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Diffstat (limited to 'Master')
| -rw-r--r-- | Master/texmf-dist/doc/latex/dynkin-diagrams/README | 8 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.bib | 171 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.pdf | bin | 233754 -> 516988 bytes | |||
| -rw-r--r-- | Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.tex | 1711 | ||||
| -rw-r--r-- | Master/texmf-dist/tex/latex/dynkin-diagrams/dynkin-diagrams.sty | 4315 |
5 files changed, 4585 insertions, 1620 deletions
diff --git a/Master/texmf-dist/doc/latex/dynkin-diagrams/README b/Master/texmf-dist/doc/latex/dynkin-diagrams/README index 26cba041b7c..e7c2e42116b 100644 --- a/Master/texmf-dist/doc/latex/dynkin-diagrams/README +++ b/Master/texmf-dist/doc/latex/dynkin-diagrams/README @@ -2,9 +2,9 @@ ___________________________________ Dynkin diagrams - v2.0 + v3.1 - 18 November 2017 + 11 February 2018 ___________________________________ Authors : Ben McKay @@ -15,5 +15,5 @@ Licence : Released under the LaTeX Project Public License v1.3c or ---------------------------------------------------------------------- -Provides Dynkin diagrams drawn in TikZ. - +Draws Dynkin diagrams in LaTeX documents, using the TikZ package. +Version 3.1 improves the documentation to give code for all examples.
\ No newline at end of file diff --git a/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.bib b/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.bib index a72cb1dade1..30fc8b08f5c 100644 --- a/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.bib +++ b/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.bib @@ -2,6 +2,25 @@ % Encoding: ISO8859_1 +@Article{Baba:2009, + Title = {Satake diagrams and restricted root systems of semisimple pseudo-{R}iemannian symmetric spaces}, + Author = {Baba, Kurando}, + Journal = {Tokyo J. Math.}, + Year = {2009}, + Number = {1}, + Pages = {127--158}, + Volume = {32}, + + Fjournal = {Tokyo Journal of Mathematics}, + ISSN = {0387-3870}, + Mrclass = {17B20 (17B22 53C35)}, + Mrnumber = {2541161}, + Mrreviewer = {Oksana S. Yakimova}, + Owner = {user}, + Timestamp = {2017.12.04}, + Url = {https://doi.org/10.3836/tjm/1249648414} +} + @Book{Bourbaki:2002, Title = {Lie groups and {L}ie algebras. {C}hapters 4--6}, Author = {Bourbaki, Nicolas}, @@ -37,20 +56,23 @@ Url = {https://doi.org/10.1017/CBO9780511614910} } -@Book{Dynkin:2000, - Title = {Selected papers of {E}. {B}. {D}ynkin with commentary}, - Author = {Dynkin, E. B.}, - Publisher = {American Mathematical Society, Providence, RI; International Press, Cambridge, MA}, - Year = {2000}, - Note = {Edited by A. A. Yushkevich, G. M. Seitz and A. L. Onishchik}, +@Article{Chuah:2013, + Title = {Cartan automorphisms and {V}ogan superdiagrams}, + Author = {Chuah, Meng-Kiat}, + Journal = {Math. Z.}, + Year = {2013}, + Number = {3-4}, + Pages = {793--800}, + Volume = {273}, - ISBN = {0-8218-1065-0}, - Mrclass = {01A75 (60Jxx)}, - Mrnumber = {1757976}, - Mrreviewer = {William M. McGovern}, + Fjournal = {Mathematische Zeitschrift}, + ISSN = {0025-5874}, + Mrclass = {17B20 (17B40)}, + Mrnumber = {3030677}, + Mrreviewer = {Zi-Xin Hou}, Owner = {user}, - Pages = {xxviii+796}, - Timestamp = {2017.11.15} + Timestamp = {2017.12.04}, + Url = {https://doi.org/10.1007/s00209-012-1030-z} } @Article{Dynkin:1952, @@ -69,6 +91,41 @@ Timestamp = {2017.11.15} } +@Book{Dynkin:2000, + Title = {Selected papers of {E}. {B}. {D}ynkin with commentary}, + Author = {Dynkin, E. B.}, + Publisher = {American Mathematical Society, Providence, RI; International Press, Cambridge, MA}, + Year = {2000}, + Note = {Edited by A. A. Yushkevich, G. M. Seitz and A. L. Onishchik}, + + ISBN = {0-8218-1065-0}, + Mrclass = {01A75 (60Jxx)}, + Mrnumber = {1757976}, + Mrreviewer = {William M. McGovern}, + Owner = {user}, + Pages = {xxviii+796}, + Timestamp = {2017.11.15} +} + +@Article{Frappat/Sciarrino/Sorba:1989, + Title = {Structure of basic {L}ie superalgebras and of their affine extensions}, + Author = {Frappat, L. and Sciarrino, A. and Sorba, P.}, + Journal = {Comm. Math. Phys.}, + Year = {1989}, + Number = {3}, + Pages = {457--500}, + Volume = {121}, + + Fjournal = {Communications in Mathematical Physics}, + ISSN = {0010-3616}, + Mrclass = {17B70 (17A70 17B40)}, + Mrnumber = {990776}, + Mrreviewer = {A. Pianzola}, + Owner = {user}, + Timestamp = {2017.12.18}, + Url = {http://0-projecteuclid.org.library.ucc.ie/euclid.cmp/1104178142} +} + @Book{Grove/Benson:1985, Title = {Finite reflection groups}, Author = {Grove, L. C. and Benson, C. T.}, @@ -139,6 +196,25 @@ Url = {https://doi.org/10.1017/CBO9780511626234} } +@Article{Khastgir/Sasaki:1996, + Title = {Non-canonical folding of {D}ynkin diagrams and reduction of affine {T}oda theories}, + Author = {Khastgir, S. Pratik and Sasaki, Ryu}, + Journal = {Progr. Theoret. Phys.}, + Year = {1996}, + Number = {3}, + Pages = {503--518}, + Volume = {95}, + + Fjournal = {Progress of Theoretical Physics}, + ISSN = {0033-068X}, + Mrclass = {81T10 (17B81 58F07 81R10)}, + Mrnumber = {1388245}, + Mrreviewer = {Mehmet Koca}, + Owner = {user}, + Timestamp = {2017.12.18}, + Url = {https://doi.org/10.1143/PTP.95.503} +} + @Book{OnishchikVinberg:1990, Title = {Lie groups and algebraic groups}, Author = {Onishchik, A. L. and Vinberg, {\`E}. B.}, @@ -176,6 +252,60 @@ Url = {https://doi.org/10.1007/978-3-642-74334-4} } +@Article{Ransingh:2013, + Title = {Vogan diagrams of untwisted affine {K}ac-{M}oody superalgebras}, + Author = {Ransingh, Biswajit}, + Journal = {Asian-Eur. J. Math.}, + Year = {2013}, + Number = {4}, + Pages = {1350062, 10}, + Volume = {6}, + + Fjournal = {Asian-European Journal of Mathematics}, + ISSN = {1793-5571}, + Mrclass = {17B67 (17B05 17B22 17B40)}, + Mrnumber = {3149279}, + Mrreviewer = {Xiangqian Guo}, + Owner = {user}, + Timestamp = {2018.01.11} +} + +@Article{Ransingh:unpub, + Title = {{Vogan diagrams of affine twisted Lie superalgebras}}, + Author = {Ransingh, B.}, + Journal = {ArXiv e-prints}, + Year = {2013}, + + Month = mar, + + Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, + Adsurl = {http://adsabs.harvard.edu/abs/2013arXiv1303.0092R}, + Archiveprefix = {arXiv}, + Eprint = {1303.0092}, + Keywords = {Mathematical Physics, Mathematics - Representation Theory}, + Owner = {user}, + Primaryclass = {math-ph}, + Timestamp = {2018.01.11} +} + +@Article{Regelskis/Vlaar:2016, + Title = {{Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type}}, + Author = {{Regelskis}, V. and {Vlaar}, B.}, + Journal = {ArXiv e-prints}, + Year = {2016}, + + Month = feb, + + Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, + Adsurl = {http://adsabs.harvard.edu/abs/2016arXiv160208471R}, + Archiveprefix = {arXiv}, + Eprint = {1602.08471}, + Keywords = {Mathematical Physics, Mathematics - Quantum Algebra, Mathematics - Representation Theory, Nonlinear Sciences - Exactly Solvable and Integrable Systems}, + Owner = {user}, + Primaryclass = {math-ph}, + Timestamp = {2017.12.04} +} + @Book{Satake:1980, Title = {Algebraic structures of symmetric domains}, Author = {Satake, Ichir\^o}, @@ -192,6 +322,23 @@ Timestamp = {2017.11.15} } +@InCollection{Zuber:1998, + Title = {Generalized {D}ynkin diagrams and root systems and their folding}, + Author = {Zuber, Jean-Bernard}, + Booktitle = {Topological field theory, primitive forms and related topics ({K}yoto, 1996)}, + Publisher = {Birkh\"auser Boston, Boston, MA}, + Year = {1998}, + Pages = {453--493}, + Series = {Progr. Math.}, + Volume = {160}, + + Mrclass = {17B20 (05C25 20F55)}, + Mrnumber = {1653035}, + Mrreviewer = {Saeid Azam}, + Owner = {user}, + Timestamp = {2017.12.18} +} + @Book{Vinberg:1994, Title = {Lie groups and {L}ie algebras, {III}}, Editor = {Vinberg, \`E. B.}, diff --git a/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.pdf b/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.pdf Binary files differindex 851c6ae2200..90af26049f5 100644 --- a/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.pdf +++ b/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.pdf diff --git a/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.tex b/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.tex index f6566c0be0e..afe3a99eea5 100644 --- a/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.tex +++ b/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.tex @@ -1,688 +1,1303 @@ \documentclass{amsart} -\title{The Dynkin diagrams package} +\title{The Dynkin diagrams package \\ Version 3.1} \author{Ben McKay} -\date{\today} +\date{11 February 2018} +\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{etoolbox} \usepackage{longtable} -\usepackage{showexpl} +\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{backgrounds} \usetikzlibrary{decorations.markings} +\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} -\renewcommand\ResultBox{\fcolorbox{gray!50}{gray!30}} +\NewDocumentCommand\wdtA{}{.7cm} +\NewDocumentCommand\wdtD{}{3cm} +\NewDocumentCommand\wdtL{}{3cm} +\newcolumntype{A}{@{}>{\columncolor[gray]{.9}$}m{\wdtA}<{$}} +\newcolumntype{D}{>{\columncolor[gray]{.9}}m{\wdtD}} +\newcolumntype{L}{>{\columncolor[gray]{.9}}p{\wdtL}} +\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}{}} & \dynk[#1]{#2}[#3]{#4}% +}% + +\NewDocumentEnvironment{dynkinTable}{mmm}% +{% +\RenewDocumentCommand\wdtD{}{#2} +\RenewDocumentCommand\wdtL{}{#3} +\begin{longtable}{ADL} +\caption{#1}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{2}{c}{continued \dots}\\ +\endfoot +\endlastfoot +}% +{% +\end{longtable} +}% + + +\definecolor{example-color}{gray}{1} +\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 \begin{document} \maketitle +\begin{center} +\begin{varwidth}{\textwidth} \tableofcontents +\end{varwidth} +\end{center} + + +\setlength{\arrayrulewidth}{1.5pt} + \section{Quick introduction} -This is a test of the Dynkin diagram package. -Load the package via + + +\begin{tcolorbox}[title={Load the Dynkin diagram package (see options below)}] \begin{verbatim} \usepackage{dynkin-diagrams} \end{verbatim} -(see below for options) and invoke it directly: +\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}} +\tikz \dynkin{B}{3}; +\end{tcblisting} +\begin{tcblisting}{title={Inside a \TikZ environment}} +\begin{tikzpicture} + \dynkin{B}{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,edgeLength=.5cm,foldradius=.5cm} +\end{verbatim} +\end{tcolorbox} +\begin{tcolorbox}[title={\dots or pass to the package}] +\begin{verbatim} +\usepackage[ + ordering=Kac, + edge/.style=blue, + mark=o, + radius=.06cm] + {dynkin-diagrams} +\end{verbatim} +\end{tcolorbox} -\begin{LTXexample} -The flag variety of pointed lines in -projective 3-space is associated to -the Dynkin diagram \dynkin[parabolic=3]{A}{3}. -\end{LTXexample} -or use the long form inside a \verb!\tikz! statement: -\begin{LTXexample} -\tikz \dynkin[parabolic=3]{A}{3}; -\end{LTXexample} +\section{Coxeter diagrams} -or a TikZ environment: -\begin{LTXexample} -\begin{tikzpicture} -\dynkin[parabolic=3,label]{A}{3} -\end{tikzpicture} -\end{LTXexample} -With labels for the roots: -\begin{LTXexample} -\dynkin[parabolic=3,label]{A}{3} -\end{LTXexample} -\newpage\noindent% -Make up your own labels for the roots: -\begin{LTXexample} -\begin{tikzpicture} -\dynkin[parabolic=3]{A}{3} -\rootlabel{2}{\alpha_2} -\end{tikzpicture} -\end{LTXexample} -Use any text scale you like: -\begin{LTXexample} +\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{Labels for the roots} + +\begin{tcblisting}{title={Label the roots by root number}} +\dynkin[label]{B}{3} +\end{tcblisting} +\begin{tcblisting}{title={Make a macro to assign labels to roots}} +\dynkin[label,labelMacro/.code={\alpha_{#1}}]{D}{5} +\end{tcblisting} +\begin{tcblisting}{title={Label a single root}} \begin{tikzpicture} -\dynkin[parabolic=3,textscale=1.2]{A}{3}; -\rootlabel{2}{\alpha_2} + \dynkin{B}{3} + \dynkinLabelRoot{2}{\alpha_2} \end{tikzpicture} -\end{LTXexample} -and access root labels via TikZ: -\begin{LTXexample} +\end{tcblisting} +\begin{tcblisting}{title={Use a text style}} \begin{tikzpicture} -\dynkin[parabolic=3]{A}{3}; -\node at (root label 2) {\(\alpha_2\)}; + \dynkin[text/.style={scale=1.2}]{B}{3}; + \dynkinLabelRoot{2}{\alpha_2} \end{tikzpicture} -\end{LTXexample} -The labels have default locations: -\begin{LTXexample} +\end{tcblisting} +\begin{tcblisting}{title={Access root labels via TikZ}} \begin{tikzpicture} -\dynkin{E}{8}; -\rootlabel{1}{\alpha_1} -\rootlabel{2}{\alpha_2} -\rootlabel{3}{\alpha_3} + \dynkin{B}{3}; + \node[below] at (root 2) {\(\alpha_2\)}; \end{tikzpicture} -\end{LTXexample} -You can use a starred form to flip labels to alternate locations: -\begin{LTXexample} +\end{tcblisting} +\begin{tcblisting}{title={The labels have default locations}} \begin{tikzpicture} -\dynkin{E}{8}; -\rootlabel*{1}{\alpha_1} -\rootlabel*{2}{\alpha_2} -\rootlabel*{3}{\alpha_3} + \dynkin{E}{8}; + \dynkinLabelRoot{1}{\alpha_1} + \dynkinLabelRoot{2}{\alpha_2} + \dynkinLabelRoot{3}{\alpha_3} \end{tikzpicture} -\end{LTXexample} -TikZ can access the roots themselves: -\typeout{AAAAAAA} -\begin{LTXexample} +\end{tcblisting} +\begin{tcblisting}{title={The starred form flips labels to alternate locations}} \begin{tikzpicture} -\dynkin{A}{4}; -\fill[white,draw=black] (root 2) circle (.1cm); -\draw[black] (root 2) circle (.05cm); + \dynkin{E}{8}; + \dynkinLabelRoot*{1}{\alpha_1} + \dynkinLabelRoot*{2}{\alpha_2} + \dynkinLabelRoot*{3}{\alpha_3} \end{tikzpicture} -\end{LTXexample} -Some diagrams will have double edges: -\begin{LTXexample} +\end{tcblisting} + +\section{Style} + +\begin{tcblisting}{title={Colours}} +\dynkin[edge/.style={blue!50,thick},*/.style=blue!50!red]{F}{4} +\end{tcblisting} +\begin{tcblisting}{title={Edge lengths}} +\dynkin[edgeLength=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}} +\dynkin[parabolic=124,x/.style={brown,very thick}]{E}{8} +\end{tcblisting} +\begin{tcblisting}{title={Sizes of root marks}} +\dynkin[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} -\end{LTXexample} -or triple edges: -\begin{LTXexample} \dynkin{G}{2} -\end{LTXexample} -\newpage\noindent% -Draw curves between the roots: -\begin{LTXexample} +\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[reverseArrows]{F}{4} +\dynkin[reverseArrows]{G}{2} +\end{tcblisting} + + +\section{Drawing on top of a Dynkin diagram} + +\begin{tcblisting}{title={TikZ can access the roots themselves}} \begin{tikzpicture} -\dynkin[parabolic=429]{E}{8} -\draw[very thick, black!50,-latex] (root 3.south) to [out=-45, in=-135] (root 6.south); + \dynkin{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{tikzpicture} -\end{LTXexample} -Draw dots on the roots: -\begin{LTXexample} +\end{tcblisting} +\begin{tcblisting}{title={Draw curves between the roots}} \begin{tikzpicture} -\dynkin[label]{C}{8} -\dynkinopendot{3} -\dynkinopendot{7} + \dynkin[label]{E}{8} + \draw[very thick, black!50,-latex] + (root 3.south) to [out=-45, in=-135] (root 6.south); \end{tikzpicture} -\end{LTXexample} -Colours: -\begin{LTXexample} -\dynkin[color=blue!50,backgroundcolor=red!20]{G}{2} -\end{LTXexample} -Edge lengths: -\begin{LTXexample} -\dynkin[edgelength=1.2,parabolic=3]{A}{3} -\end{LTXexample} -Sizes of dots and crosses: -\begin{LTXexample} -\dynkin[dotradius=.08cm,parabolic=3]{A}{3} -\end{LTXexample} -Edge styles: -\begin{LTXexample} -\dynkin[edge=very thick,parabolic=3]{A}{3} -\end{LTXexample} -Open circles instead of closed dots: -\begin{LTXexample} -\dynkin[open]{E}{8} -\end{LTXexample} -Add closed dots to the open circles, at roots in the current ordering: -\begin{LTXexample} +\end{tcblisting} +\begin{tcblisting}{title={Change marks}} \begin{tikzpicture} -\dynkin[open]{E}{8}; -\dynkincloseddot{5} -\dynkincloseddot{8} -\end{tikzpicture} -\end{LTXexample} -More colouring: -\begin{LTXexample} -\begin{tikzpicture}[show background rectangle, - background rectangle/.style={fill=red!10}] -\dynkin[parabolic=1,backgroundcolor=blue!20]{G}{2} + \dynkin[mark=o,label]{E}{8}; + \dynkinRootMark{*}{5} + \dynkinRootMark{*}{8} \end{tikzpicture} -\end{LTXexample} -Cross styles: -\begin{LTXexample} -\dynkin[parabolic=124,cross=thin]{E}{8} -\end{LTXexample} -\newpage\noindent{} -Suppress arrows: -\begin{LTXexample} -\dynkin[arrows=false]{F}{4} -\end{LTXexample} -\begin{LTXexample} -\dynkin[arrows=false]{G}{2} -\end{LTXexample} - -\section{Syntax} - -The syntax is \verb!\dynkin[<options>]{<letter>}{<rank>}! where \verb!<letter>! is \(A,B,C,D,E,F\) or \(G\), the family of root system for the Dynkin diagram, and \verb!<rank>! is an integer representing the rank, or is the symbol \verb!*! to represent an indefinite rank: -\begin{LTXexample} -\dynkin[edge=thick,edgelength=.5cm]{A}{*} -\end{LTXexample} -\begin{LTXexample} -\dynkin[edge=thick,edgelength=.5cm]{B}{*} -\end{LTXexample} -\begin{LTXexample} -\dynkin[edge=thick,edgelength=.5cm]{C}{*} -\end{LTXexample} -\begin{LTXexample} -\dynkin[edge=thick,edgelength=.5cm]{D}{*} -\end{LTXexample} -Outside a TikZ environment, the command builds its own TikZ environment. - +\end{tcblisting} -\newcommand*{\typ}[1]{\(\left<\texttt{#1}\right>\)} -\newcommand*{\optionLabel}[3]{%% -\multicolumn{2}{l}{\(\texttt{#1}=\texttt{#2}, \texttt{default}=\texttt{#3}\)} \\ -}%% -\section{Options} -\par\noindent{}All \verb!\dynkin! options (except \texttt{affine}, \texttt{folded}, \texttt{label} and \texttt{parabolic} ) can also be passed to the package to force a global default option: -\par\noindent% -\begin{verbatim} -\usepackage[ - ordering=Kac, - color=blue, - open, - dotradius=.06cm, - backgroundcolor=red] - {dynkin-diagrams} -\end{verbatim} -\par\noindent% -\begin{tabular}{p{1cm}p{10cm}} -\optionLabel{parabolic}{\typ{integer}}{0} -& A parabolic subgroup with specified integer, where the integer -is computed as \(n=\sum 2^i a_i\), \(a_i=0\) or \(1\), to say that root \(i\) is crossed, i.e. a noncompact root. \\ -\optionLabel{color}{\typ{color name}}{black} \\ -\optionLabel{backgroundcolor}{\typ{color name}}{white} -& This only says what color you have already set for the background rectangle. It is needed precisely for the \(G_2\) root system, to draw the triple line correctly, and only when your background color is not white. \\ -\optionLabel{dotradius}{\typ{number}cm}{.05cm} -& size of the dots and of the crosses in the Dynkin diagram \\ -\optionLabel{edgelength}{\typ{number}cm}{.35cm} -& distance between nodes in the Dynkin diagram \\ -\optionLabel{edge}{\typ{TikZ style data}}{thin} -& style of edges in the Dynkin diagram \\ -\optionLabel{open}{\typ{true or false}}{false} -& use open circles rather than solid dots as default \\ -\optionLabel{label}{true or false}{false} -& whether to label the roots by their root numbers. \\ -\optionLabel{arrows}{\typ{true or false}}{true} -& whether to draw the arrows that arise along the edges. \\ -\optionLabel{folded}{\typ{true or false}}{true} -& whether, when drawing \(A\), \(D\) or \(E_6\) diagrams, to draw them folded. \\ -\optionLabel{foldarrowstyle}{\typ{TikZ style}}{stealth-stealth} -& when drawing folded diagrams, style for the fold arrows. \\ -\optionLabel{foldarrowcolor}{\typ{colour}}{black!50} -& when drawing folded diagrams, colour for the fold arrows. \\ -\optionLabel{Coxeter}{\typ{true or false}}{false} -& whether to draw a Coxeter diagram, rather than a Dynkin diagram. \\ +\section{Mark lists} -\optionLabel{ordering}{\typ{Adams, Bourbaki, Carter, Dynkin, Kac}}{Bourbaki} -& which ordering of the roots to use in exceptional root systems as follows: -\end{tabular} +The package allows a list of root marks instead of a rank: -\newpage +\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.) -\NewDocumentCommand\tablerow{mm}% +\NewDocumentCommand\ClassicalLieSuperalgebras{m}% {% -\(#1_{#2}\) -& -\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{dynkinTable}{Classical Lie superalgebras \cite{Frappat/Sciarrino/Sorba:1989}. #1}{3.5cm}{6.5cm} +A_{mn} & \dynk{A}{ooo.oto.oo} +B_{mn} & \dynk{B}{ooo.oto.oo} +B_{0n} & \dynk{B}{ooo.ooo.o*} +C_{n} & \dynk{C}{too.oto.oo} +D_{mn} & \dynk{D}{ooo.oto.oooo} +D_{21\alpha} & \dynk{A}{oto} +F_4 & \dynk{F}{ooot} +G_3 & \dynk[extended,affineMark=t]{G}{2} +\end{dynkinTable} }% -\begin{center} -\begin{longtable}{@{}llllll@{}} -\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} +\begingroup +\tikzset{/Dynkin diagram,radius=.07cm} +\ClassicalLieSuperalgebras{We need a slightly larger radius parameter to distinguish the tensor product symbols from the solid dots.