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.)

\begin{scope}[local bounding box=E6]
 \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);
\begin{scope}[xshift=5.5cm,yshift=-0.6cm,rotate=30,local bounding box=D4]
 \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);
\node[anchor=south] at (E6.north) {$\mathrm{E}_6$};
\node[anchor=south] at (D4.north) {$\mathrm{D}_4$};

enter image description here

  • Thank you for help! I shall learn more about this useful package. – sawdada Nov 27 '18 at 0:12

This is but a quick sketch for D_4.

\tikzstyle{vertex}=[circle, draw, minimum size=0pt]


\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);



enter image description here

  • 1
    \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.

Dynkin diagrams variously folded

\newcommand{\invol}[2]{\draw[latex-latex] (root #1) to [out=-60,in=-120] 
node[midway,below]{$\sigma$} (root #2);}
\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}
\begin{dynkinDiagram}[edge length=.75cm]{A}{oo.o**.**o.oo}
\end{dynkinDiagram} \\

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