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-rw-r--r--graphics/pgf/contrib/dynkin-diagrams/dynkin-diagrams.tex1247
1 files changed, 637 insertions, 610 deletions
diff --git a/graphics/pgf/contrib/dynkin-diagrams/dynkin-diagrams.tex b/graphics/pgf/contrib/dynkin-diagrams/dynkin-diagrams.tex
index 4969361082..22d55fdbd5 100644
--- a/graphics/pgf/contrib/dynkin-diagrams/dynkin-diagrams.tex
+++ b/graphics/pgf/contrib/dynkin-diagrams/dynkin-diagrams.tex
@@ -1,7 +1,7 @@
\documentclass{amsart}
-
-\title{The Dynkin diagrams package \\ Version 3.141592653}
-
+\title[The Dynkin diagrams package]%
+{The Dynkin diagrams package \\ Version 3.1415926535}
+%% My name:
\makeatletter
\DeclareRobustCommand{\scotsMc}{\scotsMcx{c}}
\DeclareRobustCommand{\scotsMC}{\scotsMcx{\textsc{c}}}
@@ -16,12 +16,10 @@
\@uclclist\scotsMc\scotsMC
}
\makeatother
-
\author{Ben \scotsMc{}Kay}
\address{School of Mathematical Sciences, University College Cork, Cork, Ireland}
\email{b.mckay@ucc.ie}
-\date{4 December 2019}
-
+\date{2 February 2020}
\usepackage{etex}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenx}
@@ -55,6 +53,7 @@
\usepackage{filecontents}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{decorations.pathmorphing}
+%% Use white rulings in tables.
\arrayrulecolor{white}
\makeatletter
\def\rulecolor#1#{\CT@arc{#1}}
@@ -80,13 +79,28 @@
\newcolumntype{P}{>{\columncolor[gray]{.9}}p{10cm}}
\NewDocumentCommand\textleftcurly{}{\texttt{\char'173}}%
\NewDocumentCommand\textrightcurly{}{\texttt{\char'175}}%
+\newcount\seriesLength
+\newcount\rankLength
\NewDocumentCommand\csDynkin{omom}%
{%
- \texttt{\detokenize{\dynkin}\!\!\!%
+ \texttt{\detokenize{\dynkin}\!\!%
\IfNoValueTF{#1}{}{[#1]}%
- \textleftcurly#2\textrightcurly%
+ \StrLen{#2}[\thatseriesLength]%
+ \seriesLength\thatseriesLength\relax%
+ \ifnum\seriesLength=1\relax%
+ \IfNoValueT{#1}{\ }%
+ #2%
+ \else%
+ \textleftcurly#2\textrightcurly%
+ \fi%
\IfNoValueTF{#3}{}{[#3]}%
- \textleftcurly#4\textrightcurly%
+ \StrLen{#4}[\thatrankLength]%
+ \rankLength\thatrankLength\relax%
+ \ifnum\rankLength=1\relax%
+ #4%
+ \else%
+ \textleftcurly#4\textrightcurly%
+ \fi%
}%
}%
@@ -167,20 +181,20 @@ before upper={\widowpenalties=3 10000 10000 150}}
\documentclass{amsart}
\usepackage{dynkin-diagrams}
\begin{document}
-The Dynkin diagram of \(B_3\) is \dynkin{B}{3}.
+The Dynkin diagram of \(B_3\) is \dynkin B3.
\end{document}
\end{verbatim}
\end{tcolorbox}
\begin{tcblisting}{title={Invoke it}}
-The Dynkin diagram of \(B_3\) is \dynkin{B}{3}.
+The Dynkin diagram of \(B_3\) is \dynkin B3.
\end{tcblisting}
\begin{tcblisting}{title={Inside a \TikZ statement}}
The Dynkin diagram of \(B_3\) is
-\tikz \dynkin{B}{3};
+\tikz \dynkin B3;
\end{tcblisting}
\begin{tcblisting}{title={Inside a Dynkin diagram environment}}
The Dynkin diagram of \(B_3\) is
-\begin{dynkinDiagram}{B}{3}
+\begin{dynkinDiagram}B3
\draw[very thick,red] (root 1) to [out=-45, in=-135] (root 3);
\end{dynkinDiagram}
\end{tcblisting}
@@ -188,23 +202,30 @@ The Dynkin diagram of \(B_3\) is
Baseline controls vertical alignment:
the Dynkin diagram of \(B_3\) is
\begin{tikzpicture}[baseline=(origin.base)]
-\dynkin{B}{3}
+\dynkin B3
\draw[very thick,red] (root 1) to [out=-45, in=-135] (root 3);
\end{tikzpicture}
\end{tcblisting}
+In a TikZ picture, you might need to kill the default vertical shift (needed to allow inline Dynkin diagrams):
+\begin{tcblisting}{title={Inside TikZ pictures}}
+\begin{tikzpicture}
+\draw (0,0) -- (.5,1) -- (1,0);
+\dynkin[vertical shift=0,edge length=1cm]G2
+\end{tikzpicture}
+\end{tcblisting}
\begin{tcblisting}{title={Indefinite rank Dynkin diagrams}}
-\dynkin{B}{}
+\dynkin B{}
\end{tcblisting}
\begin{dynkinTable}{The Dynkin diagrams of the reduced simple root systems \cite{Bourbaki:2002} pp. 265--290, plates I--IX}{2.25cm}{2.5cm}
-\dyn{A}{}
-\dyn{C}{}
-\dyn{D}{}
-\dyn{E}{6}
-\dyn{E}{7}
-\dyn{E}{8}
-\dyn{F}{4}
-\dyn{G}{2}
+\dyn A{}
+\dyn C{}
+\dyn D{}
+\dyn E6
+\dyn E7
+\dyn E8
+\dyn F4
+\dyn G2
\end{dynkinTable}
@@ -243,87 +264,87 @@ You can also pass options to the package in \verb!\usepackage!.
\end{tcblisting}
\begin{tcblisting}{title={gonality option for \(G_2\) and \(I_n\) Coxeter diagrams}}
-\(G_2=\dynkin[Coxeter,gonality=n]{G}{2}\), \
-\(I_n=\dynkin[Coxeter,gonality=n]{I}{}\)
+\(G_2=\dynkin[Coxeter,gonality=n]G2\), \
+\(I_n=\dynkin[Coxeter,gonality=n]I{}\)
\end{tcblisting}
\begin{dynkinTable}{The Coxeter diagrams of the simple reflection groups}{2.25cm}{6cm}
-\dyn[Coxeter]{A}{}
-\dyn[Coxeter]{B}{}
-\dyn[Coxeter]{C}{}
-\dyn[Coxeter]{E}{6}
-\dyn[Coxeter]{E}{7}
-\dyn[Coxeter]{E}{8}
-\dyn[Coxeter]{F}{4}
-\dyn[Coxeter,gonality=n]{G}{2}
-\dyn[Coxeter]{H}{3}
-\dyn[Coxeter]{H}{4}
-\dyn[Coxeter,gonality=n]{I}{}
+\dyn[Coxeter]A{}
+\dyn[Coxeter]B{}
+\dyn[Coxeter]C{}
+\dyn[Coxeter]E6
+\dyn[Coxeter]E7
+\dyn[Coxeter]E8
+\dyn[Coxeter]F4
+\dyn[Coxeter,gonality=n]G2
+\dyn[Coxeter]H3
+\dyn[Coxeter]H4
+\dyn[Coxeter,gonality=n]I{}
\end{dynkinTable}
\section{Satake diagrams}\label{section:Satake}
\begin{tcblisting}{title={Satake diagrams use the standard name instead of a rank}}
-\(A_{IIIb}=\dynkin{A}{IIIb}\)
+\(A_{IIIb}=\dynkin A{IIIb}\)
\end{tcblisting}
We use a solid gray bar to denote the folding of a Dynkin diagram, rather than the usual double arrow, since the diagrams turn out simpler and easier to read.
\begin{dynkinTable}{The Satake diagrams of the real simple Lie algebras \cite{Helgason:2001} p. 532--534}{2.75cm}{3cm}
-\dyn{A}{I}
-\dyn{A}{II}
-\dyn{A}{IIIa}
-\dyn{A}{IIIb}
-\dyn{A}{IV}
-\dyn{B}{I}
-\dyn{B}{II}
-\dyn{C}{I}
-\dyn{C}{IIa}
-\dyn{C}{IIb}
-\dyn{D}{Ia}
-\dyn{D}{Ib}
-\dyn{D}{Ic}
-\dyn{D}{II}
-\dyn{D}{IIIa}
-\dyn{D}{IIIb}
-\dyn{E}{I}
-\dyn{E}{II}
-\dyn{E}{III}
-\dyn{E}{IV}
-\dyn{E}{V}
-\dyn{E}{VI}
-\dyn{E}{VII}
-\dyn{E}{VIII}
-\dyn{E}{IX}
-\dyn{F}{I}
-\dyn{F}{II}
-\dyn{G}{I}
+\dyn A{I}
+\dyn A{II}
+\dyn A{IIIa}
+\dyn A{IIIb}
+\dyn A{IV}
+\dyn B{I}
+\dyn B{II}
+\dyn C{I}
+\dyn C{IIa}
+\dyn C{IIb}
+\dyn D{Ia}
+\dyn D{Ib}
+\dyn D{Ic}
+\dyn D{II}
+\dyn D{IIIa}
+\dyn D{IIIb}
+\dyn E{I}
+\dyn E{II}
+\dyn E{III}
+\dyn E{IV}
+\dyn E{V}
+\dyn E{VI}
+\dyn E{VII}
+\dyn E{VIII}
+\dyn E{IX}
+\dyn F{I}
+\dyn F{II}
+\dyn GI
\end{dynkinTable}
\section{How to fold}
\begin{tcblisting}{title={If you don't like the solid gray ``folding bar'', most people use arrows. Here is \(E_{II}\)}}
\newcommand{\invol}[2]{\draw[latex-latex] (root #1) to
[out=-60,in=-120] node[midway,below]{$\sigma$} (root #2);}
-\begin{dynkinDiagram}[edge length=.75cm,labels*={1,...,6}]{E}{6}
-\invol{1}{6}\invol{3}{5}
+\begin{dynkinDiagram}[edge length=.75cm,labels*={1,...,6}]E6
+\invol 16\invol 35
\end{dynkinDiagram}
\end{tcblisting}
\begin{tcblisting}{title={The double arrows for \(A_{IIIa}\) are big}}
\newcommand{\invol}[2]{\draw[latex-latex] (root #1) to
[out=-60,in=-120] node[midway,below]{$\sigma$} (root #2);}
\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}
+\invol 1{10}\invol 29\invol 38\invol 47\invol 56
\end{dynkinDiagram}
\end{tcblisting}
\begin{tcblisting}{title={If you don't like the solid gray ``folding bar'', most people use arrows \dots}}
\tikzset{/Dynkin diagram/fold style/.style={stealth-stealth,thick,
shorten <=1mm,shorten >=1mm,}}
-\dynkin[ply=3,edge length=.75cm]{D}{4}
-\begin{dynkinDiagram}[ply=4]{D}[1]%
+\dynkin[ply=3,edge length=.75cm]D4
+\begin{dynkinDiagram}[ply=4]D[1]%
{****.*****.*****}
- \dynkinFold{1}{13}
- \dynkinFold[bend right=90]{0}{14}
+ \dynkinFold 1{13}
+ \dynkinFold[bend right=90] 0{14}
\end{dynkinDiagram}
\end{tcblisting}
@@ -331,65 +352,63 @@ shorten <=1mm,shorten >=1mm,}}
\tikzset{/Dynkin diagram/fold style/.style=
{decorate,decoration={name=coil,aspect=0.5,
segment length=1mm,amplitude=.6mm}}}
-\dynkin[ply=3,edge length=.75cm]{D}{4}
-\begin{dynkinDiagram}[ply=4]{D}[1]%
+\dynkin[ply=3,edge length=.75cm]D4
+\begin{dynkinDiagram}[ply=4]D[1]%
{****.*****.*****}
- \dynkinFold{1}{13}
- \dynkinFold[bend right=90]{0}{14}
+ \dynkinFold 1{13}
+ \dynkinFold[bend right=90]0{14}
\end{dynkinDiagram}
\end{tcblisting}
-
\section{Labels for the roots}
-
