20220504-extended-dynkin-diagrams.tex

\documentclass[10pt,a4paper]{article}  % -*- coding: utf-8 -*-
\usepackage[a4paper,margin=1.5cm]{geometry}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{times}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
%
\usepackage{mathrsfs}
%\usepackage{bm}
%\usepackage{stmaryrd}
\usepackage{wasysym}
\usepackage{url}
\usepackage{graphicx}
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{tikz}
\usetikzlibrary{matrix,arrows,decorations.markings}
%\usepackage{hyperref}
%
%
%
\mathchardef\emdash="07C\relax
\mathchardef\hyphen="02D\relax
\DeclareUnicodeCharacter{00A0}{~}
%
%
%
\newcommand{\dynkinradius}{.08cm}
\newcommand{\dynkinstep}{.70cm}
\newcommand{\dynkinXsize}{1.5pt}
\newcommand{\dynkindoublesep}{1pt}
\newcommand{\dynkinarrowsize}{.12cm}
\newcommand{\dynkinshortendim}{1pt}
\newcommand{\dynkindot}[2]{\begin{scope}[shift={(\dynkinstep*#1,\dynkinstep*#2)}]\fill[fill=white,draw=black] (0,0) circle (\dynkinradius);\end{scope}}
\newcommand{\dynkinline}[3]{%
\begin{scope}[shift={(\dynkinstep*#1,\dynkinstep*#2)},rotate=#3]%
\draw (0,0) -- (\dynkinstep,0);%
\end{scope}%
}
\newcommand{\dynkinanyline}[4]{%
\draw (\dynkinstep*#1,\dynkinstep*#2) -- (\dynkinstep*#3,\dynkinstep*#4);%
}
\newcommand{\dynkindots}[3]{%
\begin{scope}[shift={(\dynkinstep*#1,\dynkinstep*#2)},rotate=#3]%
\draw[dotted] (0,0) -- (\dynkinstep,0);%
\end{scope}%
}
\newcommand{\dynkindoubleline}[3]{%
\begin{scope}[shift={(\dynkinstep*#1,\dynkinstep*#2)},rotate=#3]%
\draw (0,\dynkindoublesep) -- (\dynkinstep,\dynkindoublesep);%
\draw (0,-\dynkindoublesep) -- (\dynkinstep,-\dynkindoublesep);%
\draw (\dynkinstep*0.5-\dynkinarrowsize*0.5,-\dynkinarrowsize) -- (\dynkinstep*0.5+\dynkinarrowsize*0.5,0) -- (\dynkinstep*0.5-\dynkinarrowsize*0.5,\dynkinarrowsize);%
\end{scope}%
}
\newcommand{\dynkintripleline}[3]{%
\begin{scope}[shift={(\dynkinstep*#1,\dynkinstep*#2)},rotate=#3]%
\draw (0,0) -- (\dynkinstep,0);%
\draw (0,\dynkindoublesep*1.5) -- (\dynkinstep,\dynkindoublesep*1.5);%
\draw (0,-\dynkindoublesep*1.5) -- (\dynkinstep,-\dynkindoublesep*1.5);%
\draw (\dynkinstep*0.5-\dynkinarrowsize*0.5,-\dynkinarrowsize) -- (\dynkinstep*0.5+\dynkinarrowsize*0.5,0) -- (\dynkinstep*0.5-\dynkinarrowsize*0.5,\dynkinarrowsize);%
\end{scope}%
}
\newcommand{\dynkinquadrupleline}[3]{%
\begin{scope}[shift={(\dynkinstep*#1,\dynkinstep*#2)},rotate=#3]%
\draw (0,\dynkindoublesep*0.5) -- (\dynkinstep,\dynkindoublesep*0.5);%
\draw (0,-\dynkindoublesep*0.5) -- (\dynkinstep,-\dynkindoublesep*0.5);%
\draw (0,\dynkindoublesep*1.5) -- (\dynkinstep,\dynkindoublesep*1.5);%
\draw (0,-\dynkindoublesep*1.5) -- (\dynkinstep,-\dynkindoublesep*1.5);%
\draw (\dynkinstep*0.5-\dynkinarrowsize*0.5,-\dynkinarrowsize) -- (\dynkinstep*0.5+\dynkinarrowsize*0.5,0) -- (\dynkinstep*0.5-\dynkinarrowsize*0.5,\dynkinarrowsize);%
\end{scope}%
}
\newcommand{\dynkintextnorth}[3]{%
\node[anchor=south] at (\dynkinstep*#1,\dynkinstep*#2) {\footnotesize #3};%
}
\newcommand{\dynkintextsouth}[3]{%
\node[anchor=north] at (\dynkinstep*#1,\dynkinstep*#2) {\footnotesize #3};%
}
\newcommand{\dynkintextwest}[3]{%
\node[anchor=east] at (\dynkinstep*#1,\dynkinstep*#2) {\footnotesize #3};%
}
\newcommand{\dynkintexteast}[3]{%
\node[anchor=west] at (\dynkinstep*#1,\dynkinstep*#2) {\footnotesize #3};%
}
%\newcommand{\dynkindoubleline}[4]{
%\draw[postaction={decorate},decoration={markings,mark=at position 0.6 with {\arrow{>}}}] (\dynkinstep*#1,\dynkinstep*#2) -- (\dynkinstep*#3,\dynkinstep*#4);
%}
%\newcommand{\dynkintripleline}[4]{\draw[postaction={decorate},decoration={markings,mark=at position 0.6 with {\arrow{>}}}] (\dynkinstep*#1,\dynkinstep*#2) -- (\dynkinstep*#3,\dynkinstep*#4);}
\newenvironment{dynkin}{\begin{tikzpicture}[baseline=-\dynkinradius]}{\end{tikzpicture}}
%
%
%
\begin{document}
\pretolerance=8000
\tolerance=50000

