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author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
commit | e0c6872cf40896c7be36b11dcc744620f10adf1d (patch) | |
tree | 60335e10d2f4354b0674ec22d7b53f0f8abee672 /graphics/pgf/contrib/dynkin-diagrams/dynkin-diagrams.tex |
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diff --git a/graphics/pgf/contrib/dynkin-diagrams/dynkin-diagrams.tex b/graphics/pgf/contrib/dynkin-diagrams/dynkin-diagrams.tex new file mode 100644 index 0000000000..6d84d1606d --- /dev/null +++ b/graphics/pgf/contrib/dynkin-diagrams/dynkin-diagrams.tex @@ -0,0 +1,1757 @@ +\documentclass{amsart} + +\title{The Dynkin diagrams package \\ Version 3.141592} + +\makeatletter +\DeclareRobustCommand{\scotsMc}{\scotsMcx{c}} +\DeclareRobustCommand{\scotsMC}{\scotsMcx{\textsc{c}}} +\DeclareRobustCommand{\scotsMcx}[1]{% + M% + \raisebox{\dimexpr\fontcharht\font`M-\height}{% + \check@mathfonts\fontsize{\sf@size}{0}\selectfont + \kern.3ex\underline{\kern-.3ex #1\kern-.3ex}\kern.3ex + }% +} +\expandafter\def\expandafter\@uclclist\expandafter{% + \@uclclist\scotsMc\scotsMC +} +\makeatother + +\author{Ben \scotsMc{}Kay} +\address{School of Mathematical Sciences, University College Cork, Cork, Ireland} +\email{b.mckay@ucc.ie} +\date{24 January 2019} + +\usepackage{etex} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenx} +\usepackage{etoolbox} +\usepackage{lmodern} +\usepackage[kerning=true,tracking=true]{microtype} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{array} +\usepackage{xstring} +\usepackage{longtable} +\usepackage[listings]{tcolorbox} +\tcbuselibrary{breakable} +\tcbuselibrary{skins} +\usepackage[pdftex]{hyperref} +\hypersetup{ + colorlinks = true, %Colours links instead of ugly boxes + urlcolor = black, %Colour for external hyperlinks + linkcolor = black, %Colour of internal links + citecolor = black %Colour of citations +} +\usepackage{booktabs} +\usepackage{colortbl} +\usepackage{varwidth} +\usepackage{dynkin-diagrams} +\usepackage{fancyvrb} +\usepackage{xspace} +\newcommand{\TikZ}{Ti\textit{k}Z\xspace} +\usepackage{filecontents} +\usetikzlibrary{decorations.markings} +\usetikzlibrary{decorations.pathmorphing} +\arrayrulecolor{white} +\makeatletter + \def\rulecolor#1#{\CT@arc{#1}} + \def\CT@arc#1#2{% + \ifdim\baselineskip=\z@\noalign\fi + {\gdef\CT@arc@{\color#1{#2}}}} + \let\CT@arc@\relax +\rulecolor{white} +\makeatother + +\newcommand{\C}[1]{\mathbb{C}^{#1}} +\renewcommand*{\arraystretch}{1.5} +\NewDocumentCommand\wdtA{}{.7cm} +\NewDocumentCommand\wdtD{}{3cm} +\NewDocumentCommand\wdtE{}{6cm} +\NewDocumentCommand\wdtL{}{3cm} +\newcolumntype{A}{@{}>{\columncolor[gray]{.9}$}m{\wdtA}<{$}} +\newcolumntype{B}{@{}>{\columncolor[gray]{.9}}m{\wdtA}} +\newcolumntype{D}{>{\columncolor[gray]{.9}}m{\wdtD}} +\newcolumntype{E}{>{\columncolor[gray]{.9}}m{\wdtE}} +\newcolumntype{L}{>{\columncolor[gray]{.9}}p{\wdtL}} +\newcolumntype{M}{>{\columncolor[gray]{.9}}l} +\newcolumntype{P}{>{\columncolor[gray]{.9}}p{10cm}} +\NewDocumentCommand\textleftcurly{}{\texttt{\char'173}}% +\NewDocumentCommand\textrightcurly{}{\texttt{\char'175}}% +\NewDocumentCommand\csDynkin{omom}% +{% + \texttt{\detokenize{\dynkin}\!\!\!% + \IfNoValueTF{#1}{}{[#1]}% + \textleftcurly#2\textrightcurly% + \IfNoValueTF{#3}{}{[#3]}% + \textleftcurly#4\textrightcurly% + }% +}% + +\NewDocumentCommand\dynk{omom}% +{% + \dynkin[#1]{#2}[#3]{#4}&\csDynkin[#1]{#2}[#3]{#4}\\ +}% + +\NewDocumentCommand\typesetSubseries{m}% +{% + \IfInteger{#1}{#1}{\IfStrEq{#1}{}{n}{#1}} +}% + +\NewDocumentCommand\dyn{omom}% +{% + {#2}_{\typesetSubseries{#4}}^{\IfInteger{#3}{#3}{\IfStrEq{#1}{extended}{1}{}}} & \dynk[#1]{#2}[#3]{#4}% +}% + + +\NewDocumentEnvironment{dynkinTable}{mmm}% +{% +\RenewDocumentCommand\wdtD{}{#2} +\RenewDocumentCommand\wdtL{}{#3} +\begin{longtable}{ADM} +\caption{#1}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{2}{c}{continued \dots}\\ +\endfoot +\endlastfoot +}% +{% +\end{longtable} +}% + + +\definecolor{example-color}{gray}{.85} +\definecolor{example-border-color}{gray}{.7} + +\tcbset{coltitle=black,colback=example-color,colframe=example-border-color,enhanced,breakable,pad at break*=1mm, +toprule=1.2mm,bottomrule=1.2mm,leftrule=1mm,rightrule=1mm,toprule at break=-1mm,bottomrule at break=-1mm, +before upper={\widowpenalties=3 10000 10000 150}} + +\makeatletter +\def\@tocline#1#2#3#4#5#6#7{\relax + \ifnum #1>\c@tocdepth% + \else + \par \addpenalty\@secpenalty\addvspace{#2}% + \begingroup \hyphenpenalty\@M + \@ifempty{#4}{% + \@tempdima\csname