} +\endgroup +\ClassicalLieSuperalgebras{Here we see the problem with using the default radius parameter, which is too small for tensor product symbols.} -\par\noindent{}All other options are passed to TikZ. -\section{Finding the roots} -The roots are labelled from \(1\) to \(r\), where \(r\) is the rank. -The command sets up TikZ nodes \texttt{(root 1)}, \texttt{(root 2)}, and so on. -Affine extended Dynkin diagrams have affine root are at \texttt{(root 0)}. -Use these tikz nodes to draw on the Dynkin diagram, as above. -It also sets up TikZ nodes \texttt{(root label 0)}, \texttt{(root label 1)}, and so on for the labels, and TikZ nodes \texttt{(root label swap 0)}, \texttt{(root label swap 1)}, and so on as alternative label locations, in case you want two labels on the same root, or the default choice doesn't look the way you like. -\begin{LTXexample} -\begin{tikzpicture} -\dynkin{E}{6}; -\rootlabel{2}{\alpha_2} -\rootlabel{5}{\alpha_5} -\end{tikzpicture} -\end{LTXexample} +\section{Indefinite edges} -\section{Example: some parabolic subgroups} +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} -\newcommand{\drawparabolic}[3]{#1_{#2,#3} & \tikz \dynkin[parabolic=#3]{#1}{#2}; \\} +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[makeIndefiniteEdge={3-5},label]{D}{5} +\end{tcblisting} +\begin{tcblisting}{title={Give a list of edges to become indefinite}} +\dynkin[makeIndefiniteEdge/.list={1-2,3-5},label]{D}{5} +\end{tcblisting} -\begin{center} -\begin{longtable}{@{}>{$}r<{$}m{2cm}m{2cm}@{}} -\endfirsthead -\endhead -\endfoot -\endlastfoot -\drawparabolic{A}{1}{0} -\drawparabolic{A}{1}{2} -\drawparabolic{A}{2}{0} -\drawparabolic{A}{2}{2} -\drawparabolic{A}{2}{4} -\drawparabolic{A}{2}{6} -\drawparabolic{B}{2}{6} -\drawparabolic{C}{3}{10} -\drawparabolic{D}{5}{8} -\drawparabolic{E}{6}{10} -\drawparabolic{E}{7}{202} -\drawparabolic{E}{8}{246} -\drawparabolic{F}{4}{26} -\drawparabolic{G}{2}{0} -\drawparabolic{G}{2}{2} -\drawparabolic{G}{2}{4} -\drawparabolic{G}{2}{6} -\end{longtable} -\end{center} +\begin{tcblisting}{title={Indefinite edge style}} +\dynkin[indefiniteEdge/.style={draw=black,fill=white,thin,densely dashed},% + edgeLength=1cm,% + makeIndefiniteEdge={3-5}] + {D}{5} +\end{tcblisting} +\begin{tcblisting}{title={The ratio of the lengths of indefinite edges to those of other edges}} +\dynkin[edgeLength = .5cm,% + indefiniteEdgeRatio=3,% + makeIndefiniteEdge={3-5}] + {D}{5} +\end{tcblisting} -\section{Example: the Hermitian symmetric spaces} +\section{Parabolic subgroups} - \renewcommand*{\arraystretch}{1.5} -\begin{center} -\begin{longtable}{@{}>{$}r<{$}m{2.2cm}m{5cm}@{}} +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} + +\NewDocumentCommand\HSS{mommm}% +{% + \begingroup + \renewcommand*{\arraystretch}{1.2} + \begin{tabular}{@{}>{$}r<{$}@{}m{6cm}@{}} + \\ + \IfNoValueTF{#2}% + {% + #1 & \dynkin{#3}{#4} \\ + & \csDynkin{#3}{#4} \\ + }% + {% + #1 & \dynkin[#2]{#3}{#4} \\ + & \csDynkin[#2]{#3}{#4} \\ + }% + & #5% + \\[.75em] + \end{tabular} + \endgroup + \\ +}% + +\renewcommand*{\arraystretch}{1} +\begin{longtable}{>{\columncolor[gray]{.9}}p{7cm}} +\caption{The Hermitian symmetric spaces} \endfirsthead +\caption{\dots continued}\\ \endhead +\caption{continued \dots}\\ \endfoot \endlastfoot - A_n & - \dynkin[parabolic=16]{A}{*} & - Grassmannian of $k$-planes in $\C{n+1}$ - \\ - B_n & - \dynkin[parabolic=2]{B}{*} & - $(2n-1)$-dimensional hyperquadric, i.e. the variety of null lines in $\C{2n+1}$ - \\ - C_n & - \dynkin[parabolic=32]{C}{*} & - space of Lagrangian $n$-planes in $\C{2n}$ - \\ - D_n & - \dynkin[parabolic=2]{D}{*} & - $(2n-2)$-dimensional hyperquadric, i.e. the variety of null lines in $\C{2n}$ - \\ - D_n & - \dynkin[parabolic=64]{D}{*} & - one component of the variety of maximal dimension null subspaces of $\C{2n}$ \\ - D_n & - \dynkin[parabolic=32]{D}{*} & - the other component\\ - E_6 & - \dynkin[parabolic=2]{E}{6} & - complexified octave projective plane\\ - E_6 & - \dynkin[parabolic=64]{E}{6}&its dual plane\\ - E_7 & - \dynkin[parabolic=128]{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}[parabolic=1]{B}{}{$(2n-1)$-dimensional hyperquadric, i.e. the variety of null lines in $\C{2n+1}$} +\HSS{C_n}[parabolic=16]{C}{}{space of Lagrangian $n$-planes in $\C{2n}$} +\HSS{D_n}[parabolic=1]{D}{}{$(2n-2)$-dimensional hyperquadric, i.e. the variety of null lines in $\C{2n}$} +\HSS{D_n}[parabolic=32]{D}{}{one component of the variety of maximal dimension null subspaces of $\C{2n}$} +\HSS{D_n}[parabolic=16]{D}{}{the other component} +\HSS{E_6}[parabolic=1]{E}{6}{complexified octave projective plane} +\HSS{E_6}[parabolic=32]{E}{6}{its dual plane} +\HSS{E_7}[parabolic=64]{E}{7}{the space of null octave 3-planes in octave 6-space} \end{longtable} -\end{center} - - -\section{Affine extended Dynkin diagrams} - -\begin{LTXexample} -\dynkin[affine,edge=thick]{A}{*} -\end{LTXexample} - -\begin{LTXexample} -\dynkin[edgelength=1cm,edge=thick,affine]{A}{*} -\end{LTXexample} - -\begin{LTXexample} -\dynkin[scale=1.5,edge=thick,affine]{A}{*} -\end{LTXexample} - - -\begin{LTXexample} -\begin{tikzpicture} -\dynkin[affine,label]{A}{8}; -\end{tikzpicture} -\end{LTXexample} - - -\begin{LTXexample} -\begin{tikzpicture} -\dynkin[affine]{A}{*}; -\node at (root label 0) {\(\alpha_0\)}; -\end{tikzpicture} -\end{LTXexample} -\begin{LTXexample} -\begin{tikzpicture} -\dynkin[affine]{A}{9} -\node at (root label 0) {\(\alpha_0\)}; -\end{tikzpicture} -\end{LTXexample} -You can use TikZ to put in labels: -\begin{LTXexample} -\begin{tikzpicture} -\dynkin[affine]{A}{9}; -\node at (root label 0) {\(\alpha_0\)}; -\node at (root label 1) {\(\alpha_1\)}; -\node at (root label 2) {\(\alpha_2\)}; -\node at (root label 3) {\(\alpha_3\)}; -\end{tikzpicture} -\end{LTXexample} -\begin{LTXexample} -\dynkin[affine,label]{A}{1} -\end{LTXexample} +\section{Extended Dynkin diagrams} -\begin{LTXexample} -\dynkin[affine,label]{B}{8} -\end{LTXexample} +\begin{tcblisting}{title={Extended Dynkin diagrams}} +\dynkin[extended]{A}{7} +\end{tcblisting} -\begin{LTXexample} -\dynkin[affine,label]{B}{*} -\end{LTXexample} -\begin{LTXexample} -\dynkin[affine,label]{C}{8} -\end{LTXexample} +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} -\begin{LTXexample} -\dynkin[affine,label]{C}{*} -\end{LTXexample} -\begin{LTXexample} -\dynkin[affine,label]{D}{8} -\end{LTXexample} +\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} +\newcolumntype{D}{>{\columncolor[gray]{1}}m{\wdtD}} +\begin{dynkinTable}{The Dynkin diagrams of the extended simple root systems in Kac style. At the moment, it only works on a white background.}{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} +\endgroup -\begin{LTXexample} -\dynkin[affine,label]{D}{*} -\end{LTXexample} -\begin{LTXexample} -\dynkin[affine,label]{E}{6} -\end{LTXexample} -\begin{LTXexample} -\dynkin[affine,label]{E}{7} -\end{LTXexample} +\section{Folded Dynkin diagrams} -\begin{LTXexample} -\dynkin[affine,label]{E}{8} -\end{LTXexample} +The Dynkin diagrams package has limited support for folding Dynkin diagrams. -Open circles instead of closed dots: -\begin{LTXexample} -\dynkin[affine,open,label]{E}{8} -\end{LTXexample} +\begin{tcblisting}{title={Folding}} +\dynkin[fold]{A}{13} +\end{tcblisting} -\begin{LTXexample} -\dynkin[affine,label]{F}{4} -\end{LTXexample} +\begin{tcblisting}{title={Big fold radius}} +\dynkin[fold,foldradius=1cm]{A}{13} +\end{tcblisting} -\begin{LTXexample} -\dynkin[affine,label]{G}{2} -\end{LTXexample} +\begin{tcblisting}{title={Small fold radius}} +\dynkin[fold,foldradius=.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 form \verb!ply=2!. -\section{Coxeter diagrams} +\begin{tcblisting}{title={3-ply}} +\dynkin[ply=3]{D}{4} +\dynkin[ply=3]{D}[1]{4} +\end{tcblisting} -\begin{LTXexample} -\dynkin[Coxeter]{B}{7} -\end{LTXexample} +\begin{tcblisting}{title={4-ply}} +\dynkin[ply=4]{D}[1]{4} +\end{tcblisting} -\begin{LTXexample} -\dynkin[Coxeter]{F}{4} -\end{LTXexample} +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[foldleft]{D}[1]{} \ +\dynkin[foldright]{D}[1]{} \ +\dynkin[fold]{D}[1]{} +\end{tcblisting} -\begin{LTXexample} -\dynkin[Coxeter]{G}{2} -\end{LTXexample} +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}} +\begin{tikzpicture} + \dynkin[ply=4]{D}[1]{****.*****.*****}% +\end{tikzpicture} \ +\begin{tikzpicture} + \dynkin[ply=4]{D}[1]{****.*****.*****}% + \dynkinFold[bend right=65]{1}{13}% + \dynkinFold[bend right=65]{0}{14}% +\end{tikzpicture} \ +\begin{tikzpicture} + \dynkin[ply=4]{D}[1]{****.*****.*****}% + \dynkinFold{0}{1}% + \dynkinFold{1}{13}% + \dynkinFold{13}{14}% +\end{tikzpicture} +\end{tcblisting} -\begin{LTXexample} -\dynkin[Coxeter]{H}{7} -\end{LTXexample} -\begin{LTXexample} -\dynkin[Coxeter]{I}{7} -\end{LTXexample} +\begingroup +\RenewDocumentCommand\wdtD{}{3.5cm} +\RenewDocumentCommand\wdtL{}{7cm} +\NewDocumentCommand\seriesName{mmm}% +{% + \IfStrEq{#2}{0}{#1_{#3}}{#1^{#2}_{#3}}% +}% -\section{Folded Dynkin diagrams} +\NewDocumentCommand\foldingTable{smmmmmmmm}% +{% +\begin{tabular}{ADL}% +\seriesName{#2}{#3}{#4} +\seriesName{#6}{#7}{#8}&\IfBooleanTF{#1}{\reflectbox{#9}}{#9}% +\end{tabular}% +\\ \hline +}% -\begin{LTXexample} -\dynkin[folded]{E}{6} -\end{LTXexample} -\begin{LTXexample} -\dynkin[folded,label]{E}{6} -\end{LTXexample} +\NewDocumentCommand\fold{smmmmmm}% +{% + \IfBooleanTF{#1}% + {% + \foldingTable% + {#2}{#3}{#4}{\dynk[fold]{#2}[#3]{#4}}% + {#5}{#6}{#7}{\dynk[reverseArrows]{#5}[#6]{#7}}% + }% + {% + \foldingTable% + {#2}{#3}{#4}{\dynk[fold]{#2}[#3]{#4}}% + {#5}{#6}{#7}{\dynk{#5}[#6]{#7}}% + }% +}% -\begin{LTXexample} -\dynkin[folded]{A}{*} -\end{LTXexample} +\begin{filecontents*}{DoneTwoElBendy.tex} +\begin{tikzpicture} + \dynkin[ply=4]{D}[1]{****.*****.*****} + \dynkinFold[bend right=65]{1}{13} + \dynkinFold[bend right=65]{0}{14} +\end{tikzpicture} +\end{filecontents*} -\begin{LTXexample} -\dynkin[folded,label]{A}{1} -\end{LTXexample} -\begin{LTXexample} -\dynkin[folded,label]{A}{2} -\end{LTXexample} +\begin{filecontents*}{DoneTwoElStraight.tex} +\begin{tikzpicture} + \dynkin[ply=4]{D}[1]{****.*****.*****} + \dynkinFold{0}{1} + \dynkinFold{1}{13} + \dynkinFold{13}{14} +\end{tikzpicture} +\end{filecontents*} -\begin{LTXexample} -\dynkin[folded,label]{A}{3} -\end{LTXexample} +\pgfkeys{/Dynkin diagram,foldradius=.35cm} +\begin{longtable}{@{}p{15cm}@{}} +\caption{Some foldings of Dynkin diagrams}\\ +\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]{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[foldright]{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 -\begin{LTXexample} -\dynkin[folded,label]{A}{4} -\end{LTXexample} -\begin{LTXexample} -\dynkin[folded,label]{A}{10} -\end{LTXexample} -\begin{LTXexample} -\dynkin[folded,label]{A}{11} -\end{LTXexample} +\section{Root ordering}\label{section:order} -\begin{LTXexample} -\dynkin[folded,label,arrows=false]{A}{11} -\end{LTXexample} +\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. -\begin{LTXexample} -\dynkin[folded]{D}{*} -\end{LTXexample} +\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{LTXexample} -\dynkin[folded,label]{D}{1} -\end{LTXexample} +\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} -\begin{LTXexample} -\dynkin[folded,label]{D}{2} -\end{LTXexample} -\begin{LTXexample} -\dynkin[folded,label]{D}{3} -\end{LTXexample} +\section{Connecting Dynkin diagrams}\label{section:name} -\begin{LTXexample} -\dynkin[folded,label]{D}{4} -\end{LTXexample} +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{tikzpicture} + \dynkin[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/foldStyle] + ($(upper root \i)$) -- ($(lower root \i)$);% + }% + \end{scope} +\end{tikzpicture} +\end{tcblisting} -\begin{LTXexample} -\dynkin[folded,label]{D}{10} -\end{LTXexample} +The following diagrams arise in the Satake diagrams of the pseudo-Riemannian symmetric spaces \cite{Baba:2009}. -\begin{LTXexample} -\dynkin[folded,label]{D}{11} -\end{LTXexample} +\begin{tcblisting}{} +\pgfkeys{/Dynkin diagram,edgeLength=.5cm,foldradius=.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/foldStyle] + ($(1 root \i)$) + -- + ($(2 root \i)$);% + }% + \end{scope} +\end{tikzpicture} +\end{tcblisting} +\begin{tcblisting}{} +\pgfkeys{/Dynkin diagram/edgeLength=.75cm,/Dynkin diagram/edge/.style={draw=white,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/foldStyle] ($(1 root \i)$) -- ($(2 root \i)$);% + \draw[/Dynkin diagram/foldStyle] ($(2 root \i)$) -- ($(3 root \i)$);% + \draw[/Dynkin diagram/foldStyle] ($(3 root \i)$) -- ($(4 root \i)$);% + }% + \end{scope} +\end{tikzpicture} +\end{tcblisting} -\section{Satake diagrams} +\section{Other examples} -We have incomplete support for Satake diagrams as yet, following the conventions of \cite{Helgason:2001}. +Below we draw the Vogan diagrams of some affine Lie superalgebras \cite{Ransingh:2013,Ransingh:unpub}. -\begin{LTXexample} -\dynkin{A}{I} -\end{LTXexample} +\begingroup -\begin{LTXexample} -\dynkin{A}{II} -\end{LTXexample} +\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% +}% -\begin{LTXexample} -\dynkin{E}{I} -\end{LTXexample} +\tikzset{/Dynkin diagram,labelMacro/.code=\labls{#1},label,radius=.06cm} -\begin{LTXexample} -\dynkin{E}{II} -\end{LTXexample} -\begin{LTXexample} -\dynkin{E}{III} -\end{LTXexample} +\tcbset{text width=10cm} +\RenewDocumentCommand\wdtA{}{2cm} -\begin{LTXexample} -\dynkin{E}{IV} -\end{LTXexample} +\NewDocumentEnvironment{Category}{m}% +{% +\begin{tcolorbox}[title={\(#1\)},breakable]{} +}% +{% +\end{tcolorbox} +}% -\begin{LTXexample} -\dynkin{E}{V} -\end{LTXexample} +\begin{Category}{\mathfrak{sl}\left(2m|2n\right)^{(2)}} +\begin{tcblisting}{} +\begin{tikzpicture} + \dynkin[ply=2,label]{B}[1]{oo.oto.oo} + \dynkinLabelRoot*{7}{1} +\end{tikzpicture} +\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,doubleEdges]{B}[1]{oo.Oto.Oo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[ply=2,label,doubleFold]{B}[1]{oo.Oto.Oo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[ply=2,label,doubleEdges]{B}[1]{oo.OtO.oo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[ply=2,label,doubleFold]{B}[1]{oo.OtO.oo} +\end{tcblisting} +\end{Category} + +\begin{Category}{\mathfrak{sl}\left(2|2n\right)^{(2)}} +\begin{tcblisting}{} +\dynkin[ply=2,label,doubleEdges]{D}[1]{oo.oto.ooo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[ply=2,label,doubleFoldLeft]{D}[1]{oo.oto.ooo} +\end{tcblisting} +\end{Category} + +\begin{Category}{\mathfrak{osp}\left(2m|2n\right)^{(2)}} +\begin{tcblisting}{} +\dynkin[label,labelMacro/.code={1}]{D}[2]{o.oto.oo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[label,labelMacro/.code={1}]{D}[2]{o.Oto.Oo} +\end{tcblisting} +\end{Category} + +\begin{Category}{\mathfrak{osp}\left(2|2n\right)^{(2)}} +\begin{tcblisting}{} +\dynkin[label,labelMacro/.code=\lablIt{#1}, + affineMark=*] + {D}[2]{o.o.o.o*} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[label,labelMacro/.code=\lablIt{#1}, + affineMark=*] + {D}[2]{o.O.o.o*} +\end{tcblisting} +\end{Category} + +\begin{Category}{\mathfrak{sl}\left(1|2n+1\right)^{4}} +\begin{tcblisting}{} +\dynkin[label,labelMacro/.code={1}]{D}[2]{o.o.o.o*} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[label,labelMacro/.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/foldStyle] + ($(upper root \i)$) -- ($(lower root \i)$); + } + \end{scope} +\end{tikzpicture} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[fold]{A}[1]{oo.t.ooooo.t.oo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[fold,affineMark=t]{A}[1]{oo.o.ootoo.o.oo} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[affineMark=t]{A}[1]{o*.t.*o} +\end{tcblisting} +\end{Category} + +\begin{Category}{B^1} +\begin{tcblisting}{} +\dynkin[affineMark=*]{A}[2]{o.oto.o*} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[affineMark=*]{A}[2]{o.oto.o*} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[affineMark=*]{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[doubleEdges,fold,affineMark=t,odd]{A}[2]{to.o*} +\end{tcblisting} +\begin{tcblisting}{} +\dynkin[doubleEdges,fold,affineMark=t,odd]{A}[2]{t*.oo} +\end{tcblisting} +\end{Category} + +\begin{Category}{F^1} +\begin{tcblisting}{} +\begin{tikzpicture}% + \dynkin{A}{oto*}% + \dynkinQuadrupleEdge{1}{2}% + \dynkinTripleEdge{4}{3}% +\end{tikzpicture}% +\end{tcblisting} +\begin{tcblisting}{} +\begin{tikzpicture}% + \dynkin{A}{*too}% + \dynkinQuadrupleEdge{1}{2}% + \dynkinTripleEdge{4}{3}% +\end{tikzpicture}% +\end{tcblisting} +\end{Category} + +\begin{Category}{G^1} +\begin{tcblisting}{} +\begin{tikzpicture}% + \dynkin{A}{ot*oo}% + \dynkinQuadrupleEdge{1}{2}% + \dynkinDefiniteDoubleEdge{4}{3}% +\end{tikzpicture}% +\end{tcblisting} +\begin{tcblisting}{} +\begin{tikzpicture}% + \dynkin{A}{oto*o}% + \dynkinQuadrupleEdge{1}{2}% + \dynkinDefiniteDoubleEdge{4}{3}% +\end{tikzpicture}% +\end{tcblisting} +\begin{tcblisting}{} +\begin{tikzpicture}% + \dynkin{A}{*too*}% + \dynkinQuadrupleEdge{1}{2}% + \dynkinDefiniteDoubleEdge{4}{3}% +\end{tikzpicture}% +\end{tcblisting} +\begin{tcblisting}{} +\begin{tikzpicture}% + \dynkin{A}{*tooo}% + \dynkinQuadrupleEdge{1}{2}% + \dynkinDefiniteDoubleEdge{4}{3}% +\end{tikzpicture}% +\end{tcblisting} +\end{Category} -\begin{LTXexample} -\dynkin{E}{VI} -\end{LTXexample} -\begin{LTXexample} -\dynkin{E}{VII} -\end{LTXexample} -\begin{LTXexample} -\dynkin{E}{VIII} -\end{LTXexample} -\begin{LTXexample} -\dynkin{E}{XI} -\end{LTXexample} -\begin{LTXexample} -\dynkin{F}{I} -\end{LTXexample} +\section{Syntax} -\begin{LTXexample} -\dynkin{F}{II} -\end{LTXexample} +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} -\begin{LTXexample} -\dynkin{G}{I} -\end{LTXexample} -\begin{LTXexample} -\begin{tikzpicture} -\dynkin[open]{E}{6} -\draw[\dynkinfoldarrowstyle,\dynkinfoldarrowcolor] - (root 1.south) to [out=-45, in=-135] (root 6.south); -\draw[\dynkinfoldarrowstyle,\dynkinfoldarrowcolor] - (root 3.south) to [out=-45, in=-135] (root 5.south); -\end{tikzpicture} -\end{LTXexample} -\begin{LTXexample} -\begin{tikzpicture} -\dynkin[open]{E}{6} -\dynkincloseddot{3} -\dynkincloseddot{4} -\dynkincloseddot{5} -\draw[\dynkinfoldarrowstyle,\dynkinfoldarrowcolor] - (root 1.south) to [out=-45, in=-135] (root 6.south); -\end{tikzpicture} -\end{LTXexample} +\section{Options} -\section{Other stuff} +\newcommand*{\typ}[1]{\(\left<\texttt{#1}\right>\)} +\newcommand*{\optionLabel}[3]{%% +\multicolumn{2}{l}{\(\texttt{#1}=\texttt{#2}\),} \\ +\multicolumn{2}{l}{\(\textrm{default}: \texttt{#3}\)} \\ +}%% -Some sophisticated diagrams: -\begin{center} -\begin{tikzpicture} -\dynkin[folded]{D}{9} -\foreach \i in {2,6,8,9} { - \dynkinopendot{\i} -} -\dynkinline[white]{4}{5} -\dynkindots{4}{5} -\dynkinopendot{4} -\dynkincloseddot{5} -\end{tikzpicture} -\end{center} -can be drawn using sending TikZ options to \verb!\dynkinline! to erase the old edge, \verb!