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diff --git a/Master/texmf-dist/doc/latex/dynkin-diagrams/README b/Master/texmf-dist/doc/latex/dynkin-diagrams/README index 0a9ec82779f..26cba041b7c 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 - v1.0 + v2.0 - 8 September 2017 + 18 November 2017 ___________________________________ Authors : Ben McKay diff --git a/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.bib b/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.bib new file mode 100644 index 00000000000..a72cb1dade1 --- /dev/null +++ b/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.bib @@ -0,0 +1,212 @@ +% This file was created with JabRef 2.10b2. +% Encoding: ISO8859_1 + + +@Book{Bourbaki:2002, + Title = {Lie groups and {L}ie algebras. {C}hapters 4--6}, + Author = {Bourbaki, Nicolas}, + Publisher = {Springer-Verlag, Berlin}, + Year = {2002}, + Note = {Translated from the 1968 French original by Andrew Pressley}, + Series = {Elements of Mathematics (Berlin)}, + + ISBN = {3-540-42650-7}, + Mrclass = {17-01 (00A05 20E42 20F55 22-01)}, + Mrnumber = {1890629}, + Owner = {user}, + Pages = {xii+300}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1007/978-3-540-89394-3} +} + +@Book{Carter:2005, + Title = {Lie algebras of finite and affine type}, + Author = {Carter, R. W.}, + Publisher = {Cambridge University Press, Cambridge}, + Year = {2005}, + Series = {Cambridge Studies in Advanced Mathematics}, + Volume = {96}, + + ISBN = {978-0-521-85138-1; 0-521-85138-6}, + Mrclass = {17-02 (17B67)}, + Mrnumber = {2188930}, + Mrreviewer = {Stephen Slebarski}, + Owner = {user}, + Pages = {xviii+632}, + Timestamp = {2017.11.15}, + 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}, + + ISBN = {0-8218-1065-0}, + Mrclass = {01A75 (60Jxx)}, + Mrnumber = {1757976}, + Mrreviewer = {William M. McGovern}, + Owner = {user}, + Pages = {xxviii+796}, + Timestamp = {2017.11.15} +} + +@Article{Dynkin:1952, + Title = {Semisimple subalgebras of semisimple {L}ie algebras}, + Author = {Dynkin, E. B.}, + Journal = {Mat. Sbornik N.S.}, + Year = {1952}, + Note = {Reprinted in English translation in \cite{Dynkin:2000}.}, + Pages = {349--462 (3 plates)}, + Volume = {30(72)}, + + Mrclass = {09.1X}, + Mrnumber = {0047629}, + Mrreviewer = {I. Kaplansky}, + Owner = {user}, + Timestamp = {2017.11.15} +} + +@Book{Grove/Benson:1985, + Title = {Finite reflection groups}, + Author = {Grove, L. C. and Benson, C. T.}, + Publisher = {Springer-Verlag, New York}, + Year = {1985}, + Edition = {Second}, + Series = {Graduate Texts in Mathematics}, + Volume = {99}, + + ISBN = {0-387-96082-1}, + Mrclass = {20-01 (20B25 20H15)}, + Mrnumber = {777684}, + Owner = {user}, + Pages = {x+133}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1007/978-1-4757-1869-0} +} + +@Book{Helgason:2001, + Title = {Differential geometry, {L}ie groups, and symmetric spaces}, + Author = {Helgason, Sigurdur}, + Publisher = {American Mathematical Society, Providence, RI}, + Year = {2001}, + Note = {Corrected reprint of the 1978 original}, + Series = {Graduate Studies in Mathematics}, + Volume = {34}, + + ISBN = {0-8218-2848-7}, + Mrclass = {53C35 (22E10 22E46 22E60)}, + Mrnumber = {1834454}, + Owner = {user}, + Pages = {xxvi+641}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1090/gsm/034} +} + +@Book{Humphreys:1990, + Title = {Reflection groups and {C}oxeter groups}, + Author = {Humphreys, James E.