r@tocindent\number#1\endcsname\relax + }{% + \@tempdima#4\relax + }% + \parindent\z@ \leftskip#3\relax \advance\leftskip\@tempdima\relax + #5\leavevmode\hskip-\@tempdima #6\nobreak\relax + ,~#7\par + \endgroup + \fi} +\makeatother + +\fvset{fontsize=\small} + +\begin{document} + +\maketitle +\begin{center} +\begin{varwidth}{\textwidth} +\tableofcontents +\end{varwidth} +\end{center} + + +\setlength{\arrayrulewidth}{1.5pt} + +\section{Quick introduction} +\begin{tcolorbox}[title={Load the Dynkin diagram package (see options below)}] +\begin{verbatim} +\documentclass{amsart} +\usepackage{dynkin-diagrams} +\begin{document} +The Dynkin diagram of \(B_3\) is \dynkin{B}{3}. +\end{document} +\end{verbatim} +\end{tcolorbox} +\begin{tcblisting}{title={Invoke it}} +The Dynkin diagram of \(B_3\) is \dynkin{B}{3}. +\end{tcblisting} +\begin{tcblisting}{title={Inside a \TikZ statement}} +The Dynkin diagram of \(B_3\) is +\tikz \dynkin{B}{3}; +\end{tcblisting} +\begin{tcblisting}{title={Inside a Dynkin diagram environment}} +The Dynkin diagram of \(B_3\) is +\begin{dynkinDiagram}{B}{3} +\draw[very thick,red] (root 1) to [out=-45, in=-135] (root 3); +\end{dynkinDiagram} +\end{tcblisting} +\begin{tcblisting}{title={Inside a \TikZ environment}} +The baseline controls the vertical alignment: +the Dynkin diagram of \(B_3\) is +\begin{tikzpicture}[baseline=(origin.base)] +\dynkin{B}{3} +\draw[very thick,red] (root 1) to [out=-45, in=-135] (root 3); +\end{tikzpicture} +\end{tcblisting} +\begin{tcblisting}{title={Indefinite rank Dynkin diagrams}} +\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} +\end{dynkinTable} + + +\section{Set options globally} + +\begin{tcolorbox}[title={Most options set globally \dots}] +\begin{verbatim} +\pgfkeys{/Dynkin diagram,edge length=.5cm,fold radius=.5cm, +indefinite edge/.style={ + draw=black,fill=white,thin,densely dashed}} +\end{verbatim} +\end{tcolorbox} +You can also pass options to the package in \verb!\usepackage!. +\emph{Danger:} spaces in option names are replaced with hyphens: \texttt{edge length=1cm} is \texttt{edge-length=1cm} as a global option; moreover you should drop the extension \verb!/.style! on any option with spaces in its name (but not otherwise). For example, +\begin{tcolorbox}[title={\dots or pass global options to the package}] +\begin{verbatim} +\usepackage[ + ordering=Kac, + edge/.style=blue, + indefinite-edge={draw=green,fill=white,densely dashed}, + indefinite-edge-ratio=5, + mark=o, + root-radius=.06cm] + {dynkin-diagrams} +\end{verbatim} +\end{tcolorbox} + + + +\section{Coxeter diagrams} + +\begin{tcblisting}{title={Coxeter diagram option}} +\dynkin[Coxeter]{F}{4} +\end{tcblisting} + +\begin{tcblisting}{title={gonality option for \(G_2\) and \(I_n\) Coxeter diagrams}} +\(G_2=\dynkin[Coxeter,gonality=n]{G}{2}\), \ +\(I_n=\dynkin[Coxeter,gonality=n]{I}{}\) +\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}{} +\end{dynkinTable} + +\section{Satake diagrams}\label{section:Satake} + +\begin{tcblisting}{title={Satake diagrams use the standard name instead of a rank}} +\(A_{IIIb}=\dynkin{A}{IIIb}\) +\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} +\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} +\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} +\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]% +{****.*****.*****} + \dynkinFold{1}{13} + \dynkinFold[bend right=90]{0}{14} +\end{dynkinDiagram} +\end{tcblisting} + +\begin{tcblisting}{title={\dots but you could try springs pulling roots together}} +\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]% +{****.*****.*****} + \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_{#1}},edge length=.75cm]{D}{5} +\end{tcblisting} +\begin{tcblisting}{title={Labelling several roots}} +\dynkin[labels={,2,...,5,,7},label macro/.code={\alpha_{#1}}]{A}{7} +\end{tcblisting} +\begin{tcblisting}{title={The \texttt{foreach} notation I}} +\dynkin[labels={1,3,...,7},]{A}{9} +\end{tcblisting} +\begin{tcblisting}{title={The \texttt{foreach} notation II}} +\dynkin[labels={,\alpha_2,\alpha_...,\alpha_7},]{A}{7} +\end{tcblisting} +\begin{tcblisting}{title={The \texttt{foreach} notation III}} +\dynkin[label macro/.code={\beta_{#1}},labels={,2,...,7},]{A}{7} +\end{tcblisting} +\begin{tcblisting}{title={Label the roots individually by root number}} +\dynkin[label]{B}{3} +\end{tcblisting} +\begin{tcblisting}{title={Label a single root}} +\begin{dynkinDiagram}{B}{3} +\dynkinLabelRoot{2}{\alpha_2} +\end{dynkinDiagram} +\end{tcblisting} +\begin{tcblisting}{title={Use