\begin{tcblisting}{title={Make a macro to assign labels to roots}}
-\dynkin[label,label macro/.code={\alpha_{\drlap{#1}}},edge length=.75cm]{D}{5}
+\dynkin[label,label macro/.code={\alpha_{\drlap{#1}}},edge length=.75cm]D5
\end{tcblisting}
\begin{tcblisting}{title={Labelling several roots}}
-\dynkin[labels={,2,...,5,,7},label macro/.code={\alpha_{\drlap#1}}]{A}{7}
+\dynkin[labels={,2,...,5,,7},label macro/.code={\alpha_{\drlap#1}}]A7
\end{tcblisting}
\begin{tcblisting}{title={The \texttt{foreach} notation I}}
-\dynkin[labels={1,3,...,7},]{A}{9}
+\dynkin[labels={1,3,...,7},]A9
\end{tcblisting}
\begin{tcblisting}{title={The \texttt{foreach} notation II}}
-\dynkin[labels={,\alpha_2,\alpha_...,\alpha_7},]{A}{7}
+\dynkin[labels={,\alpha_2,\alpha_...,\alpha_7},]A7
\end{tcblisting}
\begin{tcblisting}{title={The \texttt{foreach} notation III}}
-\dynkin[label macro/.code={\beta_{\drlap{#1}}},labels={,2,...,7},]{A}{7}
+\dynkin[label macro/.code={\beta_{\drlap{#1}}},labels={,2,...,7},]A7
\end{tcblisting}
\begin{tcblisting}{title={Label the roots individually by root number}}
-\dynkin[label]{B}{3}
+\dynkin[label]B3
\end{tcblisting}
\begin{tcblisting}{title={Label a single root}}
-\begin{dynkinDiagram}{B}{3}
-\dynkinLabelRoot{2}{\alpha_{\drlap{2}}}
+\begin{dynkinDiagram}B3
+\dynkinLabelRoot 2{\alpha_{\drlap{2}}}
\end{dynkinDiagram}
\end{tcblisting}
\begin{tcblisting}{title={Access root labels via TikZ}}
-\begin{dynkinDiagram}{B}{3}
+\begin{dynkinDiagram}B3
\node[below] at (root 2) {\(\alpha_{\drlap{2}}\)};
\end{dynkinDiagram}
\end{tcblisting}
\begin{tcblisting}{title={Commands to label several roots}}
-\begin{dynkinDiagram}{A}{7}
+\begin{dynkinDiagram}A7
\dynkinLabelRoots{,\alpha_2,\alpha_3,\alpha_4,\alpha_5,,\alpha_7}
\end{dynkinDiagram}
\end{tcblisting}
\begin{tcblisting}{title={The labels have default locations, mostly below roots}}
-\dynkin[edge length=.75cm,labels={1,2,3}]{E}{8}
+\dynkin[edge length=.75cm,labels={1,2,3}]E8
\end{tcblisting}
\begin{tcblisting}{title={The starred form flips labels to alternate locations, mostly above roots}}
-\dynkin[edge length=.75cm,labels*={1,2,3}]{E}{8}
+\dynkin[edge length=.75cm,labels*={1,2,3}]E8
\end{tcblisting}
\begin{tcblisting}{title={Labelling several roots and alternates}}
\dynkin[%
label macro/.code={\alpha_{\drlap{#1}}},
label macro*/.code={\gamma_{\drlap{#1}}},
labels={,2,...,5,,7},
-labels*={1,3,4,5,6}]{A}{7}
+labels*={1,3,4,5,6}]A7
\end{tcblisting}
\begin{tcblisting}{title={Commands to label several roots}}
-\begin{dynkinDiagram}{A}{7}
+\begin{dynkinDiagram}A7
\dynkinLabelRoots{,\alpha_2,\alpha_3,\alpha_4,\alpha_5,,\alpha_7}
\dynkinLabelRoots*{a,b,c,d,e,f,g}
\end{dynkinDiagram}
@@ -399,44 +418,44 @@ labels*={1,3,4,5,6}]{A}{7}
Note the slight improvement that \verb!\drlap! makes: the labels are centered on the middle of the letter \(\alpha\), ignoring the space taken up by the subscripts, using the \verb!mathtools! command \verb!\mathrlap!, but only for labels which are \emph{not} placed to the left or right of a root.
\begin{tcblisting}{title={Label subscript spacing}}
\dynkin[label,label macro/.code={\alpha_{#1}},
- edge length=.75cm]{D}{15}
+ edge length=.75cm]D{15}
\par\noindent{}%
\dynkin[label,label macro/.code={\alpha_{\drlap{#1}}},
- edge length=.75cm]{D}{15}
+ edge length=.75cm]D{15}
\end{tcblisting}
\begin{tcblisting}{title={Label subscript spacing}}
\dynkin[label,label macro/.code={\alpha_{#1}},
- edge length=.75cm]{E}{8}
+ edge length=.75cm]E8
\dynkin[label,label macro/.code={\alpha_{#1}},backwards,
- edge length=.75cm]{E}{8}
+ edge length=.75cm]E8
\par\noindent{}%
\dynkin[label,label macro/.code={\alpha_{\mathrlap{#1}}},
- edge length=.75cm]{E}{8}
+ edge length=.75cm]E8
\dynkin[label,label macro/.code={\alpha_{\mathrlap{#1}}},backwards,
- edge length=.75cm]{E}{8}
+ edge length=.75cm]E8
\par\noindent{}%
\dynkin[label,label macro/.code={\alpha_{\drlap{#1}}},
- edge length=.75cm]{E}{8}
+ edge length=.75cm]E8
\dynkin[label,label macro/.code={\alpha_{\drlap{#1}}},backwards,
- edge length=.75cm]{E}{8}
+ edge length=.75cm]E8
\end{tcblisting}
\newpage
\section{Height and depth of labels}
Labels are set with default maximum height the height of the character \(b\), and default maximum depth the depth of the character \(g\).
To change these, set \verb!label height! and \verb!label depth!:
-\begin{tcblisting}{title={Change height and dept of characters}}
-\dynkin[labels={a,b,c,d}]{F}{4}
-\dynkin[labels*={a,b,c,d}]{F}{4}
+\begin{tcblisting}{title={Change height and depth of characters}}
+\dynkin[labels={a,b,c,d},label height=d,label depth=d]F4
+\dynkin[labels*={a,b,c,d},label height=d,label depth=d]F4
\dynkin[%
label macro/.code={\alpha_{\drlap{#1}}},
label macro*/.code={\gamma_{\drlap{#1}}},
label height=$\alpha_1$,
label depth=$\alpha_1$,
labels={,2,...,5,,7},
-labels*={1,3,4,5,6}]{A}{7}
-\dynkin[labels={A,B,C,D},label height=$A$,label depth=$A$]{F}{4}
-\dynkin[labels={a^1,b^2,c^3,d^4},label height=$X^X$]{F}{4}
+labels*={1,3,4,5,6}]A7
+\dynkin[labels={A,B,C,D},label height=$A$,label depth=$A$]F4
+\dynkin[labels={a^1,b^2,c^3,d^4},label height=$X^X$]F4
\end{tcblisting}
\section{Text style for the labels}
@@ -445,14 +464,14 @@ labels*={1,3,4,5,6}]{A}{7}
edge length=.75cm,
labels={1,2,n-1,n},
label macro/.code={\alpha_{\drlap{#1}}}
-]{A}{}
+]A{}
\end{dynkinDiagram}
\end{tcblisting}
\begin{tcblisting}{title={Use a text style; font selection is in the label macro}}
\begin{dynkinDiagram}[text style={scale=1.2,blue},
edge length=.75cm,
labels={1,2,n-1,n},
-label macro/.code={\mathbb{A}_{\drlap{#1}}}]{A}{}
+label macro/.code={\mathbb{A}_{\drlap{#1}}}]A{}
\end{dynkinDiagram}
\end{tcblisting}
@@ -460,23 +479,23 @@ label macro/.code={\mathbb{A}_{\drlap{#1}}}]{A}{}
\section{Bracing roots}
\begin{tcblisting}{title={Bracing roots}}
-\begin{dynkinDiagram}{A}{*.*x*.*}
-\dynkinBrace[p]{1}{2}
-\dynkinBrace[q]{4}{5}
+\begin{dynkinDiagram}A{*.*x*.*}
+\dynkinBrace[p]12
+\dynkinBrace[q]45
\end{dynkinDiagram}
\end{tcblisting}
\begin{tcblisting}{title={Bracing roots, and a starred form}}
-\begin{dynkinDiagram}{A}{10}
-\dynkinBrace[\text{Roots 2 to 9}]{2}{9}
-\dynkinBrace*[\text{Roots 3 to 8}]{3}{8}
+\begin{dynkinDiagram}A{10}
+\dynkinBrace[\text{Roots 2 to 9}]29
+\dynkinBrace*[\text{Roots 3 to 8}]38
\end{dynkinDiagram}
\end{tcblisting}
\begin{tcblisting}{title={Bracing roots}}
\newcommand\circleRoot[1]{\draw (root #1) circle (3pt);}
-\begin{dynkinDiagram}{A}{**.***.***.***.***.**}
-\circleRoot{4}\circleRoot{7}\circleRoot{10}\circleRoot{13}
-\dynkinBrace[y-1]{1}{3}
-\dynkinBrace[z-1]{5}{6}
+\begin{dynkinDiagram}A{**.***.***.***.***.**}
+\circleRoot 4\circleRoot 7\circleRoot 10\circleRoot 13
+\dynkinBrace[y-1]13
+\dynkinBrace[z-1]56
\dynkinBrace[t-1]{11}{12}
\dynkinBrace[x-1]{14}{16}
\end{dynkinDiagram}
@@ -488,21 +507,21 @@ label macro/.code={\mathbb{A}_{\drlap{#1}}}]{A}{}
\setcounter{EPNo}{0}
\NewDocumentCommand\EP{smmmm}%
{%
-\stepcounter{EPNo}\roman{EPNo}. &
-\def\eL{.6cm}
+\stepcounter{EPNo}\roman{EPNo}. &%
+\def\eL{.6cm}%
\IfStrEqCase{#2}%
{%
-{D}{\gdef\eL{1cm}}%
-{E}{\gdef\eL{.75cm}}%
-{F}{\gdef\eL{.35cm}}%
-{G}{\gdef\eL{.35cm}}%
+D{\gdef\eL{1cm}}%
+E{\gdef\eL{.75cm}}%
+F{\gdef\eL{.35cm}}%
+G{\gdef\eL{.35cm}}%
}%
-\tikzset{/Dynkin diagram,edge length=\eL}
\IfBooleanTF{#1}%
-{\dynkin[backwards,labels*={#4},labels={#5}]{#2}{#3}}
-{\dynkin[labels*={#4},labels={#5}]{#2}{#3}}
+{\dynkin[edge length=\eL,backwards,labels*={#4},labels={#5}]{#2}{#3}}
+{\dynkin[edge length=\eL,labels*={#4},labels={#5}]{#2}{#3}}
\\
}%
+\renewcommand*\do[1]{\EP#1}%
\begin{longtable}{MM}
\caption{Dynkin diagrams from Euler products \cite{Langlands:1967}}\\
\endfirsthead
@@ -511,38 +530,39 @@ label macro/.code={\mathbb{A}_{\drlap{#1}}}]{A}{}
\multicolumn{2}{c}{continued \dots}\\
\endfoot
\endlastfoot
-\EP{A}{***.**}{1,1,1,1,1}{,1,2,n-1,n}
-\EP{A}{***.**}{1,1,1,1,1}{1,2,n-1,n}
-\EP{A}{**.***.*}{1,1,1,1,1,1}{1,2,m-1,,m,n}
-\EP{B}{**.***}{2,2,2,2,1}{1,2,n-1,n}
-\EP*{B}{***.**}{2,2,2,2,1}{n,n-1,2,1,}
-\EP{C}{**.***}{1,1,1,1,2}{1,2,n-1,}
-\EP*{C}{***.**}{1,1,1,1,2}{n,n-1,2,1,}
