%% boi-6r.tex 
%% Tables of rootsystems
%% Input file for Chapter 6 of the Book of Involutions

\subsection{Classification of irreducible root systems}
\label{gr.classroot}
%
There are four infinite families $A_n$, $B_n$, $C_n$,~$D_n$ and five
exceptional irreducible root systems $E_6$, $E_7$, $E_8$,
$F_4$,~$G_2$. We refer to Bourbaki~\cite{Bourbaki3} for the following
datas about root systems.
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\nopagebreak
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\begin{roottable}{Type $A_n$, $n \geq 1$}
%
{\index{B}{a1n@{$A_n$ (root system)}}%
  Let $V = \LR^{n+1}/(e_1 + e_2 + \dots +e_{n+1})\LR$ where
  $\{e_1,\dots, e_{n+1}\}$ is the canonical basis of $\LR^{n+1}$.  We
  denote by~$\overline{e}_i$ the class of~$e_i$ in~$V$.}
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{$\Phi = \bmidb{\overline{e}_i-\overline{e}_j}{i\neq j}$,\quad
  $n(n+1)$ roots.\\}
%
{$\Lambda_r = \bmidb{\sum a_i\overline{e}_i}{\sum a_i = 0}$.\\}
%
{$\Lambda = \sum\overline{e}_i\LZ$,\quad $\Lambda/\Lambda_r \simeq \LZ
  / (n+1) \LZ$.\\}
%
{$ \Pi = \{\overline{e}_1 - \overline{e}_2, \overline{e}_2 -
  \overline{e}_3, \dots, \overline{e}_n - \overline{e}_{n+1}
  \}$.\\[1ex]}
%
{$\kern1.5\unitlength %
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      \end{picture}}}$\\[2ex]}
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{$\{1\}$ if $n= 1$,\quad $\{1,\tau\}$ if $n\geq 2$.\\}
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{$ \Lambda_+ = \bmidb{ \sum a_i \cdot \overline{e}_i \in \Lambda} {a_1
    \geq a_2 \geq \dots\geq a_{n+1} }$.\\}
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{$\overline{e}_1 + \overline{e}_2 + \dots + \overline{e}_i$,\quad $i = 1$, $2$, \dots,
  $n+1$.\\}
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\end{roottable}
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\begin{roottable}{Type $B_n$, $n \geq 1$}
%
{\index{B}{b1n@{$B_n$ (root system)}}%
  Let $V = \LR^n$ with canonical basis $\{e_i\}$.}
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{$\Phi=\bmidb{\pm e_i,\pm e_i\pm e_j}{i>j}$,\quad $2n^2$ roots.\\}
%
{$\Lambda_r = \LZ^n$.\\}
%
{$ \Lambda = \Lambda_r + \frac{1}{2}(e_1 + e_2 + \dots + e_n)\LZ$,
  \quad $\Lambda/\Lambda_r \simeq \LZ/2 \LZ$.\\}
%
{$\Pi = \{ e_1 - e_2, e_2 - e_3, \dots, e_{n-1} - e_n, e_n
  \}$.\\[1ex]}
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{$\kern1.5\unitlength %
\begin{picture}(0,0)\put(44,0){\hcenter{${>}$}}
\end{picture}
\vcenter{\hbox{\begin{picture}(0,0)%
      \put(0,0){\circle{2}}%
      \put(0,-5){\hcenter{$\scriptstyle 1$}}
        \put(1,0){\line(1,0){10}}%
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        \put(12,-5){\hcenter{$\scriptstyle 2$}}%
        \put(13,0){\line(1,0){5}}% 
        \put(22,0){\circle*{0}}%
        \put(25,0){\circle*{0}}% 
        \put(28,0){\circle*{0}}% 
        \put(37,0){\line(-1,0){5}}%
        \put(38,0){\circle{2}}%
        \put(38,-5){\hcenter{$\scriptstyle n-1$}}% 
        \put(39,0.5){\line(1,0){10}}%
        \put(39,-0.5){\line(1,0){10}}%
        \put(50,0){\circle{2}}%
        \put(50,-5){\hcenter{$\scriptstyle n$}}% 
      \end{picture}}}$\\[2ex]}
%
{$\{1\}$.\\}