\dynkindots! to make indefinite edges, and then redrawing the roots next to any edge we draw: -\begin{LTXexample} -\begin{tikzpicture}[show background rectangle, - background rectangle/.style={fill=red!10}] -\dynkin[folded]{D}{9}; -\foreach \i in {2,6,8,9} { - \dynkinopendot{\i} -} -\dynkinline[red!10]{4}{5} -\dynkindots{4}{5} -\dynkinopendot{4} -\dynkincloseddot{5} -\end{tikzpicture} -\end{LTXexample} +\renewcommand*{\arraystretch}{1} +\par\noindent% +\begin{longtable}{p{1cm}p{10cm}} +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{2}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\optionLabel{text/.style}{\typ{TikZ style data}}{scale=.7} +& Style for any labels on the roots. \\ +\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{radius}{\typ{number}cm}{.05cm} +& size of the dots and of the crosses in the Dynkin diagram \\ +\optionLabel{edgeLength}{\typ{number}cm}{.35cm} +& distance between nodes in the Dynkin diagram \\ +\optionLabel{edge/.style}{TikZ style data}{thin} +& style of edges in the Dynkin diagram \\ +\optionLabel{mark}{\typ{o,O,t,x,X,*}}{*} +& default root mark \\ +\optionLabel{affineMark}{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{labelMacro}{\typ{1-parameter \TeX{} macro}}{\texttt{\#1}} +& the current labelling scheme. \\ +\optionLabel{makeIndefiniteEdge}{\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{indefiniteEdgeRatio}{\typ{float}}{1.6} +& ratio of indefinite edge lengths to other edge lengths. \\ +\optionLabel{indefiniteEdge/.style}{\typ{TikZ style data}}{draw=black,fill=white,thin,densely dotted} +& style of the dotted or dashed middle third of each indefinite edge. \\ +\optionLabel{arrows}{\typ{true or false}}{true} +& whether to draw the arrows that arise along the edges. \\ +\optionLabel{reverseArrows}{\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{foldleft}{\typ{true or false}}{true} +& whether to fold the roots on the left side of a Dynkin diagram. \\ +\optionLabel{foldright}{\typ{true or false}}{true} +& whether to fold the roots on the right side of a Dynkin diagram. \\ +\optionLabel{foldradius}{\typ{length}}{.3cm} +& the radius of circular arcs used in curved edges of folded Dynkin diagrams. \\ +\optionLabel{foldStyle}{\typ{TikZ style data}}{draw=black!40,fill=none,line width=radius} +& when drawing folded diagrams, style for the fold indicators. \\ +\optionLabel{*/.style}{\typ{TikZ style data}}{draw=black,fill=black} +& style for roots like \dynkin{A}{*} \\ +\optionLabel{o/.style}{\typ{TikZ style data}}{draw=black,fill=black} +& style for roots like \dynkin{A}{o} \\ +\optionLabel{O/.style}{\typ{TikZ style data}}{draw=black,fill=black} +& style for roots like \dynkin{A}{O} \\ +\optionLabel{t/.style}{\typ{TikZ style data}}{draw=black,fill=black} +& style for roots like \dynkin{A}{t} \\ +\optionLabel{x/.style}{\typ{TikZ style data}}{draw=black} +& style for roots like \dynkin{A}{x} \\ +\optionLabel{X/.style}{\typ{TikZ style data}}{draw=black,thick} +& style for roots like \dynkin{A}{X} \\ +\optionLabel{leftFold/.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{rightFold/.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{doubleEdges}{\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{doubleFold}{\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{doubleLeft}{\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{doubleFoldLeft}{\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{doubleRight}{\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{doubleFoldRight}{\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{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. -Always draw roots after edges. \nocite{*} \bibliographystyle{amsplain} diff --git a/Master/texmf-dist/tex/latex/dynkin-diagrams/dynkin-diagrams.sty b/Master/texmf-dist/tex/latex/dynkin-diagrams/dynkin-diagrams.sty index 8ed53464f8f..aec4b70689c 100644 --- a/Master/texmf-dist/tex/latex/dynkin-diagrams/dynkin-diagrams.sty +++ b/Master/texmf-dist/tex/latex/dynkin-diagrams/dynkin-diagrams.sty @@ -2,7 +2,7 @@ % % The Dynkin Diagrams package. % -% Version 2 +% Version 3.1 % % % This package draws Dynkin diagrams in LaTeX documents, using the TikZ package. @@ -18,7 +18,7 @@ % % \NeedsTeXFormat{LaTeX2e}[1994/06/01] -\ProvidesPackage{dynkin-diagrams}[2017/11/14 Dynkin diagrams] +\ProvidesPackage{dynkin-diagrams}[2018/02/11 Dynkin diagrams] \RequirePackage{tikz} \RequirePackage{xstring} \RequirePackage{xparse} @@ -29,122 +29,225 @@ \usetikzlibrary{decorations.markings} \usetikzlibrary{arrows,arrows.meta} \usetikzlibrary{calc} +\usetikzlibrary{fit} %% %% Application programming interface: %% See dynkin-diagrams.tex file for examples of use. %% -\NewDocumentCommand\dynkin{O{}mm}% +\NewDocumentCommand\dynkin{O{}mO{0}m}% {% \ifdefined\filldraw% - \@dynkin[#1]{#2}{#3}% + \@dynkin[#1]{#2}[#3]{#4}% \else% - \tikz[baseline=-\the\dimexpr\fontdimen22\textfont2\relax ]{\@dynkin[#1]{#2}{#3}}% + \tikz[baseline=-0.5ex]{\@dynkin[#1]{#2}[#3]{#4}}% \fi% }% -%% \convertRootNumber{<n>} -%% -> -%% Converts <n> from Bourbaki ordering to the current ordering, storing the result in a count called \RootNumber. -\NewDocumentCommand\convertRootNumber{m}% +\NewDocumentCommand\dynkinRefreshRoots{}% {% - \IfStrEq{#1}{0} - { - \global\RootNumber=0 - } - { - \IfStrEqCase{\dynkinseries}% + \dynkin@draw@all@roots{}% + \ifdynkin@label@the@roots\dynkinPrintLabels{}\fi% +}% + + +%% \dynkinLabelRoot{<r>}{<s>} or \dynkinLabelRoot*{<r>}{<s>} +%% Prints the label string <s> on the Dynkin diagram at root number <r>, in the current ordering convention. +%% Starred form uses the opposite label location. +\NewDocumentCommand\dynkinLabelRoot{smm}% +{% + \ifnum\dynkin@nodes<#2% + \ClassError{Dynkin diagrams}{Unrecognized root: ``#2'' found when labelling Dynkin diagram \dynkin@user@series{\dynkin@user@string}. Allowed values are up to \the\dynkin@nodes}{}% + \fi% + \newcount\rpo% + \rpo=#2% + \advance\rpo by 1% + \StrMid{\dynkin@label@directions}{\the\rpo}{\the\rpo}[\temp]% + \IfBooleanTF{#1}% {% - {E}% - {% - \ifnum\dynkinrank=6% - \IfStrEqCase{\dynkinordering}% - {% - {Adams}{\RootNumber=\stringcharacterinposition{152436}{#1}}% - {Carter}{\RootNumber=\stringcharacterinposition{142356}{#1}}% - {Dynkin}{\RootNumber=\stringcharacterinposition{162345}{#1}}% - {Kac}{\RootNumber=\stringcharacterinposition{162345}{#1}}% + \IfStrEqCase{\temp}{% + {l}{% + \node[inner sep=\dynkin@root@radius,% + label={% + [/Dynkin diagram,/Dynkin diagram/text]% + right:% + \(\pgfkeys{/Dynkin diagram/labelMacro=#3}\)% }% - [\RootNumber=#1]% - \else% - \ifnum\dynkinrank=7% - \IfStrEqCase{\dynkinordering}% - {% - {Adams}{\RootNumber=\stringcharacterinposition{6354217}{#1}}% - {Carter}{\RootNumber=\stringcharacterinposition{7564321}{#1}}% - {Dynkin}{\RootNumber=\stringcharacterinposition{1723456}{#1}}% - {Kac}{\RootNumber=\stringcharacterinposition{1723456}{#1}}% + ]% + at (\dynkin@root@name #2){};% + }% + {r}{% + \node[inner sep=\dynkin@root@radius,% + label={% + [/Dynkin diagram,/Dynkin diagram/text]% + left:% + \(\pgfkeys{/Dynkin diagram/labelMacro=#3}\)% }% - [\RootNumber=#1]% - \else% - \ifnum\dynkinrank=8% - \IfStrEqCase{\dynkinordering}% - {% - {Adams}{\RootNumber=\stringcharacterinposition{13245678}{#1}}% - {Carter}{\RootNumber=\stringcharacterinposition{86754321}{#1}}% - {Dynkin}{\RootNumber=\stringcharacterinposition{18234567}{#1}}% - {Kac}{\RootNumber=\stringcharacterinposition{78654321}{#1}}% - }% - [\RootNumber=#1]% - \else% - \fi% - \fi% - \fi% - }% - {F}% - {% - \IfStrEqCase{\dynkinordering}% - {% - {Adams}{\RootNumber=\stringcharacterinposition{4321}{#1}}% + ]% + at (\dynkin@root@name #2){};% + }% + {a}{% + \node[inner sep=\dynkin@root@radius,% + label={% + [/Dynkin diagram,/Dynkin diagram/text]% + below:% + \(\pgfkeys{/Dynkin diagram/labelMacro=#3}\)% + }% + ]% + at (\dynkin@root@name #2){};% + }% + {b}{% + \node[inner sep=\dynkin@root@radius,% + label={% + [/Dynkin diagram,/Dynkin diagram/text]% + above:% + \(\pgfkeys{/Dynkin diagram/labelMacro=#3}\)% + }% + ]% + at (\dynkin@root@name #2){};% + }% + {d}{% + \node[inner sep=\dynkin@root@radius,% + label={% + [/Dynkin diagram,/Dynkin diagram/text]% + above right:% + \(\pgfkeys{/Dynkin diagram/labelMacro=#3}\)% + }% + ]% + at (\dynkin@root@name #2){};% }% - [\RootNumber=#1]% }% - {G}% - {% - \IfStrEqCase{\dynkinordering}% - {% - {Carter}{\RootNumber=\stringcharacterinposition{21}{#1}}% - {Dynkin}{\RootNumber=\stringcharacterinposition{21}{#1}}% - {Kac}{\RootNumber=\stringcharacterinposition{21}{#1}}% + [\ClassError% + {Dynkin diagrams}% + {Unrecognized root label direction: + ``\temp'' in Dynkin diagram \dynkin@user@series{\dynkin@user@string} for root #2}% + {}] + }% + {% + \IfStrEqCase{\temp}{% + {l}{% + \node[inner sep=\dynkin@root@radius,% + label={% + [/Dynkin diagram,/Dynkin diagram/text]% + left:% + \(\pgfkeys{/Dynkin diagram/labelMacro=#3}\)% + }% + ]% + at (\dynkin@root@name #2){};% + }% + {r}{% + \node[inner sep=\dynkin@root@radius,% + label={% + [/Dynkin diagram,/Dynkin diagram/text]% + right:% + \(\pgfkeys{/Dynkin diagram/labelMacro=#3}\)% + }% + ]% + at (\dynkin@root@name #2){};% + }% + {a}{% + \node[inner sep=\dynkin@root@radius,% + label={% + [/Dynkin diagram,/Dynkin diagram/text]% + above:% + \(\pgfkeys{/Dynkin diagram/labelMacro=#3}\)% + }% + ]% + at (\dynkin@root@name #2){};% + }% + {b}{ % + \node[inner sep=\dynkin@root@radius,% + label={% + [/Dynkin diagram,/Dynkin diagram/text]% + below:% + \(\pgfkeys{/Dynkin diagram/labelMacro=#3}\)% + }% + ]% + at (\dynkin@root@name #2){};% + }% + {d}{% + \node[inner sep=\dynkin@root@radius,% + label={% + [/Dynkin diagram,/Dynkin diagram/text]% + below right:% + \(\pgfkeys{/Dynkin diagram/labelMacro=#3}\)% + }% + ]% + at (\dynkin@root@name #2){};% }% - [\RootNumber=#1]% }% + [\ClassError% + {Dynkin diagrams}% + {Unrecognized root label direction: + ``\temp'' in Dynkin diagram \dynkin@user@series{\dynkin@user@string} for root #2}% + {}] }% - [\RootNumber=#1]% - } }% -\NewDocumentCommand\dynkinprint{m}% +%% \dynkinPrintLabels +%% Prints the default labels on the Dynkin diagram, in the given ordering. +\newcommand{\dynkinPrintLabels}% {% - \scalebox{\dynkintextscale}{\(#1\)}% + \foreach \i in {1,...,\the\dynkin@nodes}{\dynkinLabelRoot{\i}{\i}}% + \ifdynkin@is@extended% + \dynkinLabelRoot{0}{0}% + \else% + \ifdynkin@is@twisted% + \dynkinLabelRoot{0}{0}% + \fi% + \fi% }% -%% \rootlabel{<n>}{<s>} or \rootlabel*{<n>}{<s>} -%% -> -%% Prints the label string <s> on the Dynkin diagram at root number <n>, in the current ordering convention. -\NewDocumentCommand\rootlabel{smm}% +%% \dynkinCrossRootMark{<n>} +%% Prints a cross at root <n> on the current Dynkin diagram. +%% The starred form accepts <n> in the Bourbaki ordering. +\NewDocumentCommand\dynkinCrossRootMark{sO{}m}% {% \IfBooleanTF{#1}% - {\node at (root label swap #2) {\dynkinprint{#3}};}% - {\node at (root label #2) {\dynkinprint{#3}};}% + {% + \convertRootNumber{#3}% + }% + {% + \RootNumber=#3% + }% + \draw[/Dynkin diagram,/Dynkin diagram/x,#2]% + ($(\dynkin@root@name \the\RootNumber)+(\dynkin@root@radius,\dynkin@root@radius)$)% + --% + ($(\dynkin@root@name \the\RootNumber)-(\dynkin@root@radius,\dynkin@root@radius)$);% + \draw[/Dynkin diagram,/Dynkin diagram/x,#2]% + ($(\dynkin@root@name \the\RootNumber)+(-\dynkin@root@radius,\dynkin@root@radius)$)% + --% + ($(\dynkin@root@name \the\RootNumber)+(\dynkin@root@radius,-\dynkin@root@radius)$);% }% -%% \dynkinprintlabels -%% -> -%% Prints the default labels on the Dynkin diagram, in the given ordering. -\newcommand{\dynkinprintlabels}% +%% \dynkinHeavyCrossRootMark{<n>} +%% Prints a heavy cross at root <n> on the current Dynkin diagram. +%% The starred form accepts <n> in the Bourbaki ordering. +\NewDocumentCommand\dynkinHeavyCrossRootMark{sO{}m}% {% - \foreach \i in {1,...,\the\dynkinrank}% - {\rootlabel{\i}{\i}}% - \ifisaffine\rootlabel{0}{0}\fi% + \IfBooleanTF{#1}% + {% + \convertRootNumber{#3}% + }% + {% + \RootNumber=#3% + }% + \draw[/Dynkin diagram,/Dynkin diagram/X,#2]% + ($(\dynkin@root@name \the\RootNumber)+(\dynkin@root@radius,\dynkin@root@radius)$)% + --% + ($(\dynkin@root@name \the\RootNumber)-(\dynkin@root@radius,\dynkin@root@radius)$);% + \draw[/Dynkin diagram,/Dynkin diagram/X,#2]% + ($(\dynkin@root@name \the\RootNumber)+(-\dynkin@root@radius,\dynkin@root@radius)$)% + --% + ($(\dynkin@root@name \the\RootNumber)+(\dynkin@root@radius,-\dynkin@root@radius)$);% }% -%% \dynkincross{<n>} -%% -> -%% Prints a cross at root <n> on the current Dynkin diagram. + +%% \dynkinHollowRootMark{<n>} +%% Prints an hollow dot at root <n> on the current Dynkin diagram. %% The starred form accepts <n> in the Bourbaki ordering. -\NewDocumentCommand\dynkincross{sO{}m}% +\NewDocumentCommand\dynkinHollowRootMark{sO{}m}% {% \IfBooleanTF{#1}% {% @@ -153,21 +256,13 @@ {% \RootNumber=#3% }% - \draw[\dynkincrossstyle,\dynkincolor,#2]% - ($(root \the\RootNumber)+(\dynkinradius,\dynkinradius)$)% - --% - ($(root \the\RootNumber)-(\dynkinradius,\dynkinradius)$);% - \draw[\dynkincrossstyle,\dynkincolor]% - ($(root \the\RootNumber)+(-\dynkinradius,\dynkinradius)$)% - --% - ($(root \the\RootNumber)+(\dynkinradius,-\dynkinradius)$);% + \fill[/Dynkin diagram,/Dynkin diagram/o,#2] (\dynkin@root@name \the\RootNumber) circle (\dynkin@root@radius);% }% -%% \dynkinopendot{<n>} -%% -> -%% Prints an open dot at root <n> on the current Dynkin diagram. +%% \dynkinDoubleHollowRootMark{<n>} +%% Prints a double hollow dot at root <n> on the current Dynkin diagram. %% The starred form accepts <n> in the Bourbaki ordering. -\NewDocumentCommand\dynkinopendot{sO{}m}% +\NewDocumentCommand\dynkinDoubleHollowRootMark{sO{}m}% {% \IfBooleanTF{#1}% {% @@ -176,14 +271,14 @@ {% \RootNumber=#3% }% - \fill[\dynkinbackcolor,draw=\dynkincolor,#2] (root \the\RootNumber) circle (\dynkinradius);% + \fill[/Dynkin diagram,/Dynkin diagram/o,#2] (\dynkin@root@name \the\RootNumber) circle (2*\dynkin@root@radius);% + \fill[/Dynkin diagram,/Dynkin diagram/o,#2] (\dynkin@root@name \the\RootNumber) circle (\dynkin@root@radius);% }% -%% \dynkincloseddot{<n>} -%% -> -%% Prints a closed dot at root <n> on the current Dynkin diagram. +%% \dynkinSolidRootMark{<n>} +%% Prints a solid dot at root <n> on the current Dynkin diagram. %% The starred form accepts <n> in the Bourbaki ordering. -\NewDocumentCommand\dynkincloseddot{sO{}m}% +\NewDocumentCommand\dynkinSolidRootMark{sO{}m}% {% \IfBooleanTF{#1}% {% @@ -192,53 +287,154 @@ {% \RootNumber=#3% }% - \fill[\dynkincolor,draw=\dynkincolor,#2] (root \the\RootNumber) circle (\dynkinradius);% + \fill[/Dynkin diagram,/Dynkin diagram/*,#2] (\dynkin@root@name \the\RootNumber) circle (\dynkin@root@radius);% }% -%% \dynkindot{<n>} -%% -> -%% Prints a dot at root <n> on the current Dynkin diagram in the default style. +%% \dynkinTensorRootMark{<n>} +%% Prints a tensor product symbol at root <n> on the current Dynkin diagram. %% The starred form accepts <n> in the Bourbaki ordering. -\NewDocumentCommand\dynkindot{sO{}m}% +\NewDocumentCommand\dynkinTensorRootMark{sO{}m}% {% \IfBooleanTF{#1}% {% - \ifnum#3=0% - \ifdynkinopendots% - \dynkincloseddot*[#2]{0}% - \else% - \dynkinopendot*[#2]{0}% - \fi% - \else% - \ifdynkinopendots% - \dynkinopendot*[#2]{#3}% - \else% - \dynkincloseddot*[#2]{#3}% - \fi% - \fi% + \convertRootNumber{#3}% }% {% - \ifnum#3=0% - \ifdynkinopendots% - \dynkincloseddot[#2]{0}% - \else% - \dynkinopendot[#2]{0}% - \fi% - \else% - \ifdynkinopendots% - \dynkinopendot[#2]{#3}% - \else% - \dynkincloseddot[#2]{#3}% - \fi% - \fi% + \RootNumber=#3% }% + \fill[/Dynkin diagram,/Dynkin diagram/o,#2] (\dynkin@root@name \the\RootNumber) circle ({\dynkin@root@radius});% + \draw[/Dynkin diagram,/Dynkin diagram/x,#2]% + ($(\dynkin@root@name \the\RootNumber)+({\dynkin@root@radius/sqrt(2)},{\dynkin@root@radius/sqrt(2)})$)% + --% + ($(\dynkin@root@name \the\RootNumber)-({\dynkin@root@radius/sqrt(2)},{\dynkin@root@radius/sqrt(2)})$);% + \draw[/Dynkin diagram,/Dynkin diagram/x,#2]% + ($(\dynkin@root@name \the\RootNumber)+({-\dynkin@root@radius/sqrt(2)},{\dynkin@root@radius/sqrt(2)})$)% + --% + ($(\dynkin@root@name \the\RootNumber)+({\dynkin@root@radius/sqrt(2)},{-\dynkin@root@radius/sqrt(2)})$);% }% -%% \dynkinline{<p>}{<q>} -%% -> +%% \dynkinRootMark{<s>}{<n>} +%% Prints a dot at root <n> on the current Dynkin diagram using mark style <s>. +%% Use <s> empty to get the default mark style. +%% The starred form accepts <n> in the Bourbaki ordering. +\NewDocumentCommand\dynkinRootMark{smm}% +{% + \IfBooleanTF{#1}% + {% + \IfStrEqCase{#2}% + {% + {}{\dynkinRootMark*{\dynkin@root@mark}{#3}}% + {*}{\dynkinSolidRootMark*{#3}}% + {O}{\dynkinDoubleHollowRootMark*{#3}}% + {X}{\dynkinHeavyCrossRootMark*{#3}}% + {o}{\dynkinHollowRootMark*{#3}}% + {t}{\dynkinTensorRootMark*{#3}}% + {x}{\dynkinCrossRootMark*{#3}}% + }% + [\ClassError% + {Dynkin diagrams}% + {Unrecognized root mark: ``#2'' in Dynkin diagram% + \dynkin@user@series{\dynkin@user@string}}% + {}] + }% + {% + \IfStrEqCase{#2}% + {% + {}{\dynkinRootMark{\dynkin@root@mark}{#3}}% + {*}{\dynkinSolidRootMark{#3}}% + {O}{\dynkinDoubleHollowRootMark{#3}}% + {X}{\dynkinHeavyCrossRootMark{#3}}% + {o}{\dynkinHollowRootMark{#3}}% + {t}{\dynkinTensorRootMark{#3}}% + {x}{\dynkinCrossRootMark{#3}}% + }% + [\ClassError{Dynkin diagrams}{Unrecognized root mark: ``#2'' in Dynkin diagram \dynkin@user@series{\dynkin@user@string}}{}] + }% +}% + +%% \dynkinDefiniteSingleEdge{<p>}{<q>} %% Draws a single line from root <p> to root <q> on the current Dynkin diagram in the current label ordering. %% The starred form accepts <p> and <q> in the Bourbaki ordering. -\NewDocumentCommand\dynkinline{sO{}mm}% +\NewDocumentCommand\dynkinDefiniteSingleEdge{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,edge,#2] + ($(\dynkin@root@name \the\@fromRoot)$) + -- + ($(\dynkin@root@name \the\@toRoot)$);% + \end{scope}% +}% + +%% \dynkinIndefiniteSingleEdge{<p>}{<q>} +%% Draws a single line from root <p> to root <q> on the current Dynkin diagram in the current label ordering, +%% drawn as dashed to indicate an edge containing an indefinite number of roots. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinIndefiniteSingleEdge{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,edge,#2] + ($(\dynkin@root@name \the\@fromRoot)$) + -- + (${(2/3)}*(\dynkin@root@name \the\@fromRoot)+{(1/3)}*(\dynkin@root@name \the\@toRoot)$); + \draw[/Dynkin diagram,/Dynkin diagram/indefiniteEdge,#2] + (${(2/3)}*(\dynkin@root@name \the\@fromRoot)+{(1/3)}*(\dynkin@root@name \the\@toRoot)$) + -- + (${(1/3)}*(\dynkin@root@name \the\@fromRoot)+{(2/3)}*(\dynkin@root@name \the\@toRoot)$); + \draw[/Dynkin diagram,/Dynkin diagram/edge,#2] + (${(1/3)}*(\dynkin@root@name \the\@fromRoot)+{(2/3)}*(\dynkin@root@name \the\@toRoot)$) + -- + ($(\dynkin@root@name \the\@toRoot)$); + \end{scope}% +}% + +%%% \dynkinRightFold{<p>}{<q>} +%%% Draws an arrow to represent folding from root <p> to root <q> on the current Dynkin diagram in the current label ordering, curving to the right. +%%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinRightFold{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \dynkinFold*[/Dynkin diagram/rightFold,#2]{#3}{#4}% + }% + {% + \dynkinFold[/Dynkin diagram/rightFold,#2]{#3}{#4}% + }% +}% + +%%% \dynkinLeftFold{<p>}{<q>} +%%% Draws an arrow to represent folding from root <p> to root <q> on the current Dynkin diagram in the current label ordering, curving to the left. +%%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinLeftFold{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \dynkinFold*[/Dynkin diagram/leftFold,#2]{#3}{#4}% + }% + {% + \dynkinFold[/Dynkin diagram/leftFold,#2]{#3}{#4}% + }% +}% + +%% \dynkinFold{<p>}{<q>} +%% Draws some colouring to indicate which roots are being folded together, including roots <p> and <q>. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinFold{sO{}mm}% {% \IfBooleanTF{#1}% {% @@ -248,14 +444,41 @@ \@fromRoot=#3% \@toRoot=#4% }% - \draw[\dynkincolor,\dynkinedgestyle,#2] ($(root \the\@fromRoot)$) -- ($(root \the\@toRoot)$);% + \convertRootPair{\@fromRoot}{\@toRoot}% + \begin{scope}[on background layer] + \draw + [/Dynkin diagram/foldStyle,#2] + ($(\dynkin@root@name \the\@fromRoot)$) + to + ($(\dynkin@root@name \the\@toRoot)$); + \end{scope}% }% -%% \dynkinfoldarrow{<p>}{<q>} -%% -> -%% Draws an arrow to represent folding from root <p> to root <q> on the current Dynkin diagram in the current label ordering. + +%% \dynkinDefiniteRightDownArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteRightDownArc{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (90:0:\dynkin@fold@radius) -- ($(\dynkin@root@name \the\@toRoot)$);% + \end{scope}% +}% + +%% \dynkinIndefiniteRightDownArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering. %% The starred form accepts <p> and <q> in the Bourbaki ordering. -\NewDocumentCommand\dynkinfoldarrow{sO{}mm}% +\NewDocumentCommand\dynkinIndefiniteRightDownArc{sO{}mm}% {% \IfBooleanTF{#1}% {% @@ -265,14 +488,27 @@ \@fromRoot=#3% \@toRoot=#4% }% - \draw[\dynkinfoldarrowstyle,\dynkinfoldarrowcolor,#2] (root \the\@fromRoot) -- (root \the\@toRoot);% + \node (center) at ($(\dynkin@root@name \the\@fromRoot)-(0,\dynkin@fold@radius)$) {};% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + (center) + ++(90:\dynkin@fold@radius) + arc [start angle=90, end angle=60, radius=\dynkin@fold@radius];% + \draw[/Dynkin diagram,/Dynkin diagram/indefiniteEdge,fill=none,#2] + (center) + ++(60:\dynkin@fold@radius) + arc [start angle=60, end angle=30, radius=\dynkin@fold@radius];% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + (center) + ++(30:\dynkin@fold@radius) + arc [start angle=30, end angle=0, radius=\dynkin@fold@radius];% + \end{scope}% }% -%% \dynkindownarc{<p>}{<q>} -%% -> +%% \dynkinDefiniteRightUpArc{<p>}{<q>} %% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering. %% The starred form accepts <p> and <q> in the Bourbaki ordering. -\NewDocumentCommand\dynkindownarc{sO{}mm}% +\NewDocumentCommand\dynkinDefiniteRightUpArc{sO{}mm}% {% \IfBooleanTF{#1}% {% @@ -282,14 +518,17 @@ \@fromRoot=#3% \@toRoot=#4% }% - \draw[\dynkincolor,\dynkinedgestyle,#2] ($(root \the\@fromRoot)$) arc (90:0:\dynkinedgelength);% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + ($(\dynkin@root@name \the\@fromRoot)$) + arc (-90:0:\dynkin@fold@radius) -- ($(\dynkin@root@name \the\@toRoot)$);% + \end{scope}% }% -%% \dynkinuparc{<p>}{<q>} -%% -> +%% \dynkinIndefiniteRightUpArc{<p>}{<q>} %% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering. %% The starred form accepts <p> and <q> in the Bourbaki ordering. -\NewDocumentCommand\dynkinuparc{sO{}mm}% +\NewDocumentCommand\dynkinIndefiniteRightUpArc{sO{}mm}% {% \IfBooleanTF{#1}% {% @@ -299,14 +538,282 @@ \@fromRoot=#3% \@toRoot=#4% }% - \draw[\dynkincolor,\dynkinedgestyle,#2] ($(root \the\@fromRoot)$) arc (0:-90:\dynkinedgelength);% + \node (center) at ($(\dynkin@root@name \the\@fromRoot)+(0,\dynkin@fold@radius)$) {};% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + (center) + ++(-90:\dynkin@fold@radius) + arc [start angle=-90, end angle=-60, radius=\dynkin@fold@radius];% + \draw[/Dynkin diagram,/Dynkin diagram/indefiniteEdge,fill=none,#2] + (center) + ++(-60:\dynkin@fold@radius) + arc [start angle=-60, end angle=-30, radius=\dynkin@fold@radius];% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + (center) + ++(-30:\dynkin@fold@radius) + arc [start angle=-30, end angle=0, radius=\dynkin@fold@radius] -- ($(\dynkin@root@name \the\@toRoot)$);% + \end{scope}% }% -%% \dynkinsemicircle{<p>}{<q>} -%% -> + +%% \dynkinDefiniteLeftDownArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteLeftDownArc{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (90:180:\dynkin@fold@radius) -- ($(\dynkin@root@name \the\@toRoot)$);% + \end{scope}% +}% + +%% \dynkinIndefiniteLeftDownArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinIndefiniteLeftDownArc{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \node (center) at ($(\dynkin@root@name \the\@fromRoot)-(0,\dynkin@fold@radius)$) {};% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + (center) + ++(90:\dynkin@fold@radius) + arc [start angle=90, end angle=120, radius=\dynkin@fold@radius];% + \draw[/Dynkin diagram,/Dynkin diagram/indefiniteEdge,fill=none,#2] + (center) + ++(120:\dynkin@fold@radius) + arc [start angle=120, end angle=150, radius=\dynkin@fold@radius];% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + (center) + ++(150:\dynkin@fold@radius) + arc [start angle=150, end angle=180, radius=\dynkin@fold@radius] -- ($(\dynkin@root@name \the\@toRoot)$);% + \end{scope}% +}% + +%% \dynkinDefiniteLeftUpArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteLeftUpArc{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + ($(\dynkin@root@name \the\@fromRoot)$) + arc (-90:-180:\dynkin@fold@radius) -- ($(\dynkin@root@name \the\@toRoot)$);% + \end{scope}% +}% + +%% \dynkinIndefiniteLeftUpArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinIndefiniteLeftUpArc{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \node (center) at ($(\dynkin@root@name \the\@fromRoot)+(0,\dynkin@fold@radius)$) {};% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + (center) + ++(-90:\dynkin@fold@radius) + arc [start angle=-90, end angle=-120, radius=\dynkin@fold@radius];% + \draw[/Dynkin diagram,/Dynkin diagram/indefiniteEdge,fill=none,#2] + (center) + ++(-120:\dynkin@fold@radius) + arc [start angle=-120, end angle=-150, radius=\dynkin@fold@radius];% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + (center) + ++(-150:\dynkin@fold@radius) + arc [start angle=-150, end angle=-180, radius=\dynkin@fold@radius] -- ($(\dynkin@root@name \the\@toRoot)$);% + \end{scope}% +}% + + +%% \dynkinDefiniteSemiCircle{<p>}{<q>} +%% Draws a half circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteSemiCircle{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + ($(\dynkin@root@name \the\@fromRoot)$) + arc (90:-90:\dynkin@fold@radius) + -- ($(\dynkin@root@name \the\@toRoot)$);% + \end{scope}% +}% + +%% \dynkinIndefiniteSemiCircle{<p>}{<q>} %% Draws a half circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering. %% The starred form accepts <p> and <q> in the Bourbaki ordering. -\NewDocumentCommand\dynkinsemicircle{sO{}mm}% +\NewDocumentCommand\dynkinIndefiniteSemiCircle{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \node (center) at ($(\dynkin@root@name \the\@fromRoot)-(0,\dynkin@fold@radius)$) {};% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + (center) + ++(90:\dynkin@fold@radius) + arc [start angle=90, end angle=30, radius=\dynkin@fold@radius];% + \draw[/Dynkin diagram,/Dynkin diagram/indefiniteEdge,fill=none,#2] + (center) + ++(30:\dynkin@fold@radius) + arc [start angle=30, end angle=-30, radius=\dynkin@fold@radius];% + \draw[/Dynkin diagram,/Dynkin diagram/edge,fill=none,#2] + (center) + ++(-30:\dynkin@fold@radius) + arc [start angle=-30, end angle=-90, radius=\dynkin@fold@radius] -- ($(\dynkin@root@name \the\@toRoot)$);% + \end{scope}% +}% + +%% \dynkinDefiniteDoubleRightDownArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering +%% as a double path. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteDoubleRightDownArc{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,double,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (90:0:{\dynkin@fold@radius}) -- ($(\dynkin@root@name \the\@toRoot)$);% + \ifdynkin@arrows% + \ifdynkin@reverse@arrows% + \path[-<,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (90:45:{\dynkin@fold@radius});% + \else% + \path[->,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (90:45:{\dynkin@fold@radius});% + \fi% + \fi% + \end{scope}% +}% + + +%% \dynkinDefiniteDoubleUpRightArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering +%% as a double path. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteDoubleUpRightArc{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,double,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (180:90:{\dynkin@fold@radius}) -- ($(\dynkin@root@name \the\@toRoot)$);% + \ifdynkin@arrows% + \ifdynkin@reverse@arrows% + \path[-<,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (180:135:{\dynkin@fold@radius});% + \else% + \path[->,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (180:135:{\dynkin@fold@radius});% + \fi% + \fi% + \end{scope}% +}% + + +%% \dynkinDefiniteDoubleUpLeftArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering +%% as a double path. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteDoubleUpLeftArc{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,double,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (-90:0:{\dynkin@fold@radius}) -- ($(\dynkin@root@name \the\@toRoot)$);% + \ifdynkin@arrows% + \ifdynkin@reverse@arrows% + \path[-<,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (-90:-45:{\dynkin@fold@radius});% + \else% + \path[->,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (-90:-45:{\dynkin@fold@radius});% + \fi% + \fi% + \end{scope}% +}% + + + + +%% \dynkinDefiniteDoubleDownRightArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering +%% as a double path. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteDoubleDownRightArc{sO{}mm}% {% \IfBooleanTF{#1}% {% @@ -316,14 +823,32 @@ \@fromRoot=#3% \@toRoot=#4% }% - \draw[\dynkincolor,\dynkinedgestyle,#2] ($(root \the\@fromRoot)$) arc (90:-90:\dynkinedgelength);% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,double,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + -- + ($(\dynkin@root@name \the\@toRoot)+(-\dynkin@fold@radius,\dynkin@fold@radius)$)% + arc (-180:-90:{\dynkin@fold@radius}) -- ($(\dynkin@root@name \the\@toRoot)$);% + \ifdynkin@arrows% + \ifdynkin@reverse@arrows% + \path[-<,tips] + ($(\dynkin@root@name \the\@toRoot)+(-\dynkin@fold@radius,\dynkin@fold@radius)$)% + arc (-180:-135:{\dynkin@fold@radius});% + \else% + \path[->,tips] + ($(\dynkin@root@name \the\@toRoot)+(-\dynkin@fold@radius,\dynkin@fold@radius)$)% + arc (-180:-135:{\dynkin@fold@radius});% + \fi% + \fi% + \end{scope}% }% -%% \dynkindots{<p>}{s<q>} -%% -> -%% Draws a dotted line from root <p> to root <q> on the current Dynkin diagram. + +%% \dynkinDefiniteDoubleRightUpArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering +%% as a double path. %% The starred form accepts <p> and <q> in the Bourbaki ordering. -\NewDocumentCommand\dynkindots{sO{}mm}% +\NewDocumentCommand\dynkinDefiniteDoubleRightUpArc{sO{}mm}% {% \IfBooleanTF{#1}% {% @@ -333,14 +858,245 @@ \@fromRoot=#3% \@toRoot=#4% }% - \draw[densely dotted,\dynkincolor,#2] ($(root \the\@fromRoot)$) -- ($(root \the\@toRoot)$);% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,double,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (270:360:{\dynkin@fold@radius}) -- ($(\dynkin@root@name \the\@toRoot)$);% + \ifdynkin@arrows% + \path[->,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (270:315:\dynkin@fold@radius);% + \fi% + \end{scope}% +}% + +%% \dynkinDefiniteDoubleLeftDownArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering +%% as a double path. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteDoubleLeftDownArc{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,double,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (90:180:{\dynkin@fold@radius}) -- ($(\dynkin@root@name \the\@toRoot)$);% + \ifdynkin@arrows% + \ifdynkin@reverse@arrows% + \path[-<,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (90:135:{\dynkin@fold@radius});% + \else% + \path[->,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (90:135:{\dynkin@fold@radius});% + \fi% + \fi% + \end{scope}% }% -%% \dynkindoubleline{<p>}{<q>} -%% -> + +%% \dynkinDefiniteDoubleDownLeftArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering +%% as a double path. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteDoubleDownLeftArc{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,double,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (360:270:{\dynkin@fold@radius}) -- ($(\dynkin@root@name \the\@toRoot)$);% + \ifdynkin@arrows% + \ifdynkin@reverse@arrows% + \path[-<,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (360:315:{\dynkin@fold@radius});% + \else% + \path[->,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (360:315:{\dynkin@fold@radius});% + \fi% + \fi% + \end{scope}% +}% + + + +%% \dynkinDefiniteDoubleLeftUpArc{<p>}{<q>} +%% Draws a quarter circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering +%% as a double path. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteDoubleLeftUpArc{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,double,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (-90:-180:{\dynkin@fold@radius}) -- ($(\dynkin@root@name \the\@toRoot)$);% + \ifdynkin@arrows% + \ifdynkin@reverse@arrows% + \path[-<,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (-90:-135:\dynkin@fold@radius);% + \else% + \path[->,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (-90:-135:\dynkin@fold@radius);% + \fi% + \fi% + \end{scope}% +}% + + +%% \dynkinDefiniteDoubleDownRightSemiCircle{<p>}{<q>} +%% Draws a semi circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering +%% as a double path. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteDoubleDownRightSemiCircle{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,double,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (90:-90:{\dynkin@fold@radius}) -- ($(\dynkin@root@name \the\@toRoot)$);% + \ifdynkin@arrows% + \ifdynkin@reverse@arrows% + \path[-<,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (90:0:\dynkin@fold@radius);% + \else% + \path[->,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (90:0:\dynkin@fold@radius);% + \fi% + \fi% + \end{scope}% +}% + + + +%% \dynkinDefiniteDoubleUpRightSemiCircle{<p>}{<q>} +%% Draws a semi circle from root <p> to root <q> on the current Dynkin diagram in the current label ordering +%% as a double path. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinDefiniteDoubleUpRightSemiCircle{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,double,fill=none,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (-90:90:{\dynkin@fold@radius}) -- ($(\dynkin@root@name \the\@toRoot)$);% + \ifdynkin@arrows% + \ifdynkin@reverse@arrows% + \path[-<,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (-90:0:\dynkin@fold@radius);% + \else% + \path[->,tips] + ($(\dynkin@root@name \the\@fromRoot)$)% + arc (-90:0:\dynkin@fold@radius);% + \fi% + \fi% + \end{scope}% +}% + + +%% \dynkinEdge[<o>]{<f>}{<p>}{<q>} +%% Applies \dynkinDefinite<f>[<o>]{<p>}{<q>} if the edge <p><q> is definite, +%% otherwise applies \dynkinIndefinite<f>[<o>]{<p>}{<q>} +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinEdge{sO{}mmm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#4}{#5}% + \dynkin@is@edge@indefinite{\@fromRoot}{\@toRoot}% + \ifdynkin@is@indefinite@edge% + \csname dynkinIndefinite#3\endcsname[#2]{\@fromRoot}{\@toRoot}% + \else% + \csname dynkinDefinite#3\endcsname[#2]{\@fromRoot}{\@toRoot}% + \fi% + }% + {% + \dynkin@is@edge@indefinite{#4}{#5}% + \ifdynkin@is@indefinite@edge% + \csname dynkinIndefinite#3\endcsname[#2]{#4}{#5}% + \else% + \csname dynkinDefinite#3\endcsname[#2]{#4}{#5}% + \fi% + }% +}% + +%% \dynkinEdgeArrow{<p>}{<q>} +%% Draws an arrow head on the edge from root <p> to root <q>. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinEdgeArrow{sO{}mm}% +{% + \ifdynkin@arrows% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \ifdynkin@reverse@arrows% + \path[-<,tips] + ($(\dynkin@root@name \the\@fromRoot)$) + -- + ($.3*(\dynkin@root@name \the\@fromRoot)+.7*(\dynkin@root@name \the\@toRoot)$);% + \else% + \path[->,tips] + ($(\dynkin@root@name \the\@fromRoot)$) + -- + ($.3*(\dynkin@root@name \the\@fromRoot)+.7*(\dynkin@root@name \the\@toRoot)$);% + \fi% + \end{scope}% + \fi% +}% + +%% \dynkinDefiniteDoubleEdge{<p>}{<q>} %% Draws an oriented double line from root <p> to root <q> on the current Dynkin diagram. %% The starred form accepts <p> and <q> in the Bourbaki ordering. -\NewDocumentCommand\dynkindoubleline{sO{}mm}% +\NewDocumentCommand\dynkinDefiniteDoubleEdge{sO{}mm}% {% \IfBooleanTF{#1}% {% @@ -350,20 +1106,50 @@ \@fromRoot=#3% \@toRoot=#4% }% - \ifdynkinarrows% - \draw[double,postaction={decorate},\dynkincolor,\dynkinedgestyle,#2]% - ($(root \the\@fromRoot)$) -- ($(root \the\@toRoot)$);% - \else% - \draw[double,\dynkincolor,\dynkinedgestyle,#2]% - ($(root \the\@fromRoot)$) -- ($(root \the\@toRoot)$);% + \newcount\onesbit% + \newcount\twosbit% + \StrChar{\dynkin@roots}{\the\@fromRoot}[\my@root@marker]% + \IfStrEq{\my@root@marker}{x}% + {% + \global\onesbit=1% + }% + {% + \global\onesbit=0% + }% + \StrChar{\dynkin@roots}{\the\@toRoot}[\my@root@marker]% + \IfStrEq{\my@root@marker}{x}% + {% + \global\twosbit=1% + }% + {% + \global\twosbit=0% + }% + \def\LL{.5*\dynkin@root@radius} + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + --% + +({\the\onesbit*\LL},{\LL})% + --% + ($(\dynkin@root@name \the\@toRoot)+(-\the\twosbit*\LL,\LL)$)% + --% + ($(\dynkin@root@name \the\@toRoot)$)% + --% + ($(\dynkin@root@name \the\@toRoot)-(\the\twosbit*\LL,\LL)$)% + --% + ($(\dynkin@root@name \the\@fromRoot)+(\the\onesbit*\LL,-\LL)$)% + --% + cycle;% + \end{scope}% + \ifdynkin@arrows% + \dynkinEdgeArrow[#2]{\the\@fromRoot}{\the\@toRoot}% \fi% }% -%% \dynkintripleline{<p><q>} -%% -> +%% \dynkinTripleEdge{<p><q>} %% Draws an oriented triple line from root <p> to root <q> on the current Dynkin diagram. %% The starred form accepts <p> and <q> in the Bourbaki ordering. -\NewDocumentCommand\dynkintripleline{sO{}mm}% +\NewDocumentCommand\dynkinTripleEdge{sO{}mm}% {% \IfBooleanTF{#1}% {% @@ -373,351 +1159,1148 @@ \@fromRoot=#3% \@toRoot=#4% }% - \pgfmathparse{mod(div(\dynkinparabolic,2),2)}% - \let\onesbit\pgfmathresult% - \pgfmathparse{mod(div(\dynkinparabolic,4),2)}% - \let\twosbit\pgfmathresult% - \draw[\dynkincolor,fill=\dynkinbackcolor,\dynkinedgestyle,#2] % - ($(root \the\@fromRoot)$)% - --% - +(\onesbit*\dynkinradius,\dynkinradius)% - --% - ($(root \the\@toRoot)+(-\twosbit*\dynkinradius,\dynkinradius)$)% - --% - ($(root \the\@toRoot)$)% - --% - ($(root \the\@toRoot)-(\twosbit*\dynkinradius,\dynkinradius)$)% - --% - ($(root \the\@fromRoot)+(\onesbit*\dynkinradius,-\dynkinradius)$)% - --% - cycle;% - \ifdynkinarrows% - \draw[% - \dynkincolor,% - \dynkinedgestyle,% - -{Classical TikZ Rightarrow[length={3*\dynkinradius}]},% - #2% + \newcount\onesbit + \newcount\twosbit + \StrChar{\dynkin@roots}{\the\@fromRoot}[\my@root@marker]% + \IfStrEq{\my@root@marker}{x}% + {% + \global\onesbit=1% + }% + {% + \global\onesbit=0% + }% + \StrChar{\dynkin@roots}{\the\@toRoot}[\my@root@marker]% + \IfStrEq{\my@root@marker}{x}% + {% + \global\twosbit=1% + }% + {% + \global\twosbit=0% + }% + \begin{scope}[on background layer]% + \draw[/Dynkin diagram,/Dynkin diagram/edge,#2]% + ($(\dynkin@root@name \the\@fromRoot)$)% + --% + +({\the\onesbit*\dynkin@root@radius},{\dynkin@root@radius})% + --% + ($(\dynkin@root@name \the\@toRoot)+(-\twosbit*\dynkin@root@radius,\dynkin@root@radius)$)% + --% + ($(\dynkin@root@name \the\@toRoot)$)% + --% + ($(\dynkin@root@name \the\@toRoot)-(\twosbit*\dynkin@root@radius,\dynkin@root@radius)$)% + --% + ($(\dynkin@root@name \the\@fromRoot)+(\onesbit*\dynkin@root@radius,-\dynkin@root@radius)$)% + --% + cycle;% + \draw[/Dynkin diagram,/Dynkin diagram/edge,#2] + ($(\dynkin@root@name \the\@fromRoot)$) + -- + ($(\dynkin@root@name \the\@toRoot)$);% + \end{scope}% + \ifdynkin@arrows% + \dynkinEdgeArrow[#2]{\the\@fromRoot}{\the\@toRoot}% + \fi% +}% + + +%% \dynkinQuadrupleEdge{<p>}{<q>} +%% \dynkinQuadrupleEdge*{<p>}{<q>} +%% Draws an oriented edge of valence 4 from root <p> to root <q> on the current Dynkin diagram. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinQuadrupleEdge{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \begin{scope}[on background layer]% + \draw[% + /Dynkin diagram, + /Dynkin diagram/edge, + #2, ]% - ($(root \the\@toRoot)$) --% - ($.65*(root \the\@fromRoot)+.35*(root \the\@toRoot)$);% + ($(\dynkin@root@name \the\@fromRoot)+(0,\dynkin@root@radius)$)--% + ($(\dynkin@root@name \the\@toRoot)+(0,\dynkin@root@radius)$)--% + ($(\dynkin@root@name \the\@toRoot)+(0,-\dynkin@root@radius)$)--% + ($(\dynkin@root@name \the\@fromRoot)+(0,-\dynkin@root@radius)$)--% + cycle; + \draw[% + /Dynkin diagram,/Dynkin diagram/edge, + #2, + ]% + ($(\dynkin@root@name \the\@fromRoot)+(0,\dynkin@root@radius/3)$)--% + ($(\dynkin@root@name \the\@toRoot)+(0,\dynkin@root@radius/3)$)--% + ($(\dynkin@root@name \the\@toRoot)+(0,-\dynkin@root@radius/3)$)--% + ($(\dynkin@root@name \the\@fromRoot)+(0,-\dynkin@root@radius/3)$)--% + cycle; + \end{scope}% + \ifdynkin@arrows% + \dynkinEdgeArrow[#2]{\the\@fromRoot}{\the\@toRoot}% \fi% - \draw[\dynkincolor,#2] ($(root \the\@fromRoot)$) -- ($(root \the\@toRoot)$);% }% +%% \repeatCharacter{<n>}{<s>} +%% Outputs <n> copies of the string <s> +\ExplSyntaxOn +\DeclareExpandableDocumentCommand{\repeatCharacter}{O{}mm} + { + \int_compare:nT { #2 > 0 } + { + #3 \prg_replicate:nn { #2 - 1 } { #1#3 } + } + } +\ExplSyntaxOff + +%% \stringCharacterInPosition{<s>}{<n>} +%% Outputs the element of string <s> in position <n>. +\ExplSyntaxOn +\cs_new:Npn \stringCharacterInPosition #1 #2 +{ +\str_item:fn { #1 } { #2 } +} +\cs_generate_variant:Nn \str_item:nn {f} +\ExplSyntaxOff + + + + %%% %%% Implementation: %%% -\def\dynkinseries{A} % Which series of root system: A,B,C,D,E,F,G -\newcount\dynkinrank % Which rank of root system: 1,2,... -\newif\ifisaffine % Is this an affine extended root system? -\newif\iflabeltheroots % Should we label the roots by the current root ordering convention? -\newif\ifdynkinopendots % Should we draw the roots using open circles or closed dots? -\newif\ifdynkinarrows % Should we draw arrows on Dynkin diagrams? -\newif\ifdynkincoxeter % Should we draw Coxeter diagrams? -\newif\ifdynkinfolded % Should we fold our Dynkin diagrams? - -\pgfkeys{% - /dynkin/.is family,% - /tikz/decoration={markings,mark=at position 0.7 with {\arrow{>}}},% - /dynkin,% - open/.is if = dynkinopendots,% - open=false,% - Coxeter/.is if = dynkincoxeter,% - Coxeter=false,% - arrows/.is if = dynkinarrows,% - arrows=true,% - dotradius/.estore in = \dynkinradius,% - dotradius=.05cm,% - color/.store in =\dynkincolor,% - backgroundcolor/.store in =\dynkinbackcolor,% - color = black,% - backgroundcolor = white,% - edge/.store in = \dynkinedgestyle,% - edge = thin,% - cross/.store in = \dynkincrossstyle,% - cross = thick,% - edgelength/.estore in = \dynkinedgelength,% - edgelength = .35cm,% - ordering/.store in = \dynkinordering,% - ordering = Bourbaki,% - textscale/.estore in = \dynkintextscale,% - textscale = 0.7,% - foldarrowstyle/.estore in = \dynkinfoldarrowstyle,% - foldarrowstyle = stealth-stealth,% - foldarrowcolor/.estore in = \dynkinfoldarrowcolor,% - foldarrowcolor = black!50,% - default/.style = {% - label/.is if = labeltheroots,% - label = false,% - parabolic = 0,% - affine/.is if = isaffine,% - affine = false,% - folded/.is if = dynkinfolded,% - folded=false,% - },% - parabolic/.estore in = \dynkinparabolic,% - .search also={/tikz},% -}% - -\ProcessPgfPackageOptions{/dynkin}\relax - -% *=not a Satake diagram -% Anything else is the Roman numeral of the diagram, i.e. EVIII diagrams have numeral VIII. -\gdef\dynkinSatake{*} - -\NewDocumentCommand\@dynkin{O{}mm}{% - \pgfkeys{/dynkin, default, #1}% - \xdef\dynkinseries{#2}% - \IfSubStr{ABCDEFGHI}{#2}{}{\errorSeries}% - \global\dynkinrank=0% - \xdef\dynkinSatake{#3}% - \newif\ifwerefolded - \ifdynkinfolded - \global\werefoldedtrue - \else - \global\werefoldedfalse - \fi - \IfInteger{#3}% +\def\dynkin@diagram@name{anonymous} +% Default diagram name + +\def\dynkin@root@mark{*} +% Default mark + +\def\dynkin@affine@root@mark{o} +% Default affine root mark + +\def\dynkin@roots{} +% List of marks for each root. + +\def\dynkin@user@series{} +% Series string passed from user. +% For example: +% \dynkin{A}{3} passes the string A, +% \dynkin{A2}{*o*} passes the string A2, +% \dynkin{E2}{} passes the string E2. + +\def\dynkin@user@string{} +% Control string passed from user. +% For example: +% \dynkin{A}{3} passes the string 3, +% \dynkin{A}{*o*} passes the string *o*, +% \dynkin{A}{III} passes the string III. + +\def\dynkin@string{} +% \dynkin@user@string{} with some modifications to it to expand it out. + +\def\dynkin@series{A} +% Which series of root system: A,B,C,D,E,F,G + +\newcount\dynkin@rank +% Which rank of root system: 1,2,... + +\newcount\dynkin@nodes +% How many nodes (besides the zero node for affine diagrams) are there? + +\newif\ifdynkin@is@extended +% Is this an extended extended root system? + +\newif\ifdynkin@is@twisted +% Is this a twisted extended root system? + +\def\dynkin@twisted@series{0} +% Which Kac series? 0=finite, 1,2,3->infinite + +\newif\ifdynkin@label@the@roots +% Should we label the roots by the current root ordering convention? + +\newif\ifdynkin@reverse@arrows +% Should we reverse the directions of all arrows? + +\newif\ifdynkin@arrows +% Should we draw arrows on Dynkin diagrams? + +\newif\ifdynkin@left@fold +% Is the left side of the Dynkin diagram folded? + +\newif\ifdynkin@right@fold +% Is the right side of the Dynkin diagram folded? + +\newif\ifdynkin@Coxeter +% Should we draw Coxeter diagrams? + +\newif\ifdynkin@odd +% For twisted A series diagrams, is the rank odd? + +\newcount\dynkin@ply +% Maximum number of nodes arranged vertically in the folding of the Dynkin diagram + +\def\dynkin@ply@value{1} +% Default maximum number of nodes arranged vertically in the folding of the Dynkin diagram + +\def\dynkin@label@directions{} +% List of directions in which to draw the labels attached to the roots: a=above, b=below, l=left, r=right. + +\def\dynkin@current@location{(0,0)} + +\NewDocumentCommand\regurgitate{m}{#1} + +\pgfkeys{ + /Dynkin diagram/.is family, + /Dynkin diagram, + name/.estore in = \dynkin@diagram@name, + name = anonymous, + mark/.estore in = \dynkin@root@mark, + mark = *, + affineMark/.estore in = \dynkin@affine@root@mark, + affineMark = o, + edgeLength/.estore in = \dynkin@edge@length, + edgeLength = .35cm, + edge/.style={draw=black,fill=white,thin}, + makeIndefiniteEdge/.code={\dynkin@set@edge@indefinite@pair{#1}}, + indefiniteEdgeRatio/.estore in = \dynkin@indefinite@edge@ratio, + indefiniteEdgeRatio = 1.6, + indefiniteEdge/.style={draw=black,fill=white,thin,densely dotted}, + arrows/.is if = dynkin@arrows, + arrows = true, + reverseArrows/.is if = dynkin@reverse@arrows, + reverseArrows = false, + foldStyle/.style = {draw=black!40,fill=none,line width=\dynkin@root@radius}, + leftFold/.style = {}, + rightFold/.style = {}, + doubleEdges/.style = { + foldStyle/.style = { + draw=black, + double=white, + fill=none, + double distance=\dynkin@root@radius, + line width=\defaultpgflinewidth} + }, + doubleFold/.style = { + foldStyle/.style = { + draw=black, + double=black!40, + fill=none, + double distance=\dynkin@root@radius, + line width=\defaultpgflinewidth} + }, + doubleLeft/.style = { + leftFold/.style = { + draw=black, + double=white, + fill=none, + double distance=\dynkin@root@radius, + line width=\defaultpgflinewidth} + }, + doubleFoldLeft/.style = { + leftFold/.style = { + draw=black, + double=black!40, + fill=none, + double distance=\dynkin@root@radius, + line width=\defaultpgflinewidth} + }, + doubleRight/.style = { + rightFold/.style = { + draw=black, + double=white, + fill=none, + double distance=\dynkin@root@radius, + line width=\defaultpgflinewidth} + }, + doubleFoldRight/.style = { + rightFold/.style = { + draw=black, + double=black!40, + fill=none, + double distance=\dynkin@root@radius, + line width=\defaultpgflinewidth} + }, + radius/.estore in = \dynkin@root@radius, + radius=.05cm, + foldradius/.estore in = \dynkin@fold@radius, + foldradius=.3cm, + */.style = { + draw=black, + fill=black, + }, + O/.style = { + draw=black, + fill=white, + }, + X/.style = { + draw=black, + thick + }, + o/.style = { + draw=black, + fill=white, + }, + t/.style = { + draw=black, + fill=white, + }, + x/.style = { + draw=black, + }, + Coxeter/.is if = dynkin@Coxeter, + Coxeter=false, + ordering/.store in = \dynkin@ordering, + ordering = Bourbaki, + text/.style={scale=.7}, + labelMacro/.code = {\regurgitate{#1}}, + odd/.is if = dynkin@odd, + odd=false, + Kac/.style={ + ordering=Kac, + radius=.05cm, + edgeLength=.66cm, + indefiniteEdgeRatio = 3, + o/.style = + { + draw=black, + fill=white, + preaction={ + draw=white, + line width=.9mm + } + }, + mark=o, + indefiniteEdge/.style={draw=black,fill=white,thin,loosely dotted}, + }, + default/.style = { + label/.is if = dynkin@label@the@roots, + label = false, + at/.estore in = \dynkin@current@location, + at = {(0,0)}, + parabolic/.estore in = \dynkin@parabolic, + parabolic = 0, + gonality/.estore in = \dynkin@gonality, + gonality = 0, + extended/.is if = dynkin@is@extended, + extended = false, + twisted/.is if = dynkin@is@twisted, + twisted = false, + twistedSeries/.estore in = \dynkin@twisted@series, + twistedSeries = 0, + ply/.estore in = \dynkin@ply@value, + ply = 1, + fold/.style = {ply=2}, + foldleft/.is if = dynkin@left@fold, + foldleft = false, + foldright/.is if = dynkin@right@fold, + foldright = false, + }, + .search also={/tikz}, +} + +\ProcessPgfPackageOptions{/Dynkin diagram}\relax + +%% \dynkin@put@direction{<r>}{<d>} +%% Assigns to \dynkin@label@directions the direction that the label of root <r> (in default ordering) should sit from the root node location, <d>=left, right, above, below or diagonal. +\NewDocumentCommand\dynkin@put@direction{mm}% +{% + \newcount\drpo% + \drpo=\the\dynkin@nodes% + \advance\drpo by 1% + \newcount\dynkin@where% + \dynkin@where=#1% + \StrMid{\dynkin@label@directions}{1}{\the\dynkin@where}[\dynkin@start]% + \advance\dynkin@where by 2 + \StrMid{\dynkin@label@directions}{\the\dynkin@where}{\the\drpo}[\dynkin@end]% + \IfStrEqCase{#2}{% + {left}{\xdef\dynkin@label@directions{\dynkin@start l\dynkin@end}}% + {right}{\xdef\dynkin@label@directions{\dynkin@start r\dynkin@end}}% + {above}{\xdef\dynkin@label@directions{\dynkin@start a\dynkin@end}}% + {below}{\xdef\dynkin@label@directions{\dynkin@start b\dynkin@end}}% + {diagonal}{\xdef\dynkin@label@directions{\dynkin@start d\dynkin@end}}% + }% + [\ClassError{Dynkin diagrams}{Unrecognized direction: ``#2'' in Dynkin diagram \dynkin@user@series{\dynkin@user@string}}{}]% +}% + + +\xdef\replace@DR{} + +% \expand@Dynkin@Roots@By@Char{<c>}, +% for example if <c> is the letter x, expands out any expression like +% x7 in \dynkin@string into 7 copies of the letter x. +\NewDocumentCommand\expand@Dynkin@Roots@By@Char{m}% +{% + \xdef\replace@DR{} + \foreach \i in {0,...,9}% {% - \global\dynkinrank=#3% - \gdef\dynkinSatake{*}% + \StrSubstitute[0]{\dynkin@string}{#1\i}{\replace@DR}[\temp@DR]% + \xdef\dynkin@string{\temp@DR}% + \xdef\replace@DR{\replace@DR #1}% }% +}% + +% \expand@Dynkin@Roots@Digits{} expands out any expression like x7 in \dynkin@roots into 7 copies of the letter x, and so on for any letter which is not a digit. +\NewDocumentCommand\expand@Dynkin@Roots@Digits{}% +{% + \edef\current@string{\dynkin@string} + \StrLen{\current@string}[\string@len] + \foreach \j in {1,...,\string@len}% {% - \IfStrEqCase{#2}% + \StrChar{\current@string}{\j}[\cccc]% + \IfInteger{\cccc}% + {}% + {% + \expand@Dynkin@Roots@By@Char{\cccc}% + }% + }% +}% + +% \dynkin@integer@rank{} expands a \dynkin@string 3 into ***, i.e. +% writes the given number <n> of copies of the default root mark into the string \dynkin@string. +\NewDocumentCommand\dynkin@integer@rank{}% +{% + \global\dynkin@rank=\dynkin@string% + \global\dynkin@nodes=\dynkin@string% + \ifdynkin@is@twisted% + \IfStrEqCase{\dynkin@series}% {% {A}% {% - \IfStrEqCase{#3}% - {% - {*}{ }% - {I}{ }% - {II}{}% - {III}{}% - {IV} {}% - }% - [\errorRank]% - }% - {B}% - {% - \IfStrEqCase{#3}% - {% - {*}{ }% - {I}{}% - {II} {}% - }% - [\errorRank]% - }% - {C}% - {% - \IfStrEqCase{#3}% - {% - {*}{ }% - {I}{}% - {II} {}% - }% - [\errorRank]% + \divide\dynkin@nodes by 2% + \ifodd\dynkin@rank% + \global\dynkin@oddtrue% + \advance\dynkin@nodes by 1% + \else% + \global\dynkin@oddfalse% + \fi% }% {D}% {% - \IfStrEqCase{#3}% + \IfStrEqCase{\dynkin@twisted@series}% {% - {*}{ }% - {I}{ }% - {II} {}% - {III}{}% + {2}% + {% + \global\advance\dynkin@nodes by -1% + }% + {3}% + {% + \IfStrEq{\dynkin@string}{4}% + {% + \global\dynkin@nodes=2% + }% + {% + \dynkin@error@series% + }% + }% }% - [\errorRank]% + [\dynkin@error@series]% }% {E}% {% - \IfStrEqCase{#3}% + \IfStrEq{\dynkin@twisted@series}{2}% {% - {I}{ \global\dynkinrank=6}% - {II}% + \IfStrEq{\dynkin@string}{6}% {% - \global\dynkinfoldedtrue% - \global\dynkinrank=6% + \global\dynkin@nodes=4% }% - {III}% {% - \global\dynkinfoldedtrue% - \global\dynkinrank=6% + \dynkin@error@series% }% - {IV}% + }% + {% + \dynkin@error@series% + }% + }% + }% + \fi% + \xdef\dynkin@string{\repeatCharacter{\the\dynkin@nodes}{\dynkin@root@mark}}% +}% + +\NewDocumentCommand\dynkin@clear@indefinite@edge@list{}% +{% + \xdef\dynkin@indefinite@edge@list{}% +}% + +\NewDocumentCommand\dynkin@set@edge@indefinite{mm}% +{% + \newcount\first% + \first=#1\relax% + \newcount\second% + \second=#2\relax% + \ifnum\the\first<\the\second% + \listxadd\dynkin@indefinite@edge@list{\the\first,\the\second}% + \else% + \listxadd\dynkin@indefinite@edge@list{\the\second,\the\first}% + \fi% +}% + +\NewDocumentCommand\dynkin@set@edge@indefinite@pair{>{\SplitArgument{1}{-}}m}% +{% +\dynkin@set@edge@indefinite#1 +}% + +\newif\ifdynkin@is@indefinite@edge + +\NewDocumentCommand\dynkin@typeout@indefinite@edge@list{}% +{% + \renewcommand*{\do}[1]{\typeout{##1}}% + \typeout{Indefinite edges: [}\dolistloop{\dynkin@indefinite@edge@list}\typeout{]}% +}% + + +%% \dynkin@is@edge@indefinite{<p>}{<q>} sets the global if \ifdynkin@is@indefinite@edge to true or false +%% depending on whether there is an indefinite edge between roots <p> and <q>. +%% The starred form uses Bourbaki ordering. +\NewDocumentCommand\dynkin@is@edge@indefinite{smm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#2}{#3}% + }% + {% + \@fromRoot=#2% + \@toRoot=#3% + }% + % Next we sort the order, since edges are stored as undirected edges. + \newcount\first% + \global\first=\@fromRoot\relax% + \newcount\second% + \global\second=\@toRoot\relax% + \ifnum\the\second<\the\first% + \global\first=\@toRoot\relax% + \global\second=\@fromRoot\relax% + \fi% + \global\dynkin@is@indefinite@edgefalse\relax% + \renewcommand*{\do}[1]{% + \IfStrEq{##1}{\the\first,\the\second}% + {\global\dynkin@is@indefinite@edgetrue\listbreak}% + {}}% + \dolistloop{\dynkin@indefinite@edge@list}% +}% + +% \dynkin@grok@indefinite@edges{} reads the input string <s> found when you write \dynkin{<c>}{<s>}, and +% interprets it to say which edges are indefinite edges. +\NewDocumentCommand\dynkin@grok@indefinite@edges{}% +{% + \newcount\rootnum + \rootnum=1 + \newcount\dynkin@string@length + \StrLen{\dynkin@string}[\temp]% + \dynkin@string@length=\temp + \foreach \i in {2,...,\the\dynkin@string@length}% + {% + \StrChar{\dynkin@string}{\i}[\c]% + \IfStrEq{\c}{.}% + {% + \newcount\rootnumpo% + \rootnumpo=\rootnum% + \advance\rootnumpo by 1\relax% + \ifnum\the\rootnum<\the\dynkin@nodes% + \dynkin@set@edge@indefinite{\rootnum}{\rootnumpo}% + \fi% + }% + {% + \global\advance\rootnum by 1% + }% + }% +}% + +\xdef\spacy{ } + +\xdef\questionMarks{} + +\NewDocumentCommand\dynkin@clear@label@directions{}% +{% + \xdef\dynkin@label@directions{}% +}% + + +\NewDocumentCommand\dynkin@set@default@label@directions{}% +{% + \newcount\drpo% + \drpo=\the\dynkin@nodes% + \advance\drpo by 1\relax% + \xdef\dynkin@label@directions{\repeatCharacter{\the\drpo}{?}}% +}% + +\newlength{\defaultpgflinewidth}% + + +% \@dynkin[<s>]{<X>}[<sb>]{<Y>} +% Draws a complete Dynkin diagram of +% series <X> and +% subseries <sb>, +% described by the string <Y> +% with TikZ options specified by <s>. +\NewDocumentCommand\@dynkin{O{}mO{0}m}% +{% + \setlength{\defaultpgflinewidth}{\pgflinewidth}% + \global\defaultpgflinewidth=\defaultpgflinewidth\relax% + \dynkin@clear@indefinite@edge@list% + \xdef\dynkin@parabolic{0}% + \pgfkeys{/Dynkin diagram, default, #1}% + \xdef\dynkin@user@series{#2}% + \xdef\dynkin@twisted@series{#3}% + \xdef\dynkin@user@string{#4}% + \global\dynkin@ply=\dynkin@ply@value\relax% + \xdef\dynkin@indefinite@edge@length{(\dynkin@edge@length*\dynkin@indefinite@edge@ratio)}\relax% + \xdef\dynkin@series{#2}% + \IfStrEq{\dynkin@diagram@name}{anonymous}% + {% + \xdef\dynkin@root@name{root\spacy}% + }% + {% + \xdef\dynkin@root@name{\dynkin@diagram@name\spacy root\spacy}% + }% + \dynkin@grok@series% + \IfSubStr{ABCDEFGHI}{\dynkin@series}{}{\dynkin@error@series}% + \xdef\dynkin@string{#4} + \IfInteger{\dynkin@string}% + {% + \dynkin@integer@rank% + }% + {% + % Turn Satake codes into Dynkin diagram expressions in \dynkin@string. + \dynkin@grok@Satake@codes% + }% + % Expand out any digits in \dynkin@string into multiples of the various root marks. + \expand@Dynkin@Roots@Digits% + % Assign to \dynkin@roots the input string \dynkin@string with all . symbols removed, + % so we only get the symbols representing the marks for the various roots. + \StrDel{\dynkin@string}{.}[\temp]% + \xdef\dynkin@roots{\temp}% + \StrLen{\dynkin@roots}[\temp]% + \global\dynkin@nodes=\temp\relax% + \dynkin@grok@indefinite@edges% + \dynkin@find@rank{}% + \dynkin@cross@out@parabolics{}% + \dynkin@set@default@label@directions{}% + \check@Dynkin@diagram{}% + \node (Dynkin current) at \dynkin@current@location{};% + \ifdynkin@is@twisted% + \csname twisted\dynkin@series dynkin\endcsname% + \else% + \ifdynkin@is@extended% + \csname extended\dynkin@series dynkin\endcsname% + \else% + \csname\dynkin@series dynkin\endcsname% + \fi% + \fi% + \dynkinRefreshRoots% +}% + +%% We know the number of nodes; lets find the rank. +\NewDocumentCommand\dynkin@find@rank{}% +{% + \global\dynkin@rank=\the\dynkin@nodes% + \ifdynkin@is@twisted% + \IfStrEqCase{\dynkin@series}% + {% + {A}% + {% + \multiply\dynkin@rank by 2% + \ifdynkin@odd% + \advance\dynkin@rank by -1% + \fi% + }% + {D}% + {% + \IfStrEqCase{\dynkin@twisted@series}% + {% + {2} {% - \global\dynkinrank=6% + \advance\dynkin@rank by 1% }% - {V}% + {3} {% - \global\dynkinrank=7% + \advance\dynkin@rank by 2% }% - {VI}% - {% - \global\dynkinrank=7% + }% + }% + {E}% + {% + \advance\dynkin@rank by 2% + }% + }% + \fi% +}% + +%% \dynkin@grok@series +%% Interprets the dynkin@series, to see if it is extended, twisted, and what twisted series it is. +\NewDocumentCommand\dynkin@grok@series{}% +{% + \newcount\lenny + \StrLen{\dynkin@series}[\lenny] + \ifnum\lenny>1% + \dynkin@error@series% + \fi + \edef\series{\dynkin@series} + \IfStrEqCase{\dynkin@twisted@series}% + {% + {0}{}% + {1}{ \global\dynkin@is@extendedtrue}% + {2}{% + \IfSubStr{ADE}{\dynkin@series}% + {% + \global\dynkin@is@twistedtrue% + }% + {% + \dynkin@error@series% + }% + }% + {3}{% + \IfStrEq{\dynkin@series}{D}% + {% + \global\dynkin@is@twistedtrue% + }% + {% + \dynkin@error@series% + }% + }% + }% + [\dynkin@error@series]% +}% + + +\newif\ifdynkin@Satake@diagram + +\NewDocumentCommand\dynkin@grok@Satake@codes{}% +{% + \ifdynkin@is@extended% + \else% + \ifdynkin@is@twisted% + \else% + \global\dynkin@Satake@diagramtrue% + \fi% + \fi% + \IfStrEqCase{\dynkin@series}% + {% + {A}% + {% + \IfStrEqCase{\dynkin@string}% + {% + {even}{\gdef\dynkin@string{***.***}\global\dynkin@oddfalse\global\dynkin@Satake@diagramfalse}% + {odd}{\gdef\dynkin@string{****.***}\global\dynkin@oddtrue\global\dynkin@Satake@diagramfalse}% + {}{\gdef\dynkin@string{**.**}\global\dynkin@Satake@diagramfalse}% + {I}{ \gdef\dynkin@string{oo.oo}}% + {II}{\gdef\dynkin@string{*o*.o*}}% + {IIIa}{\global\dynkin@ply=2\gdef\dynkin@string{oo.o**.**o.oo}}% + {IIIb}{\global\dynkin@ply=2\gdef\dynkin@string{oo.ooo.oo}}% + {IV} {\global\dynkin@ply=2\gdef\dynkin@string{o*.*o}}% + }% + [\global\dynkin@Satake@diagramfalse]% + }% + {B}% + {% + \IfStrEqCase{\dynkin@string}% + {% + {}{% + \global\dynkin@Satake@diagramfalse% + \ifdynkin@Coxeter% + \gdef\dynkin@string{***.***}% + \else% + \ifdynkin@is@extended% + \gdef\dynkin@string{***.***}% + \else% + \gdef\dynkin@string{**.***}% + \fi% + \fi% }% - {VII}% - {% - \global\dynkinrank=7% + {I}{\gdef\dynkin@string{oo.o*.**}}% + {II}{\gdef\dynkin@string{o*.**}}% + }% + [\global\dynkin@Satake@diagramfalse]% + }% + {C}% + {% + \IfStrEqCase{\dynkin@string}% + {% + {}{% + \global\dynkin@Satake@diagramfalse% + \ifdynkin@Coxeter% + \gdef\dynkin@string{***.***}% + \else% + \gdef\dynkin@string{**.***}% + \fi% }% - {VIII}% + {I}{\gdef\dynkin@string{oo.oo}}% + {IIa}{\gdef\dynkin@string{*o*.o*.**}}% + {IIb}{\gdef\dynkin@string{*o*.o*o}}% + }% + [\global\dynkin@Satake@diagramfalse]% + }% + {D}% + {% + \IfStrEqCase{\dynkin@string}% + {% + {}{% + \global\dynkin@Satake@diagramfalse% + \ifdynkin@is@extended% + \ifnum\dynkin@ply=4% + \gdef\dynkin@string{****.*.*****} + \else% + \gdef\dynkin@string{***.****}% + \fi% + \else% + \ifdynkin@is@twisted% + \IfStrEqCase{\dynkin@twisted@series}% + {% + {2}{ \gdef\dynkin@string{**.***}}% + {3}{\gdef\dynkin@string{***}}% + }% + [\dynkin@error@series]% + \else% + \gdef\dynkin@string{**.****}% + \fi% + \fi% + }% + {Ia}{\gdef\dynkin@string{oo.o*.***}}% + {Ib}{\global\dynkin@ply=2\gdef\dynkin@string{o.ooo}}% + {Ic}{\gdef\dynkin@string{o.ooo}}% + {II} {\gdef\dynkin@string{o*.