\include{preamble}
\begin{document}
\tikzstyle{C}=[draw, thick, fill =white, circle, inner sep=0pt, minimum size=6pt]
\paragraph{\bf Exercise 6:} Some things about classification.
\begin{enumerate}
\item Let $E$ be a euclidean space/$\RR$ with inner product $\<,\>$ (of any big dimension). Call a finite subset $A=\{\alpha_1, \dots, \alpha_r\}\subset E$ \emph{admissible} if
\begin{enumerate}[(i)]
\item $A$ is a set of linearly independent unit vectors ($\<\alpha_i, \alpha_i\>=1$),
\item $\<\alpha_i, \alpha_j\> \leq 0$ whenever $i \neq j$, and
\item $4\<\alpha_i,\alpha_j\>^2 \in \{0,1,2,3\}$ whenever $i \neq j$.
\end{enumerate}
Associate to any admissible set $A$ a graph $\Gamma(A)$ (called the Coxeter diagram) with vertices labeled by elements of $A$ (or $i$ short for $\alpha_i$), with $m_{i,j} = 4\<\alpha_i,\alpha_j\>^2$ edges connecting $i$ to $j$:
$$\begin{array}{c@{\qquad}l}
\begin{matrix}\begin{tikzpicture}
\node[C, label=above:{$i$}] (i) at (0,0) {};
\node[C, label=above:{$j$}] (j) at (1,0) {};
\end{tikzpicture}\end{matrix} & \text{if } 4\<\alpha_i,\alpha_j\>^2 = 0,\\
\begin{matrix}\begin{tikzpicture}
\node[C, label=above:{$i$}] (i) at (0,0) {};
\node[C, label=above:{$j$}] (j) at (1,0) {};
\draw[thick] (i) to (j);
\end{tikzpicture}\end{matrix} & \text{if } 4\<\alpha_i,\alpha_j\>^2 = 1, \\
\begin{matrix}\begin{tikzpicture}
\node[C, label=above:{$i$}] (i) at (0,0) {};
\node[C, label=above:{$j$}] (j) at (1,0) {};
\draw[thick, double distance=2pt] (i) to (j);
\end{tikzpicture}\end{matrix} & \text{if } 4\<\alpha_i,\alpha_j\>^2 = 2,\\
\begin{matrix}\begin{tikzpicture}
\node[C, label=above:{$i$}] (i) at (0,0) {};
\node[C, label=above:{$j$}] (j) at (1,0) {};
\draw[thick, double distance=3pt] (i) to (j);
\draw[thick] (i) to (j);
\end{tikzpicture}\end{matrix} & \text{if } 4\<\alpha_i,\alpha_j\>^2 = 3.
\end{array}$$
Let $A=\{\alpha_1, \dots, \alpha_r\} $ be an admissible set yielding a connected graph $\Gamma(A)$.
\begin{enumerate}
\item Show that the number of pairs of vertices connected by at least one edge strictly less than $r$. \\
{}[What is the condition on vertices being adjacent? Consider $\<\alpha, \alpha\>$ where $\alpha = \sum_{A} \alpha_i$.]
\item Show that $\Gamma(A)$ contains no cycles. [Note that any subset of an admissible set is admissible.
]
\item Show that the degree (counting multiple edges) of any vertex in $\Gamma(A)$ is no more than three.\\
{}[Take a vertex $\alpha \in A$, and let $S$ be the set containing $\alpha$ together with its neighborhood (the vertices adjacent to it). Note that in the span of $S$ is a unit vector $\beta$ which is orthogonal to $S - \{\alpha\}$, so that $\alpha = \sum_{\gamma \in S - \{\alpha\} + \{\beta\}} \<\alpha, \gamma\> \gamma$ and $\<\alpha, \beta\> \neq 0$ (why??).]
\item Show that if $S \subseteq A$ has graph
$$\Gamma(S) =
\begin{matrix}\begin{tikzpicture}
\foreach \x in {0, 1, 2, 4, 5}{\node[C] (\x) at (\x,0) {};}
\draw[thick] (0) to (1) to (2) to (2.5,0) (3.5,0) to (4) to (5);
\node at (3,0) {$\cdots$};
\end{tikzpicture}\end{matrix},$$
then $A' = A - S + \{\sum_S \alpha\}$ is admissible (with graph $\Gamma(A')$ obtained by collapsing the subgraph $\Gamma(S)$ to a single vertex).