a text style}} +\begin{dynkinDiagram}[text/.style={scale=1.2}]{B}{3}; +\dynkinLabelRoot{2}{\alpha_2} +\end{dynkinDiagram} +\end{tcblisting} +\begin{tcblisting}{title={Access root labels via TikZ}} +\begin{dynkinDiagram}{B}{3} +\node[below] at (root 2) {\(\alpha_2\)}; +\end{dynkinDiagram} +\end{tcblisting} +\begin{tcblisting}{title={Commands to label several roots}} +\begin{dynkinDiagram}{A}{7} +\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} +\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} +\end{tcblisting} +\begin{tcblisting}{title={Labelling several roots and alternates}} +\dynkin[% +label macro/.code={\alpha_{#1}}, +label macro*/.code={\gamma_{#1}}, +labels={,2,...,5,,7}, +labels*={1,3,4,5,6}]{A}{7} +\end{tcblisting} +\begin{tcblisting}{title={Commands to label several roots}} +\begin{dynkinDiagram}{A}{7} +\dynkinLabelRoots{,\alpha_2,\alpha_3,\alpha_4,\alpha_5,,\alpha_7} +\dynkinLabelRoots*{a,b,c,d,e,f,g} +\end{dynkinDiagram} +\end{tcblisting} + + +\section{Bracing roots} +\begin{tcblisting}{title={Bracing roots}} +\begin{dynkinDiagram}{A}{*.*x*.*} +\dynkinBrace[p]{1}{2} +\dynkinBrace[q]{4}{5} +\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} +\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} +\dynkinBrace[t-1]{11}{12} +\dynkinBrace[x-1]{14}{16} +\end{dynkinDiagram} +\end{tcblisting} + +\begin{filecontents*}{EulerProducts.tex} +\tikzset{/Dynkin diagram,ordering=Dynkin,label macro/.code={\alpha_{#1}}} +\newcounter{EPNo} +\setcounter{EPNo}{0} +\NewDocumentCommand\EP{smmmm}% +{% +\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}}% +}% +\tikzset{/Dynkin diagram,edge length=\eL} +\IfBooleanTF{#1}% +{\dynkin[backwards,labels*={#4},labels={#5}]{#2}{#3}} +{\dynkin[labels*={#4},labels={#5}]{#2}{#3}} +\\ +}% +\begin{longtable}{MM} +\caption{Dynkin diagrams from Euler products \cite{Langlands:1967}}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\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} +\end{longtable} +\end{filecontents*} +{\input{EulerProducts}}\VerbatimInput{EulerProducts.tex} + +\section{Style} +\begin{tcblisting}{title={Colours}} +\dynkin[ + edge/.style={blue!50,thick}, + */.style=blue!50!red, + 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} +\end{tcblisting} +\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} +\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 +\end{tabular} +\begin{tcblisting}{title={Mark styles}} +The parabolic subgroup \(E_{8,124}\) is \dynkin[parabolic=124,x/.style={brown,very thick}]{E}{8} +\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} +\end{tcblisting} + + +\section{Suppress or reverse arrows} +\begin{tcblisting}{title={Some diagrams have double or triple edges}} +\dynkin{F}{4} +\dynkin{G}{2} +\end{tcblisting} +\begin{tcblisting}{title={Suppress arrows}} +\dynkin[arrows=false]{F}{4} +\dynkin[arrows=false]{G}{2} +\end{tcblisting} +\begin{tcblisting}{title={Reverse arrows}} +\dynkin[reverse arrows]{F}{4} +\dynkin[reverse arrows]{G}{2} +\end{tcblisting} + + +\section{Backwards and upside down} + +\begin{tcblisting}{title={Default}} +\dynkin{E}{8} +\dynkin{F}{4} +\dynkin{G}{2} +\end{tcblisting} +\begin{tcblisting}{title={Backwards}} +\dynkin[backwards]{E}{8} +\dynkin[backwards]{F}{4} +\dynkin[backwards]{G}{2} +\end{tcblisting} +\begin{tcblisting}{title={Reverse arrows}} +\dynkin[reverse arrows]{F}{4} +\dynkin[reverse arrows]{G}{2} +\end{tcblisting} +\begin{tcblisting}{title={Backwards, reverse arrows}} +\dynkin[backwards,reverse arrows]{F}{4} +\dynkin[backwards,reverse arrows]{G}{2} +\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} +\end{tcblisting} + + +\section{Drawing on top of a Dynkin diagram} + +\begin{tcblisting}{title={TikZ can access the roots themselves}} +\begin{dynkinDiagram}{A}{4} + \fill[white,draw=black] (root 2) circle (.15cm); + \fill[white,draw=black] (root 2) circle (.1cm); + \draw[black] (root 2) circle (.05cm); +\end{dynkinDiagram} +\end{tcblisting} +\begin{tcblisting}{title={Draw curves between the roots}} +\begin{dynkinDiagram}[label]{E}{8} + \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} +\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} +\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. +(Careful: in an affine root system, a mark list will \emph{not} contain a mark for root zero.) + +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} +\end{tcblisting} + +\NewDocumentCommand\ClassicalLieSuperalgebras{om}% +{% +\IfValueT{#1}{\tikzset{/Dynkin diagram,root radius=#1}} +\RenewDocumentCommand\wdtE{}{10cm} +\begin{dynkinTable}{Classical Lie superalgebras \cite{Frappat/Sciarrino/Sorba:1989}. #2}{3.5cm}{6.5cm} +\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} +G_3 & \dynk[extended,affine mark=t, +reverse arrows]{G}{2} +\end{dynkinTable} +\IfValueT{#1}{\tikzset{/Dynkin diagram,root radius=.05cm}} +}% + +\ClassicalLieSuperalgebras[.07cm]{We need a slightly larger root radius parameter to distinguish the tensor product symbols from the solid dots.