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\newcommand{\Z}{\mathbb Z}
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\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
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\begin{document}
\title{Properties of simple roots}
\author{Your name here}
\date{\today}
\maketitle
%\thispagestyle{empty}
{\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 for a root system $\Delta$ in
$V$,
a subset $\Pi \subset \Delta$ is a set of simple roots (a simple root system)
if \begin{enumerate}
\item{ $\Pi$ is a basis in $V$;}
\item{Each root $\beta \in \Delta$ can be written as a linear
combination of elements of $\Pi$ with integer coefficients of the same sign,
i.e.
$$ \beta = \sum_{\alpha \in \Pi} m_\alpha \alpha $$
with all $m_\alpha \geq 0$ or all $m_\alpha \leq 0$.}
\end{enumerate}
The root $\beta$ is
positive if the coefficients are nonnegative, and negative otherwise.
The set of all positive roots (positive root system)
associated to $\Pi$ is denoted $\Delta^+$.
Below we will assume that the root system $\Delta$ is reduced, that is,
for any $\alpha \in \Delta, 2 \alpha \notin \Delta$.
\begin{theorem} \label{sim-pos}
In a given $\Delta$, a set of simple roots $\Pi \subset \Delta$
and the associated set of positive roots
$\Delta^+ \subset \Delta$ determine each other uniquely.
\end{theorem}
Hint: easy. Use the explicit construction of $\Pi \subset \Delta^+$ given
in \cite{3}.
The question of existence of sets of simple roots for any abstract root
system $\Delta$ is settled in \cite{3}. Theorem \ref{sim-pos}
shows that once $\Pi$ is chosen $\Delta^+$ is unique.
In this paper we want to address the question of the possible choices for
$\Pi \subset \Delta$. We start with a couple of examples.
\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 $\R^3$.
Present the vectors of this root system in a standard orthonormal basis of
the plane. Find possible simple root systems $\Pi \subset \Delta$ and
the associated sets of positive roots $\Delta^+$,
$\Pi \subset \Delta^+ \subset \Delta$.
Check that any two simple root systems
$\Pi \subset \Delta$ can be mapped to each other by an orthogonal
transformation (see \cite{1} for definition) of $V$, and that the same
transformation maps the associated sets of positive roots.
\end{example}
\begin{example} \label{B2}
Consider the root system of the type $B_2$ in $V= \R^2$: it consists of eight
vectors $\{\pm e_1 \pm e_2, \pm e_1, \pm e_2 \}$. Find possible simple
root systems $\Pi \subset \Delta$ and check that they can be obtained from
any chosen one by an orthogonal transformation of $\R^2$. Check that
the same transformation maps the associated sets of positive roots to each
other.
\end{example}
We start working towards a result generalizing our observations.
Recall the definition of a reflection associated to an element $\alpha \in V$
(cf. \cite{1}):
$$ s_\alpha(x) = x - \frac{2\langle x, \alpha \rangle}{\langle \alpha, \alpha
\rangle}\alpha .$$
It is an orthogonal transformation of $V$.
\begin{theorem} Let $\Pi \subset \Delta$ be a set of simple roots, associated
to the set of positive roots $\Delta^+$. For any $\alpha \in \Delta$,
the set obtained by reflection $s_\alpha(\Pi)$ is a simple root system
with the associated positive root system $s_\alpha (\Delta^+)$.
\end{theorem}
To understand better the passage from $\Delta^+$ to $s_\alpha(\Delta^+)$,
we consider the special case when $\alpha$ is a simple root. Then
$\Delta^+$ and $s_\alpha(\Delta^+)$ differ by only one root:
\begin{theorem} \label{sa}
Let $\Pi \subset \Delta$ be a simple root system,
contained in a positive root set $\Delta^+$. If $\alpha \in \Pi$, then
the reflection $s_\alpha$ maps the set $\Delta^+ \setminus \{\alpha\}$ to
itself.