-\EP{D}{**.****}{1,1,1,1,1,1}{1,2,n-2,n-1,n}
-\EP{D}{**.****}{1,1,1,1,1,1}{1,2,n-2,n-1,n}
-\EP{E}{6}{1,1,1,1,1,1}{1,...,5}
-\EP*{E}{7}{1,1,1,1,1,1,1}{6,...,1}
-\EP{E}{7}{1,1,1,1,1,1,1}{1,...,6}
-\EP*{E}{8}{1,1,1,1,1,1,1,1}{7,...,1}
-\EP{E}{8}{1,1,1,1,1,1,1,1}{1,...,7}
-\EP{G}{2}{1,3}{,1}
-\EP{G}{2}{1,3}{1}
-\EP{B}{**.*.**}{2,2,2,2,1}{,1,2,n-1,n}
-\EP{F}{4}{1,1,2,2}{,3,2,1}
-\EP{C}{3}{1,1,2}{,2,1}
-\EP{C}{**.***}{1,1,1,1,2}{,1,n-2,n-1,n}
-\EP*{B}{3}{2,2,1}{1,2}
-\EP{F}{4}{1,1,2,2}{1,2,3}
-\EP{D}{**.****}{1,1,1,1,1,1}{1,2,n-2,n-2,n,n}
-\EP{E}{6}{1,1,1,1,1,1}{1,2,3,4,,5}
-\EP{E}{6}{1,1,1,1,1,1}{1,2,3,5,,4}
-\EP*{E}{7}{1,1,1,1,1,1,1}{,5,...,1,6}
-\EP*{E}{7}{1,1,1,1,1,1,1}{,6,4,3,2,1,5}
-\EP*{E}{8}{1,1,1,1,1,1,1,1}{,6,...,1,7}
-\EP*{E}{8}{1,1,1,1,1,1,1,1}{,7,5,4,3,2,1,6}
-\EP*{E}{7}{1,1,1,1,1,1,1}{5,...,1,,6}
-\EP*{E}{7}{1,1,1,1,1,1,1}{1,...,5,,6}
-\EP*{E}{8}{1,1,1,1,1,1,1,1}{6,...,1,,7}
+\docsvlist{
+A{***.**}{1,1,1,1,1}{,1,2,n-1,n},
+A{***.**}{1,1,1,1,1}{1,2,n-1,n},
+A{**.***.*}{1,1,1,1,1,1}{1,2,m-1,,m,n},
+B{**.***}{2,2,2,2,1}{1,2,n-1,n},
+*B{***.**}{2,2,2,2,1}{n,n-1,2,1,},
+C{**.***}{1,1,1,1,2}{1,2,n-1,},
+*C{***.**}{1,1,1,1,2}{n,n-1,2,1,},
+D{**.****}{1,1,1,1,1,1}{1,2,n-2,n-1,n},
+D{**.****}{1,1,1,1,1,1}{1,2,n-2,n-1,n},
+E6{1,1,1,1,1,1}{1,...,5},
+*E7{1,1,1,1,1,1,1}{6,...,1},
+E7{1,1,1,1,1,1,1}{1,...,6},
+*E8{1,1,1,1,1,1,1,1}{7,...,1},
+E8{1,1,1,1,1,1,1,1}{1,...,7},
+G2{1,3}{,1},
+G2{1,3}{1},
+B{**.*.**}{2,2,2,2,1}{,1,2,n-1,n},
+F4{1,1,2,2}{,3,2,1},
+C3{1,1,2}{,2,1},
+C{**.***}{1,1,1,1,2}{,1,n-2,n-1,n},
+*B3{2,2,1}{1,2},
+F4{1,1,2,2}{1,2,3},
+D{**.****}{1,1,1,1,1,1}{1,2,n-2,n-2,n,n},
+E6{1,1,1,1,1,1}{1,2,3,4,,5},
+E6{1,1,1,1,1,1}{1,2,3,5,,4},
+*E7{1,1,1,1,1,1,1}{,5,...,1,6},
+*E7{1,1,1,1,1,1,1}{,6,4,3,2,1,5},
+*E8{1,1,1,1,1,1,1,1}{,6,...,1,7},
+*E8{1,1,1,1,1,1,1,1}{,7,5,4,3,2,1,6},
+*E7{1,1,1,1,1,1,1}{5,...,1,,6},
+*E7{1,1,1,1,1,1,1}{1,...,5,,6},
+*E8{1,1,1,1,1,1,1,1}{6,...,1,,7}}
\end{longtable}
\end{filecontents*}
{\input{EulerProducts}}\VerbatimInput{EulerProducts.tex}
@@ -555,81 +575,77 @@ label macro/.code={\mathbb{A}_{\drlap{#1}}}]{A}{}
arrow color=red]{F}{4}
\end{tcblisting}
\begin{tcblisting}{title={Edge lengths}}
-The Dynkin diagram of \(A_3\) is \dynkin[edge length=1.2,parabolic=3]{A}{3}
+The Dynkin diagram of \(A_3\) is \dynkin[edge length=1.2]A3
\end{tcblisting}
\newpage
\begin{tcblisting}{title={Root marks}}
-\dynkin{E}{8}
-\dynkin[mark=*]{E}{8}
-\dynkin[mark=o]{E}{8}
-\dynkin[mark=O]{E}{8}
-\dynkin[mark=t]{E}{8}
-\dynkin[mark=x]{E}{8}
-\dynkin[mark=X]{E}{8}
+\dynkin E8
+\dynkin[mark=*]E8
+\dynkin[mark=o]E8
+\dynkin[mark=O]E8
+\dynkin[mark=t]E8
+\dynkin[mark=x]E8
+\dynkin[mark=X]E8
\end{tcblisting}
At the moment, you can only use:
-\par\noindent\begin{tabular}{>{\ttfamily}cl}
-* & solid dot \\
-o & hollow circle \\
-O & double hollow circle \\
-t & tensor root \\
-x & crossed root \\
-X & thickly crossed root
+\par\noindent\begin{tabular}{>{\ttfamily}ccl}
+* &\dynkin[mark=*]A1& solid dot \\
+o &\dynkin[mark=o]A1& hollow circle \\
+O&\dynkin[mark=O]A1 & double hollow circle \\
+t &\dynkin[mark=t]A1& tensor root \\
+x &\dynkin[mark=x]A1& crossed root \\
+X &\dynkin[mark=X]A1& thickly crossed root
\end{tabular}
\begin{tcblisting}{title={Mark styles}}
-The parabolic subgroup \(E_{8,124}\) is \dynkin[parabolic=124,x/.style={brown,very thick}]{E}{8}
+The parabolic subgroup \(E_{8,124}\) is \dynkin[parabolic=124,x/.style={brown,very thick}]E8
\end{tcblisting}
\begin{tcblisting}{title={Sizes of root marks}}
-\(A_{3,3}\) with big root marks is \dynkin[root radius=.08cm,parabolic=3]{A}{3}
+\(A_{3,3}\) with big root marks is \dynkin[root radius=.08cm,parabolic=3]A3
\end{tcblisting}
-
\section{Suppress or reverse arrows}
\begin{tcblisting}{title={Some diagrams have double or triple edges}}
-\dynkin{F}{4}
-\dynkin{G}{2}
+\dynkin F4
+\dynkin G2
\end{tcblisting}
\begin{tcblisting}{title={Suppress arrows}}
-\dynkin[arrows=false]{F}{4}
-\dynkin[arrows=false]{G}{2}
+\dynkin[arrows=false]F4
+\dynkin[arrows=false]G2
\end{tcblisting}
\begin{tcblisting}{title={Reverse arrows}}
-\dynkin[reverse arrows]{F}{4}
-\dynkin[reverse arrows]{G}{2}
+\dynkin[reverse arrows]F4
+\dynkin[reverse arrows]G2
\end{tcblisting}
-
\section{Backwards and upside down}
-
\begin{tcblisting}{title={Default}}
-\dynkin{E}{8}
-\dynkin{F}{4}
-\dynkin{G}{2}
+\dynkin E8
+\dynkin F4
+\dynkin G2
\end{tcblisting}
\begin{tcblisting}{title={Backwards}}
-\dynkin[backwards]{E}{8}
-\dynkin[backwards]{F}{4}
-\dynkin[backwards]{G}{2}
+\dynkin[backwards]E8
+\dynkin[backwards]F4
+\dynkin[backwards]G2
\end{tcblisting}
\begin{tcblisting}{title={Reverse arrows}}
-\dynkin[reverse arrows]{F}{4}
-\dynkin[reverse arrows]{G}{2}
+\dynkin[reverse arrows]F4
+\dynkin[reverse arrows]G2
\end{tcblisting}
\begin{tcblisting}{title={Backwards, reverse arrows}}
-\dynkin[backwards,reverse arrows]{F}{4}
-\dynkin[backwards,reverse arrows]{G}{2}
+\dynkin[backwards,reverse arrows]F4
+\dynkin[backwards,reverse arrows]G2
\end{tcblisting}
\begin{tcblisting}{title={Backwards versus upside down}}
-\dynkin[label]{E}{8}
-\dynkin[label,backwards]{E}{8}
-\dynkin[label,upside down]{E}{8}
-\dynkin[label,backwards,upside down]{E}{8}
+\dynkin[label]E8
+\dynkin[label,backwards]E8
+\dynkin[label,upside down]E8
+\dynkin[label,backwards,upside down]E8
\end{tcblisting}
-
\section{Drawing on top of a Dynkin diagram}
\begin{tcblisting}{title={TikZ can access the roots themselves}}
-\begin{dynkinDiagram}{A}{4}
+\begin{dynkinDiagram}A4
\fill[white,draw=black] (root 2) circle (.15cm);
\fill[white,draw=black] (root 2) circle (.1cm);
\draw[black] (root 2) circle (.05cm);
@@ -637,25 +653,22 @@ The parabolic subgroup \(E_{8,124}\) is \dynkin[parabolic=124,x/.style={brown,ve
\end{tcblisting}
\newpage
\begin{tcblisting}{title={Draw curves between the roots}}
-\begin{dynkinDiagram}[label]{E}{8}
+\begin{dynkinDiagram}[label]E8
\draw[very thick, black!50,-latex]
(root 3.south) to [out=-45, in=-135] (root 6.south);
\end{dynkinDiagram}
\end{tcblisting}
\begin{tcblisting}{title={Change marks}}
-\begin{dynkinDiagram}[mark=o,label]{E}{8}
- \dynkinRootMark{*}{5}
- \dynkinRootMark{*}{8}
+\begin{dynkinDiagram}[mark=o,label]E8
+ \dynkinRootMark{*}5
+ \dynkinRootMark{*}8
\end{dynkinDiagram}
\end{tcblisting}
-
\section{Mark lists}
-
The package allows a list of root marks instead of a rank:
-
\begin{tcblisting}{title={A mark list}}
-\dynkin{E}{oo**ttxx}
+\dynkin E{oo**ttxx}
\end{tcblisting}
The mark list \verb!oo**ttxx! has one mark for each root: \verb!o!, \verb!o!, \dots, \verb!x!.
Roots are listed in the current default ordering.
@@ -663,7 +676,7 @@ Roots are listed in the current default ordering.
If you need to repeat a mark, you can give a \emph{single digit} positive integer to indicate how many times to repeat it.
\begin{tcblisting}{title={A mark list with repetitions}}
-\dynkin{A}{x4o3t4}
+\dynkin A{x4o3t4}
\end{tcblisting}
\NewDocumentCommand\ClassicalLieSuperalgebras{om}%
@@ -674,15 +687,15 @@ If you need to repeat a mark, you can give a \emph{single digit} positive intege
\IfValueT{#1}{
& & \texttt{\textbackslash{}tikzset\{/Dynkin diagram,root radius=#1\}} \\
}
-A_{mn} & \dynk{A}{o3.oto.oo}
-B_{mn} & \dynk{B}{o3.oto.oo}
-B_{0n} & \dynk{B}{o3.o3.o*}
-C_{n} & \dynk{C}{too.oto.oo}
-D_{mn} & \dynk{D}{o3.oto.o4}
-D_{21\alpha} & \dynk{A}{oto}
-F_4 & \dynk{F}{ooot}
+A_{mn} & \dynk A{o3.oto.oo}
+B_{mn} & \dynk B{o3.oto.oo}
+B_{0n} & \dynk B{o3.o3.o*}
+C_{n} & \dynk C{too.oto.oo}
+D_{mn} & \dynk D{o3.oto.o4}
+D_{21\alpha} & \dynk A{oto}
+F_4 & \dynk F{ooot}
G_3 & \dynk[extended,affine mark=t,
-reverse arrows]{G}{2}
+reverse arrows]G2
\end{dynkinTable}
\IfValueT{#1}{\tikzset{/Dynkin diagram,root radius=.05cm}}
}%
@@ -692,49 +705,47 @@ reverse arrows]{G}{2}
\ClassicalLieSuperalgebras{Here we see the problem with using the default root radius parameter, which is too small for tensor product symbols.}
-
\section{Indefinite edges}
-An \emph{indefinite edge} is a dashed edge between two roots, \dynkin{A}{*.*} indicating that an indefinite number of roots have been omitted from the Dynkin diagram.