\item Show that $\Gamma(A)$ cannot contain any of the following graphs as subgraphs:
$$\begin{matrix}\begin{tikzpicture}
\foreach \x in {0, 1, 2, 4, 5,6}{\node[C] (\x) at (\x,0) {};}
\draw[thick] (1) to (2) to (2.5,0) (3.5,0) to (4) to (5);
\draw[thick, double distance=2pt] (0) to (1) (5) to (6);
\node at (3,0) {$\cdots$};
\end{tikzpicture}\end{matrix}$$
$$\begin{matrix}\begin{tikzpicture}
\foreach \x in {0, 1, 2, 4, 5}{\node[C] (\x) at (\x,0) {};}
\node[C] (6a) at (6,-.5) {}; \node[C] (6b) at (6,.5) {};
\draw[thick] (1) to (2) to (2.5,0) (3.5,0) to (4) to (5) to (6a) (5) to (6b);
\draw[thick, double distance=2pt] (0) to (1);
\node at (3,0) {$\cdots$};
\end{tikzpicture}\end{matrix}$$
$$\begin{matrix}\begin{tikzpicture}
\foreach \x in {1, 2, 4, 5}{\node[C] (\x) at (\x,0) {};}
\node[C] (6a) at (6,-.5) {}; \node[C] (6b) at (6,.5) {};
\node[C] (0a) at (0,-.5) {}; \node[C] (0b) at (0,.5) {};
\draw[thick] (0b) to (1) (0a) to (1) to (2) to (2.5,0) (3.5,0) to (4) to (5) to (6a) (5) to (6b);
\node at (3,0) {$\cdots$};
\end{tikzpicture}\end{matrix}$$
[Use the previous part]
\item Show that the only remaining possible graphs associated to admissible sets are of one of the following four forms:
$$\begin{matrix}\begin{tikzpicture}
\node[C] (i) at (0,0) {};
\node[C] (j) at (1,0) {};
\draw[thick, double distance=3pt] (i) to (j);
\draw[thick] (i) to (j);
\end{tikzpicture}\end{matrix} $$
$$\begin{matrix}\begin{tikzpicture}
\foreach \x in {0, 1, 2, 4, 5}{\node[C] (\x) at (\x,0) {};}
\draw[thick] (0) to (1) to (2) to (2.5,0) (3.5,0) to (4) to (5);
\node at (3,0) {$\cdots$};
\end{tikzpicture}\end{matrix}$$
$$\begin{matrix}\begin{tikzpicture}
\foreach \x in {1, 2, 3, 4,...,8}{\node[C] (\x) at (\x,0) {};}
\draw[thick] (1) to (2) to (2.2,0) (2.8,0) to (3) to (4)
(5) to (6) to (6.2,0) (6.8,0) to (7) to (8);
\draw[thick, double distance=2pt] (4) to (5);
\node at (2.5,0) {$\cdots$}; \node at (6.5,0) {$\cdots$};
\end{tikzpicture}\end{matrix}$$
$$\begin{matrix}\begin{tikzpicture}
\foreach \x in {1, 2, 3, 4}{\node[C] (\x) at (\x-4,0) {};}
\foreach \x in {1, 2, 3}{
\node[C] (u\x) at (15:\x) {};
\node[C] (d\x) at (-15:\x) {};}
\draw[thick] (1) to (2) to (2.2-4,0) (2.8-4,0) to (3) to (4)
(4) to (u1) to (15:1.2) (15:1.8) to (u2) to (u3)
(4) to (d1) to (-15:1.2) (-15:1.8) to (d2) to (d3);
\node at (2.5-4,0) {$\cdots$};
\node[rotate=15] at (15:1.5) {$\cdots$};
\node[rotate=-15] at (-15:1.5) {$\cdots$};
\end{tikzpicture}\end{matrix}$$
\item Show the only possible graphs of the third type are
$$\begin{matrix}\begin{tikzpicture}
\foreach \x in {1, 2, 3, 4,5}{\node[C] (\x) at (\x,0) {};}
\draw[thick] (1) to (2) to (2.2,0) (2.8,0) to (3) to (4);
\draw[thick, double distance=2pt] (4) to (5);
\node at (2.5,0) {$\cdots$};
\end{tikzpicture}\end{matrix}
\qquad \text{and} \qquad
\begin{matrix}\begin{tikzpicture}
\foreach \x in {1, 2, 3, 4}{\node[C] (\x) at (\x,0) {};}
\draw[thick] (1) to (2) (3) to (4);
\draw[thick, double distance=2pt] (2) to (3);
\end{tikzpicture}\end{matrix}$$
[Suppose the vectors corresponding to the vertices to the left of the double bond are $\lambda_1, \dots, \lambda_\ell$ (from left to right) and the vertices to the rights of the double bond are $\mu_1, \dots, \mu_m$ (from right to left). Let $\lambda = \sum_i i \lambda_i$ and $\mu = \sum_i i \mu_i$. Show that
$\<\lambda, \lambda\> = \ell(\ell+1)/2$, $\< \mu, \mu\> = m(m+1)/2$, and $\<\lambda,\mu\>^2 = \ell^2 m^2/2$, and use the Cauchy-Schwarz inequality for inner products.]