} + +\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. +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} +\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} +\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} +\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} +\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} +\end{tcblisting} + + +\begingroup +\RenewDocumentCommand\wdtA{}{.35cm} +\RenewDocumentCommand\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} +\end{dynkinDiagram} +} +\\ +% 2 +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{12}{n-d} +\end{dynkinDiagram} +} +\\ +% 3 +B_n & +\multicolumn{2}{E}{ +\begin{dynkinDiagram}{B}{**.*.o.oo} +\dynkinLabelRoot{3}{r} +\end{dynkinDiagram} +} +\\ +% 4 +C_n & +\multicolumn{2}{E}{ +\begin{dynkinDiagram}{C}{o.o*o.o*o.oo} +\dynkinLabelRoot{3}{d} +\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} +\end{dynkinDiagram} +} +\\ +% 6 +E_6 & +\dynk{E}{*oooo*} +% 7 +E_6 & +\dynk{E}{o*o*oo} +% 8 +E_6 & +\dynk{E}{o*oooo} +% 9 +E_6 & +\dynk{E}{**ooo*} +% 10 +E_7 & +\dynk{E}{*oooooo} +% 11 +E_7 & +\dynk{E}{ooooo*o} +% 12 +E_7 & +\dynk{E}{oooooo*} +% 13 +E_7 & +\dynk{E}{*oooo*o} +% 14 - corrected from Springer. +E_7 & +\dynk{E}{*oooo**} +% 15 +E_7 & +\dynk{E}{*o**o*o} +% 16 +E_8 & +\dynk{E}{*ooooooo} +% 17 +E_8 & +\dynk{E}{ooooooo*} +% 18 +E_8 & +\dynk{E}{*oooooo*} +% 19 +E_8 & +\dynk{E}{oooooo**} +% 20 +E_8 & +\dynk{E}{*oooo***} +% 21 +F_4 & +\dynk{F}{ooo*} +% 22 +D_4 & +\dynk{D}{o*oo} +\end{longtable} +\endgroup + + + + +\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}. +\end{tcblisting} + +\begin{filecontents*}{hermitian-symmetric-spaces.tex} +\NewDocumentCommand\HSS{mommm} +{#1&\IfNoValueTF{#2}{\dynkin{#3}{#4}}{\dynkin[parabolic=#2]{#3}{#4}}\\} +\renewcommand*{\arraystretch}{1.5} +\begin{longtable} +{>{\columncolor[gray]{.9}}>$l<$>{\columncolor[gray]{.9}}>$l<$>{\columncolor[gray]{.9}}l} +\caption{The Hermitian symmetric spaces}\endfirsthead +\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} +\end{longtable} +\end{filecontents*} +\begingroup +\input{hermitian-symmetric-spaces.tex} +\endgroup +\VerbatimInput{hermitian-symmetric-spaces.tex} + +\begin{tcblisting}{title={Folded parabolics look bad (zoom in on a root)}} +\dynkin[fold,parabolic=3]{C}{2} +\dynkin[fold,parabolic=3]{G}{2} +\end{tcblisting} +\begin{tcblisting}{title={Folded parabolics: you can try using thicker crosses}} +\dynkin[fold,x/.style={very thick,line cap=round},parabolic=3]{C}{2} +\dynkin[fold,x/.style={ultra thick,line cap=round},parabolic=3]{G}{2} +\end{tcblisting} + + +\section{Extended Dynkin diagrams} + +\begin{tcblisting}{title={Extended Dynkin diagrams}} +\dynkin[extended]{A}{7} +\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}!: +\begin{tcblisting}{title={Extended Dynkin diagrams}} +\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} +\end{dynkinTable} + + + + + +\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}} +\(A^{(1)}_7=\dynkin{A}[1]{7}, \ +E^{(2)}_6=\dynkin{E}[2]{6}, \ +D^{(3)}_4=\dynkin{D}[3]{4}\) +\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} +\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} +\end{dynkinTable} + + + + +\section{Extended Coxeter diagrams} + +\begin{tcblisting}{title={Extended and Coxeter options together}} +\dynkin[extended,Coxeter]{F}{4} +\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} +\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} +\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} +\end{dynkinTable} +\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} +\end{dynkinTable} +\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} +\end{dynkinTable} +\endgroup + +\section{Ceref style} +We include a style called \verb!ceref! which shapes the root markers more oblongly and with shadows. +The word ``ceref'' is an old form of the word ``serif''. +\begin{tcblisting}{title={Ceref style}} +\dynkin[ceref]{F}{4} +\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} +\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} +\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} +\end{dynkinTable} +\endgroup + + +\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} +\end{tcblisting} +\begin{tcblisting}{title={Big fold radius}} +\dynkin[fold,fold radius=1cm]{A}{13} +\end{tcblisting} +\begin{tcblisting}{title={Small fold radius}} +\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} +\end{tcblisting} +\begin{tcblisting}{title={4-ply}} +\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]{} +\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]{****.