+An \emph{indefinite edge} is a dashed edge between two roots, \dynkin A{*.*} indicating that an indefinite number of roots have been omitted from the Dynkin diagram.
In between any two entries in a mark list, place a period to indicate an indefinite edge:
\begin{tcblisting}{title={Indefinite edges}}
-\dynkin{D}{o.o*.*.t.to.t}
+\dynkin D{o.o*.*.t.to.t}
\end{tcblisting}
In certain diagrams, roots may have an edge between them even though they are not subsequent in the ordering.
For such rare situations, there is an option:
\begin{tcblisting}{title={Indefinite edge option}}
-\dynkin[make indefinite edge={3-5},label]{D}{5}
+\dynkin[make indefinite edge={3-5},label]D5
\end{tcblisting}
\begin{tcblisting}{title={Give a list of edges to become indefinite}}
-\dynkin[make indefinite edge/.list={1-2,3-5},label]{D}{5}
+\dynkin[make indefinite edge/.list={1-2,3-5},label]D5
\end{tcblisting}
\begin{tcblisting}{title={Indefinite edge style}}
-\dynkin[indefinite edge/.style={draw=black,fill=white,thin,densely dashed},%
- edge length=1cm,%
- make indefinite edge={3-5}]
- {D}{5}
+\dynkin[indefinite edge/.style={
+ draw=black,fill=white,thin,densely dashed},
+ edge length=1cm,
+ make indefinite edge={3-5}]D5
\end{tcblisting}
\begin{tcblisting}{title={The ratio of the lengths of indefinite edges to those of other edges}}
-\dynkin[edge length = .5cm,%
- indefinite edge ratio=3,%
- make indefinite edge={3-5}]
- {D}{5}
+\dynkin[edge length = .5cm,
+ indefinite edge ratio=3,
+ make indefinite edge={3-5}]D5
\end{tcblisting}
-\begingroup
+%\begingroup
\renewcommand{\wdtA}{.35cm}
\renewcommand{\wdtE}{6.55cm}
\begin{dynkinTable}{Springer's table of indices \cite{Springer:2009}, pp. 320-321, with one form of \(E_7\) corrected}{2.5cm}{3.7cm}
% 1
A_n &
\multicolumn{2}{E}{
-\begin{dynkinDiagram}{A}{o.o*o.o*o.o}
-\dynkinLabelRoot{3}{d}
-\dynkinLabelRoot{6}{n-d}
+\begin{dynkinDiagram}A{o.o*o.o*o.o}
+\dynkinLabelRoot 3d
+\dynkinLabelRoot 6{n-d}
\end{dynkinDiagram}
}
\\
@@ -742,9 +753,9 @@ A_n &
A_n &
\multicolumn{2}{E}{
\begin{dynkinDiagram}{A}{o.o*o.o*o.o*o.o*o.o}
-\dynkinLabelRoot{3}{d}
-\dynkinLabelRoot{6}{rd}
-\dynkinLabelRoot{9}{n-rd}
+\dynkinLabelRoot 3d
+\dynkinLabelRoot 6{rd}
+\dynkinLabelRoot 9{n-rd}
\dynkinLabelRoot{12}{n-d}
\end{dynkinDiagram}
}
@@ -752,7 +763,7 @@ A_n &
% 3
B_n &
\multicolumn{2}{E}{
-\begin{dynkinDiagram}{B}{**.*.o.oo}
+\begin{dynkinDiagram}B{**.*.o.oo}
\dynkinLabelRoot{3}{r}
\end{dynkinDiagram}
}
@@ -760,81 +771,130 @@ B_n &
% 4
C_n &
\multicolumn{2}{E}{
-\begin{dynkinDiagram}{C}{o.o*o.o*o.oo}
-\dynkinLabelRoot{3}{d}
-\dynkinLabelRoot{6}{rd}
+\begin{dynkinDiagram}C{o.o*o.o*o.oo}
+\dynkinLabelRoot 3d
+\dynkinLabelRoot 6{rd}
\end{dynkinDiagram}
}
\\
% 5
D_n &
\multicolumn{2}{E}{
-\begin{dynkinDiagram}{D}{o.o*o.o*o.ooo}
-\dynkinLabelRoot{3}{d}
-\dynkinLabelRoot{6}{rd}
+\begin{dynkinDiagram}D{o.o*o.o*o.ooo}
+\dynkinLabelRoot 3d
+\dynkinLabelRoot 6{rd}
\end{dynkinDiagram}
}
\\
% 6
E_6 &
-\dynk{E}{*oooo*}
+\dynk E{*oooo*}
% 7
E_6 &
-\dynk{E}{o*o*oo}
+\dynk E{o*o*oo}
% 8
E_6 &
-\dynk{E}{o*oooo}
+\dynk E{o*oooo}
% 9
E_6 &
-\dynk{E}{**ooo*}
+\dynk E{**ooo*}
% 10
E_7 &
-\dynk{E}{*oooooo}
+\dynk E{*oooooo}
% 11
E_7 &
-\dynk{E}{ooooo*o}
+\dynk E{ooooo*o}
% 12
E_7 &
-\dynk{E}{oooooo*}
+\dynk E{oooooo*}
% 13
E_7 &
-\dynk{E}{*oooo*o}
+\dynk E{*oooo*o}
% 14 - corrected from Springer.
E_7 &
-\dynk{E}{*oooo**}
+\dynk E{*oooo**}
% 15
E_7 &
-\dynk{E}{*o**o*o}
+\dynk E{*o**o*o}
% 16
E_8 &
-\dynk{E}{*ooooooo}
+\dynk E{*ooooooo}
% 17
E_8 &
-\dynk{E}{ooooooo*}
+\dynk E{ooooooo*}
% 18
E_8 &
-\dynk{E}{*oooooo*}
+\dynk E{*oooooo*}
% 19
E_8 &
-\dynk{E}{oooooo**}
+\dynk E{oooooo**}
% 20
E_8 &
-\dynk{E}{*oooo***}
+\dynk E{*oooo***}
% 21
F_4 &
-\dynk{F}{ooo*}
+\dynk F{ooo*}
% 22
D_4 &
-\dynk{D}{o*oo}
+\dynk D{o*oo}
\end{longtable}
+
\endgroup
+\section{Root ordering}\label{section:order}
+\begin{tcblisting}{title={Root ordering}}
+\dynkin[label,ordering=Adams]E6
+\dynkin[label,ordering=Bourbaki]E6
+\dynkin[label,ordering=Carter]E6
+\dynkin[label,ordering=Dynkin]E6
+\dynkin[label,ordering=Kac]E6
+\end{tcblisting}
+Default is Bourbaki.
+Sources are Adams \cite{Adams:1996} p. 56--57, Bourbaki \cite{Bourbaki:2002} p. pp. 265--290 plates I-IX, Carter \cite{Carter:2005} p. 540--609, Dynkin \cite{Dynkin:1952}, Kac \cite{Kac:1990} p. 43.
+\NewDocumentCommand\tablerow{mm}%
+{%
+#1_{#2}&
+\dynkin[label,ordering=Adams]{#1}{#2}&
+\dynkin[label]{#1}{#2}&
+\dynkin[label,ordering=Carter]{#1}{#2}&
+\dynkin[label,ordering=Dynkin]{#1}{#2}&
+\dynkin[label,ordering=Kac]{#1}{#2}\\
+}%
+\begin{center}
+\renewcommand{\wdtA}{.7cm}
+\renewcommand{\wdtL}{2.2cm}
+\begin{longtable}{@{}ALLLLL@{}}
+\toprule
+& Adams & Bourbaki & Carter & Dynkin & Kac \\ \midrule
+\endfirsthead
+\toprule
+& Adams & Bourbaki & Carter & Dynkin & Kac \\ \midrule
+\endhead
+\bottomrule
+\endfoot
+\bottomrule
+\endlastfoot
+\tablerow E6\tablerow E7\tablerow E8\tablerow F4\tablerow G2
+\end{longtable}
+\end{center}
+The marks are set down in order according to the current root ordering:
+\begin{tcblisting}{}
+\dynkin[label]E{*otxXOt*}
+\dynkin[label,ordering=Carter]E{*otxXOt*}
+\dynkin[label,ordering=Kac]E{*otxXOt*}
+\end{tcblisting}
+\begin{tcblisting}{title={Convert between orderings}}
+\newcount\r
+\dynkinOrder E8.Carter::6->Bourbaki.{\r}
+In \(E_8\), root 6 in Carter's ordering is root \the\r{} in Bourbaki's ordering.
+\end{tcblisting}
+
\section{Parabolic subgroups}
Each set of roots is assigned a number, with each binary digit zero or one to say whether the corresponding root is crossed or not:
\begin{tcblisting}{}
The flag variety of pointed lines in
projective 3-space is associated to
-the Dynkin diagram \dynkin[parabolic=3]{A}{3}.
+the Dynkin diagram \dynkin[parabolic=3]A3.
\end{tcblisting}
\begin{filecontents*}{hermitian-symmetric-spaces.tex}
@@ -847,15 +907,15 @@ the Dynkin diagram \dynkin[parabolic=3]{A}{3}.
\caption{\dots continued}\\ \endhead
\caption{continued \dots}\\ \endfoot
\endlastfoot
-\HSS{A_n}{A}{**.*x*.**}{Grassmannian of $k$-planes in $\C{n+1}$}
-\HSS{B_n}[1]{B}{}{$(2n-1)$-dimensional hyperquadric, i.e. the variety of null lines in $\C{2n+1}$}
-\HSS{C_n}[16]{C}{}{space of Lagrangian $n$-planes in $\C{2n}$}
-\HSS{D_n}[1]{D}{}{$(2n-2)$-dimensional hyperquadric, i.e. the variety of null lines in $\C{2n}$}
-\HSS{D_n}[32]{D}{}{one component of the variety of maximal dimension null subspaces of $\C{2n}$}
-\HSS{D_n}[16]{D}{}{the other component}
-\HSS{E_6}[1]{E}{6}{complexified octave projective plane}
-\HSS{E_6}[32]{E}{6}{its dual plane}
-\HSS{E_7}[64]{E}{7}{the space of null octave 3-planes in octave 6-space}
+\HSS{A_n}A{**.*x*.**}{Grassmannian of $k$-planes in $\C{n+1}$}
+\HSS{B_n}[1]B{}{$(2n-1)$-dimensional hyperquadric, i.e. the variety of null lines in $\C{2n+1}$}
+\HSS{C_n}[16]C{}{space of Lagrangian $n$-planes in $\C{2n}$}
+\HSS{D_n}[1]D{}{$(2n-2)$-dimensional hyperquadric, i.e. the variety of null lines in $\C{2n}$}
+\HSS{D_n}[32]D{}{one component of the variety of maximal dimension null subspaces of $\C{2n}$}
+\HSS{D_n}[16]D{}{the other component}
+\HSS{E_6}[1]E6{complexified octave projective plane}
+\HSS{E_6}[32]E6{its dual plane}
+\HSS{E_7}[64]E7{the space of null octave 3-planes in octave 6-space}
\end{longtable}
\end{filecontents*}
\begingroup
@@ -866,24 +926,24 @@ the Dynkin diagram \dynkin[parabolic=3]{A}{3}.