\pagestyle{empty}

\newcommand{\highstrut}{\vrule height 15pt depth 2pt width 0pt}

\begin{tabular}{ll}

\begin{tabular}{|r|l|}
\hline

\highstrut$\widetilde{A_1}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkindoubleline{-1}{0}{0};
    \dynkindoubleline{0}{0}{180};
    \dynkindot{-1}{0};
    \dynkindot{0}{0};
    \dynkintextnorth{-1}{0}{1};
    \dynkintextnorth{0}{0}{1};
\end{dynkin}
\end{tabular}
\\\hline

\highstrut$\widetilde{A_n}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkinline{0}{0}{0};
    \dynkinline{1}{0}{0};
    \dynkinline{2}{0}{0};
    \dynkindots{3}{0}{0};
    \dynkinline{4}{0}{0};
    \dynkinline{5}{0}{0};
    \dynkinanyline{0}{0}{3}{-1};
    \dynkinanyline{6}{0}{3}{-1};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{3}{0};
    \dynkindot{4}{0};
    \dynkindot{5}{0};
    \dynkindot{6}{0};
    \dynkindot{3}{-1};
    \dynkintextnorth{0}{0}{1};
    \dynkintextnorth{1}{0}{1};
    \dynkintextnorth{2}{0}{1};
    \dynkintextnorth{3}{0}{1};
    \dynkintextnorth{4}{0}{1};
    \dynkintextnorth{5}{0}{1};
    \dynkintextnorth{6}{0}{1};
    \dynkintextsouth{3}{-1}{1};
\end{dynkin}
\\($n+1$~nodes)\\
\end{tabular}
\\\hline

\highstrut$\widetilde{B_n}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkinline{0}{0}{120};
    \dynkinline{0}{0}{-120};
    \dynkinline{0}{0}{0};
    \dynkinline{1}{0}{0};
    \dynkindots{2}{0}{0};
    \dynkinline{3}{0}{0};
    \dynkindoubleline{4}{0}{0};
    \dynkindot{-0.5}{0.866025404};
    \dynkindot{-0.5}{-0.866025404};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{3}{0};
    \dynkindot{4}{0};
    \dynkindot{5}{0};
    \dynkintextwest{-0.5}{0.866025404}{1};
    \dynkintextwest{-0.5}{-0.866025404}{1};
    \dynkintextnorth{0}{0}{2};
    \dynkintextnorth{1}{0}{2};
    \dynkintextnorth{2}{0}{2};
    \dynkintextnorth{3}{0}{2};
    \dynkintextnorth{4}{0}{2};
    \dynkintextnorth{5}{0}{2};
\end{dynkin}
\\($n+1$~nodes)\\
\end{tabular}
\\\hline