\end{theorem}
\begin{corollary} Any two positive root systems in $\Delta$ can be obtained
from each other by a composition of reflections with respect to the roots
in $\Delta$.
\end{corollary}
Hint: Let $\Delta^+_1 $ and $\Delta^+_2$ be two positive root systems.
Recall that the negative roots $\Delta^-_i$ are the negatives of
the elements in $\Delta^+_i$, $i=1,2$ (see \cite{3}).
Use induction on the number of elements in the intersection
$\Delta^+_1 \cap \Delta^-_2$. Theorem \ref{sa} provides a way to decrease
this number by one.
The statements above show that although a set of simple roots
is not unique for a given $\Delta$, they are related to each other by
a simple orthogonal transformation of the space $V$. In particular,
the angles and relative lengths of simple roots in any two simple
root systems in $\Delta$ are the same.
The next theorem proves another useful
property of simple roots.
\begin{theorem} \label{seq}
Let $\Pi \subset \Delta^+$ be a simple and a positive
root systems in $\Delta$.
Any positive root $\beta \in \Delta^+$ can be written as a sum
$$ \beta = \alpha_1 + \alpha_2 + \ldots + \alpha_k, $$
where $\alpha_i \in \Pi$ for all $i =1, \ldots , k$ (repetitions are allowed).
Moreover,
it can be done so that each partial sum
$$ \alpha_1 + \ldots + \alpha_m, \;\; 1 \leq m \leq k $$
is also a root.
\end{theorem}
Hint: Choose $t \in V$ such that $\langle t, \alpha \rangle =1$ for
all $\alpha \in \Pi$. Prove that such $t$ exists and that the number
$r=\langle t, \beta \rangle $
is a positive integer for any $\beta \in \Delta^+$. Using Lemma 7 in
\cite{3}, show that $\langle \alpha, \beta \rangle >0$ for some
$\alpha \in \Pi$.
Then proceed by induction on $r$. Theorem 10(1) in \cite{2} allows
us to reduce $r$ by one.
\begin{example} Let $\Delta$ be the root system in $V= \R^2$
such that the angle between the simple roots is $\frac{5\pi}{6}$.
This condition determines $\Delta$ completely (this is the root system
of the type $G_2$).
Construct and sketch the simple roots,
positive roots, and the whole root system $\Delta$. Apply
Theorem \ref{seq} in this case to present each positive root
as a sum of simple roots.
\end{example}
Recall that two root systems $\Delta$ and $\Delta'$ are \emph{isomorphic}
if there exists an linear automorphism of $V$ that maps
$\Delta$ onto $\Delta'$ and preserves the integers
$\frac{2\la \beta, \alpha\ra}{\la \alpha, \alpha\ra}i$.
A root system is irreducible if it
cannot be decomposed as a disjoint union of two root systems
$\Delta = \Delta' \cup \Delta''$ of smaller dimension, so that each
element of $\Delta'$ is orthogonal to each element of $\Delta''$.
\begin{example} Up to isomorphism, there are just three reduced irreducible
root systems in $V=\R^3$, of the types $A_3$, $B_3$ and $C_3$
(see Example 5 in \cite{2}, Examples 10 and 11 in \cite{3} for
definitions). Find the other possible reduced root systems in $V= \R^3$ (they
can be represented as a union of two or more root systems in
smaller dimensions).
\end{example}
Hint: Note that the only reduced root system in $\R$
is of the type $A_1$. A classification of root systems in $\R^2$
can be carried out as indicated at the end of \cite{2}.
\begin{thebibliography}{2}
\bibitem[1]{1} Your classmate, {\it Reflections in a Euclidean space},
preprint, MIT, 2004.
\bibitem[2]{2} Your classmate, {\it Abstract root systems},
preprint, MIT, 2004.
\bibitem[3]{3} Your classmate, {\it Simple and positive roots},
preprint, MIT, 2004.
\end{thebibliography}
\end{document}