\section{Extended Dynkin diagrams}
\begin{tcblisting}{title={Extended Dynkin diagrams}}
-\dynkin[extended]{A}{7}
+\dynkin[extended]A7
\end{tcblisting}
-The extended Dynkin diagrams are also described in the notation of Kac \cite{Kac:1990} p. 55 as affine untwisted Dynkin diagrams: we extend \verb!\dynkin{A}{7}! to become \verb!\dynkin{A}[1]{7}!:
+The extended Dynkin diagrams are also described in the notation of Kac \cite{Kac:1990} p. 55 as affine untwisted Dynkin diagrams: we extend \verb!\dynkin{A}{7}! to become \verb!\dynkin A[1]7!:
\begin{tcblisting}{title={Extended Dynkin diagrams}}
-\dynkin{A}[1]{7}
+\dynkin A[1]7
\end{tcblisting}
\renewcommand*{\arraystretch}{1.5}
\begin{dynkinTable}{The Dynkin diagrams of the extended simple root systems}{3cm}{5cm}
-\dyn[extended]{A}{1}
-\dyn[extended]{A}{}
-\dyn[extended]{B}{}
-\dyn[extended]{C}{}
-\dyn[extended]{D}{}
-\dyn[extended]{E}{6}
-\dyn[extended]{E}{7}
-\dyn[extended]{E}{8}
-\dyn[extended]{F}{4}
-\dyn[extended]{G}{2}
+\dyn[extended]A{1}
+\dyn[extended]A{}
+\dyn[extended]B{}
+\dyn[extended]C{}
+\dyn[extended]D{}
+\dyn[extended]E6
+\dyn[extended]E7
+\dyn[extended]E8
+\dyn[extended]F4
+\dyn[extended]G2
\end{dynkinTable}
\newpage
@@ -891,149 +951,156 @@ The extended Dynkin diagrams are also described in the notation of Kac \cite{Kac
\section{Affine twisted and untwisted Dynkin diagrams}
The affine Dynkin diagrams are described in the notation of Kac \cite{Kac:1990} p. 55:
\begin{tcblisting}{title={Affine Dynkin diagrams}}
-\(A^{(1)}_7=\dynkin{A}[1]{7}, \
-E^{(2)}_6=\dynkin{E}[2]{6}, \
-D^{(3)}_4=\dynkin{D}[3]{4}\)
+\(A^{(1)}_7=\dynkin A[1]7, \
+E^{(2)}_6=\dynkin E[2]6, \
+D^{(3)}_4=\dynkin D[3]4\)
\end{tcblisting}
\begin{dynkinTable}{The affine Dynkin diagrams}{3cm}{3.75cm}
-\dyn{A}[1]{1}
-\dyn{A}[1]{}
-\dyn{B}[1]{}
-\dyn{C}[1]{}
-\dyn{D}[1]{}
-\dyn{E}[1]{6}
-\dyn{E}[1]{7}
-\dyn{E}[1]{8}
-\dyn{F}[1]{4}
-\dyn{G}[1]{2}
-\dyn{A}[2]{2}
-\dyn{A}[2]{even}
-\dyn{A}[2]{odd}
-\dyn{D}[2]{}
-\dyn{E}[2]{6}
-\dyn{D}[3]{4}
+\dyn A[1]1
+\dyn A[1]{}
+\dyn B[1]{}
+\dyn C[1]{}
+\dyn D[1]{}
+\dyn E[1]6
+\dyn E[1]7
+\dyn E[1]8
+\dyn F[1]4
+\dyn G[1]2
+\dyn A[2]2
+\dyn A[2]{even}
+\dyn A[2]{odd}
+\dyn D[2]{}
+\dyn E[2]6
+\dyn D[3]4
\end{dynkinTable}
\begin{dynkinTable}{Some more affine Dynkin diagrams}{3cm}{3.25cm}
-\dyn{A}[2]{4}
-\dyn{A}[2]{5}
-\dyn{A}[2]{6}
-\dyn{A}[2]{7}
-\dyn{A}[2]{8}
-\dyn{D}[2]{3}
-\dyn{D}[2]{4}
-\dyn{D}[2]{5}
-\dyn{D}[2]{6}
-\dyn{D}[2]{7}
-\dyn{D}[2]{8}
-\dyn{D}[3]{4}
-\dyn{E}[2]{6}
+\dyn A[2]4
+\dyn A[2]5
+\dyn A[2]6
+\dyn A[2]7
+\dyn A[2]8
+\dyn D[2]3
+\dyn D[2]4
+\dyn D[2]5
+\dyn D[2]6
+\dyn D[2]7
+\dyn D[2]8
+\dyn D[3]4
+\dyn E[2]6
\end{dynkinTable}
+\begin{dynkinTable}{Some more Kac--Moody Dynkin diagrams, only allowed in Kac ordering}{3cm}{3.25cm}
+\dyn[ordering=Kac,label]E6
+\dyn[ordering=Kac,label]E7
+\dyn[ordering=Kac,label]E8
+\dyn[ordering=Kac,label]E9
+\dyn[ordering=Kac,label]E{10}
+\dyn[ordering=Kac,label]E{11}
+\end{dynkinTable}
\section{Extended Coxeter diagrams}
\begin{tcblisting}{title={Extended and Coxeter options together}}
-\dynkin[extended,Coxeter]{F}{4}
+\dynkin[extended,Coxeter]F4
\end{tcblisting}
\begin{dynkinTable}{The extended (affine) Coxeter diagrams}{3cm}{6cm}
-\dyn[extended,Coxeter]{A}{}
-\dyn[extended,Coxeter]{B}{}
-\dyn[extended,Coxeter]{C}{}
-\dyn[extended,Coxeter]{D}{}
-\dyn[extended,Coxeter]{E}{6}
-\dyn[extended,Coxeter]{E}{7}
-\dyn[extended,Coxeter]{E}{8}
-\dyn[extended,Coxeter]{F}{4}
-\dyn[extended,Coxeter]{G}{2}
-\dyn[extended,Coxeter]{H}{3}
-\dyn[extended,Coxeter]{H}{4}
-\dyn[extended,Coxeter]{I}{1}
+\dyn[extended,Coxeter]A{}
+\dyn[extended,Coxeter]B{}
+\dyn[extended,Coxeter]C{}
+\dyn[extended,Coxeter]D{}
+\dyn[extended,Coxeter]E6
+\dyn[extended,Coxeter]E7
+\dyn[extended,Coxeter]E8
+\dyn[extended,Coxeter]F4
+\dyn[extended,Coxeter]G2
+\dyn[extended,Coxeter]H3
+\dyn[extended,Coxeter]H4
+\dyn[extended,Coxeter]I1
\end{dynkinTable}
\section{Kac style}
We include a style called \verb!Kac! which tries to imitate the style of \cite{Kac:1990}.
\begin{tcblisting}{title={Kac style}}
-\dynkin[Kac]{F}{4}
+\dynkin[Kac]F4
\end{tcblisting}
\begingroup
\pgfkeys{/Dynkin diagram,Kac}
\begin{dynkinTable}{The Dynkin diagrams of the simple root systems in Kac style}{5cm}{4.5cm}
-\dyn{A}{}
-\dyn{B}{}
-\dyn{C}{}
-\dyn{D}{}
-\dyn{E}{6}
-\dyn{E}{7}
-\dyn{E}{8}
-\dyn{F}{4}
-\dyn{G}{2}
+\dyn A{}
+\dyn B{}
+\dyn C{}
+\dyn D{}
+\dyn E6
+\dyn E7
+\dyn E8
+\dyn F4
+\dyn G2
\end{dynkinTable}
-\newpage
\begin{dynkinTable}{The Dynkin diagrams of the extended simple root systems in Kac style}{5cm}{4.5cm}
-\dyn[extended]{A}{1}
-\dyn[extended]{A}{}
-\dyn[extended]{B}{}
-\dyn[extended]{C}{}
-\dyn[extended]{D}{}
-\dyn[extended]{E}{6}
-\dyn[extended]{E}{7}
-\dyn[extended]{E}{8}
-\dyn[extended]{F}{4}
-\dyn[extended]{G}{2}
+\dyn[extended]A1
+\dyn[extended]A{}
+\dyn[extended]B{}
+\dyn[extended]C{}
+\dyn[extended]D{}
+\dyn[extended]E6
+\dyn[extended]E7
+\dyn[extended]E8
+\dyn[extended]F4
+\dyn[extended]G2
\end{dynkinTable}
+\newpage
\begin{dynkinTable}{The Dynkin diagrams of the twisted simple root systems in Kac style}{6cm}{4.5cm}
-\dyn{A}[2]{2}
-\dyn{A}[2]{even}
-\dyn{A}[2]{odd}
-\dyn{D}[2]{}
-\dyn{E}[2]{6}
-\dyn{D}[3]{4}
+\dyn A[2]2
+\dyn A[2]{even}
+\dyn A[2]{odd}
+\dyn D[2]{}
+\dyn E[2]6
+\dyn D[3]4
\end{dynkinTable}
\endgroup
-\newpage
\section{Ceref style}
We include a style called \verb!ceref! which paints oblong root markers with shadows.
The word ``ceref'' is an old form of the word ``serif''.
\begin{tcblisting}{title={Ceref style}}
-\dynkin[ceref]{F}{4}
+\dynkin[ceref]F4
\end{tcblisting}
\begingroup
\pgfkeys{/Dynkin diagram,ceref}
\begin{dynkinTable}{The Dynkin diagrams of the simple root systems in ceref style}{5cm}{4.5cm}
-\dyn{A}{}
-\dyn{B}{}
-\dyn{C}{}
-\dyn{D}{}
-\dyn{E}{6}
-\dyn{E}{7}
-\dyn{E}{8}
-\dyn{F}{4}
-\dyn{G}{2}
+\dyn A{}
+\dyn B{}
+\dyn C{}
+\dyn D{}
+\dyn E6
+\dyn E7
+\dyn E8
+\dyn F4
+\dyn G2
\end{dynkinTable}
\begin{dynkinTable}{The Dynkin diagrams of the extended simple root systems in ceref style}{5cm}{4.5cm}
-\dyn[extended]{A}{1}
-\dyn[extended]{A}{}
-\dyn[extended]{B}{}
-\dyn[extended]{C}{}
-\dyn[extended]{D}{}
-\dyn[extended]{E}{6}
-\dyn[extended]{E}{7}
-\dyn[extended]{E}{8}
-\dyn[extended]{F}{4}
-\dyn[extended]{G}{2}
+\dyn[extended]A1
+\dyn[extended]A{}
+\dyn[extended]B{}
+\dyn[extended]C{}
+\dyn[extended]D{}
+\dyn[extended]E6
+\dyn[extended]E7
+\dyn[extended]E8
+\dyn[extended]F4
+\dyn[extended]G2
\end{dynkinTable}
\begin{dynkinTable}{The Dynkin diagrams of the twisted simple root systems in ceref style}{6cm}{4.5cm}
-\dyn{A}[2]{2}
-\dyn{A}[2]{even}
-\dyn{A}[2]{odd}
-\dyn{D}[2]{}
-\dyn{E}[2]{6}
-\dyn{D}[3]{4}
+\dyn A[2]2
+\dyn A[2]{even}
+\dyn A[2]{odd}
+\dyn D[2]{}
+\dyn E[2]6
+\dyn D[3]4
\end{dynkinTable}
\endgroup
@@ -1041,42 +1108,42 @@ The word ``ceref'' is an old form of the word ``serif''.
\section{More on folded Dynkin diagrams}
The Dynkin diagrams package has limited support for folding Dynkin diagrams.
\begin{tcblisting}{title={Folding}}
-\dynkin[fold]{A}{13}
+\dynkin[fold]A{13}
\end{tcblisting}
\begin{tcblisting}{title={Big fold radius}}
-\dynkin[fold,fold radius=1cm]{A}{13}
+\dynkin[fold,fold radius=1cm]A{13}
\end{tcblisting}
\begin{tcblisting}{title={Small fold radius}}
-\dynkin[fold,fold radius=.2cm]{A}{13}
+\dynkin[fold,fold radius=.2cm]A{13}
\end{tcblisting}
Some Dynkin diagrams have multiple foldings, which we attempt to distinguish (not entirely successfully) by their \emph{ply}: the maximum number of roots folded together.