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{$ \Lambda_+ = \bmidb{ \sum a_i e_i \in \Lambda }{a_1 \geq a_2 \geq
    \dots \geq a_n \geq 0 }$.\\}
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{$0$, $\frac{1}{2}(e_1 + e_2 + \dots + e_n)$.\\}
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\end{roottable}
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\begin{roottable}{Type $C_n$, $n \geq 1$} 
%
{\index{B}{c1n@{$C_n$ (root system)}}%
  Let $V = \LR^n$ with canonical basis $\{e_i\}$.}
%
{$\Phi=\bmidb{\pm2e_i,\pm e_i\pm e_j}{i>j}$,\quad $2n^2$ roots.\\}
%
{$\Lambda_r = \bmidb{ \sum a_i e_i }{a_i \in \LZ,\,\sum a_i \in 2
    \LZ}$.\\}
%
{$\Lambda = \LZ^n$, \quad $\Lambda/\Lambda_r \simeq \LZ/2 \LZ$.\\}
%
{$\Pi = \{ e_1 - e_2, e_2 - e_3, \dots, e_{n-1} - e_n, 2 e_n
  \}$.\\[1ex]}
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{$\kern1.5\unitlength
\begin{picture}(0,0)\put(44,0){\hcenter{${<}$}}
\end{picture}
\vcenter{\hbox{\begin{picture}(0,0)%
      \put(0,0){\circle{2}}%
      \put(0,-5){\hcenter{$\scriptstyle 1$}}
        \put(1,0){\line(1,0){10}}%
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        \put(12,-5){\hcenter{$\scriptstyle 2$}}%
        \put(13,0){\line(1,0){5}}% 
        \put(22,0){\circle*{0}}%
        \put(25,0){\circle*{0}}% 
        \put(28,0){\circle*{0}}% 
        \put(37,0){\line(-1,0){5}}%
        \put(38,0){\circle{2}}%
        \put(38,-5){\hcenter{$\scriptstyle n-1$}}% 
        \put(39,0.5){\line(1,0){10}}%
        \put(39,-0.5){\line(1,0){10}}%
        \put(50,0){\circle{2}}%
        \put(50,-5){\hcenter{$\scriptstyle n$}}% 
      \end{picture}}}$\\[2ex]}
%
{$\{1\}$.\\}
%
{$\Lambda_+ = \bmidb{ \sum a_i e_i \in \Lambda }{a_1 \geq a_2 \geq
    \dots \geq a_n \geq 0 }$.\\}
%
{$0$, $e_1$.\\}
%
\end{roottable}
%
\begin{roottable}{Type $D_n$, $n \geq 3$}
%
{\index{B}{d1n@{$D_n$ (root system)}}%
  (For $n = 2$ the definition works but yields $A_1 + A_1$.)  Let $V =
  \LR^n$ with canonical basis $\{e_i\}$.}
%
{$\Phi = \bmidb{\pm e_i \pm e_j}{i > j}$, \quad $2n (n-1)$ roots.\\}
%
{$ \Lambda_r = \bmidb{\sum a_i e_i}{a_i \in\LZ,\, \sum a_i \in 2
    \LZ}$.\\}
%
{$ \Lambda = \LZ^n + \frac{1}{2} (e_1 + e_2 + \dots + e_n)\LZ$,\\[1ex]
%
  && $\Lambda/\Lambda_r \simeq
\begin{cases}
  \LZ/2 \LZ \oplus \LZ/2 \LZ & \text{if $n$ is even},\\ \LZ/4 \LZ &
  \text{if $n$ is odd}.
\end{cases}$\\[1ex]}
%
{$\Pi = \{ e_1 - e_2, \dots, e_{n-1} - e_n, e_{n-1} + e_n\}$.\\[4ex]}
%
{$\kern1.5\unitlength %
\begin{picture}(0,0)%
        \put(47,7){$\vcenter{\hbox{$\scriptstyle n-1$}}$}%
        \put(47,-7){$\vcenter{\hbox{$\scriptstyle n
\phantom{\scriptstyle -1}$}}$}%
\end{picture}%
\vcenter{\hbox{\begin{picture}(0,0)%
      \put(0,0){\circle{2}}%
      \put(0,-5){\hcenter{$\scriptstyle 1$}}
        \put(1,0){\line(1,0){10}}%
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        \put(12,-5){\hcenter{$\scriptstyle 2$}}%
        \put(13,0){\line(1,0){5}}% 
        \put(22,0){\circle*{0}}%
        \put(25,0){\circle*{0}}% 
        \put(28,0){\circle*{0}}% 
        \put(37,0){\line(-1,0){5}}%
        \put(38,0){\circle{2}}%
        \put(38,-5){\llap{$\scriptstyle n-2$}}% 
        \put(45,7){\circle{2}}%
        \put(45,-7){\circle{2}}%
        \put(38.71,0.71){\line(1,1){5.58}}