\highstrut$\widetilde{C_n}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkindoubleline{-1}{0}{0};
    \dynkinline{0}{0}{0};
    \dynkinline{1}{0}{0};
    \dynkindots{2}{0}{0};
    \dynkinline{3}{0}{0};
    \dynkindoubleline{5}{0}{180};
    \dynkindot{-1}{0};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{3}{0};
    \dynkindot{4}{0};
    \dynkindot{5}{0};
    \dynkintextnorth{-1}{0}{1};
    \dynkintextnorth{0}{0}{2};
    \dynkintextnorth{1}{0}{2};
    \dynkintextnorth{2}{0}{2};
    \dynkintextnorth{3}{0}{2};
    \dynkintextnorth{4}{0}{2};
    \dynkintextnorth{5}{0}{1};
\end{dynkin}
\\($n+1$~nodes)\\
\end{tabular}
\\\hline

\highstrut$\widetilde{D_n}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkinline{0}{0}{120};
    \dynkinline{0}{0}{-120};
    \dynkinline{0}{0}{0};
    \dynkinline{1}{0}{0};
    \dynkindots{2}{0}{0};
    \dynkinline{3}{0}{0};
    \dynkinline{4}{0}{60};
    \dynkinline{4}{0}{-60};
    \dynkindot{-0.5}{0.866025404};
    \dynkindot{-0.5}{-0.866025404};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{3}{0};
    \dynkindot{4}{0};
    \dynkindot{4.5}{0.866025404};
    \dynkindot{4.5}{-0.866025404};
    \dynkintextwest{-0.5}{0.866025404}{1};
    \dynkintextwest{-0.5}{-0.866025404}{1};
    \dynkintextnorth{0}{0}{2};
    \dynkintextnorth{1}{0}{2};
    \dynkintextnorth{2}{0}{2};
    \dynkintextnorth{3}{0}{2};
    \dynkintextnorth{4}{0}{2};
    \dynkintexteast{4.5}{0.866025404}{1};
    \dynkintexteast{4.5}{-0.866025404}{1};
\end{dynkin}
\\($n+1$~nodes)\\
\end{tabular}
\\\hline

\highstrut$\widetilde{G_2}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkinline{-1}{0}{0};
    \dynkintripleline{0}{0}{0};
    \dynkindot{-1}{0};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkintextnorth{-1}{0}{1};
    \dynkintextnorth{0}{0}{2};
    \dynkintextnorth{1}{0}{3};
\end{dynkin}
\end{tabular}
\\\hline

\highstrut$\widetilde{F_4}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkinline{-1}{0}{0};
    \dynkinline{0}{0}{0};
    \dynkindoubleline{1}{0}{0};
    \dynkinline{2}{0}{0};
    \dynkindot{-1}{0};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{3}{0};
    \dynkintextnorth{-1}{0}{1};
    \dynkintextnorth{0}{0}{2};
    \dynkintextnorth{1}{0}{3};
    \dynkintextnorth{2}{0}{4};
    \dynkintextnorth{3}{0}{2};
\end{dynkin}
\end{tabular}
\\\hline

\highstrut$\widetilde{E_6}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkinline{0}{0}{0};
    \dynkinline{1}{0}{0};
    \dynkinline{2}{0}{0};
    \dynkinline{2}{0}{-90};
    \dynkinline{2}{-1}{-90};
    \dynkinline{3}{0}{0};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{2}{-1};
    \dynkindot{2}{-2};
    \dynkindot{3}{0};
    \dynkindot{4}{0};
    \dynkintextnorth{0}{0}{1};
    \dynkintextnorth{1}{0}{2};
    \dynkintextnorth{2}{0}{3};
    \dynkintextnorth{3}{0}{2};
    \dynkintextnorth{4}{0}{1};
    \dynkintexteast{2}{-1}{2};
    \dynkintexteast{2}{-2}{1};
\end{dynkin}
\end{tabular}
\\\hline

\highstrut$\widetilde{E_7}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkinline{-1}{0}{0};
    \dynkinline{0}{0}{0};
    \dynkinline{1}{0}{0};
    \dynkinline{2}{0}{0};
    \dynkinline{2}{0}{-90};
    \dynkinline{3}{0}{0};
    \dynkinline{4}{0}{0};
    \dynkindot{-1}{0};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{2}{-1};
    \dynkindot{3}{0};
    \dynkindot{4}{0};
    \dynkindot{5}{0};
    \dynkintextnorth{-1}{0}{1};
    \dynkintextnorth{0}{0}{2};
    \dynkintextnorth{1}{0}{3};
    \dynkintextnorth{2}{0}{4};
    \dynkintextnorth{3}{0}{3};
    \dynkintextnorth{4}{0}{2};
    \dynkintextnorth{5}{0}{1};
    \dynkintexteast{2}{-1}{2};
\end{dynkin}
\end{tabular}
\\\hline