\item Bonus: Show the only graphs of the fourth kind are
$$\begin{matrix}\begin{tikzpicture}
\foreach \x in {1, 2, 3, 4}{\node[C] (\x) at (\x-4,0) {};}
\foreach \x in {1}{
\node[C] (u\x) at (15:\x) {};
\node[C] (d\x) at (-15:\x) {};}
\draw[thick] (1) to (2) to (2.2-4,0) (2.8-4,0) to (3) to (4)
(4) to (u1)
(4) to (d1) ;
\node at (2.5-4,0) {$\cdots$};
\end{tikzpicture}\end{matrix}
\qquad \text{ and } \qquad
\begin{matrix}\begin{tikzpicture}
\foreach \x in {1, 2, 3, 4}{\node[C] (\x) at (\x-4,0) {};}
\foreach \x in {1, 2}{\node[C] (u\x) at (15:\x) {};}
\node[C] (d1) at (-15:1) {};
\draw[thick] (1) to (2) to (2.2-4,0) (2.8-4,0) to (3) to (4)
(4) to (u1) to (u2)
(4) to (d1) ;
\node at (2.5-4,0) {$\cdots$};
\draw[|-|] (-3,-.5) to node[midway, below] {$4\leq * \leq 6$} (0, -.5);
\end{tikzpicture}\end{matrix}
$$
[This is like the previous part, only more so]
\end{enumerate}
\item Show that by normalizing the elements of any base $B$ for a set of roots $R$, you get an admissible set $A$. From part 1, what's left over? Show that there's an admissible set associated to every remaining graph by displaying existence. Namely, associate most of the remaining possible graphs to a classical root systems (showing existence), and take for granted that the remaining five are associated to the \emph{exceptional simple Lie algebras}, $E_6, E_7, E_8, F_4$, and $G_2$:
$$E_6, E_7, E_8: \begin{matrix}\begin{tikzpicture}
\foreach \x in {1, 2, 3, 4}{\node[C] (\x) at (\x-4,0) {};}
\foreach \x in {1, 2}{\node[C] (u\x) at (15:\x) {};}
\node[C] (d1) at (-15:1) {};
\draw[thick] (1) to (2) to (2.2-4,0) (2.8-4,0) to (3) to (4)
(4) to (u1) to (u2)
(4) to (d1) ;
\node at (2.5-4,0) {$\cdots$};
\end{tikzpicture}\end{matrix}
$$
$$F_4: \begin{matrix}\begin{tikzpicture}
\foreach \x in {1, 2, 3, 4}{\node[C] (\x) at (\x,0) {};}
\draw[thick] (1) to (2) (3) to (4);
\draw[thick, double distance=2pt] (2) to (3);
\end{tikzpicture}\end{matrix}$$
$$G_2: \begin{matrix}\begin{tikzpicture}
\node[C] (i) at (0,0) {};
\node[C] (j) at (1,0) {};
\draw[thick, double distance=3pt] (i) to (j);
\draw[thick] (i) to (j);
\end{tikzpicture}\end{matrix} $$
\item A \emph{Dynkin diagram} associated to a base $B$ for a root system is a decorated Coxeter graph for the associated normalized admissible set. If $\alpha_i$ is adjacent to $\alpha_j$, and the root $\beta_i$ associated to $\alpha_i$ is longer than the root $\beta_j$ associated to $\alpha_j$, decorate the $m_{i,j}$ edges connecting $\alpha_i$ to $\alpha_j$ with an arrow pointing to $\alpha_i$ (the normalization of the longer root).
Classify all (finite type) connected Dynkin diagrams.
\end{enumerate}
\end{document}