Most diagrams can only allow a 2-ply folding, so \verb!fold! is a synonym for \verb!ply=2!.
\begin{tcblisting}{title={3-ply}}
-\dynkin[ply=3]{D}{4}
-\dynkin[ply=3,fold right]{D}{4}
-\dynkin[ply=3]{D}[1]{4}
+\dynkin[ply=3]D4
+\dynkin[ply=3,fold right]D4
+\dynkin[ply=3]D[1]4
\end{tcblisting}
\begin{tcblisting}{title={4-ply}}
-\dynkin[ply=4]{D}[1]{4}
+\dynkin[ply=4]D[1]4
\end{tcblisting}
The \(D^{(1)}_{\ell}\) diagrams can be folded on their left end and separately on their right end:
\begin{tcblisting}{title={Left, right and both}}
-\dynkin{D}[1]{} \
-\dynkin[fold left]{D}[1]{} \
-\dynkin[fold right]{D}[1]{} \
-\dynkin[fold]{D}[1]{}
+\dynkin D[1]{} \
+\dynkin[fold left]D[1]{} \
+\dynkin[fold right]D[1]{} \
+\dynkin[fold]D[1]{}
\end{tcblisting}
We have to be careful about the 4-ply foldings of \(D^{(1)}_{2\ell}\), for which we can have two different patterns, so by default, the package only draws as much as it can without distinguishing the two:
\begin{tcblisting}{title={Default \(D^{(1)}_{2\ell}\) and the two ways to finish it}}
- \dynkin[ply=4]{D}[1]{****.*****.*****}%
+ \dynkin[ply=4]D[1]{****.*****.*****}%
\
\begin{dynkinDiagram}[ply=4]{D}[1]{****.*****.*****}%
- \dynkinFold[bend right=90]{1}{13}%
- \dynkinFold[bend right=90]{0}{14}%
+ \dynkinFold[bend right=90]1{13}%
+ \dynkinFold[bend right=90]0{14}%
\end{dynkinDiagram} \
\begin{dynkinDiagram}[ply=4]{D}[1]{****.*****.*****}%
- \dynkinFold{0}{1}%
- \dynkinFold{1}{13}%
+ \dynkinFold01%
+ \dynkinFold1{13}%
\dynkinFold{13}{14}%
\end{dynkinDiagram}
\end{tcblisting}
@@ -1114,16 +1181,16 @@ We have to be careful about the 4-ply foldings of \(D^{(1)}_{2\ell}\), for which
\begin{filecontents*}{DoneTwoElBendy.tex}
\begin{dynkinDiagram}[ply=4]{D}[1]%
{****.*****.*****}
- \dynkinFold[bend right=90]{1}{13}
- \dynkinFold[bend right=90]{0}{14}
+\dynkinFold[bend right=90]1{13}
+\dynkinFold[bend right=90]0{14}
\end{dynkinDiagram}
\end{filecontents*}
\begin{filecontents*}{DoneTwoElStraight.tex}
-\begin{dynkinDiagram}[ply=4]{D}[1]%
+\begin{dynkinDiagram}[ply=4]D[1]%
{****.*****.*****}
- \dynkinFold{0}{1}
- \dynkinFold{1}{13}
- \dynkinFold{13}{14}
+\dynkinFold01
+\dynkinFold1{13}
+\dynkinFold{13}{14}
\end{dynkinDiagram}
\end{filecontents*}
\pgfkeys{/Dynkin diagram,fold radius=.35cm}
@@ -1135,162 +1202,119 @@ We have to be careful about the 4-ply foldings of \(D^{(1)}_{2\ell}\), for which
\multicolumn{1}{c}{continued \dots}\\
\endfoot
\endlastfoot
-\fold{A}{0}{3}{C}{0}{2}
-\foldingTable{A}{0}{2\ell-1}{\dynk[fold]{A}{**.*****.**}}%
-{C}{0}{\ell}{\dynk{C}{}}
-\fold*{B}{0}{3}{G}{0}{2}
-\foldingTable{D}{0}{4}{\dynk[ply=3,fold right]{D}{4}}%
-{G}{0}{2}{\dynk{G}{2}}
-\foldingTable{D}{0}{\ell+1}{\dynk[fold]{D}{}}%
-{B}{0}{\ell}{\dynk{B}{}}
-\fold*{E}{0}{6}{F}{0}{4}
-\foldingTable{A}{1}{3}{\dynk[ply=4]{A}[1]{3}}%
-{A}{1}{1}{\dynk{A}[1]{1}}
-\foldingTable{A}{1}{2\ell-1}{\dynk[fold]{A}[1]{**.*****.**}}%
-{C}{1}{\ell}{\dynk{C}[1]{}}
-\foldingTable{B}{1}{3}{\dynk[ply=3]{B}[1]{3}}%
-{A}{2}{2}{\dynk{A}[2]{2}}
-\foldingTable{B}{1}{3}{\dynk[ply=2]{B}[1]{3}}%
-{G}{1}{2}{\dynk{G}[1]{2}}
-\foldingTable{B}{1}{\ell}{\dynk[fold]{B}[1]{}}{D}{2}{\ell}{\dynk{D}[2]{}}
-\foldingTable{D}{1}{4}{\dynk[ply=3]{D}[1]{4}}%
-{B}{1}{3}{\dynk{B}[1]{3}}
-\foldingTable{D}{1}{4}{\dynk[ply=3]{D}[1]{4}}%
-{G}{1}{2}{\dynk{G}[1]{2}}
-\foldingTable{D}{1}{\ell+1}{\dynk[fold]{D}[1]{}}%
-{D}{2}{\ell}{\dynk{D}[2]{}}
-\foldingTable{D}{1}{\ell+1}{%
-\dynk[fold right]{D}[1]{}}%
-{B}{1}{\ell}{\dynk{B}[1]{}}
-\foldingTable{D}{1}{2\ell}{%
+\fold A03C02
+\foldingTable A0{2\ell-1}{\dynk[fold]A{**.*****.**}}%
+C0{\ell}{\dynk C{}}
+\fold*B03G02
+\foldingTable D04{\dynk[ply=3,fold right]D4}%
+G02{\dynk G2}
+\foldingTable D0{\ell+1}{\dynk[fold]D{}}%
+B0{\ell}{\dynk B{}}
+\fold* E06F04
+\foldingTable A13{\dynk[ply=4]A[1]3}%
+A11{\dynk A[1]1}
+\foldingTable A1{2\ell-1}{\dynk[fold]A[1]{**.*****.**}}%
+C1{\ell}{\dynk C[1]{}}
+\foldingTable B13{\dynk[ply=3]B[1]3}%
+A22{\dynk A[2]2}
+\foldingTable B13{\dynk[ply=2]B[1]3}%
+G12{\dynk G[1]2}
+\foldingTable B1{\ell}{\dynk[fold]B[1]{}}D2{\ell}{\dynk D[2]{}}
+\foldingTable D14{\dynk[ply=3]D[1]4}%
+B13{\dynk B[1]3}
+\foldingTable D14{\dynk[ply=3]D[1]4}%
+G12{\dynk G[1]2}
+\foldingTable D1{\ell+1}{\dynk[fold]D[1]{}}%
+D2{\ell}{\dynk D[2]{}}
+\foldingTable D1{\ell+1}{%
+\dynk[fold right]D[1]{}}%
+B1{\ell}{\dynk B[1]{}}
+\foldingTable D1{2\ell}{%
\input{DoneTwoElStraight.tex}
&
\VerbatimInput{DoneTwoElStraight.tex} \\
}%
-{A}{2}{\text{odd}}{\dynk{A}[2]{odd}}
-\foldingTable{D}{1}{2\ell}{%
+A2{\text{odd}}{\dynk A[2]{odd}}
+\foldingTable D1{2\ell}{%
\input{DoneTwoElBendy.tex}
&
\VerbatimInput{DoneTwoElBendy.tex} \\
}%
-{A}{2}{\text{even}}{\dynk{A}[2]{even}}
-\fold*{E}{1}{6}{F}{1}{4}
-\foldingTable{E}{1}{6}{\dynk[ply=3]{E}[1]{6}}%
-{D}{3}{4}{\dynk{D}[3]{4}}
-\fold{E}{1}{7}{E}{2}{6}
-\fold{F}{1}{4}{G}{1}{2}
-\foldingTable{A}{2}{\text{odd}}{%
-\dynk[odd,fold]{A}[2]{****.***}
+A2{\text{even}}{\dynk A[2]{even}}
+\fold* E16F14
+\foldingTable E16{\dynk[ply=3]E[1]6}%
+D34{\dynk D[3]4}
+\fold E17E26
+\fold F14G12
+\foldingTable A2{\text{odd}}{%
+\dynk[odd,fold]A[2]{****.***}
}%
-{A}{2}{\text{even}}{\dynk{A}[2]{even}}
-\foldingTable{D}{2}{3}{\dynk[fold]{D}[2]{3}}%
-{A}{2}{2}{\dynk{A}[2]{2}}
+A2{\text{even}}{\dynk A[2]{even}}
+\foldingTable D23{\dynk[fold]D[2]3}%
+A22{\dynk A[2]2}
\end{longtable}
\endgroup
\begingroup
\renewcommand{\wdtA}{.8cm}
\begin{dynkinTable}{Frobenius fixed point subgroups of finite simple groups of Lie type \cite{Carter:1995} p. 15}{3cm}{6cm}
-A_{\ell\ge 1} & \dynk{A}{}
-{}^2\!A_{\ell\ge 2} & \dynk[fold]{A}{}
-B_{\ell\ge 2} & \dynk{B}{}
-{}^2\!B_2 & \dynk[fold]{B}{2}
-C_{\ell\ge3} & \dynk{C}{}
-D_{\ell\ge4} & \dynk{D}{}
-{}^2\!D_{\ell\ge4} & \dynk[fold]{D}{}
-{}^3\!D_4 & \dynk[ply=3]{D}{4}
-E_6 & \dynk{E}{6}
-{}^2\!E_6 & \dynk[fold]{E}{6}
-E_7 & \dynk{E}{7}
-E_8 & \dynk{E}{8}
-F_4 & \dynk{F}{4}
-{}^2\!F_4 & \dynk[fold]{F}{4}
-G_2 & \dynk{G}{2}
-{}^2G_2 & \dynk[fold]{G}{2}
+A_{\ell\ge 1} & \dynk A{}
+{}^2\!A_{\ell\ge 2} & \dynk[fold]A{}
+B_{\ell\ge 2} & \dynk B{}
+{}^2\!B_2 & \dynk[fold]B2
+C_{\ell\ge3} & \dynk C{}
+D_{\ell\ge4} & \dynk D{}
+{}^2\!D_{\ell\ge4} & \dynk[fold]D{}
+{}^3\!D_4 & \dynk[ply=3]D4
+E_6 & \dynk E6
+{}^2\!E_6 & \dynk[fold]E6
+E_7 & \dynk E7
+E_8 & \dynk E8
+F_4 & \dynk F4
+{}^2\!F_4 & \dynk[fold]F4
+G_2 & \dynk G2
+{}^2G_2 & \dynk[fold]G2
\end{dynkinTable}
\endgroup
-\section{Root ordering}\label{section:order}
-\begin{tcblisting}{title={Root ordering}}
-\dynkin[label,ordering=Adams]{E}{6}
-\dynkin[label,ordering=Bourbaki]{E}{6}
-\dynkin[label,ordering=Carter]{E}{6}
-\dynkin[label,ordering=Dynkin]{E}{6}
-\dynkin[label,ordering=Kac]{E}{6}
-\end{tcblisting}
-Default is Bourbaki.
-Sources are Adams \cite{Adams:1996} p. 56--57, Bourbaki \cite{Bourbaki:2002} p. pp. 265--290 plates I-IX, Carter \cite{Carter:2005} p. 540--609, Dynkin \cite{Dynkin:1952}, Kac \cite{Kac:1990} p. 43.