***}}% + {IIIa}{\gdef\dynkin@string{*o*.o*o}}% + {IIIb}{\global\dynkin@ply=2\gdef\dynkin@string{*o*.o*oo}}% + }% + [\global\dynkin@Satake@diagramfalse]% + }% + {E}% + {% + \IfStrEqCase{\dynkin@string}% + {% + {}% + {% + \global\dynkin@Satake@diagramfalse% + \IfStrEq{\dynkin@twisted@series}{2}% {% - \global\dynkinrank=8% + \gdef\dynkin@string{*****}% }% - {XI}% {% - \global\dynkinrank=8% + \dynkin@error@series% }% }% - [\errorRank]% + {I}{ \global\dynkin@rank=6\gdef\dynkin@string{oooooo}}% + {II} {\global\dynkin@ply=2\gdef\dynkin@string{oooooo}}% + {III}{\global\dynkin@ply=2\gdef\dynkin@string{oo***o}}% + {IV} {\gdef\dynkin@string{oo***o}}% + {V}{ \gdef\dynkin@string{ooooooo}}% + {VI} {\gdef\dynkin@string{o*oo*o*} }% + {VII}{\gdef\dynkin@string{o****oo}}% + {VIII}{\gdef\dynkin@string{oooooooo}}% + {IX} {\gdef\dynkin@string{o****ooo}}% }% - {F}% + [\global\dynkin@Satake@diagramfalse]% + }% + {F}% + {% + \global\dynkin@rank=4% + \IfStrEqCase{\dynkin@string}% {% - \global\dynkinrank=4% - \IfStrEqCase{#3}% - {% - {I}{ }% - {II} {}% - }% - [\errorRank]% + {I}{ \gdef\dynkin@string{oooo}}% + {II} {\gdef\dynkin@string{***o}}% }% - {G}% + [\global\dynkin@Satake@diagramfalse]% + }% + {G}% + {% + \IfStrEqCase{\dynkin@string}% {% - \global\dynkinrank=2% - \IfStrEqCase{#3}% - {% - {I}{ }% - }% - [\errorRank]% + {I}{\gdef\dynkin@string{oo}}% }% - {H}% + [\global\dynkin@Satake@diagramfalse]% + }% + {H}% + {% + \IfStrEqCase{\dynkin@string}% {% - \IfStrEqCase{#3}% - {% - {*}% - {% - }% - }% - [\errorRank]% + {}{\gdef\dynkin@string{**}}% }% - {I}% + [\global\dynkin@Satake@diagramfalse]% + }% + {I}% + {% + \IfStrEqCase{\dynkin@string}% {% - \IfStrEqCase{#3}% + {}{\gdef\dynkin@string{**}}% {% - {*}% - {% - }% }% - [\errorRank]% }% + [\global\dynkin@Satake@diagramfalse]% }% - [\errorSeries]% }% - \checkDynkinDiagram% - \ifisaffine% - \csname affine#2dynkin\endcsname% + [\dynkin@error@series]% + \ifdynkin@Satake@diagram% \else% - \csname#2dynkin\endcsname% + \StrSubstitute{\dynkin@string}{*}{\dynkin@root@mark}[\temp]% + \xdef\dynkin@string{\temp}% \fi% - \iflabeltheroots\dynkinprintlabels\fi% - \ifwerefolded - \global\dynkinfoldedtrue - \else - \global\dynkinfoldedfalse - \fi }% -%% \stringcharacterinposition{<s>}{<n>} -%% -> the element of string <s> in position <n>. -\ExplSyntaxOn -\cs_new:Npn \stringcharacterinposition #1 #2 -{ -\str_item:fn { #1 } { #2 } -} -\cs_generate_variant:Nn \str_item:nn {f} -\ExplSyntaxOff +\NewDocumentCommand\dynkin@error@root@ordering{} +{% + \ClassError% + {Dynkin diagrams}% + {Unrecognized root ordering: ``\dynkin@ordering'' + in Dynkin diagram \dynkin@user@series{\dynkin@user@string}}% + {}% +}% + +\NewDocumentCommand\dynkin@error@rank{}% +{% + \ClassError% + {Dynkin diagrams}% + {Unrecognized \dynkin@user@series\spacy series rank: + ``\the\dynkin@rank'' in Dynkin diagram \dynkin@user@series{\dynkin@user@string}}% + {}% +}% -\NewDocumentCommand\errorRootOrdering{} +\NewDocumentCommand\dynkin@error@series{}% {% - \ClassWarning{Unrecognized root ordering: ``\dynkinordering'' in Dynkin diagram}% + \ClassError% + {Dynkin diagrams}% + {Unrecognized series ``\dynkin@user@series'' + in Dynkin diagram \dynkin@user@series{\dynkin@user@string}}% + {}% }% -\NewDocumentCommand\errorRank{}% + +\NewDocumentCommand\dynkin@error@ply{} {% - \ClassWarning{Unrecognized \dynkinseries{} series rank: ``\the\dynkinrank'' in Dynkin diagram}% + \ClassError% + {Dynkin diagrams}% + {Unrecognized ply: ``\the\dynkin@ply'' + in Dynkin diagram \dynkin@user@series{\dynkin@user@string}}% + {}% }% -\NewDocumentCommand\errorSeries{}% + +%% \check@Dynkin@Roots +%% Raises error messages for erroneous input in the list of Dynkin roots. +\NewDocumentCommand\check@Dynkin@Roots{}% {% - \ClassWarning{Unrecognized series ``\dynkinseries{}'' in Dynkin diagram}% + \foreach \i in {1,...,\the\dynkin@nodes}% + {% + \StrChar{\dynkin@roots}{\i}[\cccc]% + \IfSubStr{*OXotx}{\cccc}% + {% + }% + {%else + \ClassError% + {Dynkin diagrams}% + {Unrecognized Dynkin diagram root mark: + ``\cccc'' in Dynkin diagram \dynkin@user@series{\dynkin@user@string}}% + {}% + }% + }% }% -%% \checkDynkinDiagram -%% -> +%% \check@Dynkin@diagram %% Raises error messages for erroneous inputs. -\NewDocumentCommand\checkDynkinDiagram{}% +\NewDocumentCommand\check@Dynkin@diagram{}% {% - \IfStrEqCase{\dynkinordering}% + \IfSubStr{1234}{\the\dynkin@ply}{}{\dynkin@error@ply}% + \check@Dynkin@Roots% + \IfStrEqCase{\dynkin@ordering}% {% {Adams}{}% {Bourbaki}{}% {Carter}{}% {Dynkin}{}% {Kac}{}% + {TestOrder}{}% }% - [\ClassWarning{Unrecognized label ordering: ``\dynkinordering'' in Dynkin diagram}]% - \IfStrEqCase{\dynkinseries}% + [\ClassError% + {Dynkin diagrams}% + {Unrecognized label ordering: ``\dynkin@ordering'' + in Dynkin diagram \dynkin@user@series{\dynkin@user@string}}% + {}]% + \IfStrEqCase{\dynkin@series}% {% {A}{}% {B}{}% {C}{}% - {D}{}% + {D}{}% {E}% {% - \ifnum\dynkinrank=6% - \else% - \ifnum\dynkinrank=7% + \ifnum\dynkin@nodes=5% + \ifnum\dynkin@rank=6% + \IfStrEq{\dynkin@twisted@series}{2}% + {% + }% + {% + \dynkin@error@rank% + }% + \else% + \dynkin@error@rank% + \fi% + \else + \ifnum\dynkin@rank=6% \else% - \ifnum\dynkinrank=8% + \ifnum\dynkin@rank=7% \else% - \errorRank% + \ifnum\dynkin@rank=8% + \else% + \dynkin@error@rank% + \fi% \fi% \fi% \fi% }% {F}% {% - \ifnum\dynkinrank=4% + \ifnum\dynkin@rank=4% \else% - \errorRank% + \dynkin@error@rank% \fi% }% {G}% {% - \ifnum\dynkinrank=2% + \ifnum\dynkin@rank=2% \else% - \errorRank% + \dynkin@error@rank% \fi% }% {H}{}% {I}{}% }% - [\errorSeries]% + [\dynkin@error@series]% }% -% We store the number of a root, converted to the current root ordering convention, here. -\newcount\RootNumber % A slight headache: all of the routines that draw Dynkin diagrams are written % in Bourbaki ordering. We store the roots in the current ordering. % So when we draw edges, we need to convert from the Bourbaki ordering each time. % We store the conversions here. +\newcount\RootNumber \newcount\@fromRoot \newcount\@toRoot +%% \swapRootIfInLastTwoRoots{<r>} +%% If the input root <r> is one of the last two roots, then put the other in \RootNumber, otherwise +%% let \RootNumber be <r>. +\NewDocumentCommand\swapRootIfInLastTwoRoots{m}% +{% + \ifnum\dynkin@rank>1% + \newcount\drmo\relax% + \drmo=\dynkin@rank\relax% + \advance\drmo by -1\relax% + \ifnum\dynkin@rank=#1% + \global\RootNumber=\the\drmo\relax% + \else% + \ifnum\drmo=#1% + \global\RootNumber=\the\dynkin@rank\relax% + \else% + \global\RootNumber=#1\relax% + \fi% + \fi% + \else% + \global\RootNumber=#1\relax% + \fi% +}% + +%% \convertRootNumber{<n>} +%% Converts <n> from Bourbaki ordering to the current ordering, storing the result in a count called \RootNumber. +\NewDocumentCommand\convertRootNumber{m}% +{% + \IfStrEq{#1}{0}% + {% + \global\RootNumber=0% + }% + {% + \IfStrEqCase{\dynkin@series}% + {% + {A}% + {% + \IfStrEqCase{\dynkin@ordering}% + {% + {TestOrder}% + {% + \RootNumber=#1 + \advance\RootNumber by 1 + \ifnum\RootNumber>\the\dynkin@rank% + \RootNumber=1% + \fi% + }% + }% + [\global\RootNumber=#1]% + }% + {D}% + {% + \IfStrEqCase{\dynkin@ordering}% + {% + {Adams}{\swapRootIfInLastTwoRoots{#1}}% + {Dynkin}{\swapRootIfInLastTwoRoots{#1}}% + {Kac}{\swapRootIfInLastTwoRoots{#1}}% + }% + [\global\RootNumber=#1]% + }% + {E}% + {% + \ifdynkin@is@twisted% + \global\RootNumber=#1% + \else% + \ifnum\dynkin@rank=6% + \IfStrEqCase{\dynkin@ordering}% + {% + {Adams}{\global\RootNumber=\stringCharacterInPosition{152436}{#1}}% + {Carter}{\global\RootNumber=\stringCharacterInPosition{142356}{#1}}% + {Dynkin}{\global\RootNumber=\stringCharacterInPosition{162345}{#1}}% + {Kac}{\global\RootNumber=\stringCharacterInPosition{162345}{#1}}% + }% + [\global\RootNumber=#1]% + \else% + \ifnum\dynkin@rank=7% + \IfStrEqCase{\dynkin@ordering}% + {% + {Adams}{\global\RootNumber=\stringCharacterInPosition{6354217}{#1}}% + {Carter}{\global\RootNumber=\stringCharacterInPosition{7564321}{#1}}% + {Dynkin}{\global\RootNumber=\stringCharacterInPosition{1723456}{#1}}% + {Kac}{\global\RootNumber=\stringCharacterInPosition{1723456}{#1}}% + }% + [\global\RootNumber=#1]% + \else% + \ifnum\dynkin@rank=8% + \IfStrEqCase{\dynkin@ordering}% + {% + {Adams}{\global\RootNumber=\stringCharacterInPosition{13245678}{#1}}% + {Carter}{\global\RootNumber=\stringCharacterInPosition{86754321}{#1}}% + {Dynkin}{\global\RootNumber=\stringCharacterInPosition{18234567}{#1}}% + {Kac}{\global\RootNumber=\stringCharacterInPosition{78654321}{#1}}% + }% + [\global\RootNumber=#1]% + \else% + \fi% + \fi% + \fi% + \fi% + }% + {F}% + {% + \IfStrEqCase{\dynkin@ordering}% + {% + {Adams}{\global\RootNumber=\stringCharacterInPosition{4321}{#1}}% + }% + [\global\RootNumber=#1]% + }% + {G}% + {% + \IfStrEqCase{\dynkin@ordering}% + {% + {Carter}{\global\RootNumber=\stringCharacterInPosition{21}{#1}}% + {Dynkin}{\global\RootNumber=\stringCharacterInPosition{21}{#1}}% + }% + [\global\RootNumber=#1]% + }% + }% + [\global\RootNumber=#1]% + }% +}% + %% \convertRootPair{<p>}{<q>} -%% -> %% Stores conversions in \@fromRoot and \@toRoot. \NewDocumentCommand\convertRootPair{mm} {% @@ -727,759 +2310,1379 @@ \@toRoot=\RootNumber% }% +\ExplSyntaxOn +\NewDocumentCommand\moduloInt{mm}{\int_mod:nn{#1}{#2}} +\ExplSyntaxOff + %% \testbit{<n>}{<b>}{<f>}{<g>} %% If bit number <b> of <n> is 1 then expand <f> else expand <g>. -\newcommand*{\testbit}[4]% +\NewDocumentCommand\testbit{mmmm}% +{% + \newcount\x\relax% + \x=#1\relax% + \newcount\whichbit\relax% + \whichbit=#2\relax% + \ifnum\whichbit>0% + \foreach \i in {1,...,#2}% + {% + \global\divide \x by 2% + }% + \fi% + \xdef\temp{\moduloInt{\the\x}{2}}% + \x=\temp\relax% + \ifnum\the\x=1 #3\else #4\fi% +}% + +\NewDocumentCommand\dynkin@put@cross{m}% {% - \pgfmathparse{int(mod(div(#1,2^(#2)),2))}% - \let\tf\pgfmathresult% - \IfStrEq{\tf}{1}{#3}{#4}% + \newcount\dynkin@where% + \dynkin@where=#1% + \StrMid{\dynkin@roots}{1}{#1}[\dynkin@start]% + \advance\dynkin@where by 1% + \StrMid{\dynkin@roots}{\the\dynkin@where}{\the\dynkin@nodes}[\dynkin@end]% + \xdef\dynkin@roots{\dynkin@start x\dynkin@end}% }% -%% \placeRoot{<n>}{<x>}{<y>} -%% -> -%% Tell TikZ where to place node <n> (in Bourbaki ordering) for a root of a Dynkin diagram. Draws nothing. -%% Starred form swaps label positions. -\NewDocumentCommand\placeRoot{smmm}% +\NewDocumentCommand\dynkin@cross@out@parabolics{}% +{% + \IfInteger{\dynkin@parabolic}% + {% + \IfStrEq{\dynkin@parabolic}{0}% + {% + }% + {% + \newcount\drmo\relax% + \drmo=\the\dynkin@nodes\relax% + \advance\drmo by -1\relax% + \foreach \b in {0,...,\the\drmo}% + {% + \testbit{\dynkin@parabolic}{\b}{\dynkin@put@cross{\b}}{}% + }% + }% + }% +}% + +\NewDocumentCommand\dynkinMoveToRoot{sm}% {% - \convertRootNumber{#2}% - \node (root \the\RootNumber) at ({(#3)*\dynkinedgelength},{(#4)*\dynkinedgelength}) {};% \IfBooleanTF{#1}% {% - \node[above] (root label \the\RootNumber)% - at ({(#3)*\dynkinedgelength},{((#4)*\dynkinedgelength)+2*\dynkinradius}) {};% - \node[below] (root label swap \the\RootNumber)% - at ({(#3)*\dynkinedgelength},{((#4)*\dynkinedgelength)-2*\dynkinradius}) {};% + \convertRootNumber{#2}% }% {% - \node[above] (root label swap \the\RootNumber)% - at ({(#3)*\dynkinedgelength},{((#4)*\dynkinedgelength)+2*\dynkinradius}) {};% - \node[below] (root label \the\RootNumber)% - at ({(#3)*\dynkinedgelength},{((#4)*\dynkinedgelength)-2*\dynkinradius}) {};% + \global\RootNumber=#2 }% + \node (Dynkin current) at (\dynkin@root@name \the\RootNumber){};% }% -%% \placeRootHorizontalLabels{<n>}{<x>}{<y>} -%% -> -%% Tell TikZ where to place node <n> (in Bourbaki ordering) for a root of a Dynkin diagram. Draws nothing. -%% Places labels to the left or right of the root. -%% Starred form swaps label positions. -\NewDocumentCommand\placeRootHorizontalLabels{smmm}% +%% \dynkinPlaceRootHere{<n>}{<L>} +%% \dynkinPlaceRootHere*{<n>}{<L>} +%% Tell TikZ to place node <n> for a root of a Dynkin diagram at the current +%% cursor location. Draws nothing. +%% <L>=label positioning: above, below, left, right +%% Starred form converts <n> from Bourbaki ordering to default ordering. +\NewDocumentCommand\dynkinPlaceRootHere{smm}% {% - \convertRootNumber{#2}% - \node (root \the\RootNumber) at ({(#3)*\dynkinedgelength},{(#4)*\dynkinedgelength}) {};% \IfBooleanTF{#1}% {% - \node[left] (root label \the\RootNumber)% - at ({((#3)*\dynkinedgelength)-\dynkinradius},{(#4)*\dynkinedgelength}) {};% - \node[right] (root label swap \the\RootNumber)% - at ({((#3)*\dynkinedgelength)+\dynkinradius},{(#4)*\dynkinedgelength}) {};% + \convertRootNumber{#2}% }% {% - \node[left] (root label swap \the\RootNumber)% - at ({((#3)*\dynkinedgelength)-\dynkinradius},{(#4)*\dynkinedgelength}) {};% - \node[right] (root label \the\RootNumber)% - at ({((#3)*\dynkinedgelength)+\dynkinradius},{(#4)*\dynkinedgelength}) {};% + \global\RootNumber=#2 }% + \node (\dynkin@root@name \the\RootNumber) at (Dynkin current) {};% + \dynkin@put@direction{\the\RootNumber}{#3}% }% -%% \Adynkinnodes -%% -> -%% Tell TikZ where to place the nodes for an A series Dynkin diagram. Draws nothing. -\newcommand*{\Adynkinnodes}% +%% \dynkinPlaceRootRelativeTo{<p>}{<q>}{<d>}{<L>} +%% \dynkinPlaceRootRelativeTo*{<p>}{<q>}{<d>}{<L>} +%% Tell TikZ to place node <p> for a root of a Dynkin diagram at a location +%% in direction <d> from root <q>. Draws nothing. +%% <L> is the label position: above, below, left, right. +%% <d> is the direction from <q>: +%% west,east,south,north, +%% northeast,northwest,southeast,southwest, +%% southfold,northfold, +%% southeastfold,southwestfold,northeastfold,northwestfold. +%% Starred form is in Bourbaki root ordering; otherwise default ordering. +\NewDocumentCommand\dynkinPlaceRootRelativeTo{smmmm}% {% - \ifdynkinfolded% - \newcount\halfrank% - \halfrank=\dynkinrank% - \divide\halfrank by 2% - \newcount\countdown% - \countdown=\dynkinrank% - \ifodd\dynkinrank% - \foreach \b in {1,...,\the\halfrank}% - {% - \placeRoot*{\b}{\b}{1}% - \placeRoot{\the\countdown}{\b}{-1}% - \ifdynkinarrows% - \ifnum\dynkinrank>1% - \dynkinfoldarrow*{\b}{\the\countdown}% - \fi% - \fi% - \global\advance\countdown by -1% - }% - \advance\halfrank by 1% - \placeRootHorizontalLabels{\the\halfrank}{\the\halfrank}{0}% - \else% - \foreach \b in {1,...,\the\halfrank}% - {% - \placeRoot*{\b}{\b}{1}% - \placeRoot{\the\countdown}{\b}{-1}% - \ifdynkinarrows% - \dynkinfoldarrow*{\b}{\the\countdown} % - \fi% - \global\advance\countdown by -1% - }% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#2}% + }% + {% + \global\@fromRoot=#3% + \global\@toRoot=#2% + }% + \dynkin@is@edge@indefinite{\@fromRoot}{\@toRoot}% + \ifdynkin@is@indefinite@edge% + \xdef\dynkin@distance{\dynkin@indefinite@edge@length} + \else + \xdef\dynkin@distance{\dynkin@edge@length} + \fi + \IfStrEqCase{#4}% + {% + {west}{\xdef\x{-\dynkin@distance}\xdef\y{0}}% + {east}{\xdef\x{\dynkin@distance}\xdef\y{0}}% + {south}{\xdef\x{0}\xdef\y{-\dynkin@distance}}% + {north}{\xdef\x{0}\xdef\y{\dynkin@distance}}% + {southeast}{\xdef\x{cos(-60)*\dynkin@distance}\xdef\y{sin(-60)*\dynkin@distance}}% + {southwest}{\xdef\x{cos(240)*\dynkin@distance}\xdef\y{sin(240)*\dynkin@distance}}% + {northeast}{\xdef\x{cos(60)*\dynkin@distance}\xdef\y{sin(60)*\dynkin@distance}}% + {northwest}{\xdef\x{cos(120)*\dynkin@distance}\xdef\y{sin(120)*\dynkin@distance}}% + {southeastfold}{\xdef\x{\dynkin@fold@radius}\xdef\y{-\dynkin@fold@radius}}% + {southwestfold}{\xdef\x{-\dynkin@fold@radius}\xdef\y{-\dynkin@fold@radius}}% + {northeastfold}{\xdef\x{\dynkin@fold@radius}\xdef\y{\dynkin@fold@radius}}% + {northwestfold}{\xdef\x{-\dynkin@fold@radius}\xdef\y{\dynkin@fold@radius}}% + {northfold}{\xdef\x{0}\xdef\y{2*\dynkin@fold@radius}}% + {southfold}{\xdef\x{0}\xdef\y{-2*\dynkin@fold@radius}}% + }% + \node (Dynkin current) at ($(\dynkin@root@name \the\@fromRoot)+({\x},{\y})$){}; + \dynkinPlaceRootHere{\@toRoot}{#5}% +}% + +%% \dynkinEast +%% Moves the TikZ cursor one edge to the right. +%% Starred form for an indefinite edge. +\NewDocumentCommand\dynkinEast{s}% +{% + \xdef\distance{\IfBooleanTF{#1}{\dynkin@indefinite@edge@length}{\dynkin@edge@length}} + \node (Dynkin current) at ($(Dynkin current)+({\distance},0)$) {};% +}% + + + +%% \dynkinWest +%% Moves the TikZ cursor one edge to the left. +%% Starred form for an indefinite edge. +\NewDocumentCommand\dynkinWest{s}% +{% + \xdef\distance{\IfBooleanTF{#1}{\dynkin@indefinite@edge@length}{\dynkin@edge@length}} + \node (Dynkin current) at ($(Dynkin current)+({-\distance},0)$) {};% +}% + +%% \dynkinNorth +%% Moves the TikZ cursor one edge up. +%% Starred form for an indefinite edge. +\NewDocumentCommand\dynkinNorth{s}% +{% + \xdef\distance{\IfBooleanTF{#1}{\dynkin@indefinite@edge@length}{\dynkin@edge@length}} + \node (Dynkin current) at ($(Dynkin current)+(0,{\distance})$) {};% +}% + +%% \dynkinSouth +%% Moves the TikZ cursor one edge to the left. +%% Starred form for an indefinite edge. +\NewDocumentCommand\dynkinSouth{s}% +{% + \xdef\distance{\IfBooleanTF{#1}{\dynkin@indefinite@edge@length}{\dynkin@edge@length}} + \node (Dynkin current) at ($(Dynkin current)+(0,{-\distance})$) {};% +}% + +%% \dynkinNorthEast +%% Moves the TikZ cursor one edge to the north east. +%% Starred form for an indefinite edge. +\NewDocumentCommand\dynkinNorthEast{s}% +{% + \xdef\distance{\IfBooleanTF{#1}{\dynkin@indefinite@edge@length}{\dynkin@edge@length}} + \node (Dynkin current) at + ($(Dynkin current)+ + ({cos(60)*\distance},{sin(60)*\distance})$) {};% +}% + +%% \dynkinSouthEast +%% Moves the TikZ cursor one edge to the south east. +%% Starred form for an indefinite edge. +\NewDocumentCommand\dynkinSouthEast{s}% +{% + \xdef\distance{\IfBooleanTF{#1}{\dynkin@indefinite@edge@length}{\dynkin@edge@length}} + \node (Dynkin current) at + ($(Dynkin current)+ + ({cos(-60)*\distance},{sin(-60)*\distance})$) {};% +}% + +%% \dynkinNorthWest +%% Moves the TikZ cursor one edge to the north west. +%% Starred form for an indefinite edge. +\NewDocumentCommand\dynkinNorthWest{s}% +{% + \xdef\distance{\IfBooleanTF{#1}{\dynkin@indefinite@edge@length}{\dynkin@edge@length}} + \node (Dynkin current) at + ($(Dynkin current)+ + ({cos(120)*\distance},{sin(120)*\distance})$) {};% +}% + +%% \dynkinSouthWest +%% Moves the TikZ cursor one edge to the south west. +%% Starred form for an indefinite edge. +\NewDocumentCommand\dynkinSouthWest{s}% +{% + \xdef\distance{\IfBooleanTF{#1}{\dynkin@indefinite@edge@length}{\dynkin@edge@length}} + \node (Dynkin current) at + ($(Dynkin current)+ + ({cos(240)*\distance},{sin(240)*\distance})$) {};% +}% + + +%% \dynkinSouthEastFold +%% Moves the TikZ cursor one edge to the south east in the middle of a fold. +\NewDocumentCommand\dynkinSouthEastFold{}% +{% + \node (Dynkin current) at ($(Dynkin current)+({\dynkin@fold@radius},{-\dynkin@fold@radius})$) {};% +}% + +%% \dynkinSouthWestFold +%% Moves the TikZ cursor one edge to the south west in the middle of a fold. +\NewDocumentCommand\dynkinSouthWestFold{}% +{% + \node (Dynkin current) at ($(Dynkin current)+({-\dynkin@fold@radius},{-\dynkin@fold@radius})$) {};% +}% + +%% \dynkinSouthFold +%% Moves the TikZ cursor one edge to the south in the middle of a fold. +\NewDocumentCommand\dynkinSouthFold{}% +{% + \node (Dynkin current) at ($(Dynkin current)+(0,{-2*\dynkin@fold@radius})$) {};% +}% + +\NewDocumentCommand\find@mark@of@root{m}% +{% + \StrChar{\dynkin@roots}{#1}[\my@root@marker]% + \my@root@marker +}% + +\NewDocumentCommand\dynkin@draw@all@roots{}% +{% + \foreach \b in {1,...,\the\dynkin@nodes}% + {% + \StrChar{\dynkin@roots}{\b}[\c]% + \dynkinRootMark*{\c}{\b}% + }% + \ifdynkin@is@extended% + \dynkinRootMark*{\dynkin@affine@root@mark}{0}% + \else% + \ifdynkin@is@twisted% + \dynkinRootMark*{\dynkin@affine@root@mark}{0}% \fi% + \fi% +}% + + +%% \dynkin@fold@arrow@if@oo{<p>}{<q>} +%% Inputs are roots (in Bourbaki ordering). +%% If we are working on a Satake diagram, and both roots are +%% marked with hollow circles o, then draws a fold arrow between them. +\NewDocumentCommand\dynkin@fold@arrow@if@oo{mm}% +{% + \convertRootPair{#1}{#2}% + \ifdynkin@Satake@diagram% + \StrChar{\dynkin@roots}{\the\@fromRoot}[\my@root@marker]% + \IfStrEq{\my@root@marker}{o}% + {% + \StrChar{\dynkin@roots}{\the\@toRoot}[\my@other@root@marker]% + \IfStrEq{\my@other@root@marker}{o}% + {% + \dynkinFold{\the\@fromRoot}{\the\@toRoot}% + }% + {}% + }{}% \else% - \foreach \b in {1,...,\the\dynkinrank}% + \dynkinFold{\the\@fromRoot}{\the\@toRoot}% + \fi% +}% + +%% \dynkin@pipe{<f>}{<t>}{<D>}{<L>} +%% Layout the roots (as TikZ nodes) <f>, <f>+1, \dots, <t> in the Bourbaki ordering, in a straight line, +%% starting at the current position (Dynkin current), moving in the direction <D>=east, west, north, south, with labels placed according to <L>=left,right,above,below. +%% Assumes that the root <f> is already created as a node in TikZ, but the