}, + Publisher = {Cambridge University Press, Cambridge}, + Year = {1990}, + Series = {Cambridge Studies in Advanced Mathematics}, + Volume = {29}, + + ISBN = {0-521-37510-X}, + Mrclass = {20-02 (20F32 20F55 20G15 20H15)}, + Mrnumber = {1066460}, + Mrreviewer = {Louis Solomon}, + Owner = {user}, + Pages = {xii+204}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1017/CBO9780511623646} +} + +@Book{Kac:1990, + Title = {Infinite-dimensional {L}ie algebras}, + Author = {Kac, Victor G.}, + Publisher = {Cambridge University Press, Cambridge}, + Year = {1990}, + Edition = {Third}, + + ISBN = {0-521-37215-1; 0-521-46693-8}, + Mrclass = {17B65 (17B67 17B68 58F07)}, + Mrnumber = {1104219}, + Owner = {user}, + Pages = {xxii+400}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1017/CBO9780511626234} +} + +@Book{OnishchikVinberg:1990, + Title = {Lie groups and algebraic groups}, + Author = {Onishchik, A. L. and Vinberg, {\`E}. B.}, + Publisher = {Springer-Verlag}, + Year = {1990}, + + Address = {Berlin}, + Note = {Translated from the Russian and with a preface by D. A. Leites}, + Series = {Springer Series in Soviet Mathematics}, + + ISBN = {3-540-50614-4}, + Mrclass = {22-01 (17B20 20G20 22E10 22E15)}, + Mrnumber = {91g:22001}, + Mrreviewer = {James E. Humphreys}, + Owner = {user}, + Pages = {xx+328}, + Timestamp = {2017.11.15} +} + +@Book{Onishchik/Vinberg:1990, + Title = {Lie groups and algebraic groups}, + Author = {Onishchik, A. L. and Vinberg, \`E. B.}, + Publisher = {Springer-Verlag, Berlin}, + Year = {1990}, + Note = {Translated from the Russian and with a preface by D. A. Leites}, + Series = {Springer Series in Soviet Mathematics}, + + ISBN = {3-540-50614-4}, + Mrclass = {22-01 (17B20 20G20 22E10 22E15)}, + Mrnumber = {1064110}, + Mrreviewer = {James E. Humphreys}, + Owner = {user}, + Pages = {xx+328}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1007/978-3-642-74334-4} +} + +@Book{Satake:1980, + Title = {Algebraic structures of symmetric domains}, + Author = {Satake, Ichir\^o}, + Publisher = {Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J.}, + Year = {1980}, + Series = {Kan\^o Memorial Lectures}, + Volume = {4}, + + Mrclass = {32-02 (17C35 32Mxx 53C35)}, + Mrnumber = {591460}, + Mrreviewer = {S. Murakami}, + Owner = {user}, + Pages = {xvi+321}, + Timestamp = {2017.11.15} +} + +@Book{Vinberg:1994, + Title = {Lie groups and {L}ie algebras, {III}}, + Editor = {Vinberg, \`E. B.}, + Publisher = {Springer-Verlag, Berlin}, + Year = {1994}, + Note = {Structure of Lie groups and Lie algebras, A translation of {{\i}t Current problems in mathematics. Fundamental directions. Vol. 41} (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990 [ MR1056485 (91b:22001)], Translation by V. Minachin [V. V. Minakhin], Translation edited by A. L. Onishchik and \`E. B. Vinberg}, + Series = {Encyclopaedia of Mathematical Sciences}, + Volume = {41}, + + ISBN = {3-540-54683-9}, + Mrclass = {22-06 (17-06 22Exx)}, + Mrnumber = {1349140}, + Owner = {user}, + Pages = {iv+248}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1007/978-3-662-03066-0} +} + 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 fa4ed5acbe9..851c6ae2200 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 7bc9eb0a18d..f6566c0be0e 