-\NewDocumentCommand\tablerow{mm}%
-{%
-#1_{#2}&
-\dynkin[label,ordering=Adams]{#1}{#2}&
-\dynkin[label]{#1}{#2}&
-\dynkin[label,ordering=Carter]{#1}{#2}&
-\dynkin[label,ordering=Dynkin]{#1}{#2}&
-\dynkin[label,ordering=Kac]{#1}{#2}\\
-}%
-\begin{center}
-\renewcommand{\wdtA}{.7cm}
-\renewcommand{\wdtL}{2.2cm}
-\begin{longtable}{@{}ALLLLL@{}}
-\toprule
-& Adams & Bourbaki & Carter & Dynkin & Kac \\ \midrule
-\endfirsthead
-\toprule
-& Adams & Bourbaki & Carter & Dynkin & Kac \\ \midrule
-\endhead
-\bottomrule
-\endfoot
-\bottomrule
-\endlastfoot
-\tablerow{E}{6}\tablerow{E}{7}\tablerow{E}{8}\tablerow{F}{4}\tablerow{G}{2}
-\end{longtable}
-\end{center}
-The marks are set down in order according to the current root ordering:
-\begin{tcblisting}{}
-\dynkin[label]{E}{*otxXOt*}
-\dynkin[label,ordering=Carter]{E}{*otxXOt*}
-\dynkin[label,ordering=Kac]{E}{*otxXOt*}
-\end{tcblisting}
-
\section{Typesetting mathematical names of Dynkin diagrams}
The \verb!\dynkinName! command, with the same syntax as \verb!\dynkin!, typesets a default name of your diagram in \LaTeX.
It is perhaps only useful when automatically generating a large collection of Dynkin diagrams in a computer program.
\begin{tcblisting}{title={Name of a diagram}}
-\dynkinName[label,extended]{B}{7}
-\dynkinName{A}[2]{even}
-\dynkinName[Coxeter]{B}{7}
-\dynkinName[label,extended]{B}{*}
-\dynkinName{D}[3]{4}
+\dynkinName[label,extended]B7
+\dynkinName A[2]{even}
+\dynkinName[Coxeter]B7
+\dynkinName[label,extended]B*
+\dynkinName D[3]4
\end{tcblisting}
\section{Connecting Dynkin diagrams}\label{section:name}
We can make some sophisticated folded diagrams by drawing multiple diagrams, each with a name:
\begin{tcblisting}{title={Name a diagram}}
-\dynkin[name=Bob]{D}{6}
+\dynkin[name=Bob]D6
\end{tcblisting}
We can then connect the two with folding edges:
\begin{tcblisting}{title={Connect diagrams}}
-\begin{dynkinDiagram}[name=upper]{A}{3}
+\begin{dynkinDiagram}[name=upper]A3
\node (current) at ($(upper root 1)+(0,-.3cm)$) {};
- \dynkin[at=(current),name=lower]{A}{3}
- \begin{scope}[on background layer]
+ \dynkin[at=(current),name=lower]A3
+ \begin{pgfonlayer}{Dynkin behind}
\foreach \i in {1,...,3}%
{%
\draw[/Dynkin diagram/fold style]
($(upper root \i)$)
-- ($(lower root \i)$);%
}%
- \end{scope}
+ \end{pgfonlayer}
\end{dynkinDiagram}
\end{tcblisting}
The following diagrams arise in the Satake diagrams of the pseudo-Riemannian symmetric spaces \cite{Baba:2009}.
\begin{tcblisting}{}
\pgfkeys{/Dynkin diagram,edge length=.5cm,fold radius=.5cm}
\begin{tikzpicture}
- \dynkin[name=1]{A}{IIIb}
+ \dynkin[name=1]A{IIIb}
\node (a) at (-.3,-.4){};
- \dynkin[name=2,at=(a)]{A}{IIIb}
- \begin{scope}[on background layer]
+ \dynkin[name=2,at=(a)]A{IIIb}
+ \begin{pgfonlayer}{Dynkin behind}
\foreach \i in {1,...,7}%
{%
\draw[/Dynkin diagram/fold style]
@@ -1298,7 +1322,7 @@ The following diagrams arise in the Satake diagrams of the pseudo-Riemannian sym
--
($(2 root \i)$);%
}%
- \end{scope}
+ \end{pgfonlayer}
\end{tikzpicture}
\end{tcblisting}
\begin{tcblisting}{}
@@ -1309,16 +1333,16 @@ edge/.style={draw=example-color,double=black,very thick}}
\foreach \d in {1,...,4}
{
\node (current) at ($(\d*.05,\d*.3)$){};
- \dynkin[name=\d,at=(current)]{D}{oo.oooo}
+ \dynkin[name=\d,at=(current)]D{oo.oooo}
}
- \begin{scope}[on background layer]
+ \begin{pgfonlayer}{Dynkin behind}
\foreach \i in {1,...,6}%
{%
\draw[/Dynkin diagram/fold style] ($(1 root \i)$) -- ($(2 root \i)$);%
\draw[/Dynkin diagram/fold style] ($(2 root \i)$) -- ($(3 root \i)$);%
\draw[/Dynkin diagram/fold style] ($(3 root \i)$) -- ($(4 root \i)$);%
}%
- \end{scope}
+ \end{pgfonlayer}
\end{tikzpicture}
\end{tcblisting}
@@ -1327,19 +1351,19 @@ edge/.style={draw=example-color,double=black,very thick}}
\tikzset{/Dynkin diagram,edge length=1cm,fold radius=1cm}
\tikzset{/Dynkin diagram,label macro/.code={\alpha_{#1}},label macro*/.code={\beta_{#1}}}
\({}^1 D_4\) 4-ply tied straight:
-\begin{dynkinDiagram}[ply=4]{D}[1]%
+\begin{dynkinDiagram}[ply=4]D[1]%
{****.*****.*****}
- \dynkinFold{0}{1}
- \dynkinFold{1}{13}
+ \dynkinFold 01
+ \dynkinFold 1{13}
\dynkinFold{13}{14}
\dynkinLabelRoots{0,...,14}
\dynkinLabelRoots*{0,...,14}
\end{dynkinDiagram}
\({}^1 D_4\) 4-ply tied bending:
-\begin{dynkinDiagram}[ply=4]{D}[1]%
+\begin{dynkinDiagram}[ply=4]D[1]%
{****.*****.*****}
- \dynkinFold{1}{13}
- \dynkinFold[bend right=65]{0}{14}
+\dynkinFold1{13}
+\dynkinFold[bend right=65]0{14}
\dynkinLabelRoots{0,...,14}
\dynkinLabelRoots*{0,...,14}
\end{dynkinDiagram}
@@ -1349,6 +1373,8 @@ edge/.style={draw=example-color,double=black,very thick}}
Below we draw the Vogan diagrams of some affine Lie superalgebras \cite{Ransingh:2013,Ransingh:unpub}.
\begingroup
\tikzset{/Dynkin diagram,edge length=.35cm,fold radius=.3cm}
+\tikzset{/Dynkin diagram,label macro/.code=\labls{#1},label,root radius=.06cm}
+\tcbset{text width=10cm}
\NewDocumentCommand\labls{m}%
{%
\ifcase#1%
@@ -1361,7 +1387,11 @@ Below we draw the Vogan diagrams of some affine Lie superalgebras \cite{Ransingh
{2}\or%
{1}\or%
{1}\or%
- \else\typeout{What?}%
+ {1}\or%
+ {1}\or%
+ {1}\or%
+ {1}\or%
+ \else\typeout{What? `#1'}%
\fi%
}%
\NewDocumentCommand\lablIt{m}%
@@ -1372,9 +1402,6 @@ Below we draw the Vogan diagrams of some affine Lie superalgebras \cite{Ransingh
2%
\fi%
}%
-\begingroup
-\tikzset{/Dynkin diagram,label macro/.code=\labls{#1},label,root radius=.06cm}
-\tcbset{text width=10cm}
\renewcommand{\wdtA}{2cm}
\NewDocumentEnvironment{Category}{m}%
{%
@@ -1383,84 +1410,83 @@ Below we draw the Vogan diagrams of some affine Lie superalgebras \cite{Ransingh
{%
\end{tcolorbox}
}%
-
\begin{Category}{\mathfrak{sl}\left(2m|2n\right)^{(2)}}
\begin{tcblisting}{}
\begin{dynkinDiagram}[ply=2,label]{B}[1]{oo.oto.oo}
- \dynkinLabelRoot*{7}{1}
+ \dynkinLabelRoot*71
\end{dynkinDiagram}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[label]{B}[1]{oo.oto.oo}
+\dynkin[label]B[1]{oo.oto.oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[ply=2,label]{B}[1]{oo.Oto.Oo}
+\dynkin[ply=2,label]B[1]{oo.Oto.Oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[label]{B}[1]{oo.Oto.Oo}
+\dynkin[label]B[1]{oo.Oto.Oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[label]{D}[1]{oo.oto.ooo}
+\dynkin[label]D[1]{oo.oto.ooo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[label]{D}[1]{oO.otO.ooo}
+\dynkin[label]D[1]{oO.otO.ooo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[label,fold]{D}[1]{oo.oto.ooo}
+\dynkin[label,fold]D[1]{oo.oto.ooo}
\end{tcblisting}
\end{Category}
\begin{Category}{\mathfrak{sl}\left(2m+1|2n\right)^2}
\begin{tcblisting}{}
-\dynkin[label]{B}[1]{oo.oto.oo}
+\dynkin[label]B[1]{oo.oto.oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[label]{B}[1]{oO.oto.oO}
+\dynkin[label]B[1]{oO.oto.oO}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[label,fold]{B}[1]{oo.oto.oo}
+\dynkin[label,fold]B[1]{oo.oto.oo}
\end{tcblisting}
\end{Category}
\begin{Category}{\mathfrak{sl}\left(2m+1|2n+1\right)^2}
\begin{tcblisting}{}
-\dynkin[label]{D}[2]{o.oto.oo}
+\dynkin[label]D[2]{o.oto.oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[label]{D}[2]{o.OtO.oo}
+\dynkin[label]D[2]{o.OtO.oo}
\end{tcblisting}
\end{Category}
\begin{Category}{\mathfrak{sl}\left(2|2n+1\right)^{(2)}}
\begin{tcblisting}{}
-\dynkin[ply=2,label,double edges]{B}[1]{oo.Oto.Oo}
+\dynkin[ply=2,label,double edges]B[1]{oo.Oto.Oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[ply=2,label,double fold]{B}[1]{oo.Oto.Oo}
+\dynkin[ply=2,label,double fold]B[1]{oo.Oto.Oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[ply=2,label,double edges]{B}[1]{oo.OtO.oo}
+\dynkin[ply=2,label,double edges]B[1]{oo.OtO.oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[ply=2,label,double fold]{B}[1]{oo.OtO.oo}
+\dynkin[ply=2,label,double fold]B[1]{oo.OtO.oo}
\end{tcblisting}
\end{Category}
\begin{Category}{\mathfrak{sl}\left(2|2n\right)^{(2)}}
\begin{tcblisting}{}
-\dynkin[ply=2,label,double edges]{D}[1]{oo.oto.ooo}
+\dynkin[ply=2,label,double edges]D[1]{oo.oto.ooo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[ply=2,label,double fold left]{D}[1]{oo.oto.ooo}
+\dynkin[ply=2,label,double fold left]D[1]{oo.oto.ooo}
\end{tcblisting}
\end{Category}
\begin{Category}{\mathfrak{osp}\left(2m|2n\right)^{(2)}}
\begin{tcblisting}{}
-\dynkin[label,label macro/.code={1}]{D}[2]{o.oto.oo}
+\dynkin[label,label macro/.code={1}]D[2]{o.oto.oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[label,label macro/.code={1}]{D}[2]{o.Oto.Oo}
