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\begin{document}
\title{Simple and positive roots}
\author{Your name here}
\date{\today}
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{\Large 18.099 - 18.06 CI.}
{Due on Monday, May 10 in class.}
\vspace{1cm}
{\it Write a paper proving the statements and working through the
examples formulated below. Add your own
examples, asides and discussions whenever needed. }
Let $V$ be a Euclidean space, that is
a finite dimensional real linear space with a symmetric
positive definite inner product $\la, \ra$.
Recall that a root system in
$V$ is a finite set $\Delta$ of nonzero
elements of $V$ such that
\begin{enumerate}
\item{$\Delta$ spans $V$;}
\item{for all $\alpha \in \Delta$, the reflections
$$ s_\alpha (\beta) =
\beta -\frac{2 \la \beta, \alpha \ra }{\la \alpha, \alpha \ra}\alpha $$
map the set $\Delta$ to itself;}
\item{the number $\frac{2 \la \beta, \alpha \ra }{\la \alpha, \alpha \ra}$
is an integer for any $\alpha, \beta \in \Delta$.}
\end{enumerate}
A root is an element of $\Delta$.
Here are two examples of root systems in $\R^2$:
\begin{example} \label{A1}
The root system of the type $A_1 \oplus A_1$ consists of
the four vectors $\{ \pm e_1, \pm e_2 \}$ where $\{e_1,e_2 \}$ is an
orthonormal basis in $\R^2$.
\end{example}
\begin{example} \label{A2}
The root system of the type $A_2$ consists of the six vectors
$\{e_i -e_j \}_{i\neq j}$ in the plane orthogonal to the line $e_1 +e_2 +e_3$
where $\{e_1, e_2, e_3\}$ is an orthonormal basis in $\R^3$.
Rewrite the vectors of this root system in a standard orthonormal basis of
the plane and sketch it.
\end{example}
Since for any $\alpha \in \Delta$, $-\alpha$ is also in $\Delta$,
(see \cite{1}, Thm.8(1)),
the number of elements in $\Delta$ is always
greater than the dimension of $V$. The
example of type $A_2$ above shows that even a subset of mutually
noncollinear vectors in $\Delta$ might be too big to be linearly independent.
In the present paper we would like
to define a subset of $\Delta$ small enough to be a basis for $V$, yet
large enough to contain the essential
information about the geometric properties of $\Delta$. Here is a formal
definition.
\begin{definition} A subset $\Pi$ of $\Delta$ is a \emph{set of simple roots}
(a \emph{simple root system}) in $\Delta$ if \begin{enumerate}
\item{ $\Pi$ is a basis for $V$;}
\item{Each root $\beta \in \Delta$ can be written as a linear
combination of the elements of $\Pi$ with integer coefficients of the
same sign, that is,
$$ \beta = \sum_{\alpha \in \Pi} m_\alpha \alpha $$
with all $m_\alpha \geq 0$ or all $m_\alpha \leq 0$.}
\end{enumerate}
The root $\beta$ is
\emph{positive} if the coefficients are nonnegative,
and \emph{negative} otherwise.
The set of all positive roots (the \emph{positive root system}) associated to
$\Pi$ will be denoted $\Delta^+$.
\end{definition}
Below we construct a set $\Pi_t$ associated to an element $t \in V$ and
a root system $\Delta$, and
show that it satisfies the definition of a simple root system in $\Delta$.
Let $\Delta $ be a root system in $V$, and
let $t \in V$ be a vector such that $\langle t, \alpha \rangle \neq 0$ for
all $\alpha \in \Delta$ (Check that such an element always exists).
Set
$$\Delta_t^+ = \{ \alpha \in \Delta : \langle t, \alpha \rangle >0 \}.$$
Let $\Delta_t^- = \{ -\alpha, \alpha \in \Delta_t^+ \}$.
Check that $\Delta = \Delta_t^+ \cup \Delta_t^-$.
\begin{definition} An element $\alpha \in \Delta_t^+$ is \emph{decomposable}
if there exist $\beta, \gamma \in \Delta_t^+$ such that
$\alpha = \beta +\gamma$. Otherwise $\alpha \in \Delta_t^+$ is
\emph{indecomposable}.
\end{definition}
Let $\Pi_t \subset \Delta_t^+$ be the set of all indecomposable elements
in $\Delta_t^+$.
The next three Lemmas prove the properties of $\Delta_t^+$ and $\Pi_t$.
\begin{lemma} \label{pos}
Any element in $\Delta_t^+$ can be written as a linear combination
of elements in $\Pi_t$ with nonnegative integer coefficients.
\end{lemma}
Hint: By contradiction. Suppose $\gamma$ is an element of $\Delta_t^+$
for which the Lemma is false and $\langle t, \gamma \rangle>0$ is minimal,
and use that $\gamma$ is decomposable to get a contradiction.
\begin{lemma} \label{angle}
If $\alpha, \beta \in \Pi_t$, then $\langle \alpha, \beta \rangle \leq 0$.
\end{lemma}
Hint: Use Thm. 10(1) in \cite{1} : if $\langle \alpha, \beta \rangle >0$,
then $\alpha - \beta$ is a root or $0$.
Add discussion: what does this result mean for the relative position
of two simple roots?
\begin{lemma} \label{basis}
Let $A$ be a subset of $V$ such that
\begin{enumerate}
\item{$\langle t, \alpha \rangle >0$ for all $\alpha \in A$;}
\item{$\langle \alpha, \beta \rangle \leq 0$ for all $\alpha, \beta \in A$.}
\end{enumerate}
Then the elements of $A$ are linearly independent.
\end{lemma}
Hint: Assume the elements of $A$ are linearly dependent and split the
nontrivial linear combination into two sums,
with positive and negative coefficients.
Let $\lambda = \sum m_\beta \beta = \sum n_\gamma \gamma$ with
$\beta, \gamma \in A$ and all $ m_\beta , n_\gamma >0$. Show that
$\langle \lambda, \lambda \rangle =0$.
Now we are ready to prove the existence of a simple root set
in any abstract root system.
\begin{theorem} \label{simple}
For any $t \in V$ such that $\langle t, \alpha \rangle \neq 0$ for all
$\alpha \in \Delta$, the set $\Pi_t$ constructed above is a set of simple
roots, and $\Delta_t^+$ the associated set of positive roots.
\end{theorem}
Hint: Use lemmas \ref{pos}, \ref{angle}, \ref{basis}.
The converse statement is also true (and much easier to prove):
\begin{theorem} Let $\Pi$ be a set of simple roots in $\Delta$, and
suppose that $t \in V$ is such that $\langle t, \alpha \rangle >0$ for all
$\alpha \in \Pi$. Then $\Pi = \Pi_t$, and the associated set of positive
roots $\Delta^+ = \Delta_t^+$.
\end{theorem}
\begin{example} Let $V$ be the
$n$-dimensional subspace
of $\R^{n+1}$ ($n \geq 1$) orthogonal to the line
$e_1 + e_2 + \ldots + e_{n+1}$,
where $\{e_i\}_{i=1}^{n+1}$ is an orthonormal basis in $\R^{n+1}$.
The root system $\Delta$ of the type $A_n$ in $V$
consists of all
vectors $\{ e_i - e_j\}_{ i \neq j }$.
Check that $\Pi = \{ e_1 - e_2, e_2 - e_3, \ldots e_n -e_{n+1} \}$ is
a set of simple roots, and $\Delta^+ = \{ e_i - e_j\}_{i