# How to draw automorphism on Dynkin diagram?

I know how to draw a Dynkin diagram and add labels on the vertices using Tikz-cd or the package for Dynkin diagram, but how to draw an automorphism intuitively as in the following picture? I don't know how to draw those curly arrows on the diagrams.

• Welcome to TeX.SE! Since you are saying "I know how to draw a Dynkin diagram a ..." could you please post the code? The answer will depend on what you are doing. – user121799 Nov 26 '18 at 22:17
• I decided that these diagrams are easier to read if you bend the Dynkin diagram around the axis of symmetry, and then just draw thicker bars between the roots t that are being interchanged. I referred to these as folded Dynkin diagrams. If you compare my list of Satake diagrams in my dynkin-diagrams package to Satake's, I hope you will agree that it is easier to read. – Benjamin McKay Dec 7 '18 at 13:49

There is a full-fledged package for that: dynkin-diagrams. (And the outer automorphism group of D_4 is S_3.)

\documentclass{amsart}
\begin{document}
\begin{tikzpicture}
\begin{scope}[local bounding box=E6]
\dynkin{E}{6}
\foreach \X in {1,...,6}
{\node at (root \X) {\X};}
\draw[latex-latex,shorten >=2mm,shorten <=2mm] (root 1) to[out=-60,in=-120]
node[midway,below]{$\sigma$} (root 6);
\draw[latex-latex,shorten >=2mm,shorten <=2mm] (root 3) to[out=-60,in=-120]
node[midway,below]{$\sigma$} (root 5);
\end{scope}
\begin{scope}[xshift=5.5cm,yshift=-0.6cm,rotate=30,local bounding box=D4]
\dynkin{D}{4}
\foreach \X in {1,...,4}
{\node at (root \X) {\X};}
\draw[latex-latex,shorten >=2mm,shorten <=2mm] (root 4) to[bend left]
node[midway,auto]{$\sigma_{14}$} (root 1);
\draw[latex-latex,shorten >=2mm,shorten <=2mm] (root 1) to[bend left]
node[midway,auto]{$\sigma_{13}$} (root 3);
\draw[latex-latex,shorten >=2mm,shorten <=2mm] (root 3) to[bend left]
node[midway,auto]{$\sigma_{34}$} (root 4);
\end{scope}
\node[anchor=south] at (E6.north) {$\mathrm{E}_6$};
\node[anchor=south] at (D4.north) {$\mathrm{D}_4$};
\end{tikzpicture}
\end{document}


This is but a quick sketch for D_4.

\documentclass{article}
\usepackage{tikz}
\tikzstyle{vertex}=[circle, draw, minimum size=0pt]
\newcommand{\vertex}{\node[vertex]}

\begin{document}

\begin{tikzpicture}
\node at (0,0) {$D_4$};

\vertex (a) at (0,-1) {4};
\vertex (b) at (0,-3) {2};
\vertex (c) at (-2,-3) {1};
\vertex (d) at (2,-3) {3};
\draw[->] (b) -- (a);
\draw[->] (b) -- (c);
\draw[->] (b) --(d);
\draw[<->] (c) edge[bend right=80] node[below] {$\sigma$} (d);

\end{tikzpicture}

\end{document}


• \tikzstyle is deprecated; it should be \tikzset{vertex/.style={circle, draw, minimum size=0pt}} – egreg Nov 26 '18 at 22:53
• That's pretty good, though for E_6 it's more complicated.. – sawdada Nov 27 '18 at 0:13

Using the latest version (3.141) of the dynkin-diagrams package, you can compare the way that the package handles Satake diagrams by default (using a thick grey bar to show which roots are identified by an involution), with the way that you have described them (using arrows with $\sigma$ written under them). These examples are included in the package documentation.

\documentclass{amsart}
\usepackage{dynkin-diagrams}
\newcommand{\invol}[2]{\draw[latex-latex] (root #1) to [out=-60,in=-120]
node[midway,below]{$\sigma$} (root #2);}
\begin{document}
\begin{tabular}{ccc}
\dynkin[ply=3]{D}{4} &
\dynkin{E}{II} &
\dynkin{A}{IIIa} \\
{\tikzset{/Dynkin diagram/fold style/.style={stealth-stealth,thick,
shorten <=1mm,shorten >=1mm,}}
\dynkin[ply=3,edge length=.75cm]{D}{4}}
&
\begin{dynkinDiagram}[edge length=.75cm,labels*={1,...,6}]{E}{6}
\invol{1}{6}\invol{3}{5}
\end{dynkinDiagram}
&
\begin{dynkinDiagram}[edge length=.75cm]{A}{oo.o**.**o.oo}
\invol{1}{10}\invol{2}{9}\invol{3}{8}\invol{4}{7}\invol{5}{6}
\end{dynkinDiagram} \\
\end{tabular}
\end{document}