\highstrut$\widetilde{E_8}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkinline{0}{0}{0};
    \dynkinline{1}{0}{0};
    \dynkinline{2}{0}{0};
    \dynkinline{2}{0}{-90};
    \dynkinline{3}{0}{0};
    \dynkinline{4}{0}{0};
    \dynkinline{5}{0}{0};
    \dynkinline{6}{0}{0};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{2}{-1};
    \dynkindot{3}{0};
    \dynkindot{4}{0};
    \dynkindot{5}{0};
    \dynkindot{6}{0};
    \dynkindot{7}{0};
    \dynkintextnorth{0}{0}{2};
    \dynkintextnorth{1}{0}{4};
    \dynkintextnorth{2}{0}{6};
    \dynkintextnorth{3}{0}{5};
    \dynkintextnorth{4}{0}{4};
    \dynkintextnorth{5}{0}{3};
    \dynkintextnorth{6}{0}{2};
    \dynkintextnorth{7}{0}{1};
    \dynkintexteast{2}{-1}{3};
\end{dynkin}
\end{tabular}
\\\hline

\end{tabular}

&

\begin{tabular}{|r|l|}
\hline

\highstrut$\widetilde{^2A_2}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkinquadrupleline{-1}{0}{0};
    \dynkindot{-1}{0};
    \dynkindot{0}{0};
    \dynkintextnorth{-1}{0}{1};
    \dynkintextnorth{0}{0}{2};
\end{dynkin}
\end{tabular}
\\\hline

\highstrut$\widetilde{^2A_n}$\footnotesize{~($n$~odd)}&
\begin{tabular}{l}
\begin{dynkin}
    \dynkinline{0}{0}{120};
    \dynkinline{0}{0}{-120};
    \dynkinline{0}{0}{0};
    \dynkinline{1}{0}{0};
    \dynkindots{2}{0}{0};
    \dynkinline{3}{0}{0};
    \dynkindoubleline{5}{0}{180};
    \dynkindot{-0.5}{0.866025404};
    \dynkindot{-0.5}{-0.866025404};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{3}{0};
    \dynkindot{4}{0};
    \dynkindot{5}{0};
    \dynkintextwest{-0.5}{0.866025404}{1};
    \dynkintextwest{-0.5}{-0.866025404}{1};
    \dynkintextnorth{0}{0}{2};
    \dynkintextnorth{1}{0}{2};
    \dynkintextnorth{2}{0}{2};
    \dynkintextnorth{3}{0}{2};
    \dynkintextnorth{4}{0}{2};
    \dynkintextnorth{5}{0}{1};
\end{dynkin}
\\($(n+3)/2$~nodes)\\
\end{tabular}
\\\hline

\highstrut$\widetilde{^2A_n}$\footnotesize{~($n$~even)}&
\begin{tabular}{l}
\begin{dynkin}
    \dynkindoubleline{-1}{0}{0};
    \dynkinline{0}{0}{0};
    \dynkinline{1}{0}{0};
    \dynkindots{2}{0}{0};
    \dynkinline{3}{0}{0};
    \dynkindoubleline{4}{0}{0};
    \dynkindot{-1}{0};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{3}{0};
    \dynkindot{4}{0};
    \dynkindot{5}{0};
    \dynkintextnorth{-1}{0}{1};
    \dynkintextnorth{0}{0}{2};
    \dynkintextnorth{1}{0}{2};
    \dynkintextnorth{2}{0}{2};
    \dynkintextnorth{3}{0}{2};
    \dynkintextnorth{4}{0}{2};
    \dynkintextnorth{5}{0}{2};
\end{dynkin}
\\($(n+2)/2$~nodes)\\
\end{tabular}
\\\hline

\highstrut$\widetilde{^2D_n}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkindoubleline{0}{0}{180};
    \dynkinline{0}{0}{0};
    \dynkinline{1}{0}{0};
    \dynkindots{2}{0}{0};
    \dynkinline{3}{0}{0};
    \dynkindoubleline{4}{0}{0};
    \dynkindot{-1}{0};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{3}{0};
    \dynkindot{4}{0};
    \dynkindot{5}{0};
    \dynkintextnorth{-1}{0}{1};
    \dynkintextnorth{0}{0}{1};
    \dynkintextnorth{1}{0}{1};
    \dynkintextnorth{2}{0}{1};
    \dynkintextnorth{3}{0}{1};
    \dynkintextnorth{4}{0}{1};
    \dynkintextnorth{5}{0}{1};
\end{dynkin}
\\($n$~nodes)\\
\end{tabular}
\\\hline