100644 --- a/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.tex +++ b/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.tex @@ -3,124 +3,197 @@ \title{The Dynkin diagrams package} \author{Ben McKay} \date{\today} - -\usepackage{dynkin-diagrams} + \usepackage{amsmath} \usepackage{amsfonts} \usepackage{array} \usepackage{xstring} -\usepackage{etoolbox} +\usepackage{etoolbox} +\usepackage{longtable} +\usepackage{showexpl} +\usepackage{booktabs} +\usepackage{dynkin-diagrams} \usetikzlibrary{backgrounds} \usetikzlibrary{decorations.markings} -\usepackage{longtable} -\usepackage{showexpl} \newcommand{\C}[1]{\mathbb{C}^{#1}} - - \renewcommand*{\arraystretch}{1.5} +\renewcommand\ResultBox{\fcolorbox{gray!50}{gray!30}} + \begin{document} + \maketitle \tableofcontents -\section{Quick introduction} +\section{Quick introduction} This is a test of the Dynkin diagram package. Load the package via \begin{verbatim} -\usepackage{dynkin-diagrams} +\usepackage{dynkin-diagrams} \end{verbatim} -and invoke it directly: +(see below for options) and invoke it directly: + \begin{LTXexample} The flag variety of pointed lines in projective 3-space is associated to -the Dykin diagram \dynk[parabolic=3]{A}{3}. +the Dynkin diagram \dynkin[parabolic=3]{A}{3}. \end{LTXexample} -or use the long form inside a \verb!\tikz! statement or environment: + +or use the long form inside a \verb!\tikz! statement: \begin{LTXexample} \tikz \dynkin[parabolic=3]{A}{3}; \end{LTXexample} + +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} -\tikz \dynkin[parabolic=3,label=true]{A}{3}; +\dynkin[parabolic=3,label]{A}{3} \end{LTXexample} - -\bigskip - -Inside an environment: +\newpage\noindent% +Make up your own labels for the roots: \begin{LTXexample} \begin{tikzpicture} -\dynkin[parabolic=3,label=true]{A}{3} +\dynkin[parabolic=3]{A}{3} +\rootlabel{2}{\alpha_2} \end{tikzpicture} \end{LTXexample} - -\bigskip - -Make up your own labels for the roots: - +Use any text scale you like: +\begin{LTXexample} +\begin{tikzpicture} +\dynkin[parabolic=3,textscale=1.2]{A}{3}; +\rootlabel{2}{\alpha_2} +\end{tikzpicture} +\end{LTXexample} +and access root labels via TikZ: \begin{LTXexample} \begin{tikzpicture} \dynkin[parabolic=3]{A}{3}; -\node at (root label 2) {\scalebox{.7}{\(\alpha_2\)}}; +\node at (root label 2) {\(\alpha_2\)}; \end{tikzpicture} \end{LTXexample} - -\newpage - -Drawing curves between the roots: - +The labels have default locations: \begin{LTXexample} \begin{tikzpicture} -\dynkin[parabolic=429]{E}{8} -\draw[brown,-latex] - (root 3.south) - to [out=-90, in=-90] - (root 6.south); +\dynkin{E}{8}; +\rootlabel{1}{\alpha_1} +\rootlabel{2}{\alpha_2} +\rootlabel{3}{\alpha_3} \end{tikzpicture} \end{LTXexample} - -Various options: - +You can use a starred form to flip labels to alternate locations: \begin{LTXexample} -\tikz \dynkin[color=brown]{G}{2}; +\begin{tikzpicture} +\dynkin{E}{8}; +\rootlabel*{1}{\alpha_1} +\rootlabel*{2}{\alpha_2} +\rootlabel*{3}{\alpha_3} +\end{tikzpicture} \end{LTXexample} - +TikZ can access the roots themselves: +\typeout{AAAAAAA} \begin{LTXexample} -\tikz \dynkin[edgelength=1.2,parabolic=3]{A}{3}; +\begin{tikzpicture} +\dynkin{A}{4}; +\fill[white,draw=black] (root 2) circle (.1cm); +\draw[black] (root 2) circle (.05cm); +\end{tikzpicture} \end{LTXexample} - +Some diagrams will have