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## LaTeX Tip: Dynkin Diagrams using Tikz

Hi everyone, I’ve include the code for drawing Dynkin diagrams using Tikz package in LaTeX

\begin{center}
\begin{tikzpicture}[scale=.4]
\draw (-1,0) node[anchor=east]  {$A_n$};
\foreach \x in {0,...,5}
\draw[xshift=\x cm,thick] (\x cm,0) circle (.3cm);
\draw[dotted,thick] (0.3 cm,0) -- +(1.4 cm,0);
\foreach \y in {1.15,...,4.15}
\draw[xshift=\y cm,thick] (\y cm,0) -- +(1.4 cm,0);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}[scale=.4]
\draw (-1,0) node[anchor=east]  {$B_n$};
\foreach \x in {0,...,4}
\draw[xshift=\x cm,thick] (\x cm,0) circle (.3cm);
\draw[xshift=5 cm,thick,fill=black] (5 cm, 0) circle (.3 cm);
\draw[dotted,thick] (0.3 cm,0) -- +(1.4 cm,0);
\foreach \y in {1.15,...,3.15}
\draw[xshift=\y cm,thick] (\y cm,0) -- +(1.4 cm,0);
\draw[thick] (8.3 cm, .1 cm) -- +(1.4 cm,0);
\draw[thick] (8.3 cm, -.1 cm) -- +(1.4 cm,0);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}[scale=.4]
\draw (-1,0) node[anchor=east]  {$C_n$};
\foreach \x in {0,...,4}
\draw[xshift=\x cm,thick,fill=black] (\x cm,0) circle (.3cm);
\draw[xshift=5 cm,thick] (5 cm, 0) circle (.3 cm);
\draw[dotted,thick] (0.3 cm,0) -- +(1.4 cm,0);
\foreach \y in {1.15,...,3.15}
\draw[xshift=\y cm,thick] (\y cm,0) -- +(1.4 cm,0);
\draw[thick] (8.3 cm, .1 cm) -- +(1.4 cm,0);
\draw[thick] (8.3 cm, -.1 cm) -- +(1.4 cm,0);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}[scale=.4]
\draw (-1,0) node[anchor=east]  {$D_n$};
\foreach \x in {0,...,4}
\draw[xshift=\x cm,thick] (\x cm,0) circle (.3cm);
\draw[xshift=8 cm,thick] (30: 17 mm) circle (.3cm);
\draw[xshift=8 cm,thick] (-30: 17 mm) circle (.3cm);
\draw[dotted,thick] (0.3 cm,0) -- +(1.4 cm,0);
\foreach \y in {1.15,...,3.15}
\draw[xshift=\y cm,thick] (\y cm,0) -- +(1.4 cm,0);
\draw[xshift=8 cm,thick] (30: 3 mm) -- (30: 14 mm);
\draw[xshift=8 cm,thick] (-30: 3 mm) -- (-30: 14 mm);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}[scale=.4]
\draw (-1,0) node[anchor=east]  {$G_2$};
\draw[thick] (0 ,0) circle (.3 cm);
\draw[thick,fill=black] (2 cm,0) circle (.3 cm);
\draw[thick] (30: 3mm) -- +(1.5 cm, 0);
\draw[thick] (0: 3 mm) -- +(1.4 cm, 0);
\draw[thick] (-30: 3 mm) -- +(1.5 cm, 0);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}[scale=.4]
\draw (-3,0) node[anchor=east]  {$F_4$};
\draw[thick] (-2 cm ,0) circle (.3 cm);
\draw[thick] (0 ,0) circle (.3 cm);
\draw[thick,fill=black] (2 cm,0) circle (.3 cm);
\draw[thick,fill=black] (4 cm,0) circle (.3 cm);
\draw[thick] (15: 3mm) -- +(1.5 cm, 0);
\draw[xshift=-2 cm,thick] (0: 3 mm) -- +(1.4 cm, 0);
\draw[thick] (-15: 3 mm) -- +(1.5 cm, 0);
\draw[xshift=2 cm,thick] (0: 3 mm) -- +(1.4 cm, 0);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}[scale=.4]
\draw (-1,1) node[anchor=east]  {$E_6$};
\foreach \x in {0,...,4}
\draw[thick,xshift=\x cm] (\x cm,0) circle (3 mm);
\foreach \y in {0,...,3}
\draw[thick,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thick] (4 cm,2 cm) circle (3 mm);
\draw[thick] (4 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}[scale=.4]
\draw (-1,1) node[anchor=east]  {$E_7$};
\foreach \x in {0,...,5}
\draw[thick,xshift=\x cm] (\x cm,0) circle (3 mm);
\foreach \y in {0,...,4}
\draw[thick,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thick] (4 cm,2 cm) circle (3 mm);
\draw[thick] (4 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}[scale=.4]
\draw (-1,1) node[anchor=east]  {$E_8$};
\foreach \x in {0,...,6}
\draw[thick,xshift=\x cm] (\x cm,0) circle (3 mm);
\foreach \y in {0,...,5}
\draw[thick,xshift=\y cm] (\y cm,0) ++(.3 cm, 0) -- +(14 mm,0);
\draw[thick] (4 cm,2 cm) circle (3 mm);
\draw[thick] (4 cm, 3mm) -- +(0, 1.4 cm);
\end{tikzpicture}
\end{center}

Bye

DOX

## Table of all possible irreps of Lie groups

Last weekend, I wrote a program in SAGE that list all possible irreps of a Lie group, one specified the group (in Cartan’s classification) and the maximum sum of the Dynkin labels.

Check the notebook in here.

Since I’m not a programmer, please feel free to leave comments, specially if you find ways to optimize the program

Cheers,

DOX