that occur in $p$,
delete these two letters from $p$ and add the sum as a new letter. 
If $p \in P_n^*$ 
then $\zeta(p)$ is a partition of $n$
and it holds $p \pfmaxverf \zeta(p)$.

The partitions $p,r$ of $n$ are called connected in $\Gn^*$ ($p\succ r$), 
if $p\in P_n^*$ and $r=\zeta(p)$
or if $r=d^k$ for some $d,k\in\N$ and $p=d^{k-1}(d-1)1$.

The relation $\succ$ is coarser than $\pfmaxverf$
and defines a spanning tree
for the ``big'' connected component of $\Gn$,
seen as an undirected graph,
that contains the node $n$. Figure \ref{fig2} shows $\Gamma_{7}^*$.
\begin{figure}
  \begin{center}
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    \begin{picture}(21.6,33.3)
      \put(8.1,32.8){\makebox(0,0){7}}
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      \put(8.1,27.4){\makebox(0,0){5 2}}
      \put(13.5,27.4){\makebox(0,0){4 3}}
      \put(0.0,22.0){\makebox(0,0){5 1 1}}
      \put(5.4,22.0){\makebox(0,0){4 2 1}}
      \put(10.8,22.0){\makebox(0,0){3 3 1}}
      \put(16.2,22.0){\makebox(0,0){3 2 2}}
      \put(2.7,16.6){\makebox(0,0){4 1 1 1}}
      \put(8.1,16.6){\makebox(0,0){3 2 1 1}}
      \put(13.5,16.6){\makebox(0,0){2 2 2 1}}
      \put(5.4,11.2){\makebox(0,0){3 1 1 1 1}}
      \put(10.8,11.2){\makebox(0,0){2 2 1 1 1}}
      \put(8.1,5.8){\makebox(0,0){2 1 1 1 1 1}}
      \put(8.1,0.4){\makebox(0,0){1 1 1 1 1 1 1}}
      \thicklines
      \put(3.2,27.9){\vector(1,1){4.5}} % von 6 1 zu 7
      \put(8.1,27.9){\vector(0,1){4.5}} % von 5 2 zu 7
      \put(13.1,27.9){\vector(-1,1){4.5}} % von 4 3 zu 7
      \put(0.2,22.5){\vector(1,2){2.3}} % von 5 1 1 zu 6 1
      \put(5.2,22.5){\vector(-1,2){2.3}} % von 4 2 1 zu 6 1
      \put(15.5,22.5){\vector(-3,2){6.8}} % von 3 2 2 zu 5 2
      \put(11.0,22.5){\vector(1,2){2.3}} % von 3 3 1 zu 4 3
      \put(2.5,17.1){\vector(-1,2){2.3}} % von 4 1 1 1 zu 5 1 1
      \put(7.4,17.1){\vector(-3,2){6.8}} % von 3 2 1 1 zu 5 1 1
      \put(13.7,17.1){\vector(1,2){2.3}} % von 2 2 2 1 zu 3 2 2
      \put(5.2,11.7){\vector(-1,2){2.3}} % von 3 1 1 1 1 zu 4 1 1 1
      \put(10.6,11.7){\vector(-1,2){2.2}} % von 2 2 1 1 1 zu 3 2 1 1
      \put(7.9,6.3){\vector(-1,2){2.3}} % von 2 1 1 1 1 1 zu 3 1 1 1 1
    \end{picture}
  \end{center}
  \caption{The edges defined by $\succ$}
  \label{fig2}
\end{figure}

We obtain
\begin{mainlemma}
\label{ThUnique}
  Let $\varphi\in\Stab_{\Aut(\Dn)}(H)$
  such that $\varphi|_{\Ny{p}\Dn\Ny{r}} = \id$
  for all partitions $p,r$ such that $r\succ p$.
  Then $\varphi|_{\Rad(\Dn)}=\id$.
\end{mainlemma}

The proof of the Main Lemma needs hard conclusions.
For this reason we give only some instructive examples 
in Section \ref{SubExample} 
that demonstrate the important ideas.