double edges: \begin{LTXexample} -\tikz \dynkin[crosssize=.1cm,parabolic=3]{A}{3}; +\dynkin{F}{4} \end{LTXexample} - +or triple edges: \begin{LTXexample} -\tikz \dynkin[dotradius=.08cm,parabolic=3]{A}{3}; +\dynkin{G}{2} \end{LTXexample} - +\newpage\noindent% +Draw curves between the roots: \begin{LTXexample} -\begin{tikzpicture}[ - show background rectangle, - background rectangle/.style={fill=lightgray}] -\dynkin[parabolic=1,background color=lightgray]{G}{2} +\begin{tikzpicture} +\dynkin[parabolic=429]{E}{8} +\draw[very thick, black!50,-latex] (root 3.south) to [out=-45, in=-135] (root 6.south); \end{tikzpicture} \end{LTXexample} - - -\section{Syntax} - -Inside a \verb!\tikz! environment, 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: - +Draw dots on the roots: \begin{LTXexample} \begin{tikzpicture} -\dynkin[parabolic=5]{D}{*} +\dynkin[label]{C}{8} +\dynkinopendot{3} +\dynkinopendot{7} \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} +\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} +\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} -Outside a \verb!\tikz! environment, use \verb!\dynk! instead of \verb!\dynkin!. +\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. -\bigskip \newcommand*{\typ}[1]{\(\left<\texttt{#1}\right>\)} \newcommand*{\optionLabel}[3]{%% @@ -128,37 +201,108 @@ Outside a \verb!\tikz! environment, use \verb!\dynk! instead of \verb!\dynkin!. }%% \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{background color}{\typ{color name}}{white} +\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}{.04cm} -& size of the dots in the Dynkin diagram \\ +\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{crosssize}{\typ{number}}{1.5} -& size of the crosses, for parabolic subgroup diagrams. \\ +\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. \\ + +\optionLabel{ordering}{\typ{Adams, Bourbaki, Carter, Dynkin, Kac}}{Bourbaki} +& which ordering of the roots to use in exceptional root systems as follows: \end{tabular} -%% All other options are passed to tikz. -\section{Finding the roots} -The roots are labelled in the Bourbaki labelling, but from \(0\) to \(r-1\), where \(r\) is the rank. -The command sets up nodes \texttt{(root 0)}, \texttt{(root 1)}, and so on. -Use these tikz nodes to draw on the Dynkin diagram. -It also sets up nodes \texttt{(root label 0)}, \texttt{(root label 1)}, and so on for the labels. +\newpage + +\NewDocumentCommand\tablerow{mm}% +{% +\(#1_{#2}\) +& +\dynkin[label,ordering=Adams]{#1}{#2} +& +\dynkin[label]{#1}{#2} +& +\dynkin[label,ordering=Carter]{#1}{#2} +& +\dynkin[label,ordering=Dynkin]{#1}{#2} +& +\dynkin[label,ordering=Kac]{#1}{#2} +\\ +}% + +\begin{center} +\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} +\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{Example: some parabolic subgroups} -\newcommand{\drawparabolic}[3]%% -{#1_{#2,#3} & \tikz \dynkin[parabolic=#3]{#1}{#2}; \\} +\newcommand{\drawparabolic}[3]{#1_{#2,#3} & \tikz \dynkin[parabolic=#3]{#1}{#2}; \\} \begin{center} \begin{longtable}{@{}>{$}r<{$}m{2cm}m{2cm}@{}} @@ -167,18 +311,22 @@ It also sets up nodes \texttt{(root label 0)}, \texttt{(root label 1)}, and so o \endfoot \endlastfoot \drawparabolic{A}{1}{0} -\drawparabolic{A}{1}{1} +\drawparabolic{A}{1}{2} \drawparabolic{A}{2}{0} \drawparabolic{A}{2}{2} -\drawparabolic{A}{2}{2} -\drawparabolic{B}{2}{3} -\drawparabolic{C}{3}{5} -\drawparabolic{D}{5}{4} -\drawparabolic{E}{6}{5} -\drawparabolic{E}{7}{101} -\drawparabolic{E}{8}{123} -\drawparabolic{F}{4}{13} +\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} @@ -188,24 +336,355 @@ It also sets up nodes \texttt{(root label 0)}, \texttt{(root label 1)}, and so o \renewcommand*{\arraystretch}{1.5} \begin{center} -\begin{longtable}{@{}>{$}r<{$}m{2cm}m{5cm}@{}} +\begin{longtable}{@{}>{$}r<{$}m{2.2cm}m{5cm}@{}} \endfirsthead \endhead \endfoot \endlastfoot - A_n &\dynk[parabolic=8]{A}{*}& Grassmannian of $k$-planes in $\C{n+1}$ \\ - B_n &\dynk[parabolic=1]{B}{*}& $(2n-1)$-dimensional hyperquadric, i.e. the variety of null lines in $\C{2n+1}$ - \\ - C_n &\dynk[parabolic=16]{C}{*}& space of Lagrangian $n$-planes in $\C{2n}$ - \\ - D_n &\dynk[parabolic=1]{D}{*}&$(2n-2)$-dimensional hyperquadric, i.e. the variety of null lines in $\C{2n}$ -\\ - D_n&\dynk[parabolic=32]{D}{*}& one component of the variety of maximal dimension null subspaces of $\C{2n}$ \\ - D_n - &\dynk[parabolic=16]{D}{*}&the other component\\ - E_6&\dynk[parabolic=1]{E}{6}&complexified octave projective plane\\ - E_6&\dynk[parabolic=32]{E}{6}&its dual plane\\ - E_7 &\dynk[parabolic=64]{E}{7}& the space of null octave 3-planes in octave 6-space + 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 \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} + +\begin{LTXexample} +\dynkin[affine,label]{B}{8} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[affine,label]{B}{*} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[affine,label]{C}{8} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[affine,label]{C}{*} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[affine,label]{D}{8} +\end{LTXexample} + +\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} + +\begin{LTXexample} +\dynkin[affine,label]{E}{8} +\end{LTXexample} + +Open circles instead of closed dots: +\begin{LTXexample} +\dynkin[affine,open,label]{E}{8} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[affine,label]{F}{4} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[affine,label]{G}{2} +\end{LTXexample} + + +\section{Coxeter diagrams} + +\begin{LTXexample} +\dynkin[Coxeter]{B}{7} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[Coxeter]{F}{4} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[Coxeter]{G}{2} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[Coxeter]{H}{7} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[Coxeter]{I}{7} +\end{LTXexample} + + +\section{Folded Dynkin diagrams} + +\begin{LTXexample} +\dynkin[folded]{E}{6} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded,label]{E}{6} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded]{A}{*} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded,label]{A}{1} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded,label]{A}{2} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded,label]{A}{3} +\end{LTXexample} + +\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} + +\begin{LTXexample} +\dynkin[folded,label,arrows=false]{A}{11} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded]{D}{*} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded,label]{D}{1} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded,label]{D}{2} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded,label]{D}{3} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded,label]{D}{4} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded,label]{D}{10} +\end{LTXexample} + +\begin{LTXexample} +\dynkin[folded,label]{D}{11} +\end{LTXexample} + + + +\section{Satake