        \put(38.71,-0.71){\line(1,-1){5.58}}
      \end{picture}}}$\\[4ex]}
%
{$S_3$ if $n=4$,\quad $\{1,\tau\}$ if $n=3$ or $n>4$.\\}
%
{$\Lambda_+ = \bmidb{ \sum a_i e_i \in \Lambda }{ a_1 \geq a_2 \geq
    \dots \geq a_n,\, a_{n-1} + a_n \geq 0} $.\\}
%
{$0$, $e_1$, $\frac{1}{2}(e_1 + e_2 + \dots + e_{n-1} \pm e_n)$.\\}
%
\end{roottable}
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\begin{roottableEXC}{Exceptional types}
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\begin{tabular}{@{}l@{}l@{\quad}l}%
  $E_6$&:&$\Aut\bigl(\Dyn(\Phi)\bigr)=\{1,\tau\}$,\quad
{\index{B}{E6@{$E_6$ (root system)}}%
}%
  $\Lambda/\Lambda_r\simeq \LZ/3 \LZ$.\\[1ex]&&
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\begin{picture}(0,0)
  \put(0,0){\circle{2}} \put(12,0){\circle{2}} \put(24,0){\circle{2}}
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%
  \put(1,0){\line(1,0){10}} \put(13,0){\line(1,0){10}}
  \put(25,0){\line(1,0){10}} \put(37,0){\line(1,0){10}}
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  \put(24,-8){\circle{2}} \put(24,-7){\line(0,1){6}}
\end{picture}\\[6ex]
%
$E_7$&:&$\Aut\bigl(\Dyn(\Phi)\bigr)=\{1\}$,\quad 
{\index{B}{E7@{$E_7$ (root system)}}%
}%
$\Lambda/\Lambda_r\simeq \LZ/2 \LZ$.\\[1ex]&&
%
\kern1.5\unitlength %
\begin{picture}(0,0)
  \put(0,0){\circle{2}} \put(12,0){\circle{2}} \put(24,0){\circle{2}}
  \put(36,0){\circle{2}} \put(48,0){\circle{2}} \put(60,0){\circle{2}}
%
  \put(1,0){\line(1,0){10}} \put(13,0){\line(1,0){10}}
  \put(25,0){\line(1,0){10}} \put(37,0){\line(1,0){10}}
  \put(49,0){\line(1,0){10}}
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  \put(24,-8){\circle{2}} \put(24,-7){\line(0,1){6}}
\end{picture}\\[6ex]
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$E_8$&:&$\Aut\bigl(\Dyn(\Phi)\bigr)=\{1\}$,\quad
{\index{B}{E8@{$E_8$ (root system)}}%
}%
$\Lambda/\Lambda_r=0$.\\[1ex]&&
%
\kern1.5\unitlength %
\begin{picture}(0,0)
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  \put(72,0){\circle{2}}
%
  \put(1,0){\line(1,0){10}} \put(13,0){\line(1,0){10}}
  \put(25,0){\line(1,0){10}} \put(37,0){\line(1,0){10}}
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%
  \put(24,-8){\circle{2}} \put(24,-7){\line(0,1){6}}
\end{picture}\\[6ex]
%
$F_4$&:&$\Aut\bigl(\Dyn(\Phi)\bigr)=\{1\}$,\quad
{\index{B}{F15@{$F_4$ (root system)}}%
}%
$\Lambda/\Lambda_r=0$.
{$\kern10\unitlength %
\begin{picture}(0,0)\put(18,0){\hcenter{${>}$}}
\end{picture}
\vcenter{\hbox{\begin{picture}(0,0)%
      \put(0,0){\circle{2}} \put(12,0){\circle{2}}
      \put(24,0){\circle{2}} \put(36,0){\circle{2}}
%
      \put(1,0){\line(1,0){10}} \put(13,0.4){\line(1,0){10}}
      \put(13,-0.6){\line(1,0){10}} \put(25,0){\line(1,0){10}}
\end{picture}}}$}\\[2ex]
%
$G_2$&:&$\Aut\bigl(\Dyn(\Phi)\bigr)=\{1\}$,\quad
{\index{B}{G2r@{$G_2$ (root system)}}%
}%
$\Lambda/\Lambda_r=0$.
%
{$\kern10\unitlength %
\begin{picture}(0,0)\put(6,0){\hcenter{${<}$}}
\end{picture}
\vcenter{\hbox{\begin{picture}(0,0)%
      \put(0,0){\circle{2}} \put(12,0){\circle{2}}
%
      \put(1,0){\line(1,0){10}} \put(1,0.7){\line(1,0){10}}
      \put(1,-0.7){\line(1,0){10}}
\end{picture}}}$}\\[2ex]
\end{tabular}
\end{roottableEXC}

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