others are not. +\NewDocumentCommand\dynkin@pipe{mmmm}% +{% + \newcount\start@root + \start@root=#1 + \ifnum\start@root<#2% + \newcount\bmo + \bmo=#1 + \newcount\fpo + \fpo=#1 + \advance\fpo by 1 + \foreach \b in {\the\fpo,...,#2}% {% - \placeRoot{\b}{\b}{0}% + \dynkinPlaceRootRelativeTo*{\b}{\the\bmo}{#3}{#4}% + \dynkinEdge*{SingleEdge}{\b}{\the\bmo}% + \global\advance\bmo by 1% }% \fi% }% -%% \Adynkin -%% -> -%% Draws an A series Dynkin diagram. -\newcommand*{\Adynkin} -{ - \newif\ifwasfolded - \ifdynkinfolded - \global\wasfoldedtrue +%% \dynkin@fold{<f>}{<t>} +%% Layout the roots (as TikZ nodes) <f>, <f>+1, \dots, <t> in the Bourbaki ordering, in a folded arrangement, +%% moving first east, then down, then west, starting at the current position (Dynkin current). +%% Assumes that the root <f> is already created as a node in TikZ, but the others are not. +\NewDocumentCommand\dynkin@fold{mm}% +{% + \newcount\h% + \h=#1% + \advance\h by #2% + \advance\h by -1% + \divide\h by 2% + \dynkin@pipe{#1}{\the\h}{east}{above} + \newcount\hpo + \hpo=\the\h + \advance\hpo by 1 + \newcount\afterfold + \global\afterfold=\the\hpo + \newcount\nrts + \nrts=#2 + \advance\nrts by 1 + \advance\nrts by -#1 + \ifodd\nrts% + \global\advance\afterfold by 1 + \dynkinPlaceRootRelativeTo*{\the\hpo}{\the\h}{southeastfold}{right} + \dynkinEdge*{RightDownArc}{\the\h}{\the\hpo}% + \dynkinPlaceRootRelativeTo*{\the\afterfold}{\the\hpo}{southwestfold}{below} + \dynkinEdge*{RightUpArc}{\the\afterfold}{\the\hpo}% \else - \global\wasfoldedfalse + \dynkinPlaceRootRelativeTo*{\the\afterfold}{\the\h}{southfold}{below} + \dynkinEdge*{SemiCircle}{\the\h}{\the\afterfold}% \fi - \ifnum\dynkinrank=0% - \global\dynkinrank=7% - % Create the nodes. - \Adynkinnodes% - % Draw the edges. - \dynkinline*{1}{2}% - \dynkindots*{2}{3}% - \ifdynkinfolded% - \dynkindownarc*{3}{4}% - \dynkinuparc*{4}{5}% - \else% - \dynkinline*{3}{4}% - \dynkinline*{4}{5}% - \fi% - \dynkindots*{5}{6}% - \dynkinline*{6}{7}% + \dynkin@pipe{\the\afterfold}{#2}{west}{below} + \ifdynkin@arrows% + \newcount\countdown% + \countdown=#2% + \foreach \b in {#1,...,\the\h}% + {% + \dynkin@fold@arrow@if@oo{\b}{\the\countdown}% + \global\advance\countdown by -1% + }% + \fi% +}% + +%% \Adynkin +%% Draws an A series Dynkin diagram. +\NewDocumentCommand\Adynkin{}% +{% + \ifnum\dynkin@rank=1% + \global\dynkin@ply=1\relax% + \fi% +% % Create the roots. + \ifnum\dynkin@ply>1% + \dynkinPlaceRootHere*{1}{above}% + \dynkin@fold{1}{\the\dynkin@rank}% \else% - \ifnum\dynkinrank=1% - \global\dynkinfoldedfalse% - \fi% - % Create the nodes. - \Adynkinnodes% - % Draw the edges. - \ifnum\dynkinrank>1% - \ifnum\dynkinrank=2% - \ifdynkinfolded% - \dynkinsemicircle*{1}{2}% - \else% - \dynkinline*{1}{2}% - \fi% - \else% - \newcount\bpo% - \bpo=2% - \newcount\drmo% - \drmo=\dynkinrank% - \advance \drmo by -1% - \ifdynkinfolded% - \newcount\halfrank% - \halfrank=\dynkinrank% - \divide\halfrank by 2% - \newcount\hrmo% - \hrmo=\halfrank% - \advance\hrmo by -1% - \ifnum\halfrank>1% - \foreach \b in {1,...,\the\hrmo}% - {% - \dynkinline*{\b}{\bpo}% - \global\advance\bpo by 1% - }% - \fi% - \newcount\hrpo% - \hrpo=\halfrank% - \advance\hrpo by 1% - \ifodd\dynkinrank% - \newcount\hrpt% - \hrpt=\hrpo% - \advance\hrpt by 1% - \dynkindownarc*{\the\halfrank}{\the\hrpo}% - \dynkinuparc*{\the\hrpo}{\the\hrpt}% - \ifdynkinarrows% - \dynkinfoldarrow*{\the\halfrank}{\the\hrpt}% - \fi% - \global\advance\bpo by 2% - \ifnum\hrpt<\dynkinrank% - \foreach \b in {\the\hrpt,...,\the\drmo}% - {% - \dynkinline*{\b}{\bpo}% - \global\advance\bpo by 1% - }% - \fi% - \else% - \dynkinsemicircle*{\the\halfrank}{\the\hrpo}% - \global\advance\bpo by 1% - \ifnum\halfrank<\drmo% - \foreach \b in {\the\hrpo,...,\the\drmo}% - {% - \dynkinline*{\b}{\bpo}% - \global\advance\bpo by 1% - }% - \fi% - \fi% - \else% - \foreach \b in {1,...,\the\drmo}% - {% - \dynkinline*{\b}{\bpo}% - \global\advance\bpo by 1% - }% - \fi% - \fi% + \dynkinPlaceRootHere*{1}{below}% + \ifnum\dynkin@rank>1% + \dynkin@pipe{1}{\the\dynkin@rank}{east}{below}% \fi% \fi% - \ifisaffine% - \dynkinline*{0}{1}% - \dynkinline*{0}{\the\dynkinrank}% - \dynkindot*{0}% - \fi% - % Draw the nodes. - \IfStrEqCase{\dynkinSatake}% - {% - {*}% - {% - \foreach \b in {1,...,\the\dynkinrank}% - {% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkindot{\b}}% - }% - }% - {I}% - {% - \ifisaffine% - \dynkinline*{0}{1}% - \dynkinline*{0}{\the\dynkinrank}% - \dynkindot*{0}% - \fi% - \foreach \b in {1,...,\the\dynkinrank}% - {% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkinopendot{\b}}% - }% - }% - {II}% - {% - \newcount\bb% - \bb=1% - \foreach \b in {1,...,\the\dynkinrank}% - {% - \ifodd\bb% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkincloseddot{\b}}% - \else% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkinopendot{\b}}% - \fi% - \global\advance \bb by 1% - }% - }% - }% - \ifwasfolded - \global\dynkinfoldedtrue - \else - \global\dynkinfoldedfalse - \fi -} +}% %% \Bdynkin -%% -> %% Draw a B series Dynkin diagram. \newcommand*{\Bdynkin} { - \ifdynkincoxeter + \ifnum\dynkin@rank<2 \Adynkin - \convertRootPair{1}{2} - \node[above] at ($.5*(root \the\@fromRoot)+.5*(root \the\@toRoot)$) {\dynkinprint{4}}; \else - \ifnum\dynkinrank=0 - \dynkinrank=5 - % Create the nodes. - \Adynkinnodes - % Draw the edges. - \dynkinline*{1}{2} - \dynkindots*{2}{3} - \dynkinline*{3}{4} - \dynkindoubleline*{4}{5} + \newcount\drmo + \drmo=\the\dynkin@rank + \advance\drmo by -1 + \ifdynkin@Coxeter + \Adynkin + \convertRootPair{\the\drmo}{\the\dynkin@rank} + \node[/Dynkin diagram/text,above] + at ($.5*(\dynkin@root@name \the\@fromRoot)+.5*(\dynkin@root@name \the\@toRoot)$) + {\(4\)}; \else - % Create the nodes. - \Adynkinnodes - % Draw the edges. - \dynkinline*{1}{\the\dynkinrank}% - \newcount\rmo - \rmo=\dynkinrank - \advance \rmo by -1 - \dynkindoubleline*{\the\rmo}{\the\dynkinrank} - \fi - % Draw the nodes. - \ifisaffine - \dynkinline*{0}{2} - \dynkindot*{0} - \fi - \foreach \b in {1,...,\the\dynkinrank} - { - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkindot{\b}} - } - \fi + % Create the roots. + \ifnum\dynkin@ply>1% + \ifnum\dynkin@rank>3% + \dynkinPlaceRootHere*{1}{above}% + \dynkinPlaceRootRelativeTo*{2}{1}{east}{above}% + \dynkin@fold{2}{\the\drmo}% + \dynkinPlaceRootRelativeTo*{\the\dynkin@rank}{\the\drmo}{west}{below}% + \dynkinEdge*{DoubleEdge}{\the\drmo}{\the\dynkin@rank}% + \dynkinEdge*{SingleEdge}{1}{2}% + \else% + \ifnum\dynkin@rank=2% + \dynkinPlaceRootHere*{1}{left}% + \dynkinPlaceRootRelativeTo*{2}{1}{southfold}{left}% + \dynkinEdge*{DoubleDownRightSemiCircle}{1}{2}% + \else% + \dynkinPlaceRootHere*{1}{left}% + \dynkinPlaceRootRelativeTo*{2}{1}{southeastfold}{right}% + \dynkinPlaceRootRelativeTo*{3}{2}{southwestfold}{left}% + \dynkinEdge*{RightDownArc}{1}{2}% + \dynkinEdge*{DoubleDownLeftArc}{2}{3}% + \fi% + \fi% + \else% + \dynkinPlaceRootHere*{1}{below} + \dynkin@pipe{1}{\the\drmo}{east}{below} + \dynkinPlaceRootRelativeTo*{\the\dynkin@rank}{\the\drmo}{east}{below} + \dynkinEdge*{DoubleEdge}{\the\drmo}{\the\dynkin@rank}% + \fi% + \ifdynkin@arrows% + \ifnum\dynkin@ply>1% + \dynkinLeftFold*{1}{\the\dynkin@rank}% + \fi% + \fi% + \fi% + \fi% } %% \Cdynkin -%% -> %% Draws a C series Dynkin diagram. \newcommand*{\Cdynkin} { - \ifdynkincoxeter - \Bdynkin - \else - \ifnum\dynkinrank=0 - \dynkinrank=5 - % Create the nodes. - \Adynkinnodes - % Draw the edges. - \dynkinline*{1}{2} - \dynkindots*{2}{3} - \dynkinline*{3}{4} - \dynkindoubleline*{5}{4} - \else - % Create the nodes. - \Adynkinnodes - % Draw the edges. - \newcount\rmo - \rmo=\dynkinrank - \advance\rmo by -1 - \dynkinline*{1}{\the\rmo}% - \dynkindoubleline*{\the\dynkinrank}{\the\rmo} - \fi - % Draw the nodes. - \ifisaffine - \dynkindoubleline*{0}{1} - \dynkindot*{0} - \fi - \foreach \b in {1,...,\the\dynkinrank} - { - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkindot{\b}} - } - \fi + \ifdynkin@reverse@arrows% + \global\dynkin@reverse@arrowsfalse% + \else% + \global\dynkin@reverse@arrowstrue% + \fi% + \Bdynkin% + \ifdynkin@reverse@arrows% + \global\dynkin@reverse@arrowsfalse% + \else% + \global\dynkin@reverse@arrowstrue% + \fi% } -%% \Ddynkinnodes -%% -> -%% Tell TikZ where to place the nodes for a D series Dynkin diagram. Draws nothing. -\newcommand*{\Ddynkinnodes} +%% \Ddynkin@roots +%% Tell TikZ where to place the @roots for a D series Dynkin diagram. Draws nothing. +\newcommand*{\Ddynkin@roots} { + % Create the roots. + \ifdynkin@is@extended% + \ifnum\dynkin@ply>1% + \ifnum\dynkin@rank=4% + \dynkinPlaceRootRelativeTo*{2}{0}{southeastfold}{right}% + \else% + \dynkinPlaceRootRelativeTo*{2}{0}{southeastfold}{below}% + \fi% + \dynkinPlaceRootRelativeTo*{1}{2}{southwestfold}{left}% + \else% + \ifdynkin@left@fold% + \dynkinPlaceRootRelativeTo*{2}{0}{southeastfold}{below}% + \dynkinPlaceRootRelativeTo*{1}{2}{southwestfold}{left}% + \else% + \dynkinPlaceRootRelativeTo*{2}{0}{southeast}{left}% + \dynkinPlaceRootRelativeTo*{1}{2}{southwest}{left}% + \fi% + \fi% + \dynkinMoveToRoot*{2}% + \else + \dynkinPlaceRootHere*{1}{below} + \ifnum\dynkin@rank=4% + \ifdynkin@right@fold% + \dynkinPlaceRootRelativeTo*{2}{1}{east}{below}% + \else% + \ifnum\dynkin@ply>1% + \dynkinPlaceRootRelativeTo*{2}{1}{east}{below}% + \else% + \dynkinPlaceRootRelativeTo*{2}{1}{east}{right}% + \fi% + \fi% + \else% + \dynkinPlaceRootRelativeTo*{2}{1}{east}{below}% + \fi% + \fi \newcount\rmo - \rmo=\dynkinrank + \rmo=\dynkin@rank \advance \rmo by -1 \newcount\rmt \rmt=\rmo \advance\rmt by -1 - % Create the nodes. - \foreach \b in {1,...,\the\rmt} - { - \placeRoot{\b}{\b}{0} - } - \pgfmathparse{subtract(\the\rmo,.5)} - \let\rmh\pgfmathresult - \ifdynkinfolded - \placeRoot{\the\rmo}{\rmh}{-.9} - \placeRoot*{\the\dynkinrank}{\rmh}{.9} - \else - \placeRootHorizontalLabels{\the\rmo}{\rmh}{-.9} - \placeRootHorizontalLabels{\the\dynkinrank}{\rmh}{.9} - \fi -} + \newcount\rmth + \rmth=\rmt + \advance\rmth by -1 + \ifnum\dynkin@rank>2 + \ifnum\dynkin@rank>5% + \dynkinPlaceRootRelativeTo*{3}{2}{east}{below}% + \else% + \ifnum\dynkin@ply>1% + \dynkinPlaceRootRelativeTo*{3}{2}{east}{below}% + \else% +% \ifdynkin@left@fold% +% \dynkinPlaceRootRelativeTo*{3}{2}{east}{below}% +% \else% + \ifnum\dynkin@rank=5% + \ifdynkin@right@fold% + \dynkinPlaceRootRelativeTo*{3}{2}{east}{below}% + \else% + \dynkinPlaceRootRelativeTo*{3}{2}{east}{right}% + \fi% + \else% + \dynkinPlaceRootRelativeTo*{3}{2}{east}{right}% + \fi% +% \fi% + \fi% + \fi% + \ifnum\rmth>3% + \dynkin@pipe{3}{\the\rmth}{east}{below}% + \fi% + \ifnum\rmt>3% + \ifnum\dynkin@ply>1% + \dynkinPlaceRootRelativeTo*{\rmt}{\rmth}{east}{below}% + \else% + \ifdynkin@right@fold% + \dynkinPlaceRootRelativeTo*{\rmt}{\rmth}{east}{below}% + \else% + \dynkinPlaceRootRelativeTo*{\rmt}{\rmth}{east}{right}% + \fi% + \fi% + \dynkinEdge*{SingleEdge}{\rmt}{\rmth}% + \fi% + \ifnum\dynkin@ply=1% + \ifdynkin@right@fold% + \dynkinPlaceRootRelativeTo*{\the\rmo}{\the\rmt}{northeastfold}{right}% + \dynkinPlaceRootRelativeTo*{\the\dynkin@rank}{\the\rmt}{southeastfold}{right}% + \else% + \dynkinPlaceRootRelativeTo*{\the\rmo}{\the\rmt}{northeast}{right}% + \dynkinPlaceRootRelativeTo*{\the\dynkin@rank}{\the\rmt}{southeast}{right}% + \fi% + \else% + \dynkinPlaceRootRelativeTo*{\the\rmo}{\the\rmt}{northeastfold}{right}% + \dynkinPlaceRootRelativeTo*{\the\dynkin@rank}{\the\rmt}{southeastfold}{right}% + \fi% + \fi% +}% + +%% \Ddynkin@edges +%% Draws edges on a D series Dynkin diagram. +\NewDocumentCommand\Ddynkin@edges{}% +{% + % Draw the edges. + \newcount\rmo + \rmo=\dynkin@rank + \advance \rmo by -1 + \newcount\rmt + \rmt=\rmo + \advance\rmt by -1 + \newcount\rmtr + \rmtr=\rmt + \advance\rmtr by -1 + \ifnum\dynkin@ply>1% + \ifdynkin@is@extended% + \dynkinEdge*{RightUpArc}{1}{2}% + \else% + \dynkinEdge*{SingleEdge}{1}{2}% + \fi% + \ifnum\dynkin@rank>4% + \dynkinEdge*{SingleEdge}{2}{3}% + \fi% + \dynkinEdge*{LeftDownArc}{\the\rmo}{\the\rmt}% + \dynkinEdge*{LeftUpArc}{\the\dynkin@rank}{\the\rmt}% + \ifdynkin@arrows% + \dynkinRightFold*{\the\rmo}{\the\dynkin@rank}% + \ifdynkin@is@extended% + \dynkinLeftFold*{0}{1}% + \fi% + \fi% + \else% + \ifnum\dynkin@rank=4% + \else% + \dynkinEdge*{SingleEdge}{2}{3}% + \fi% + \ifdynkin@is@extended% + \ifdynkin@left@fold% + \dynkinEdge*{RightUpArc}{1}{2}% + \ifdynkin@arrows% + \ifdynkin@is@extended% + \dynkinLeftFold*{0}{1}% + \fi% + \fi% + \else% + \dynkinEdge*{SingleEdge}{1}{2}% + \fi% + \else% + \dynkinEdge*{SingleEdge}{1}{2}% + \fi% + \ifdynkin@right@fold% + \dynkinEdge*{LeftDownArc}{\the\rmo}{\the\rmt}% + \dynkinEdge*{LeftUpArc}{\the\dynkin@rank}{\the\rmt}% + \dynkinRightFold*{\the\rmo}{\the\dynkin@rank}% + \else% + \dynkinEdge*{SingleEdge}{\the\rmt}{\the\rmo}% + \dynkinEdge*{SingleEdge}{\the\rmt}{\the\dynkin@rank}% + \fi% + \fi% +}% + +%% \DthreePly +%% Draws a D series Dynkin diagram of rank 4, folded over a G2. +\NewDocumentCommand\DthreePly{}% +{% + \dynkinPlaceRootHere*{2}{right}% + \xdef\old@edge@length{\dynkin@edge@length}% + \pgfmathparse{1.5*\dynkin@edge@length}% + \xdef\dynkin@edge@length{\pgfmathresult pt}% + \dynkinPlaceRootRelativeTo*{3}{2}{south}{right}% + \dynkinPlaceRootRelativeTo*{4}{3}{south}{right}% + \xdef\dynkin@edge@length{\old@edge@length}% + \dynkinPlaceRootRelativeTo*{1}{3}{west}{left}% + \edef\old@fold@radius{\dynkin@fold@radius}% + \xdef\dynkin@fold@radius{\dynkin@edge@length}% + \dynkinEdge*{SingleEdge}{1}{3}% + \dynkinEdge*{LeftDownArc}{2}{1}% + \dynkinEdge*{LeftUpArc}{4}{1}% + \xdef\dynkin@fold@radius{\old@fold@radius}% + \ifdynkin@arrows% + \dynkin@fold@arrow@if@oo{2}{3}% + \dynkin@fold@arrow@if@oo{3}{4}% + \fi% +}% %% \Ddynkin -%% -> %% Draws a D series Dynkin diagram. -\newcommand*{\Ddynkin}% -{ - \ifnum\dynkinrank=1 - \gdef\dynkinseries{A} - \Adynkin - \else - \ifnum\dynkinrank=0 - \dynkinrank=6 - \Ddynkinnodes - % Draw the edges. - \dynkinline*{1}{2} - \dynkindots*{2}{3} - \dynkinline*{3}{4} - \dynkinline*{4}{5} - \dynkinline*{4}{6} - \else - \Ddynkinnodes - % Draw the edges. - \dynkinline*{1}{\the\rmt} - \dynkinline*{\the\rmt}{\the\rmo} - \dynkinline*{\the\rmt}{\the\dynkinrank} - \fi - \ifdynkinfolded - \ifdynkinarrows - \draw[\dynkinfoldarrowstyle,\dynkinfoldarrowcolor] - (root \the\rmo.east) - to [out=45, in=-45] - (root \the\dynkinrank.east); - \fi - \fi - % Draw the nodes. - \ifisaffine - \dynkinline*{0}{2} - \dynkindot*{0} - \fi - \foreach \b in {1,...,\the\dynkinrank} - { - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkindot{\b}} - } - \fi -} +\NewDocumentCommand\Ddynkin{}% +{% + \ifnum\dynkin@rank>3% + \ifnum\dynkin@rank=4% + \ifnum\dynkin@ply=3% + \DthreePly% + \else% + \Ddynkin@roots% + \Ddynkin@edges% + \fi% + \else% + \Ddynkin@roots% + \Ddynkin@edges% + \fi% + \else% + \gdef\dynkin@series{A}% + \Adynkin% + \ifnum\dynkin@ply>1% + \ifdynkin@arrows% + \ifnum\dynkin@rank=1% + \else% + \dynkinLeftFold*{1}{\the\dynkin@rank}% + \fi% + \fi% + \fi% + \fi% +}% -%% \Edynkinunfolded -%% -> +%% \Edynkin@unfolded %% Draws an E series Dynkin diagram not folded. -\newcommand*{\Edynkinunfolded}% +\newcommand*{\Edynkin@unfolded}% { - % Create the nodes. - \placeRoot{1}{1}{0} - \ifisaffine - \ifnum\dynkinrank=6 - \placeRootHorizontalLabels{2}{3}{1} + % Create the @roots. + \dynkinPlaceRootHere*{1}{below}% + \dynkinPlaceRootRelativeTo*{3}{1}{east}{below}% + \dynkinPlaceRootRelativeTo*{4}{3}{east}{below}% + \ifdynkin@is@extended + \ifnum\dynkin@rank=6 + \dynkinPlaceRootRelativeTo*{2}{4}{north}{right}% \else - \placeRoot*{2}{3}{1} + \dynkinPlaceRootRelativeTo*{2}{4}{north}{above}% \fi \else - \placeRoot*{2}{3}{1} + \dynkinPlaceRootRelativeTo*{2}{4}{north}{above}% \fi - \foreach \b in {3,...,\dynkinrank} - { - \newcount\bmo - \bmo=\b - \advance\bmo by -1 - \placeRoot{\b}{\the\bmo}{0} - } -% % Draw the edges. - \dynkinline*{1}{\the\dynkinrank} - \dynkinline*{2}{4} -} + \newcount\bmo\relax% + \bmo=4\relax% + \foreach \b in {5,...,\dynkin@rank}% + {% + \dynkinPlaceRootRelativeTo*{\b}{\the\bmo}{east}{below}% + \dynkinEdge*{SingleEdge}{\the\bmo}{\b}% + \global\advance\bmo by 1% + }% +% % Draw the remaining edges. + \dynkinEdge*{SingleEdge}{1}{3} + \dynkinEdge*{SingleEdge}{3}{4} + \dynkinEdge*{SingleEdge}{4}{2} + \ifdynkin@is@extended% + \ifnum\dynkin@rank=6% + \dynkinPlaceRootRelativeTo*{0}{2}{north}{above}% + \dynkinEdge*{SingleEdge}{0}{2}% + \else% + \ifnum\dynkin@rank=7% + \dynkinPlaceRootRelativeTo*{0}{1}{west}{below}% + \dynkinEdge*{SingleEdge}{0}{1}% + \else% + \dynkinPlaceRootRelativeTo*{0}{8}{east}{below}% + \dynkinEdge*{SingleEdge}{0}{8}% + \fi% + \fi% + \fi% +}% -%% \Edynkinfolded -%% -> -%% Draws a folded E6 Dynkin diagram. -\newcommand*{\Edynkinfolded}% -{ - \placeRoot*{1}{0}{1} - \placeRoot*{3}{1}{1} - \placeRootHorizontalLabels*{4}{2}{0} - \placeRootHorizontalLabels{2}{3}{0} - \placeRoot{5}{1}{-1} - \placeRoot{6}{0}{-1} - \dynkinline*{1}{3} - \dynkinline*{2}{4} - \dynkinline*{5}{6} - \dynkindownarc*{3}{4} - \dynkinuparc*{4}{5} -} +%% \Edynkin@folded +%% Draws a folded E6, affine E6 or affine E7 Dynkin diagram. +\NewDocumentCommand\Edynkin@folded{}% +{% + \ifnum\dynkin@rank=6% + \ifnum\dynkin@ply=2\ESixTwoPly\else\ESixThreePly\fi% + \else% + \extendedESevenFolded% + \fi% +}% + +\NewDocumentCommand\ESixTwoPly{}% +{% + \dynkinPlaceRootHere*{1}{above}% + \dynkinPlaceRootRelativeTo*{3}{1}{east}{above}% + \dynkinPlaceRootRelativeTo*{4}{3}{southeastfold}{below}% + \dynkinPlaceRootRelativeTo*{5}{4}{southwestfold}{below}% + \dynkinPlaceRootRelativeTo*{6}{5}{west}{below}% + \ifdynkin@is@extended% + \dynkinPlaceRootRelativeTo*{2}{4}{east}{below}% + \dynkinPlaceRootRelativeTo*{0}{2}{east}{right}% + \dynkinEdge*{SingleEdge}{0}{2}% + \else% + \dynkinPlaceRootRelativeTo*{2}{4}{east}{right}% + \fi% + \dynkinEdge*{SingleEdge}{1}{3}% + \dynkinEdge*{SingleEdge}{2}{4}% + \dynkinEdge*{SingleEdge}{5}{6}% + \dynkinEdge*{RightDownArc}{3}{4}% + \dynkinEdge*{RightUpArc}{5}{4}% + \ifdynkin@arrows% + \dynkin@fold@arrow@if@oo{1}{6}% + \dynkin@fold@arrow@if@oo{3}{5}% + \fi% +}% + + +\NewDocumentCommand\ESixThreePly{}% +{% + \dynkinPlaceRootHere*{3}{above}% + \edef\old@edge@length{\dynkin@edge@length}% + \pgfmathparse{1.5*\dynkin@edge@length}% + \xdef\dynkin@edge@length{\pgfmathresult pt}% + \dynkinPlaceRootRelativeTo*{2}{3}{south}{diagonal}% + \dynkinPlaceRootRelativeTo*{5}{2}{south}{below}% + \xdef\dynkin@edge@length{\old@edge@length}% + \dynkinPlaceRootRelativeTo*{1}{3}{west}{left}% + \dynkinPlaceRootRelativeTo*{0}{2}{west}{left}% + \dynkinPlaceRootRelativeTo*{6}{5}{west}{left}% + \edef\old@fold@radius{\dynkin@fold@radius}% + \xdef\dynkin@fold@radius{\dynkin@edge@length}% + \dynkinPlaceRootRelativeTo*{4}{2}{east}{right}% + \dynkinEdge*{SingleEdge}{4}{2}% + \dynkinEdge*{SingleEdge}{3}{1}% + \dynkinEdge*{SingleEdge}{2}{0}% + \dynkinEdge*{SingleEdge}{5}{6}% + \dynkinEdge*{RightDownArc}{3}{4}% + \dynkinEdge*{RightUpArc}{5}{4}% + \xdef\dynkin@fold@radius{\old@fold@radius}% + \ifdynkin@arrows% + \dynkin@fold@arrow@if@oo{1}{0}% + \dynkin@fold@arrow@if@oo{6}{0}% + \dynkin@fold@arrow@if@oo{3}{2}% + \dynkin@fold@arrow@if@oo{2}{5}% + \fi% +}% + +\NewDocumentCommand\extendedESevenFolded{}% +{% + \dynkinPlaceRootHere*{0}{above}% + \dynkinPlaceRootRelativeTo*{1}{0}{east}{above}% + \dynkinPlaceRootRelativeTo*{3}{1}{east}{above}% + \dynkinPlaceRootRelativeTo*{4}{3}{southeastfold}{left}% + \dynkinPlaceRootRelativeTo*{5}{4}{southwestfold}{below}% + \dynkinPlaceRootRelativeTo*{6}{5}{west}{below}% + \dynkinPlaceRootRelativeTo*{7}{6}{west}{below}% + \dynkinPlaceRootRelativeTo*{2}{4}{east}{below}% + \dynkinEdge*{SingleEdge}{0}{1}% + \dynkinEdge*{SingleEdge}{1}{3}% + \dynkinEdge*{SingleEdge}{2}{4}% + \dynkinEdge*{SingleEdge}{5}{6}% + \dynkinEdge*{SingleEdge}{6}{7}% + \dynkinEdge*{RightDownArc}{3}{4}% + \dynkinEdge*{RightUpArc}{5}{4}% + \ifdynkin@arrows% + \dynkin@fold@arrow@if@oo{0}{7}% + \dynkin@fold@arrow@if@oo{1}{6}% + \dynkin@fold@arrow@if@oo{3}{5}% + \fi% +}% + %% \Edynkin -%% -> %% Draws an E6 Dynkin diagram. -\newcommand*{\Edynkin}% -{ - \ifdynkinfolded - \ifnum\dynkinrank=6 - \Edynkinfolded - \else - \ClassWarning{Can not fold a diagram of type \dynkinseries\the\dynkinrank.