diagrams} + +We have incomplete support for Satake diagrams as yet, following the conventions of \cite{Helgason:2001}. + +\begin{LTXexample} +\dynkin{A}{I} +\end{LTXexample} + +\begin{LTXexample} +\dynkin{A}{II} +\end{LTXexample} + +\begin{LTXexample} +\dynkin{E}{I} +\end{LTXexample} + +\begin{LTXexample} +\dynkin{E}{II} +\end{LTXexample} + +\begin{LTXexample} +\dynkin{E}{III} +\end{LTXexample} + +\begin{LTXexample} +\dynkin{E}{IV} +\end{LTXexample} + +\begin{LTXexample} +\dynkin{E}{V} +\end{LTXexample} + +\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} + +\begin{LTXexample} +\dynkin{F}{II} +\end{LTXexample} + +\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{Other stuff} + +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} + +Always draw roots after edges. + +\nocite{*} +\bibliographystyle{amsplain} +\bibliography{dynkin-diagrams} \end{document} 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 5626893c931..8ed53464f8f 100644 --- a/Master/texmf-dist/tex/latex/dynkin-diagrams/dynkin-diagrams.sty +++ b/Master/texmf-dist/tex/latex/dynkin-diagrams/dynkin-diagrams.sty @@ -1,347 +1,1485 @@ +% +% +% The Dynkin Diagrams package. +% +% Version 2 +% +% +% This package draws Dynkin diagrams in LaTeX documents, using the TikZ package. +% Please see the file dynkin-diagrams.tex for examples of use of this package. +% +% Benjamin McKay +% b.mckay@ucc.ie +% +% Released under the LaTeX Project Public License v1.3c or later, see +% http://www.latex-project.org/lppl.txt +% +% +% +% \NeedsTeXFormat{LaTeX2e}[1994/06/01] -\ProvidesPackage{dynkin-diagrams}[2016/06/28 Dynkin diagrams] - +\ProvidesPackage{dynkin-diagrams}[2017/11/14 Dynkin diagrams] \RequirePackage{tikz} \RequirePackage{xstring} +\RequirePackage{xparse} \RequirePackage{etoolbox} +\RequirePackage{expl3} \RequirePackage{pgfkeys} +\RequirePackage{pgfopts} \usetikzlibrary{decorations.markings} - -\ProcessOptions\relax - +\usetikzlibrary{arrows,arrows.meta} +\usetikzlibrary{calc} %% -%% Application programming interface: +%% Application programming interface: +%% See dynkin-diagrams.tex file for examples of use. %% -\newcommand*{\dynk}[3][]{%% -\tikz[baseline=-\the\dimexpr\fontdimen22\textfont2\relax ] \dynkin[#1]{#2}{#3};% -}%% - -% See test1.tex file for examples of use. - -\newcommand*{\dynkin}[3][]{ -\pgfkeys{/dynkin, default, #1}% -\IfStrEq{#3}{*}{}{\dynkinrank=#3} -\IfStrEq{#2}{A}{\Adynkin[\dynkinparabolic]{#3}}{} -\IfStrEq{#2}{B}{\Bdynkin[\dynkinparabolic]{#3}}{} -\IfStrEq{#2}{C}{\Cdynkin[\dynkinparabolic]{#3}}{} -\IfStrEq{#2}{D}{\Ddynkin[\dynkinparabolic]{#3}}{} -\IfStrEq{#2}{E}{\Edynkin[\dynkinparabolic]{#3}}{} -\IfStrEq{#2}{F}{\Ffourdynkin[\dynkinparabolic]{#3}}{} -\IfStrEq{#2}{G}{\Gtwodynkin[\dynkinparabolic]}{} -\IfStrEq{\dynkinlabeltheroots}{true}{\dynkinprintlabels}{} -} +\NewDocumentCommand\dynkin{O{}mm}% +{% + \ifdefined\filldraw% + \@dynkin[#1]{#2}{#3}% + \else% + \tikz[baseline=-\the\dimexpr\fontdimen22\textfont2\relax ]{\@dynkin[#1]{#2}{#3}}% + \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{\dynkinseries}% + {% + {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}}% + }% + [\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}}% + }% + [\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}}% + }% + [\RootNumber=#1]% + }% + {G}% + {% + \IfStrEqCase{\dynkinordering}% + {% + {Carter}{\RootNumber=\stringcharacterinposition{21}{#1}}% + {Dynkin}{\RootNumber=\stringcharacterinposition{21}{#1}}% + {Kac}{\RootNumber=\stringcharacterinposition{21}{#1}}% + }% + [\RootNumber=#1]% + }% + }% + [\RootNumber=#1]% + } +}% + +\NewDocumentCommand\dynkinprint{m}% +{% + \scalebox{\dynkintextscale}{\(#1\)}% +}% + +%% \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}% +{% + \IfBooleanTF{#1}% + {\node at (root label swap #2) {\dynkinprint{#3}};}% + {\node at (root label #2) {\dynkinprint{#3}};}% +}% + +%% \dynkinprintlabels +%% -> +%% Prints the default labels on the Dynkin diagram, in the given ordering. +\newcommand{\dynkinprintlabels}% +{% + \foreach \i in {1,...,\the\dynkinrank}% + {\rootlabel{\i}{\i}}% + \ifisaffine\rootlabel{0}{0}\fi% +}% + +%% \dynkincross{<n>} +%% -> +%% Prints a cross at root <n> on the current Dynkin diagram. +%% The starred form accepts <n> in the Bourbaki ordering. +\NewDocumentCommand\dynkincross{sO{}m}% +{% + \IfBooleanTF{#1}% + {% + \convertRootNumber{#3}% + }% + {% + \RootNumber=#3% + }% + \draw[\dynkincrossstyle,\dynkincolor,#2]% + ($(root \the\RootNumber)+(\dynkinradius,\dynkinradius)$)% + --% + ($(root \the\RootNumber)-(\dynkinradius,\dynkinradius)$);% + \draw[\dynkincrossstyle,\dynkincolor]% + ($(root \the\RootNumber)+(-\dynkinradius,\dynkinradius)$)% + --% + ($(root \the\RootNumber)+(\dynkinradius,-\dynkinradius)$);% +}% + +%% \dynkinopendot{<n>} +%% -> +%% Prints an open dot at root <n> on the current Dynkin diagram. +%% The starred form accepts <n> in the Bourbaki ordering. +\NewDocumentCommand\dynkinopendot{sO{}m}% +{% + \IfBooleanTF{#1}% + {% + \convertRootNumber{#3}% + }% + {% + \RootNumber=#3% + }% + \fill[\dynkinbackcolor,draw=\dynkincolor,#2] (root \the\RootNumber) circle (\dynkinradius);% +}% +%% \dynkincloseddot{<n>} +%% -> +%% Prints a closed dot at root <n> on the current Dynkin diagram. +%% The starred form accepts <n> in the Bourbaki ordering. +\NewDocumentCommand\dynkincloseddot{sO{}m}% +{% + \IfBooleanTF{#1}% + {% + \convertRootNumber{#3}% + }% + {% + \RootNumber=#3% + }% + \fill[\dynkincolor,draw=\dynkincolor,#2] (root \the\RootNumber) circle (\dynkinradius);% +}% + +%% \dynkindot{<n>} +%% -> +%% Prints a dot at root <n> on the current Dynkin diagram in the default style. +%% The starred form accepts <n> in the Bourbaki ordering. +\NewDocumentCommand\dynkindot{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% + }% + {% + \ifnum#3=0% + \ifdynkinopendots% + \dynkincloseddot[#2]{0}% + \else% + \dynkinopendot[#2]{0}% + \fi% + \else% + \ifdynkinopendots% + \dynkinopendot[#2]{#3}% + \else% + \dynkincloseddot[#2]{#3}% + \fi% + \fi% + }% +}% + +%% \dynkinline{<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}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \draw[\dynkincolor,\dynkinedgestyle,#2] ($(root \the\@fromRoot)$) -- ($(root \the\@toRoot)$);% +}% + +%% \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. