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\begin{document}
\thispagestyle{empty}

\begin{center}
{\bf \LARGE MAT 1120HF -- Assignment \#3}\\
\end{center}

\begin{center}
\large 
Due date: Friday, December 10, 2010.
\end{center} 
\vskip .1in 

\begin{enumerate}

\item[1.]  Let $G_2$ be the simply connected compact Lie group corresponding to 
the Dynkin diagram with two vertices, connected by a directed arrow of multiplicity 3. Pretending that we don't 
know anything about $G_2$, show how to extract 
the following information from  the Dynkin diagram. 
Give \emph{\bf brief} explanations.
 
\begin{enumerate}
\item[a)] Find the Cartan matrix of $G_2$. 
\item[b)] Draw the root system $\mf{R}\subset E$ of $G_2$. Indicate the simple roots and the corresponding Weyl chamber. 
\item[c)] Use the invariant invariant inner product on $E$ for which the long roots $\alpha$ 
satisfy $||\alpha||^2=2$, to identify 
$E^*\cong E$. Draw the co-root system $\mf{R}^\vee\subset E$. Indicate the simple co-roots. (Don't draw it into the picture from (b) - it will get too messy. But use the same scale.)  
\item[d)] Show that the Weyl group of $G_2$ is the dihedral group 
$D_6$, i.e. the symmetry group of a regular hexagon.   
\item[e)] Find the dimension of $G_2$. 
\end{enumerate}
\vskip.4in 

\item[2.] Let $G$ be a compact, semi-simple Lie group. Let
$T\subset G$ be a given maximal torus, and $\Pi\subset \mf{R}$ a given
set of simple roots. A permutation $\Phi$ of $\Pi$ is called a
 \emph{Dynkin diagram automorphism} if $\l
\Phi(\alpha^\vee),\Phi(\beta)\r=\l\alpha^\vee,\beta\r$ for all
$\alpha,\beta\in\Pi$. Let $\on{Aut}(\Pi)$ denote the group of Dynkin
diagram automorphisms. On the other hand, let $\on{Aut}(G)$ be the group of automorphisms of $G$, 
$\on{Inn}(G)$ the normal subgroup of inner automorphisms, and 
\[ \on{Out}(G)=\on{Aut}(G)/\on{Inn}(G)\]
the group of \emph{outer automorphisms}. 

\vskip.1in 
\begin{enumerate}
\item[a)] Show that any element of $\on{Out}(G)$ may be represented by an element $\phi\in \on{Aut}(G)$, such that $\phi$ preserves $T$ and the induced action $\Phi$ on $X^*(T)$ preserves the set $\Pi$ of simple roots. 
Show that this defines a group morphism 
$\on{Out}(G)\to \on{Aut}(\Pi)$. 

\item[b)] Let $\phi\in \on{Aut}(G)$ be an automorphism with $\phi(t)=t$ for all $t\in T$. Show that 
$\d\phi\in \on{Aut}(\g)$ preserves each root space $\g_\alpha,\ \alpha\in\mf{R}$. Use this to define a group morphism 
$\on{span}_\Z(\mf{R})\to \U(1)$. Show that any such morphism is of the form 
$\alpha\mapsto \alpha(h)$ for some $h\in T$, and finally that 
$\phi=\Ad_h$. This proves that the group morphism $\on{Out}(G)\to \on{Aut}(\Pi)$
is \emph{injective}. 

\item[c)] Show that for $G$ simply connected, the group morphism $\on{Out}(G)\to \on{Aut}(\Pi)$
is also \emph{surjective}. (Hint: use the Serre relations.)

\item[d)] Give an example of an automorphism of $G=\SU(n),\ n>2$ that is \emph{not} inner. Describe the corresponding Dynkin diagram automorphism. 

\item[e)] Give an example of an automorphism of $G=\SO(2m),\ m>1$ that is \emph{not} inner. (Hint: use $\on{O}(2m)$.) Describe the corresponding Dynkin diagram automorphism.
\end{enumerate}
\vskip.051in 

\end{enumerate}

\end{document}