*****.*****}% + \ +\begin{dynkinDiagram}[ply=4]{D}[1]{****.*****.*****}% + \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}% + \dynkinFold{13}{14}% +\end{dynkinDiagram} +\end{tcblisting} +\begingroup +\RenewDocumentCommand\wdtA{}{.7cm} +\RenewDocumentCommand\wdtD{}{3.5cm} +\RenewDocumentCommand\wdtL{}{7cm} +\NewDocumentCommand\seriesName{mmm}% +{% + \IfStrEq{#2}{0}{#1_{#3}}{#1^{#2}_{#3}}% +}% + +\NewDocumentCommand\foldingTable{smmmmmmmm}% +{% +\begin{tabular}{ADL}% +\seriesName{#2}{#3}{#4} +\seriesName{#6}{#7}{#8}&\IfBooleanTF{#1}{\reflectbox{#9}}{#9}% +\end{tabular}% +\\ \hline +}% +\NewDocumentCommand\fold{smmmmmm}% +{% + \IfBooleanTF{#1}% + {% + \foldingTable% + {#2}{#3}{#4}{\dynk[fold]{#2}[#3]{#4}}% + {#5}{#6}{#7}{\dynk[reverse arrows]{#5}[#6]{#7}}% + }% + {% + \foldingTable% + {#2}{#3}{#4}{\dynk[fold]{#2}[#3]{#4}}% + {#5}{#6}{#7}{\dynk{#5}[#6]{#7}}% + }% +}% +\begin{filecontents*}{DoneTwoElBendy.tex} +\begin{dynkinDiagram}[ply=4]{D}[1]% +{****.*****.*****} + \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]% +{****.*****.*****} + \dynkinFold{0}{1} + \dynkinFold{1}{13} + \dynkinFold{13}{14} +\end{dynkinDiagram} +\end{filecontents*} +\pgfkeys{/Dynkin diagram,fold radius=.35cm} +\begin{longtable}{@{}p{15cm}@{}} +\caption{Some foldings of Dynkin diagrams. For these diagrams, we want to compare a folding diagram with the diagram that results when we fold it, so it looks best to set \texttt{fold radius} and \texttt{edge length} to equal lengths.}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\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}{% +\input{DoneTwoElStraight.tex} +& +\VerbatimInput{DoneTwoElStraight.tex} \\ +}% +{A}{2}{\text{odd}}{\dynk{A}[2]{odd}} +\foldingTable{D}{1}{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]{****.***} +}% +{A}{2}{\text{even}}{\dynk{A}[2]{even}} +\foldingTable{D}{2}{3}{\dynk[fold]{D}[2]{3}}% +{A}{2}{2}{\dynk{A}[2]{2}} +\end{longtable} +\endgroup +\begingroup +\RenewDocumentCommand\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} +\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} +\RenewDocumentCommand\wdtA{}{.7cm} +\RenewDocumentCommand\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{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} +\end{tcblisting} +We can then connect the two with folding edges: +\begin{tcblisting}{title={Connect diagrams}} +\begin{dynkinDiagram}[name=upper]{A}{3} + \node (current) at ($(upper root 1)+(0,-.3cm)$) {}; + \dynkin[at=(current),name=lower]{A}{3} + \begin{scope}[on background layer] + \foreach \i in {1,...,3}% + {% + \draw[/Dynkin diagram/fold style] + ($(upper root \i)$) + -- ($(lower root \i)$);% + }% + \end{scope} +\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} + \node (a) at (-.3,-.4){}; + \dynkin[name=2,at=(a)]{A}{IIIb} + \begin{scope}[on background layer] + \foreach \i in {1,...,7}% + {% + \draw[/Dynkin diagram/fold style] + ($(1 root \i)$) + -- + ($(2 root \i)$);% + }% + \end{scope} +\end{tikzpicture} +\end{tcblisting} +\begin{tcblisting}{} +\pgfkeys{/Dynkin diagram, +edge length=.75cm, +edge/.style={draw=example-color,double=black,very thick}} +\begin{tikzpicture} + \foreach \d in {1,...,4} + { + \node (current) at ($(\d*.05,\d*.3)$){}; + \dynkin[name=\d,at=(current)]{D}{oo.oooo} + } + \begin{scope}[on background layer] + \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{tikzpicture} +\end{tcblisting} + +\section{Other examples} +\begin{filecontents*}{d44.tex} +\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]% +{****.*****.*****} + \dynkinFold{0}{1} + \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]% +{****.*****.*****} + \dynkinFold{1}{13} + \dynkinFold[bend right=65]{0}{14} +\dynkinLabelRoots{0,...,14} +\dynkinLabelRoots*{0,...,14} +\end{dynkinDiagram} +\end{filecontents*} +\begingroup\input{d44}\endgroup +\VerbatimInput{d44.tex} +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} +\NewDocumentCommand\labls{m}% +{% + \ifcase#1% + {1}\or% + {1}\or% + {2}\or% + {2}\or% + {2}\or% + {2}\or% + {2}\or% + {1}\or% + {1}\or% + \else\typeout{What?