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\begin{center}
Introduction to Lie Theory\\
Homework \#8\\
\end{center}

\begin{enumerate}
\item Suppose that $E$ is a Euclidean space with basis $\alpha,\beta$
  such that
  \begin{align*}
(\alpha,\beta^\vee) &= -1,&
    (\beta, \alpha^\vee) &= -3.
  \end{align*}
  where $\alpha^\vee := \frac{2 \alpha}{(\alpha,\alpha)}$ and $\beta^\vee :=
  \frac{2\beta}{(\beta,\beta)}$ as usual.
  Suppose that $R \subset E$ is a root system with $\alpha,\beta \in
  R$.  Prove that $|R| = 12$, indeed, it is the root system of type
  $G_2$ with Cartan matrix
  $\left(\begin{array}{rr}2&-1\\-3&2\end{array}\right)$
  with respect to the base $\Delta = \{\alpha,\beta\}$.
  \item I did not prove the classification of root systems in the
    lectures. This question should give you some idea about how you
    might go about eliminating possibilities for the Dynkin diagrams by ``finding
     enough obstructions.''
    \begin{enumerate}
      \item Suppose that $R \subset E$ is a root system with base
        $\Delta = \{\alpha_1,\dots,\alpha_\ell\}$. Let
        $\Delta'$ be a non-empty subset of $\Delta$, $E'$ be the Euclidean subspace of $E$ spanned by
        $\Delta'$,
        and $R' = R \cap E'$. Prove that $R' \subset E'$ is a
        root system with base $\Delta'$.
    \item  Show that the graph
      $$
      \begin{picture}(81, 50)%
\put(24, 25){\circle{4}}%
\put(26, 25){\line(1, 0){15}}%
\put(43, 25){\circle{4}}%
\put(45, 25){\line(1, 0){15}}%
\put(62, 25){\circle{4}}%
\put(43, 42){\line(0, -1){15}}%
\put(43, 44){\circle{4}}%
\put(43, 23){\line(0, -11){15}}%
\put(43, 6){\circle{4}}%
\end{picture}
      $$
      is not the Dynkin diagram of a root
      system. ({\em Hint.} What is the determinant of the
      corresponding Cartan matrix?)
    \item Deduce that all vertices in the Dynkin diagram of a
      root system in which all roots of the same length
      are of degree $\leq 3$.
    \end{enumerate}
  \item
    Let $W < GL(E)$ be the Weyl group of a root system $R \subset
    E$.
    Let $s_1,\dots,s_\ell \in W$ be the simple reflections
    arising from a choice of a base $\Delta=\{\alpha_1,\dots,\alpha_\ell\}$ for $R$.
    Prove that the determinant of $w \in W$ viewed as a linear
    map $w:E \rightarrow E$ is equal to $(-1)^n$ for any
    (not necessarily reduced)
    expression $w = s_{i_1} \cdots s_{i_n}$ as a product
    of $n$ simple reflections.
    Deduce that $n \equiv \ell(w)\pmod{2}$ where $\ell:W \rightarrow
    \N$ is the length function, and that the map $\operatorname{sgn}:W \rightarrow \{\pm 1\},
    w \mapsto (-1)^{\ell(w)}$ is a group homomorphism.
    \item Let $R \subset E$ be a root system with
      base
      $\Delta$.
Recall that the {\em height} of a root is the sum of its coefficients
when it is written as a linear combination of the base.
Prove that there is a unique {\em highest root } $\alpha_0 \in R^+$.
In fact, 
$\alpha_0$ is the unique element
$\alpha_0 =
\sum_{i=1}^\ell a_i \alpha_i
\in R^+$ such that $a_i \geq b_i$ for every
$\beta = \sum_{i=1}^\ell b_i \alpha_i \in R^+$.
Moreover, in the case that $R$ is indecomposable with two different root lengths,
show that $\alpha_0$ is one of the long roots.

({\em Hints.} It suffices to treat the case that $R$ is
indecomposable.
Suppose that $\alpha = \sum_{i=1}^\ell a_i \alpha_i$ is a positive root of maximal
height. Using the first lemma from L8-1, you deduce that
$(\alpha,\alpha_i) \geq 0$ for all $i=1,\dots,\ell$
and that
$(\alpha,\alpha_i) = 0$ whenever $a_i = 0$.
By considering the partition $\Delta = \Delta_1 \sqcup \Delta_2$ where
$\Delta_1 = \{\alpha_i\:|\:a_i > 0\}$ and $\Delta_2 =
\{\alpha_i\:|\:a_i= 0\}$ and using the indecomposability, you see if
$\Delta_2 \neq \varnothing$ that
there must be $\alpha_i \in \Delta_1$ and $\alpha_j \in \Delta_2$
with $(\alpha_i,\alpha_j) < 0$.
This contradicts $(\alpha,\alpha_j) \geq 0$.
Hence, $a_i > 0$ for
all $i$.
Then take another positive root $\beta$ of maximal height and show
that $(\alpha,\beta) > 0$, so $\alpha-\beta$ 
is a root, which is a contradiction since it must have some positive
and some negative coefficients.)

\item Prove that the Weyl group of a root system has a unique {\em longest}
  element $w_0$. Moreover,
$w_0$ is the reflection $s_{\alpha_0}$ in the hypoerplane
       orthogonal to the highest root $\alpha_0$ from the previous
       question, hence, $w_0^2 = 1$.
   Show also that
    $\ell(w_0) = |R^+|$.
    Is $w_0 \in Z(W)$?
\item Suppose that $\lambda= \sum_{i=1}^\ell c_i \alpha_i$ for $c_i
  \in \N$.
  Show that $\lambda$ is a multiple of a root if and only if
  $w(\lambda)$ is either a positive sum of simple roots or a negative sum of simple roots for all $w \in W$.

({\em Hint.}
  Suppose that $\lambda$ is not a multiple of any root.
  Take $\mu \in E - \bigcup_{\alpha \in R}
  \alpha^\perp$ lying on
  the hyperplane $\lambda^\perp$. Then choose $w \in W$ so that
  $w(\mu)$ lies in the fundamental chamber, i.e.,
  $(\alpha_i, w(\mu)) > 0$ for all $i \in I$.
  Finally consider $w(\lambda)$, noting that
  $(w(\lambda), w(\mu)) = 0$.)
\end{enumerate}
\end{document}