} - \fi - \else - \Edynkinunfolded - \fi - % Draw the nodes. - \ifisaffine - \ifnum\dynkinrank=6 - \dynkinline*{0}{2} - \else - \dynkinline*{0}{1} - \fi - \dynkindot{0} - \fi - \IfStrEqCase{\dynkinSatake}% - {% - {*}% - {% - \foreach \b in {1,...,\the\dynkinrank}% - {% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkindot{\b}}% - }% - \ifdynkinfolded - \ifdynkinarrows - \dynkinfoldarrow*{1}{6} - \dynkinfoldarrow*{3}{5} - \fi - \fi - }% - {I}% - {% - \foreach \b in {1,...,6}% - {% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkinopendot{\b}}% - }% - }% - {II}% - {% - \ifdynkinarrows - \dynkinfoldarrow*{1}{6}% - \dynkinfoldarrow*{3}{5}% - \fi - \foreach \b in {1,...,6}% - {% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkinopendot{\b}}% - }% - }% - {III}% - {% - \dynkinfoldarrow*{1}{6}% - \foreach \b in {1,2,6}% - {% - \dynkinopendot*{\b}% - }% - \foreach \b in {3,4,5}% - {% - \dynkincloseddot*{\b}% - }% - }% - {IV}% - {% - \foreach \b in {1,6}% - {% - \dynkinopendot*{\b}% - }% - \foreach \b in {2,3,4,5}% - {% - \dynkincloseddot*{\b}% - }% - }% - {V}% - {% - \foreach \b in {1,...,7}% - {% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkinopendot{\b}}% - }% - }% - {VI}% - {% - \foreach \b in {1,3,4,6}% - {% - \dynkinopendot*{\b}% - }% - \foreach \b in {2,5,7}% - {% - \dynkincloseddot*{\b}% - }% - }% - {VII}% - {% - \foreach \b in {1,6,7}% - {% - \dynkinopendot*{\b}% - }% - \foreach \b in {2,3,4,5}% - {% - \dynkincloseddot*{\b}% - }% - }% - {VIII}% - {% - \foreach \b in {1,...,8}% - {% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkinopendot{\b}}% - }% - }% - {XI}% - {% - \foreach \b in {1,6,7,8}% - {% - \dynkinopendot*{\b}% - }% - \foreach \b in {2,3,4,5}% - {% - \dynkincloseddot*{\b}% - }% - }% - }% -} +\NewDocumentCommand\Edynkin{}% +{% + \ifnum\dynkin@ply>1% + \ifnum\dynkin@rank=6% + \Edynkin@folded% + \else% + \ifnum\dynkin@rank=7% + \ifdynkin@is@extended% + \Edynkin@folded% + \else% + \ClassError{Dynkin diagrams}% + {Can not fold a diagram of type \dynkin@user@series{} \the\dynkin@rank.}{}% + \fi% + \fi% + \fi% + \else% + \Edynkin@unfolded% + \fi% +}% %% \Fdynkin -%% -> %% Draws an F series Dynkin diagram. \newcommand*{\Fdynkin}% { - \Adynkinnodes - \ifdynkincoxeter - \dynkinline*{1}{4} + \dynkinPlaceRootHere*{1}{below} + \dynkinPlaceRootRelativeTo*{2}{1}{east}{below}% + \dynkinPlaceRootRelativeTo*{3}{2}{east}{below}% + \dynkinPlaceRootRelativeTo*{4}{3}{east}{below}% + \ifdynkin@Coxeter + \dynkinEdge*{SingleEdge}{1}{2} + \dynkinEdge*{SingleEdge}{2}{3} + \dynkinEdge*{SingleEdge}{3}{4} \convertRootPair{2}{3} - \node[above] at ($.5*(root \the\@fromRoot)+.5*(root \the\@toRoot)$) {\dynkinprint{4}}; - \foreach \b in {1,...,4}% - {% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkindot{\b}}% - }% + \node[/Dynkin diagram/text,above] + at ($.5*(\dynkin@root@name \the\@fromRoot)+.5*(\dynkin@root@name \the\@toRoot)$) + {\(4\)}; \else - \dynkinline*{1}{2} - \dynkinline*{3}{4} - \dynkindoubleline*{2}{3} - \ifisaffine - \dynkinline*{0}{1} - \dynkindot{0} - \fi - \IfStrEqCase{\dynkinSatake} - {% - {*}% - {% - \foreach \b in {1,...,4}% - {% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkindot{\b}}% - }% - }% - {I}% - {% - \foreach \b in {1,...,4}% - {% - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkinopendot{\b}}% - }% - }% - {II}% - {% - \dynkincloseddot*{1}% - \dynkincloseddot*{2}% - \dynkincloseddot*{3}% - \dynkinopendot*{4}% - }% - }% + \dynkinEdge*{SingleEdge}{1}{2} + \dynkinEdge*{SingleEdge}{3}{4} + \dynkinEdge*{DoubleEdge}{2}{3} \fi } %% \Gdynkin -%% -> %% Draws a G series Dynkin diagram. -\newcommand*{\Gdynkin}% -{ - \newif\ifwasopen - \ifdynkinopendots - \global\wasopentrue - \else - \global\wasopenfalse - \fi - \Adynkinnodes - \ifisaffine - \dynkinline*{0}{2} - \fi - \ifdynkincoxeter - \convertRootPair{1}{2} - \node[above] at ($.5*(root \the\@fromRoot)+.5*(root \the\@toRoot)$) {\dynkinprint{6}}; - \dynkinline*{1}{2} - \else - \dynkintripleline*{1}{2} - \IfStrEq{\dynkinSatake}{I}{\global\dynkinopendotstrue}{} - \ifisaffine - \dynkindot{0} - \fi - \fi - \foreach \b in {1,...,2} - { - \testbit{\dynkinparabolic}{\b}{\dynkincross{\b}}{\dynkindot{\b}} - } - \ifwasopen - \global\dynkinopendotstrue - \else - \global\dynkinopendotsfalse - \fi -} +\NewDocumentCommand\Gdynkin{}% +{% + \ifdynkin@Coxeter% + \Idynkin% + \else% + \dynkinPlaceRootHere*{1}{below}% + \dynkinPlaceRootRelativeTo*{2}{1}{east}{below}% + \dynkinTripleEdge*{1}{2}% + \fi% +}% %% \Hdynkin -%% -> %% Draws an H series Coxeter diagram. \newcommand*{\Hdynkin}% -{ - \newcount\Hn - \Hn=\dynkinrank - \dynkinrank=2 - \Adynkin - \convertRootPair{1}{2} - \node[above] at ($.5*(root \the\@fromRoot)+.5*(root \the\@toRoot)$) {\dynkinprint{\the\Hn}}; -} +{% + \Adynkin% + \convertRootPair{1}{2}% + \node[/Dynkin diagram/text,above] at ($.5*(\dynkin@root@name \the\@fromRoot)+.5*(\dynkin@root@name \the\@toRoot)$) {\(5\)};% +}% %% \Idynkin -%% -> %% Draws an I series Coxeter diagram. \newcommand*{\Idynkin}% -{ - \Adynkin - \convertRootPair{1}{2} - \node[above] at ($.5*(root \the\@fromRoot)+.5*(root \the\@toRoot)$) {\dynkinprint{5}}; -} +{% + \newcount\In% + \In=\dynkin@rank% + \dynkin@rank=2% + \Adynkin% + \convertRootPair{1}{2}% + \node[/Dynkin diagram/text,above] at ($.5*(\dynkin@root@name \the\@fromRoot)+.5*(\dynkin@root@name \the\@toRoot)$) {\(\dynkin@gonality\)};% +}% -\newcommand*{\affineAdynkin}% -{ -\ifnum\dynkinrank=0 - \placeRoot*{0}{4}{1} - \Adynkin -\else - \ifnum\dynkinrank=1 - \placeRoot{0}{0}{0} - \placeRoot{1}{2}{0} - \convertRootNumber{1} - \draw[ - double, - \dynkincolor, - {Classical TikZ Rightarrow[length={3*\dynkinradius}]}-{Classical TikZ Rightarrow[length={3*\dynkinradius}]} - ] - ($(root 0)+(\dynkinradius,0)$) -- ($(root \the\RootNumber)-(\dynkinradius,0)$); - \else - \pgfmathparse{(.5+.5*\the\dynkinrank)}% - \let\halfway\pgfmathresult% - \placeRoot*{0}{\halfway}{1} - \Adynkin - \fi -\fi -} +%% \extendedAdynkin +%% Draws an A series affine Dynkin/Coxeter diagram. +\NewDocumentCommand\extendedAdynkin{}% +{% + \ifnum\dynkin@rank=1% + \dynkinPlaceRootHere{0}{below}% + \dynkinPlaceRootRelativeTo*{1}{0}{east}{below}% + \convertRootNumber{1}% + \begin{scope}{on background layer}% + \draw[% + /Dynkin diagram/edge, + double, + {Classical TikZ Rightarrow[length={2*\dynkin@root@radius}]}% + -{Classical TikZ Rightarrow[length={2*\dynkin@root@radius}]}% + ]% + ($(\dynkin@root@name 0)+(\dynkin@root@radius,0)$) + -- + ($(\dynkin@root@name \the\RootNumber)-(\dynkin@root@radius,0)$);% + \end{scope}% + \else% + \ifnum\dynkin@ply=4% + \dynkinPlaceRootHere*{0}{left}% + \dynkinPlaceRootRelativeTo*{1}{0}{east}{right}% + \dynkinPlaceRootRelativeTo*{2}{0}{south}{left}% + \dynkinPlaceRootRelativeTo*{3}{1}{south}{right}% + \dynkinEdge*{SingleEdge}{0}{1}% + \dynkinEdge*{SingleEdge}{1}{2}% + \dynkinEdge*{SingleEdge}{2}{3}% + \dynkinEdge*{SingleEdge}{3}{0}% + \dynkinFold*{0}{2}% + \dynkinFold*{1}{3}% + \else% + \Adynkin{}% + \ifnum\dynkin@ply>1% + \dynkinPlaceRootRelativeTo*{0}{1}{southwestfold}{right}% + \dynkinEdge*{LeftDownArc}{1}{0}% + \dynkinEdge*{LeftUpArc}{\the\dynkin@rank}{0}% + \else% + \node (Dynkin current) at ($.5*(\dynkin@root@name 1)+.5*(\dynkin@root@name \the\dynkin@rank)$){};% + \dynkinNorth% + \dynkinPlaceRootHere*{0}{above}% + \dynkinEdge*{SingleEdge}{0}{1}% + \dynkinEdge*{SingleEdge}{0}{\the\dynkin@rank}% + \fi% + \dynkinRootMark*{}{0}% + \fi% + \fi% +}% -\newcommand*{\affineBdynkin}% -{ - \placeRoot*{0}{2}{1} - \Bdynkin -} +\NewDocumentCommand\extendedBthreePly{}% +{% + \dynkinPlaceRootHere*{0}{right}% + \edef\old@edge@length{\dynkin@edge@length}% + \pgfmathparse{1.5*\dynkin@edge@length}% + \xdef\dynkin@edge@length{\pgfmathresult pt}% + \dynkinPlaceRootRelativeTo*{1}{0}{south}{right}% + \dynkinPlaceRootRelativeTo*{3}{1}{south}{right}% + \xdef\dynkin@edge@length{\old@edge@length}% + \edef\old@fold@radius{\dynkin@fold@radius}% + \xdef\dynkin@fold@radius{\dynkin@edge@length}% + \dynkinPlaceRootRelativeTo*{2}{1}{west}{left}% + \dynkinEdge*{LeftDownArc}{0}{2}% + \dynkinFold*{0}{1}% + \dynkinFold*{1}{3}% + \dynkinEdge*{SingleEdge}{1}{2}% + \dynkinEdge*{DoubleDownRightArc}{2}{3}% + \xdef\dynkin@fold@radius{\old@fold@radius}% +}% -\newcommand*{\affineCdynkin} -{ - \placeRoot{0}{0}{0} - \Cdynkin -} +%% \extendedBdynkin +%% Draws a B series affine Dynkin/Coxeter diagram. +\newcommand*{\extendedBdynkin}% +{% + \ifnum\the\dynkin@rank=1 + \extendedAdynkin% + \else% + \ifnum\the\dynkin@rank=2 + \dynkinPlaceRootHere*{0}{left}% + \dynkinPlaceRootRelativeTo*{1}{0}{east}{below}% + \dynkinPlaceRootRelativeTo*{2}{1}{east}{left}% + \dynkinEdge*{SingleEdge}{0}{1}% + \dynkinEdge*{DoubleEdge}{1}{2}% + \else% + \ifnum\dynkin@ply=3% + \extendedBthreePly% + \else% + \ifnum\dynkin@ply=2% + \dynkinPlaceRootHere*{0}{left}% + \dynkinPlaceRootRelativeTo*{2}{0}{southeastfold}{below}% + \dynkinPlaceRootRelativeTo*{1}{2}{southwestfold}{left}% + \dynkinLeftFold*{0}{1}% + \dynkinEdge*{RightDownArc}{0}{2}% + \dynkinEdge*{RightUpArc}{1}{2}% + \else% + \dynkinPlaceRootHere*{0}{left}% + \dynkinPlaceRootRelativeTo*{2}{0}{southeast}{left}% + \dynkinPlaceRootRelativeTo*{1}{2}{southwest}{left}% + \dynkinEdge*{SingleEdge}{0}{2}% + \dynkinEdge*{SingleEdge}{1}{2}% + \fi% + \newcount\drmo% + \drmo=\the\dynkin@rank\relax% + \advance\drmo by -1\relax% + \newcount\bmo% + \bmo=2% + \ifnum\dynkin@rank>3% + \foreach \b in {3,...,\the\drmo}% + {% + \dynkinPlaceRootRelativeTo*{\b}{\the\bmo}{east}{below}% + \dynkinEdge*{SingleEdge}{\b}{\the\bmo}% + \global\advance\bmo by 1\relax% + }% + \fi% + \ifnum\dynkin@ply<3% + \dynkinPlaceRootRelativeTo*{\the\dynkin@rank}{\the\drmo}{east}{below}% + \fi% + \ifdynkin@Coxeter% + \dynkinEdge*{SingleEdge}{\the\drmo}{\the\dynkin@rank}% + \convertRootPair{\the\drmo}{\the\dynkin@rank} + \node[/Dynkin diagram/text,above] at + ($.5*(\dynkin@root@name \the\@fromRoot)+.5*(\dynkin@root@name \the\@toRoot)$) {\(4\)}; + \else% + \ifnum\dynkin@ply<3% + \dynkinEdge*{DoubleEdge}{\the\drmo}{\the\dynkin@rank}% + \else% + \dynkinEdge*{DoubleDownRightArc}{\the\drmo}{\the\dynkin@rank}% + \fi% + \fi% + \fi% + \fi% + \fi% +}% + +%% \extendedCdynkin +%% Draws an C series affine Dynkin/Coxeter diagram. +\newcommand*{\extendedCdynkin}% +{% + \dynkinPlaceRootHere*{0}{below}% + \dynkinEast% + \Cdynkin{}% + \ifdynkin@Coxeter% + \dynkinEdge*{SingleEdge}{0}{1}% + \convertRootPair{0}{1} + \node[/Dynkin diagram/text,above] at + ($.5*(\dynkin@root@name \the\@fromRoot)+.5*(\dynkin@root@name \the\@toRoot)$) {\(4\)}; + \else% + \dynkinEdge*{DoubleEdge}{0}{1}% + \fi% +}% + +%% \DOneFourFourPly +%% Draws a D^1_4 series affine Dynkin diagram folded about an A^2_2. +\NewDocumentCommand\DOneFourFourPly{}% +{% + \dynkinPlaceRootHere*{0}{right}% + \edef\old@edge@length{\dynkin@edge@length}% + \pgfmathparse{1.5*\dynkin@edge@length}% + \xdef\dynkin@edge@length{\pgfmathresult pt}% + \dynkinPlaceRootRelativeTo*{1}{0}{south}{right}% + \dynkinPlaceRootRelativeTo*{3}{1}{south}{right}% + \dynkinPlaceRootRelativeTo*{4}{3}{south}{right}% + \xdef\dynkin@edge@length{\old@edge@length}% + \convertRootPair{0}{4}% + \node + (Dynkin current) + at + ($.5*(\dynkin@root@name \the\@fromRoot)+.5*(\dynkin@root@name \the\@toRoot)$){};% + \dynkinWest% + \dynkinPlaceRootHere*{2}{left}% + \dynkinEdge*{SingleEdge}{0}{2}% + \dynkinEdge*{SingleEdge}{1}{2}% + \dynkinEdge*{SingleEdge}{3}{2}% + \dynkinEdge*{SingleEdge}{4}{2}% + \dynkinFold*{0}{1}% + \dynkinFold*{1}{3}% + \dynkinFold*{3}{4}% +}% + + +%% \DfourPly +%% Draws a D series affine Dynkin diagram folded about its middle. +\NewDocumentCommand\DfourPly{}% +{% + \dynkinPlaceRootHere*{0}{left}% + \dynkinPlaceRootRelativeTo*{2}{0}{southeastfold}{above}% + \dynkinPlaceRootRelativeTo*{1}{2}{southwestfold}{left}% + \dynkinMoveToRoot*{2}% + \newcount\drmo% + \drmo=\the\dynkin@rank% + \advance\drmo by -1% + \newcount\drmt% + \drmt=\the\drmo% + \advance\drmt by -1% + \xdef\old@fold{\dynkin@fold@radius}% + \pgfmathparse{\dynkin@fold@radius+2*cos(60)*\dynkin@edge@length}% + \xdef\dynkin@fold@radius{\pgfmathresult pt}% + \dynkin@fold{2}{\the\drmt}% + \xdef\dynkin@fold@radius{\old@fold}% + \dynkinPlaceRootRelativeTo*{\the\drmo}{\the\drmt}{northwestfold}{left}% + \dynkinPlaceRootRelativeTo*{\the\dynkin@rank}{\the\drmt}{southwestfold}{left}% +% \ifdynkin@arrows% +% \dynkinLeftFold*{0}{1}% +% \dynkinLeftFold*{\the\drmo}{\the\dynkin@rank}% +% \fi% + \dynkinEdge*{RightDownArc}{0}{2}% + \dynkinEdge*{RightUpArc}{1}{2}% + \dynkinEdge*{RightDownArc}{\the\drmo}{\the\drmt}% + \dynkinEdge*{RightUpArc}{\the\dynkin@rank}{\the\drmt}% +}% + +%% \extendedDthreePly +%% Draws a D^1_4 series Dynkin diagram, folded over a B^1_3. +\NewDocumentCommand\extendedDthreePly{}% +{% + \dynkinPlaceRootHere*{2}{right}% + \edef\old@edge@length{\dynkin@edge@length}% + \pgfmathparse{1.5*\dynkin@edge@length}% + \xdef\dynkin@edge@length{\pgfmathresult pt}% + \dynkinPlaceRootRelativeTo*{3}{2}{south}{right}% + \dynkinPlaceRootRelativeTo*{4}{3}{south}{right}% + \xdef\dynkin@edge@length{\old@edge@length}% + \dynkinPlaceRootRelativeTo*{1}{3}{west}{diagonal}% + \dynkinPlaceRootRelativeTo*{0}{1}{west}{left}% + \dynkinEdge*{SingleEdge}{1}{3}% + \edef\old@fold@radius{\dynkin@fold@radius}% + \xdef\dynkin@fold@radius{\dynkin@edge@length}% + \dynkinEdge*{LeftDownArc}{2}{1}% + \dynkinEdge*{LeftUpArc}{4}{1}% + \xdef\dynkin@fold@radius{\old@fold@radius}% + \ifdynkin@arrows% + \dynkin@fold@arrow@if@oo{2}{3}% + \dynkin@fold@arrow@if@oo{3}{4}% + \fi% + \dynkinEdge*{SingleEdge}{0}{1}% +}% + + +%% \extendedDdynkin +%% Draws an D series affine Dynkin/Coxeter diagram. +\NewDocumentCommand\extendedDdynkin{}% +{% + \ifnum\dynkin@ply=4% + \ifnum\dynkin@rank=4% + \DOneFourFourPly% + \else% + \DfourPly% + \fi% + \else% + \ifnum\dynkin@ply=3% + \extendedDthreePly% + \else% + \ifnum\the\dynkin@rank=1% + \extendedAdynkin% + \else + \dynkinPlaceRootHere*{0}{left}% + \Ddynkin% + \ifnum\dynkin@ply=2% + \dynkinEdge*{RightDownArc}{0}{2}% + \else% + \ifdynkin@left@fold% + \dynkinEdge*{RightDownArc}{0}{2}% + \else% + \dynkinEdge*{SingleEdge}{0}{2}% + \fi% + \fi% + \fi% + \fi% + \fi% +}% -\newcommand*{\affineDdynkin} +%% \extendedEdynkin +%% Draws an E series affine Dynkin/Coxeter diagram. +\newcommand*{\extendedEdynkin}% +{% + \Edynkin% +}% + +%% \extendedFdynkin +%% Draws an F series affine Dynkin/Coxeter diagram. +\newcommand*{\extendedFdynkin}% +{% + \ifnum\dynkin@ply=1% + \dynkinPlaceRootHere*{0}{below}% + \dynkinEast% + \Fdynkin% + \dynkinEdge*{SingleEdge}{0}{1}% + \else% + \dynkinPlaceRootHere*{0}{above}% + \dynkinPlaceRootRelativeTo*{1}{0}{east}{above}% + \dynkinEdge*{SingleEdge}{0}{1}% + \dynkinPlaceRootRelativeTo*{2}{1}{southeastfold}{right}% + \dynkinDefiniteRightDownArc*{1}{2}% + \dynkinPlaceRootRelativeTo*{3}{2}{southwestfold}{below}% + \dynkinDefiniteDoubleDownLeftArc*{2}{3}% + \dynkinPlaceRootRelativeTo*{4}{3}{west}{below}% + \dynkinEdge*{SingleEdge}{3}{4}% + \ifdynkin@arrows% + \dynkinFold*{0}{4}% + \dynkinFold*{1}{3}% + \fi% + \fi% +}% + +%% \extendedGdynkin +%% Draws an G series affine Dynkin/Coxeter diagram. +\newcommand*{\extendedGdynkin}% +{% + \xdef\dynkin@gonality{6}% + \dynkinPlaceRootHere*{0}{below}% + \dynkinEast% + \Gdynkin% + \dynkinEdge*{SingleEdge}{0}{1}% +}% + +%% \extendedHdynkin +%% Draws an H series affine Coxeter diagram. +\newcommand*{\extendedHdynkin}% +{% + \dynkinPlaceRootHere*{0}{below}% + \dynkinEast% + \Adynkin% + \dynkinEdge*{SingleEdge}{0}{1}% + \ifnum\dynkin@rank=3% + \convertRootPair{1}{2}% + \else% + \convertRootPair{0}{1}% + \fi% + \node[/Dynkin diagram/text,above] + at + ($.5*(\dynkin@root@name \the\@fromRoot)+.5*(\dynkin@root@name \the\@toRoot)$) + {\(5\)};% +}% + + +%% \extendedIdynkin +%% Draws an I series affine Coxeter diagram. +\newcommand*{\extendedIdynkin}% { - \placeRoot*{0}{2}{1} - \Ddynkin + \dynkinPlaceRootHere*{0}{below}% + \dynkinEast% + \dynkin@rank=1% + \Adynkin% + \dynkinEdge*{SingleEdge}{0}{1}% + \convertRootPair{0}{1}% + \node[/Dynkin diagram/text,above] + at + ($.5*(\dynkin@root@name \the\@fromRoot)+.5*(\dynkin@root@name \the\@toRoot)$) + {\(\infty\)};% } -\newcommand*{\affineEdynkin} -{ - \ifnum\dynkinrank=6 - \placeRoot*{0}{3}{2} - \Edynkin - \else - \placeRoot{0}{0}{0} - \Edynkin + +%% \twistedAdynkin +%% Draws a twisted A series affine Dynkin diagram. +\NewDocumentCommand\twistedAdynkin{}% +{% + \ifnum\dynkin@rank=3 + \ClassError{Dynkin diagrams}{A2 series twisted diagrams cannot have rank \the\dynkin@rank}{}% \fi -} + \ifnum\dynkin@rank=2% + \dynkinPlaceRootHere*{0}{below}% + \dynkinPlaceRootRelativeTo*{1}{0}{east}{below}% + \dynkinQuadrupleEdge*{1}{0}% + \else% + \newcount\hmo% + \hmo=\the\dynkin@nodes% + \advance\hmo by -1% + \ifodd\dynkin@rank% + \ifnum\dynkin@ply>1% + \dynkinPlaceRootHere*{0}{above}% + \dynkinPlaceRootRelativeTo*{2}{0}{southeastfold}{below}% + \dynkinPlaceRootRelativeTo*{1}{2}{southwestfold}{below}% + \dynkinEdge*{RightDownArc}{0}{2}% + \dynkinEdge*{RightUpArc}{1}{2}% + \else% + \dynkinPlaceRootHere*{0}{left}% + \dynkinPlaceRootRelativeTo*{2}{0}{southeast}{left}% + \dynkinPlaceRootRelativeTo*{1}{2}{southwest}{left}% + \dynkinEdge*{SingleEdge}{0}{2}% + \dynkinEdge*{SingleEdge}{1}{2}% + \fi% + \dynkinMoveToRoot*{2}% + \dynkin@pipe{2}{\the\hmo}{east}{below}% + \dynkinPlaceRootRelativeTo*{\the\dynkin@nodes}{\the\hmo}{east}{below}% + \dynkinEdge*{DoubleEdge}{\the\dynkin@nodes}{\the\hmo}% + \ifnum\dynkin@ply>1% + \dynkinLeftFold*{0}{1}% + \fi% + \else% + \dynkinPlaceRootHere*{0}{below}% + \dynkinPlaceRootRelativeTo*{1}{0}{east}{below}% + \dynkinEdge*{DoubleEdge}{1}{0}% + \ifnum\dynkin@nodes>1% + \ifnum\dynkin@ply>1% + \ifnum\hmo>1% + \dynkin@fold{1}{\the\hmo}% + \fi% + \dynkinPlaceRootRelativeTo*{\the\dynkin@nodes}{\the\hmo}{west}{below}% + \else% + \ifnum\hmo>1% + \dynkin@pipe{1}{\the\hmo}{east}{below}% + \fi% + \dynkinPlaceRootRelativeTo*{\the\dynkin@nodes}{\the\hmo}{east}{below}% + \fi% + \dynkinEdge*{DoubleEdge}{\the\dynkin@nodes}{\the\hmo}% + \fi% + \fi% + \fi% +}% -\newcommand*{\affineFdynkin} -{ - \placeRoot{0}{0}{0} - \Fdynkin -} +%% \twistedDdynkin +%% Draws a twisted D series affine Dynkin diagram. +\NewDocumentCommand\twistedDdynkin{}% +{% + \IfStrEqCase{\dynkin@twisted@series}% + {% + {1}{\extendedDdynkin}% + {2}{\twistedDTwo}% + {3}% + {% + \ifnum\dynkin@rank=4% + \dynkinPlaceRootHere*{0}{below}% + \dynkinPlaceRootRelativeTo*{1}{0}{east}{below}% + \dynkinPlaceRootRelativeTo*{2}{1}{east}{below}% + \dynkinEdge*{SingleEdge}{0}{1}% + \dynkinTripleEdge*{2}{1}% + \else% + \ClassError% + {Dynkin diagrams}% + {D3 series twisted diagrams must have rank 2 and cannot have rank \the\dynkin@rank}% + {}% + \fi% + }% + }% +}% -\newcommand*{\affineGdynkin} -{ - \placeRoot{0}{3}{0} - \Gdynkin -} +\NewDocumentCommand\twistedDTwo{}% +{% + \ifnum\dynkin@rank<3% + \ClassError{Dynkin diagrams}{D2 series twisted diagrams cannot have rank \the\dynkin@rank}{}% + \fi% + \newcount\drmo% + \drmo=\the\dynkin@nodes% + \advance\drmo by -1% + \ifnum\dynkin@ply=1% + \dynkinPlaceRootHere*{0}{below}% + \dynkinPlaceRootRelativeTo*{1}{0}{east}{below}% + \else% + \ifnum\dynkin@rank=3% + \dynkinPlaceRootHere*{0}{right}% + \dynkinPlaceRootRelativeTo*{1}{0}{southwestfold}{left}% + \dynkinPlaceRootRelativeTo*{2}{1}{southeastfold}{right}% + \else% + \dynkinPlaceRootHere*{0}{above}% + \dynkinPlaceRootRelativeTo*{1}{0}{east}{above}% + \fi% + \fi% + \ifnum\dynkin@ply=2% + \dynkinEdge*{DoubleUpRightArc}{1}{0}% + \else + \dynkinEdge*{DoubleEdge}{1}{0}% + \fi% + \ifnum\dynkin@ply>1% + \ifnum\dynkin@rank>3% + \dynkin@fold{1}{\the\drmo}% + \dynkinPlaceRootRelativeTo*{\the\dynkin@nodes}{\the\drmo}{west}{below}% + \dynkinFold*{0}{\the\dynkin@nodes}% + \else% + \dynkinFold*{0}{2}% + \fi% + \else% + \ifnum\dynkin@rank>2% + \dynkin@pipe{1}{\the\drmo}{east}{below}% + \fi% + \dynkinPlaceRootRelativeTo*{\the\dynkin@nodes}{\the\drmo}{east}{below}% + \fi% + \ifnum\dynkin@ply=2% + \dynkinEdge*{DoubleDownRightArc}{\the\drmo}{\the\dynkin@nodes}% + \else + \dynkinEdge*{DoubleEdge}{\the\drmo}{\the\dynkin@nodes}% + \fi% +}% + + +%% \twistedEdynkin +%% Draws a twisted E series affine Dynkin diagram. +\NewDocumentCommand\twistedEdynkin{}% +{% + \IfStrEqCase{\dynkin@twisted@series}% + {% + {0}{\Edynkin}% + {1}{\extendedEdynkin}% + {2}% + {% + \dynkinPlaceRootHere*{0}{below}% + \dynkin@pipe{0}{2}{east}{below}% + \dynkinPlaceRootRelativeTo*{3}{2}{east}{below}% + \dynkinPlaceRootRelativeTo*{4}{3}{east}{below}% + \dynkinEdge*{SingleEdge}{3}{4}% + \dynkinEdge*{DoubleEdge}{3}{2}% + }% + }% + [\dynkin@error@series]% +}% + \endinput |