}% + \fi% +}% +\NewDocumentCommand\lablIt{m}% +{% + \ifnum#1=0\relax% + 1% + \else + 2% + \fi% +}% +\begingroup +\tikzset{/Dynkin diagram,label macro/.code=\labls{#1},label,root radius=.06cm} +\tcbset{text width=10cm} +\RenewDocumentCommand\wdtA{}{2cm} +\NewDocumentEnvironment{Category}{m}% +{% +\begin{tcolorbox}[title={\(#1\)},breakable]{} +}% +{% +\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} +\end{dynkinDiagram} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[label]{B}[1]{oo.oto.oo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[ply=2,label]{B}[1]{oo.Oto.Oo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[label]{B}[1]{oo.Oto.Oo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[label]{D}[1]{oo.oto.ooo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[label]{D}[1]{oO.otO.ooo} +\end{tcblisting} +\begin{tcblisting}{} +\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} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[label]{B}[1]{oO.oto.oO} +\end{tcblisting} +\begin{tcblisting}{} +\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} +\end{tcblisting} +\begin{tcblisting}{} +\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} +\end{tcblisting} +\begin{tcblisting}{} +\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} +\end{tcblisting} +\begin{tcblisting}{} +\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} +\end{tcblisting} +\begin{tcblisting}{} +\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} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[label,label macro/.code={1}]{D}[2]{o.Oto.Oo} +\end{tcblisting} +\end{Category} + +\begin{Category}{\mathfrak{osp}\left(2|2n\right)^{(2)}} +\begin{tcblisting}{} +\dynkin[label,label macro/.code=\lablIt{#1}, + affine mark=*] + {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*} +\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*} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[label,label macro/.code={1}]{D}[2]{o.o.O.o*} +\end{tcblisting} +\end{Category} + + +\begin{Category}{A^1} +\begin{tcblisting}{} +\begin{tikzpicture} + \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] + \foreach \i in {1,...,5}{ + \draw[/Dynkin diagram/fold style] + ($(upper root \i)$) -- ($(lower root \i)$); + } + \end{scope} +\end{tikzpicture} +\end{tcblisting} +\begin{tcblisting}{} +\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} +\end{tcblisting} +\begin{tcblisting}{} +\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*} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[affine mark=*]{A}[2]{o.oto.o*} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[affine mark=*]{A}[2]{o.ooo.oo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[odd]{A}[2]{oo.*to.*o} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[odd,fold]{A}[2]{oo.oto.oo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[odd,fold]{A}[2]{o*.oto.o*} +\end{tcblisting} +\end{Category} + +\begin{Category}{D^1} +\begin{tcblisting}{} +\dynkin{D}{otoo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin{D}{ot*o} +\end{tcblisting} +\begin{tcblisting}{} +\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*} +\end{tcblisting} +\begin{tcblisting}{} +\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}% +\end{dynkinDiagram}% +\end{tcblisting} +\begin{tcblisting}{} +\begin{dynkinDiagram}{A}{*too}% + \dynkinQuadrupleEdge{1}{2}% + \dynkinTripleEdge{4}{3}% +\end{dynkinDiagram}% +\end{tcblisting} +\end{Category} + +\begin{Category}{G^1} +\begin{tcblisting}{} +\begin{dynkinDiagram}{A}{ot*oo}% + \dynkinQuadrupleEdge{1}{2}% + \dynkinDefiniteDoubleEdge{4}{3}% +\end{dynkinDiagram}% +\end{tcblisting} +\begin{tcblisting}{} +\begin{dynkinDiagram}{A}{oto*o}% + \dynkinQuadrupleEdge{1}{2}% + \dynkinDefiniteDoubleEdge{4}{3}% +\end{dynkinDiagram}% +\end{tcblisting} +\begin{tcblisting}{} +\begin{dynkinDiagram}{A}{*too*}% + \dynkinQuadrupleEdge{1}{2}% + \dynkinDefiniteDoubleEdge{4}{3}% +\end{dynkinDiagram}% +\end{tcblisting} +\begin{tcblisting}{} +\begin{dynkinDiagram}{A}{*tooo}% + \dynkinQuadrupleEdge{1}{2}% + \dynkinDefiniteDoubleEdge{4}{3}% +\end{dynkinDiagram}% +\end{tcblisting} +\end{Category} +\endgroup + +\section{Example: the complex simple Lie algebras} +\begin{filecontents*}{simple-lie-algebras.tex} +\NewDocumentEnvironment{bunch}{}% +{\renewcommand*{\arraystretch}{1}\begin{array}{@{}ll@{}}\\ \midrule}{\\ \midrule\end{array}} +\small +\NewDocumentCommand\nct{mm}{\newcolumntype{#1}{>{\columncolor[gray]{.9}}>{$}m{#2cm}<{$}}} +\nct{G}{.3}\nct{D}{2.1}\nct{W}{3}\nct{R}{3.7}\nct{S}{3} +\NewDocumentCommand\LieG{}{\mathfrak{g}} +\NewDocumentCommand\W{om}{\ensuremath{\mathbb{Z}^{#2}\IfValueT{#1}{/\left<#1\right>}}} +\renewcommand*{\arraystretch}{1.5} +\NewDocumentCommand\quo{}{\text{quotient of } E_8} +\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 & +\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}& +\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}& +\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}& +\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}& +\begin{bunch}(-1,0,1),\\(2,-1,-1)\end{bunch} +\end{longtable} +\end{filecontents*} +\begingroup +\input{simple-lie-algebras.tex} +\endgroup +\VerbatimInput{simple-lie-algebras.tex} + +\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}} +\newcommand\catholic[2]{\draw[big arrow,green!25!white] +(root #1) to (root #2);} +\newcommand\protestant[2]{ +\begin{scope}[transparency group, opacity=.25] +\draw[big arrow,orange] (root #1) to (root #2); +\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} +\end{dynkinDiagram} +\end{filecontents*} +\begingroup +\begin{center} +\input{borovoi.tex} +\end{center} +\endgroup +\VerbatimInput{borovoi.tex} +\newpage + + +\section{Syntax} +The syntax is \verb!\dynkin[<options>]{<letter>}[<twisted rank>]{<rank>}! where \verb!<letter>! is \verb!A!, \verb!B!, \verb!C!, \verb!D!, \verb!E!, \verb!F! or \verb!G!, the family of root system for the Dynkin diagram, \verb!<twisted rank>! is \verb!0!, \verb!1!, \verb!2!, \verb!3! (default is \verb!0!) representing: +\[ +\renewcommand*{\arraystretch}{1} +\begin{array}{rp{8cm}} +0 & finite root system \\ \hline +1 & affine extended root system, i.e. of type \({}^{(1)}\) \\ +2 & affine twisted root system of type \({}^{(2)}\) \\ +3 & affine twisted root system of type \({}^{(3)}\) \\ +\end{array} +\] +and \verb!<rank>! is +\begin{enumerate} +\item +an integer representing the rank or +\item +blank to represent an indefinite rank or +\item +the name of a Satake diagram as in section~\ref{section:Satake}. +\end{enumerate} +The environment syntax is \verb!\begin{dynkinDiagram}! followed by the same parameters as \verb!\dynkin!, then various Dynkin diagram and \TikZ{} commands, and then \verb!\end{dynkinDiagram}!. + +\section{Options} +\newcommand*{\typ}[1]{\(\left<\texttt{#1}\right>\)} +\newcommand*{\optionLabel}[3]{%% +\multicolumn{2}{l}{\(\texttt{#1}=\texttt{#2}\),} \\ +\multicolumn{2}{l}{\(\textrm{default}: \texttt{#3}\)} \\ +}%% +\renewcommand*{\arraystretch}{1} +\par\noindent% +\begin{longtable}{p{1cm}p{10cm}} +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{2}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\optionLabel{ceref}{\typ{true or false}}{false} +& whether to draw roots in a ``ceref'' style. \\ +\optionLabel{edge length}{\typ{number}cm}{.35cm} +& distance between nodes in the Dynkin diagram \\ +\optionLabel{edge/.style}{TikZ style data}{solid,draw=black,fill=white,thin} +& style of edges in the Dynkin diagram \\ +\optionLabel{Kac}{\typ{true or false}}{false} +& whether to draw in the style of \cite{Kac:1990} \\ +\optionLabel{name}{\typ{string}}{anonymous} +& A name for the Dynkin diagram, with \texttt{anonymous} treated as a blank; see section~\ref{section:name}. \\ +\optionLabel{parabolic}{\typ{integer}}{0} +& A parabolic subgroup with specified integer, where the integer +is computed as \(n=\sum 2^{i-1} a_i\), \(a_i=0\) or \(1\), to say that root \(i\) is crossed, i.e. a noncompact root. \\ +\optionLabel{root radius}{\typ{number}cm}{.05cm} +& size of the dots and of the crosses in the Dynkin diagram \\ +\optionLabel{text/.style}{\typ{TikZ style data}}{scale=.7} +& Style for any labels on the roots. \\ +\optionLabel{mark}{\typ{o,O,t,x,X,*}}{*} +& default root mark \\ +\optionLabel{affine mark}{o,O,t,x,X,*}{*} +& default root mark for root zero in an affine Dynkin diagram \\ +\optionLabel{label}{true or false}{false} +& whether to label the roots according to the current labelling scheme. \\ +\optionLabel{label macro}{\typ{1-parameter \TeX{} macro}}{\texttt{\#1}} +& the current labelling scheme for roots. \\ +\optionLabel{label macro*}{\typ{1-parameter \TeX{} macro}}{\texttt{\#1}} +& the current labelling scheme for alternate roots. \\ +\optionLabel{make indefinite edge}{\typ{edge pair \(i\)-\(j\) or list of such}}{\{\}} +& edge pair or list of edge pairs to treat as having indefinitely many roots on