+\dynkin[label,label macro/.code={1}]D[2]{o.Oto.Oo}
\end{tcblisting}
\end{Category}
@@ -1468,21 +1494,21 @@ Below we draw the Vogan diagrams of some affine Lie superalgebras \cite{Ransingh
\begin{tcblisting}{}
\dynkin[label,label macro/.code=\lablIt{#1},
affine mark=*]
- {D}[2]{o.o.o.o*}
+ D[2]{o.o.o.o*}
\end{tcblisting}
\begin{tcblisting}{}
\dynkin[label,label macro/.code=\lablIt{#1},
affine mark=*]
- {D}[2]{o.O.o.o*}
+ D[2]{o.O.o.o*}
\end{tcblisting}
\end{Category}
\begin{Category}{\mathfrak{sl}\left(1|2n+1\right)^{4}}
\begin{tcblisting}{}
-\dynkin[label,label macro/.code={1}]{D}[2]{o.o.o.o*}
+\dynkin[label,label macro/.code={1}]D[2]{o.o.o.o*}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[label,label macro/.code={1}]{D}[2]{o.o.O.o*}
+\dynkin[label,label macro/.code={1}]D[2]{o.o.O.o*}
\end{tcblisting}
\end{Category}
@@ -1490,113 +1516,114 @@ Below we draw the Vogan diagrams of some affine Lie superalgebras \cite{Ransingh
\begin{Category}{A^1}
\begin{tcblisting}{}
\begin{tikzpicture}
- \dynkin[name=upper]{A}{oo.t.oo}
+ \dynkin[name=upper]A{oo.t.oo}
\node (Dynkin current) at (upper root 1){};
\dynkinSouth
- \dynkin[at=(Dynkin current),name=lower]{A}{oo.t.oo}
- \begin{scope}[on background layer]
+ \dynkin[at=(Dynkin current),name=lower]A{oo.t.oo}
+ \begin{pgfonlayer}{Dynkin behind}
\foreach \i in {1,...,5}{
\draw[/Dynkin diagram/fold style]
($(upper root \i)$) -- ($(lower root \i)$);
}
- \end{scope}
+ \end{pgfonlayer}
\end{tikzpicture}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[fold]{A}[1]{oo.t.ooooo.t.oo}
+\dynkin[fold]A[1]{oo.t.ooooo.t.oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[fold,affine mark=t]{A}[1]{oo.o.ootoo.o.oo}
+\dynkin[fold,affine mark=t]A[1]{oo.o.ootoo.o.oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[affine mark=t]{A}[1]{o*.t.*o}
+\dynkin[affine mark=t]A[1]{o*.t.*o}
\end{tcblisting}
\end{Category}
\begin{Category}{B^1}
\begin{tcblisting}{}
-\dynkin[affine mark=*]{A}[2]{o.oto.o*}
+\dynkin[affine mark=*]A[2]{o.oto.o*}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[affine mark=*]{A}[2]{o.oto.o*}
+\dynkin[affine mark=*]A[2]{o.oto.o*}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[affine mark=*]{A}[2]{o.ooo.oo}
+\dynkin[affine mark=*]A[2]{o.ooo.oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[odd]{A}[2]{oo.*to.*o}
+\dynkin[odd]A[2]{oo.*to.*o}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[odd,fold]{A}[2]{oo.oto.oo}
+\dynkin[odd,fold]A[2]{oo.oto.oo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[odd,fold]{A}[2]{o*.oto.o*}
+\dynkin[odd,fold]A[2]{o*.oto.o*}
\end{tcblisting}
\end{Category}
\begin{Category}{D^1}
\begin{tcblisting}{}
-\dynkin{D}{otoo}
+\dynkin D{otoo}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin{D}{ot*o}
+\dynkin D{ot*o}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[fold]{D}{otoo}
+\dynkin[fold]D{otoo}
\end{tcblisting}
\end{Category}
\begin{Category}{C^1}
\begin{tcblisting}{}
-\dynkin[double edges,fold,affine mark=t,odd]{A}[2]{to.o*}
+\dynkin[double edges,fold,affine mark=t,odd]A[2]{to.o*}
\end{tcblisting}
\begin{tcblisting}{}
-\dynkin[double edges,fold,affine mark=t,odd]{A}[2]{t*.oo}
+\dynkin[double edges,fold,affine mark=t,odd]A[2]{t*.oo}
\end{tcblisting}
\end{Category}
\begin{Category}{F^1}
\begin{tcblisting}{}
-\begin{dynkinDiagram}{A}{oto*}%
- \dynkinQuadrupleEdge{1}{2}%
- \dynkinTripleEdge{4}{3}%
+\begin{dynkinDiagram}A{oto*}%
+ \dynkinQuadrupleEdge 12%
+ \dynkinTripleEdge 43%
\end{dynkinDiagram}%
\end{tcblisting}
\begin{tcblisting}{}
-\begin{dynkinDiagram}{A}{*too}%
- \dynkinQuadrupleEdge{1}{2}%
- \dynkinTripleEdge{4}{3}%
+\begin{dynkinDiagram}A{*too}%
+ \dynkinQuadrupleEdge 12%
+ \dynkinTripleEdge 43%
\end{dynkinDiagram}%
\end{tcblisting}
\end{Category}
\begin{Category}{G^1}
\begin{tcblisting}{}
-\begin{dynkinDiagram}{A}{ot*oo}%
- \dynkinQuadrupleEdge{1}{2}%
- \dynkinDefiniteDoubleEdge{4}{3}%
+\begin{dynkinDiagram}A{ot*oo}%
+ \dynkinQuadrupleEdge 12%
+ \dynkinDefiniteDoubleEdge 43%
\end{dynkinDiagram}%
\end{tcblisting}
\begin{tcblisting}{}
-\begin{dynkinDiagram}{A}{oto*o}%
- \dynkinQuadrupleEdge{1}{2}%
- \dynkinDefiniteDoubleEdge{4}{3}%
+\begin{dynkinDiagram}A{oto*o}%
+ \dynkinQuadrupleEdge 12%
+ \dynkinDefiniteDoubleEdge 43%
\end{dynkinDiagram}%
\end{tcblisting}
\begin{tcblisting}{}
-\begin{dynkinDiagram}{A}{*too*}%
- \dynkinQuadrupleEdge{1}{2}%
- \dynkinDefiniteDoubleEdge{4}{3}%
+\begin{dynkinDiagram}A{*too*}%
+ \dynkinQuadrupleEdge 12%
+ \dynkinDefiniteDoubleEdge 43%
\end{dynkinDiagram}%
\end{tcblisting}
\begin{tcblisting}{}
-\begin{dynkinDiagram}{A}{*tooo}%
- \dynkinQuadrupleEdge{1}{2}%
- \dynkinDefiniteDoubleEdge{4}{3}%
+\begin{dynkinDiagram}A{*tooo}%
+ \dynkinQuadrupleEdge 12%
+ \dynkinDefiniteDoubleEdge 43%
\end{dynkinDiagram}%
\end{tcblisting}
\end{Category}
\endgroup
+\tikzset{/Dynkin diagram,label macro/.code={},label=false}
\section{Example: the complex simple Lie algebras}
\begin{filecontents*}{simple-lie-algebras.tex}
@@ -1612,24 +1639,24 @@ Below we draw the Vogan diagrams of some affine Lie superalgebras \cite{Ransingh
\begin{longtable}{@{}GDWRS@{}}
\LieG&\text{Diagram}&\text{Weights}&\text{Roots}&\text{Simple roots}\\ \midrule\endfirsthead
\LieG&\text{Diagram}&\text{Weights}&\text{Roots}&\text{Simple roots}\\ \midrule\endhead
-A_n&\dynkin{A}{}&\frac{1}{r+1}\W[\sum e_j]{n+1}&e_i-e_j&e_i-e_{i+1}\\
-B_n&\dynkin{B}{}&\frac{1}{2}\W{n}& \pm e_i, \pm e_i \pm e_j, i\ne j&e_i-e_{i+1}, e_n\\
-C_n&\dynkin{C}{}&\W{n}& \pm 2 e_i, \pm e_i \pm e_j, i\ne j&e_i-e_{i+1}, 2e_n\\
-D_n&\dynkin{D}{}&\frac{1}{2}\W{n}& \pm e_i \pm e_j, i\ne j &
+A_n&\dynkin A{}&\frac1{n+1}\W[\sum e_j]{n+1}&e_i-e_j&e_i-e_{i+1}\\
+B_n&\dynkin B{}&\frac12\W n& \pm e_i, \pm e_i \pm e_j, i\ne j&e_i-e_{i+1}, e_n\\
+C_n&\dynkin C{}&\W n& \pm 2 e_i, \pm e_i \pm e_j, i\ne j&e_i-e_{i+1}, 2e_n\\
+D_n&\dynkin D{}&\frac12\W n& \pm e_i \pm e_j, i\ne j &
\begin{bunch}e_i-e_{i+1},&i\le n-1\\e_{n-1}+e_n\end{bunch}\\
-E_8&\dynkin{E}{8}&\frac{1}{2}\W{8}&
+E_8&\dynkin E8&\frac12\W 8&
\begin{bunch}\pm2e_i\pm2e_j,&i\ne j,\\ \sum_i(-1)^{m_i}e_i,&\sum m_i \text{ even}\end{bunch}&
\begin{bunch}
2e_1-2e_2,\\2e_2-2e_3,\\2e_3-2e_4,\\2e_4-2e_5,\\2e_5-2e_6,\\2e_6+2e_7,\\
-\sum e_j,\\2e_6-2e_7
\end{bunch}\\
-E_7&\dynkin{E}{7}&\frac{1}{2}\W[e_1-e_2]{8}&\quo&\quo\\
-E_6&\dynkin{E}{6}&\frac{1}{3}\W[e_1-e_2,e_2-e_3]{8}&\quo&\quo\\
-F_4& \dynkin{F}{4}&\W{4}&
+E_7&\dynkin E7&\frac12\W[e_1-e_2]8&\quo&\quo\\
+E_6&\dynkin E6&\frac13\W[e_1-e_2,e_2-e_3]8&\quo&\quo\\
+F_4& \dynkin F4&\W4&
\begin{bunch}\pm 2e_i,\\ \pm 2e_i \pm 2e_j, \quad i \ne j,\\ \pm e_1 \pm e_2 \pm e_3 \pm e_4
\end{bunch}&
\begin{bunch}2e_2-2e_3,\\2e_3-2e_4,\\2e_4,\\e_1-e_2-e_3-e_4\end{bunch}\\
-G_2&\dynkin{G}{2}&\W[\sum e_j]{3}&
+G_2&\dynkin G2&\W[\sum e_j]3&
\begin{bunch}
\pm(1,-1,0),\\ \pm(-1,0,1),\\ \pm(0,-1,1),\\ \pm(2,-1,-1),\\ \pm(1,-2,1),\\ \pm(-1,-1,2)
\end{bunch}&
@@ -1644,8 +1671,8 @@ G_2&\dynkin{G}{2}&\W[\sum e_j]{3}&
\section{An example of Mikhail Borovoi}
\begin{filecontents*}{borovoi.tex}
\tikzset{big arrow/.style={
- -Stealth,line cap=round,line width=1mm,
- shorten <=1mm,shorten >=1mm}}
+-Stealth,line cap=round,line width=1mm,
+shorten <=1mm,shorten >=1mm}}
\newcommand\catholic[2]{\draw[big arrow,green!25!white]
(root #1) to (root #2);}
\newcommand\protestant[2]{
@@ -1654,9 +1681,9 @@ G_2&\dynkin{G}{2}&\W[\sum e_j]{3}&
\end{scope}}
\begin{dynkinDiagram}[edge length=1.2cm,
indefinite edge/.style={thick,loosely dotted},
-labels*={0,1,2,3,\ell-3,\ell-2,\ell-1,\ell}]{D}[1]{}
-\catholic{0}{6}\catholic{1}{7}
-\protestant{7}{0}\protestant{6}{1}
+labels*={0,1,2,3,\ell-3,\ell-2,\ell-1,\ell}]D[1]{}
+\catholic 06\catholic 17
+\protestant 70\protestant 61
\end{dynkinDiagram}
\end{filecontents*}
\begingroup
@@ -1764,9 +1791,9 @@ is computed as \(n=\sum 2^{i-1} a_i\), \(a_i=0\) or \(1\), to say that root \(i\
& when drawing folded diagrams, style for the fold indicators. \\
\optionLabel{*/.style}{\typ{TikZ style data}}{solid,draw=black,fill=black}
& style for roots like \dynkin{A}{*} \\
-\optionLabel{o/.style}{\typ{TikZ style data}}{solid,draw=black,fill=black}
+\optionLabel{o/.style}{\typ{TikZ style data}}{solid,draw=black,fill=white}
& style for roots like \dynkin{A}{o} \\
-\optionLabel{O/.style}{\typ{TikZ style data}}{solid,draw=black,fill=black}
+\optionLabel{O/.style}{\typ{TikZ style data}}{solid,draw=black,fill=white}
& style for roots like \dynkin{A}{O} \\
\optionLabel{t/.style}{\typ{TikZ style data}}{solid,draw=black,fill=black}
& style for roots like \dynkin{A}{t} \\