\highstrut$\widetilde{^3D_4}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkintripleline{0}{0}{0};
    \dynkinline{1}{0}{0};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkintextnorth{0}{0}{1};
    \dynkintextnorth{1}{0}{2};
    \dynkintextnorth{2}{0}{1};
\end{dynkin}
\end{tabular}
\\\hline

\highstrut$\widetilde{^2E_6}$&
\begin{tabular}{l}
\begin{dynkin}
    \dynkinline{0}{0}{0};
    \dynkindoubleline{1}{0}{0};
    \dynkinline{2}{0}{0};
    \dynkinline{3}{0}{0};
    \dynkindot{0}{0};
    \dynkindot{1}{0};
    \dynkindot{2}{0};
    \dynkindot{3}{0};
    \dynkindot{4}{0};
    \dynkintextnorth{0}{0}{1};
    \dynkintextnorth{1}{0}{2};
    \dynkintextnorth{2}{0}{3};
    \dynkintextnorth{3}{0}{2};
    \dynkintextnorth{4}{0}{1};
\end{dynkin}
\end{tabular}
\\\hline

\end{tabular}

\\

\end{tabular}

\bigskip

The coefficients are such that the coefficient of each node $n$ is
\emph{one half the sum of the coefficients of all adjacent nodes $n'$}
with multiple edges counting with the corresponding multiplicity
provided the arrow points from $n'$ to $n$.  (There is a unique way,
up to scalar multiples, of assigning such coefficients, and it is done
in the way that they are integers with gcd equal to $1$.)

The left column shows the (untwisted) extended Dynkin diagram.  Nodes
with coefficient $1$ are called “tips”.  Symmetries of the diagram act
transitively on the tips.  Removing a (single) tip gives the ordinary
(= unextended) Dynkin diagram; and the node(s) to which the tip is
attached reveal the adjoint representation among the fundamental
representations (or, in the case of $A_n$, tensor product of
fundamentals).  The number of tips equals the order of the center of
the semisimple group.

For the meaning of the right column (twisted diagrams, or Dynkin-Kac
diagrams) and why they are labeled that way, see Reeder, “Torsion
automorphisms of simple Lie algebras”, \textit{Enseign. Math.}
\textbf{56} (2010) 3–47.  Note that it is \emph{not} obtained by
simply folding the untwisted Dynkin diagram (it is, roughtly speaking,
obtained by folding the unextended untwisted Dynkin diagram,
\emph{reversing} its multiple arrows, then extending it by adding one
more node).

Borel–de~Siebenthal theory describes how to compute maximal connected
subgroups of maximal rank as centralizers.

Over the reals, applying this to classify Cartan involutions gives a
description of all noncompact real forms of the complex Lie groups,
through their maximal compact subgroups (up to isogeny), which can be
summarized as follows: to describe all real forms of a complex simple
Lie group $G$,
\begin{itemize}\setlength\itemsep{0pt}
\item either remove a single node with coefficient $2$ from the
  diagram labeled $\widetilde{G}$ in the left column (to obtain the
  unextended Dynkin diagram of the maximal compact subgroup),
\item or remove two nodes with coefficient $1$ each (i.e., tips) from
  the diagram labeled $\widetilde{G}$ in the left column (again giving
  the unextended Dynkin diagram of the maximal compact subgroup), and
  understand that the maximal compact subgroup is to be multiplied
  by $\textbf{T}$ (the unit circle),
\item or remove a single node with coefficient $1$ from the diagram
  labeled $\widetilde{^2 G}$ in the right column, if there is such
  (and again, giving the unextended Dynkin diagram of the maximal
  compact subgroup).
\end{itemize}
For example, we can read from the diagrams labeled $\widetilde{E_6}$
and $\widetilde{^2 E_6}$ above that there are five noncompact real
forms of $E_6$, having maximal compact subgroups (up to isogeny):
$A_5\times A_1$ (first case), $D_5 \times \mathbf{T}$ (second case),
$F_4$ and $C_4$ (third case).

(Vogan diagrams are very closely related to this.  \emph{Beware:} the
very confusingly similar-looking Satake(–Tits) diagrams, however, are
\emph{not} related: they are in the line of Galois cohomology,
describing real forms in relation to the split real form rather than,
as in the above, to the compact form.)

\end{document}