But now we may easily construct for each $\varphi\in\Stab_{\Aut(\Dn)}(H)$
an invertible element $h\in H$ such that
the conjugation by $h$ coincides with the action of $\varphi$ 
on the subspaces $\Ny{p}\Dn\Ny{r}$, $r \succ p$.
By the Main Lemma, we see that the action of $h$ and $\varphi$
coincide on $\Rad(\Dn)$.

Therefore the automorphism 
$\psi:\Dn\to\Dn, x\mapsto (x^\varphi)^{h^{-1}}$
centralizes $\Rad(\Dn)$
and stabilizes the set
$\{ \Ny{p} \I p\partition n\}$.
By what we have observed above,
$\psi$ induces an automorphism of $\Gn$.
Now Lemma \ref{LmAutom} 
implies, that $\psi$
is the identity or an involution.
Hence, it follows
\begin{lemma}
$$\Stab_{\Aut(\Dn)}(H)\ = 
\begin{cases}
\text{\{inner automorphisms induced by $H$\}} & \text{if $n$ odd,}  \\
\text{\{inner automorphisms induced by $H$\}} \times C_2 & \text{if $n$ even.}  
\end{cases}
$$
\end{lemma}

\section{Example for $n=7$}
\label{SubExample}

We illustrate the proof of the Main Lemma
by discussing the example of $n=7$ which provides 
a spectrum of all three typical cases which may occur in general.
This discussion therefore does not only give a flavour
of the general proof but presents all its basic elements 
in a concrete form.

In \cite[p. 718]{loewy}
a basis of $\Dn$
consisting of idempotents $\nu_q$, $q\zerl n$,
is given.
In the following, 
we consider the linear extension
$\nu: \langle q \I q\zerl n\rangle_\Q \to \Dn$
of the mapping $\{ q \I q\zerl n \} \to \Dn$, $q\mapsto \nu_q$.

We use the Lie product $\circ$ on $\Q\NS$
defined by $a\circ b:= ab-ba$ for all $a,b\in\Q\NS$.

E.g. we write
$$\nu_{1\circ 2}
  =\nu_{12-21}
  =\nu_{12} - \nu_{21}.$$

\sloppypar
In order to illustrate the proof of the Main Lemma 
we may assume $\varphi$ 
fixes elementwise the subspaces
$\Ny{7}\Dn\Ny{61}$, 
$\Ny{7}\Dn\Ny{52}$, 
$\Ny{7}\Dn\Ny{43}$, 
$\Ny{61}\Dn\Ny{511}$,
$\Ny{61}\Dn\Ny{421}$,
$\Ny{52}\Dn\Ny{322}$,
$\Ny{43}\Dn\Ny{331}$,
$\Ny{511}\Dn\Ny{4111}$,
$\Ny{511}\Dn\Ny{3211}$,
$\Ny{322}\Dn\Ny{2221}$,
$\Ny{4111}\Dn\Ny{31111}$,
$\Ny{3211}\Dn\Ny{22111}$,
$\Ny{31111}\Dn\Ny{211111}$.

We have to show that the subspaces
$\Ny{52}\Dn\Ny{421}$,
$\Ny{43}\Dn\Ny{421}$,
$\Ny{421}\Dn\Ny{3211}$,
$\Ny{331}\Dn\Ny{3211}$
are fixed elementwise by $\varphi$, too.