them. \\ +\optionLabel{indefinite edge ratio}{\typ{float}}{1.6} +& ratio of indefinite edge lengths to other edge lengths. \\ +\optionLabel{indefinite edge/.style}{\typ{TikZ style data}}{solid,draw=black,fill=white,thin,densely dotted} +& style of the dotted or dashed middle third of each indefinite edge. \\ +\optionLabel{backwards}{\typ{true or false}}{false} +& whether to reverse right to left. \\ +\optionLabel{upside down}{\typ{true or false}}{false} +& whether to reverse up to down. \\ +\optionLabel{arrows}{\typ{true or false}}{true} +& whether to draw the arrows that arise along the edges. \\ +\optionLabel{reverse arrows}{\typ{true or false}}{true} +& whether to reverse the direction of the arrows that arise along the edges. \\ +\optionLabel{fold}{\typ{true or false}}{true} +& whether, when drawing Dynkin diagrams, to draw them 2-ply. \\ +\optionLabel{ply}{\typ{0,1,2,3,4}}{0} +& how many roots get folded together, at most. \\ +\optionLabel{fold left}{\typ{true or false}}{true} +& whether to fold the roots on the left side of a Dynkin diagram. \\ +\optionLabel{fold right}{\typ{true or false}}{true} +& whether to fold the roots on the right side of a Dynkin diagram. \\ +\optionLabel{fold radius}{\typ{length}}{.3cm} +& the radius of circular arcs used in curved edges of folded Dynkin diagrams. \\ +\optionLabel{fold style/.style}{\typ{TikZ style data}}{solid,draw=black!40,fill=none,line width=radius} +& 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} +& style for roots like \dynkin{A}{o} \\ +\optionLabel{O/.style}{\typ{TikZ style data}}{solid,draw=black,fill=black} +& 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} \\ +\optionLabel{x/.style}{\typ{TikZ style data}}{solid,draw=black,line cap=round} +& style for roots like \dynkin{A}{x} \\ +\optionLabel{X/.style}{\typ{TikZ style data}}{solid,draw=black,thick,line cap=round} +& style for roots like \dynkin{A}{X} \\ +\optionLabel{fold left style/.style}{\typ{TikZ style data}}{} +& style to override the \texttt{fold} style when folding roots together on the left half of a Dynkin diagram \\ +\optionLabel{fold right style/.style}{\typ{TikZ style data}}{} +& style to override the \texttt{fold} style when folding roots together on the right half of a Dynkin diagram \\ +\optionLabel{double edges}{\typ{}}{not set} +& set to override the \texttt{fold} style when folding roots together in a Dynkin diagram, so that the foldings +are indicated with double edges (like those of an \(F_4\) Dynkin diagram without arrows). \\ +\optionLabel{double fold}{\typ{}}{not set} +& set to override the \texttt{fold} style when folding roots together in a Dynkin diagram, so that the foldings +are indicated with double edges (like those of an \(F_4\) Dynkin diagram without arrows), but filled in solidly. \\ +\optionLabel{double left}{\typ{}}{not set} +& set to override the \texttt{fold} style when folding roots together at the left side of a Dynkin diagram, so that the foldings are indicated with double edges (like those of an \(F_4\) Dynkin diagram without arrows). \\ +\optionLabel{double fold left}{\typ{}}{not set} +& set to override the \texttt{fold} style when folding roots together at the left side of a Dynkin diagram, so that the foldings are indicated with double edges (like those of an \(F_4\) Dynkin diagram without arrows), but filled in solidly. \\ +\optionLabel{double right}{\typ{}}{not set} +& set to override the \texttt{fold} style when folding roots together at the right side of a Dynkin diagram, so that the foldings are indicated with double edges (like those of an \(F_4\) Dynkin diagram without arrows). \\ +\optionLabel{double fold right}{\typ{}}{not set} +& set to override the \texttt{fold} style when folding roots together at the right side of a Dynkin diagram, so that the foldings are indicated with double edges (like those of an \(F_4\) Dynkin diagram without arrows), but filled in solidly. +\\ +\optionLabel{arrow color}{\typ{}}{black} +& set to override the default color for the arrows in nonsimply laced Dynkin diagrams. \\ +\optionLabel{Coxeter}{\typ{true or false}}{false} +& whether to draw a Coxeter diagram, rather than a Dynkin diagram. \\ +\optionLabel{ordering}{\typ{Adams, Bourbaki, Carter, Dynkin, Kac}}{Bourbaki} +& which ordering of the roots to use in exceptional root systems as in section~\ref{section:order}. \\ +\end{longtable} +\par\noindent{}All other options are passed to TikZ. + +\nocite{*} +\bibliographystyle{amsplain} +\bibliography{dynkin-diagrams} +\end{document} |