Root System

Root System

Take $\Phi$ to be a finite set of nonzero vectors in $V$ satisfying the conditions:

• $\Phi \cap \mathbb{R}\alpha =\{\alpha ,-\alpha \}\text{for all}\alpha \in \Phi$
• $s_{\alpha }\Phi =\Phi \text{for all}\alpha \in \Phi$ $W$ to be the reflections generated by all reflections $s_{\alpha },\alpha \in \Phi$. $\Phi$ is a root system with associated reflection group $W$. The elements of $\Phi$ are called roots.

Positive System $\Pi \subset \Phi$ a positive system if it consists of all those roots which are positive relative to some total ordering of $V$.

We call a subset

We can construct total ordering of $V$ by considering $\{{\lambda }_1\cdots {\lambda }_n\}$ an ordered basis of $V$, adopt the corresponding lexicographic order ${\sum }_{}{a}_i{\lambda }_i<\sum b_i{\lambda }_i$ means $a_k<b_k$ if $k$ is the least index $i$ for which $a_i\ne b_i$.

Simple System

We call a subset $\Delta$ of $\Phi$ a simple system if $\Delta$ is a vector space basis for the $\mathbb{R}$ span of $\Phi$ in $V$ and moreover, if each $\alpha \in \Phi$ is a linear combination of $\Delta$ with coefficients all of the same sign(all non negative or non positive).

Theorem

(a) If $\Delta$ is a simple system in $\Phi$, then there is a unique positive system containing $\Delta$. (b) Every positive system $\Pi$ in $\Phi$ contains a unique simple system, in particular simple system exists.

Proof (b): We consider $\Delta$ to be the smallest subset of $\Pi$ subject to the requirement that each root in $\Pi$ is a nonnegative linear combination of $\Delta$. The left to be proved is $\Delta$ is linear independent. If ${\sum }_{\alpha \in \Delta }{a}_{\alpha }\alpha =0$ with not all $a_{\alpha }=0$, then $\sum b_{\beta }\beta =\sum c_{\gamma }\gamma$ with $b_{\beta }>0,c_{\gamma }>0$, by bellow corollary $0\le \left(\sum b_{\beta }\beta ,\sum c_{\gamma }\gamma \right)\le 0,$ which means $\sum b_{\beta }\beta =\sum c_{\gamma }\gamma =0$, contradiction, so $\Delta$ is linearly independent.

Corollary

If $\Delta$ is a simple system in $\Phi$, then $(\alpha ,\beta )\le 0$ for all $\alpha \ne \beta$ in $\Delta$.

Proof The condition of $\Delta$ can be actually loosed as the smallest subset of $\Pi$ subject to the requirement that each root in $\Pi$ is a nonnegative linear combination of $\Delta$.
Assume there exist $\alpha ,\beta ,(\alpha ,\beta )>0$, then $s_{\alpha }\beta =\beta -c\alpha$ with $c=\frac{2(\beta ,\alpha )}{(\alpha ,\alpha )}>0$, $s_{\alpha }\beta \in \Phi =\Pi \bigsqcup (-\Pi )$, by the proposition below, we get $s_{\alpha }\beta \in \Pi$, then $s_{\alpha }\beta ={\sum }_{\gamma \in \Delta }{c}_{\gamma }\gamma =\beta -c\alpha$, $(1-c_{\beta })\beta =c\alpha +{\sum }_{\gamma \in \Delta \setminus \{\beta \}}{c}_{\gamma }\gamma \in \Pi ,$ furthermore $c>0$, we get $1-c_{\beta }>0$, then $\beta =\frac{c}{1-c_{\beta }}\alpha +\sum \frac{c_{\gamma }}{1-c_{\beta }}\gamma$, which means we express $\beta$ as a linear combination of element in $\Delta$ with nonnegative coefficient, contradict to minimality of $\Delta$.

Remark

Intuitively, this corollary comes from the fact that if two roots in the simple system have angle smaller than 90 degree in between, we can represent one of the root as a sum of two positive roots through reflection, which contradicts to the smallest property of $\Delta$.

Proposition

Let $\Delta$ be a simple system, contained in a positive system $\Pi$. If $\alpha \in \Delta$, then $s_{\alpha }(\Pi \setminus \{\alpha \})=\Pi \setminus \{\alpha \}$

Proof $\beta \in \Pi$, $\beta \ne \alpha$, $\beta ={\sum }_{\gamma \in \Delta }{c}_{\gamma }\gamma (c_{\gamma }>0)$, then $s_{\alpha }\beta =\beta -c\alpha ={\sum }_{\gamma \in \Delta \setminus \{\alpha \}}{c}_{\gamma }\gamma +(c_{\alpha }-c)\alpha ,$ since $c=\frac{2(\alpha ,\beta )}{(\alpha ,\alpha )}\le 0$ as $2(\alpha ,\beta )={\sum }_{\gamma \in \Delta }{c}_{\gamma }(\alpha ,\gamma )\le 0$ with $c_{\gamma }\ge 0$, $(\alpha ,\gamma )\le 0$. So $s_{\alpha }\beta \in \Pi$, if $s_{\alpha }\beta =\alpha$ we get $\beta =s_{\alpha }(s_{\alpha }\beta )=s_{\alpha }(\alpha )=-\alpha \notin \Pi$, contradiction, so $s_{\alpha }\beta \in \Pi \setminus \{\alpha \}$. $s_{\alpha }$ is injective($s_{\alpha }({\beta }_1)=s_{\alpha }({\beta }_2)$ implies ${\beta }_1={\beta }_2$) on $\Pi \setminus \{\alpha \}$, thus surjective.

Theorem

Any two positive systems in $\Phi$ are conjugate under $W$. Any two simple systems in $\Phi$ are conjugate under $W$.

Proof $\Pi$ and ${\Pi }^{\prime }$ positive systems, $r=\text{Card}(\Pi \cap (-\Pi ))$, if $r=0$, then $\Pi ={\Pi }^{\prime }$. If $r>0$, $\Delta \subset \Pi$, $\Delta \not\subset {\Pi }^{\prime }$ so $\exists \alpha \in \Delta$, $\alpha \in (-{\Pi }^{\prime })$, $-\alpha \notin (-{\Pi }^{\prime })$. $s_{\alpha }(\Pi )=(\Pi \setminus \{\alpha \})\bigsqcup \{-\alpha \},-\alpha \notin s_{\alpha }(\Pi )\cap (-{\Pi }^{\prime }),$ so $\text{Card}(s_{\alpha }\Pi \cap -\Pi )=r-1$, then inductively, we get there exist $w\in W$ than $w\Pi ={\Pi }^{\prime }$.