Figure \ref{fig3} shows this situation.
The thick edges represent the subspaces 
on which the action of $\varphi$ is assumed
as identity.
The thin edges represent the subspaces
on which the action of $\varphi$ is not known.
The eigenvalues of $\varphi$ on these eigenspaces
are denoted by $a$, $b$, $c$ and $d$.
We have to show that $a=b=c=d=1$.
\begin{figure}
  \begin{center}
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    \begin{picture}(21.6,33.3)
      \put(8.1,32.8){\makebox(0,0){7}}
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      \put(13.5,27.4){\makebox(0,0){4 3}}
      \put(0.0,22.0){\makebox(0,0){5 1 1}}
      \put(5.4,22.0){\makebox(0,0){4 2 1}}
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      \put(16.2,22.0){\makebox(0,0){3 2 2}}
      \put(2.7,16.6){\makebox(0,0){4 1 1 1}}
      \put(8.1,16.6){\makebox(0,0){3 2 1 1}}
      \put(13.5,16.6){\makebox(0,0){2 2 2 1}}
      \put(5.4,11.2){\makebox(0,0){3 1 1 1 1}}
      \put(10.8,11.2){\makebox(0,0){2 2 1 1 1}}
      \put(8.1,5.8){\makebox(0,0){2 1 1 1 1 1}}
      \put(8.1,0.4){\makebox(0,0){1 1 1 1 1 1 1}}
      \thicklines
      \put(3.2,27.9){\vector(1,1){4.5}} % von 6 1 zu 7
      \put(8.1,27.9){\vector(0,1){4.5}} % von 5 2 zu 7
      \put(13.1,27.9){\vector(-1,1){4.5}} % von 4 3 zu 7
      \put(0.2,22.5){\vector(1,2){2.3}} % von 5 1 1 zu 6 1
      \put(5.2,22.5){\vector(-1,2){2.3}} % von 4 2 1 zu 6 1
      \thinlines
      \put(5.6,22.5){\vector(1,2){2.3}} % von 4 2 1 zu 5 2
      \put(5.8,24.7){\makebox(0,0){a}}
      \thicklines
      \put(15.5,22.5){\vector(-3,2){6.8}} % von 3 2 2 zu 5 2
      \thinlines
      \put(6.1,22.5){\vector(3,2){6.8}} % von 4 2 1 zu 4 3
      \put(8.4,24.7){\makebox(0,0){b}}
      \thicklines
      \put(11.0,22.5){\vector(1,2){2.3}} % von 3 3 1 zu 4 3
      \put(2.5,17.1){\vector(-1,2){2.3}} % von 4 1 1 1 zu 5 1 1
      \put(7.4,17.1){\vector(-3,2){6.8}} % von 3 2 1 1 zu 5 1 1
      \thinlines
      \put(7.9,17.1){\vector(-1,2){2.3}} % von 3 2 1 1 zu 4 2 1
      \put(7.5,19.3){\makebox(0,0){c}}
      \put(8.3,17.1){\vector(1,2){2.2}} % von 3 2 1 1 zu 3 3 1
      \put(10,19.3){\makebox(0,0){d}}
      \thicklines
      \put(13.7,17.1){\vector(1,2){2.3}} % von 2 2 2 1 zu 3 2 2
      \put(5.2,11.7){\vector(-1,2){2.3}} % von 3 1 1 1 1 zu 4 1 1 1
      \put(10.6,11.7){\vector(-1,2){2.2}} % von 2 2 1 1 1 zu 3 2 1 1
      \put(7.9,6.3){\vector(-1,2){2.3}} % von 2 1 1 1 1 1 zu 3 1 1 1 1
    \end{picture}
  \end{center}
  \caption{$\Gamma_7$ with the eigenvalues as weights}
  \label{fig3}
\end{figure}

In the following considerations
we use rules described in \cite[1.5 Theorem, 2.1 Proposition]{loewy}.

Case 1: The calculation of $a$ and $b$.
There are the one-dimensional spaces
\begin{eqnarray*}
& & (\Ny{7}\Dn\Ny{61})(\Ny{61}\Dn\Ny{421}) \\
& = & \langle\  \Ny{7}\nu_7\nu_{61}\nu_{421} \ \rangle_\Q \\
& = & \langle\  \Ny{7}\nu_{6\circ 1}\nu_{421} \ \rangle_\Q \\
& = & \langle\  \Ny{7}\nu_{(4\circ 2)\circ 1} \ \rangle_\Q
\end{eqnarray*}
with eigenvalue $1$,
\begin{eqnarray*}
& & (\Ny{7}\Dn\Ny{52})(\Ny{52}\Dn\Ny{421}) \\
& = & \langle\  \Ny{7}\nu_7\nu_{52}\nu_{421} \ \rangle_\Q \\
& = & \langle\  \Ny{7}\nu_{5 \circ 2}\nu_{421} \ \rangle_\Q \\
& = & \langle\  \Ny{7}\nu_{(4\circ 1)\circ 2} \ \rangle_\Q \\
& = & \langle\  \Ny{7}\nu_{(1\circ 4)\circ 2} \ \rangle_\Q 
\end{eqnarray*}
with eigenvalue $a$,
\begin{eqnarray*}
& & (\Ny{7}\Dn\Ny{71})(\Ny{43}\Dn\Ny{421}) \\
& = & \langle\  \Ny{7}\nu_7\nu_{43}\nu_{421} \ \rangle_\Q \\
& = & \langle\  \Ny{7}\nu_{4\circ 3}\nu_{421} \ \rangle_\Q \\
& = & \langle\  \Ny{7}\nu_{4\circ ( 2\circ 1)} \ \rangle_\Q \\
& = & \langle\  \Ny{7}\nu_{( 2\circ 1) \circ 4} \ \rangle_\Q
\end{eqnarray*}
with eigenvalue $b$.

Applying the Jacobi identity 
$((x\circ y) \circ z + (y \circ z) \circ x + (z \circ x) \circ y = 0 )$
we get
\begin{eqnarray*}
0 & = & 0^\varphi \\
& = & (\Ny{7})^\varphi (
\overbrace{\nu_{(4\circ 2)\circ 1} + \nu_{(1\circ 4)\circ 2} + \nu_{(2\circ 1)
    \circ 4}}^{=0})^\varphi \\
& = & (\Ny{7})^\varphi (\nu_{(4\circ 2)\circ 1}
               + a\nu_{(1\circ 4)\circ 2}
               + b\nu_{(2\circ 1) \circ 4}) \\
& = & (\Ny{7})^\varphi (
       (-\nu_{(1\circ 4)\circ 2} - \nu_{(2\circ 1) \circ 4})
       + a\nu_{(1\circ 4)\circ 2}
       + b\nu_{( 2\circ 1) \circ 4}) \\
& = & (\Ny{7})^\varphi (
      (a-1)\nu_{(1\circ 4)\circ 2}
    + (b-1)\nu_{(2\circ 1)\circ 4}). 
\end{eqnarray*}

The summands in the last equation are linearly independent.
It follows that $a=1$ and $b=1$.

In a similar manner we treat the other two cases.

Case 2: The calculation of $c$.
There are the one-dimensional spaces
$$(\Ny{61}\Dn\Ny{511})(\Ny{511}\Dn\Ny{3211})(\Ny{3211}\Dn\Ny{22111}) 
\ =\ \langle\  \Ny{61}\nu_{(((2\circ 1) \circ 2)\circ 1)1)} \ \rangle_\Q$$
with eigenvalue $1$,
$$ (\Ny{61}\Dn\Ny{421})(\Ny{421}\Dn\Ny{3211})(\Ny{3211}\Dn\Ny{22111}) 
\ =\ \langle\  \Ny{61}\nu_{(((2\circ 1)\circ 1)\circ 2)1)} \ \rangle_\Q$$
with eigenvalue $c$.

The anticommutative law and the Jacobi identity imply 
that both spaces are generated by the same element.
It follows that $c=1$. 

Case 3: The calculation of $d$.
There are the one-dimensional spaces
$$ (\Ny{43}\Dn\Ny{421})(\Ny{421}\Dn\Ny{3211})
\ =\ \langle\  \Ny{43}\nu_{(3\circ 1)(2\circ 1)} \ \rangle_\Q$$
with eigenvalue $b=1$,
$$ (\Ny{43}\Dn\Ny{331})(\Ny{331}\Dn\Ny{3211})
\ =\ \langle\  \Ny{43}\nu_{(3\circ 1)(2\circ 1)} \ \rangle_\Q$$
with eigenvalue $d$.
Both spaces are generated by the same element.
It follows that $d=b=1$.

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