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkinfoldarrow{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \draw[\dynkinfoldarrowstyle,\dynkinfoldarrowcolor,#2] (root \the\@fromRoot) -- (root \the\@toRoot);% +}% + +%% \dynkindownarc{<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}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \draw[\dynkincolor,\dynkinedgestyle,#2] ($(root \the\@fromRoot)$) arc (90:0:\dynkinedgelength);% +}% + +%% \dynkinuparc{<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}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \draw[\dynkincolor,\dynkinedgestyle,#2] ($(root \the\@fromRoot)$) arc (0:-90:\dynkinedgelength);% +}% + +%% \dynkinsemicircle{<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}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \draw[\dynkincolor,\dynkinedgestyle,#2] ($(root \the\@fromRoot)$) arc (90:-90:\dynkinedgelength);% +}% + +%% \dynkindots{<p>}{s<q>} +%% -> +%% Draws a dotted line from root <p> to root <q> on the current Dynkin diagram. +%% The starred form accepts <p> and <q> in the Bourbaki ordering. +\NewDocumentCommand\dynkindots{sO{}mm}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \draw[densely dotted,\dynkincolor,#2] ($(root \the\@fromRoot)$) -- ($(root \the\@toRoot)$);% +}% + +%% \dynkindoubleline{<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}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@fromRoot=#3% + \@toRoot=#4% + }% + \ifdynkinarrows% + \draw[double,postaction={decorate},\dynkincolor,\dynkinedgestyle,#2]% + ($(root \the\@fromRoot)$) -- ($(root \the\@toRoot)$);% + \else% + \draw[double,\dynkincolor,\dynkinedgestyle,#2]% + ($(root \the\@fromRoot)$) -- ($(root \the\@toRoot)$);% + \fi% +}% + +%% \dynkintripleline{<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}% +{% + \IfBooleanTF{#1}% + {% + \convertRootPair{#3}{#4}% + }% + {% + \@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% + ]% + ($(root \the\@toRoot)$) --% + ($.65*(root \the\@fromRoot)+.35*(root \the\@toRoot)$);% + \fi% + \draw[\dynkincolor,#2] ($(root \the\@fromRoot)$) -- ($(root \the\@toRoot)$);% +}% %%% %%% Implementation: %%% -\newcount\dynkinrank - -\pgfkeys{ - /dynkin/.is family, - /tikz/decoration={markings,mark=at position 0.7 with {\arrow{>}}}, - /dynkin, - default/.style = { - label = false, - parabolic = 0, - color = black, - background color = white, - dotradius=.04cm, - edgelength=.35cm, - crosssize=.07cm - }, - label/.estore in = \dynkinlabeltheroots, - parabolic/.estore in = \dynkinparabolic, - color/.store in =\dynkincolor, - background color/.store in =\dynkinbackcolor, - dotradius/.estore in = \dynkinradius, - edgelength/.estore in = \dykinedgelength, - crosssize/.estore in = \dynkinXsize, - .search also={/tikz}, -} +\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 -\newcommand{\dynkinprintlabels} +% *=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}% + {% + \global\dynkinrank=#3% + \gdef\dynkinSatake{*}% + }% + {% + \IfStrEqCase{#2}% + {% + {A}% + {% + \IfStrEqCase{#3}% + {% + {*}{ }% + {I}{ }% + {II}{}% + {III}{}% + {IV} {}% + }% + [\errorRank]% + }% + {B}% + {% + \IfStrEqCase{#3}% + {% + {*}{ }% + {I}{}% + {II} {}% + }% + [\errorRank]% + }% + {C}% + {% + \IfStrEqCase{#3}% + {% + {*}{ }% + {I}{}% + {II} {}% + }% + [\errorRank]% + }% + {D}% + {% + \IfStrEqCase{#3}% + {% + {*}{ }% + {I}{ }% + {II} {}% + {III}{}% + }% + [\errorRank]% + }% + {E}% + {% + \IfStrEqCase{#3}% + {% + {I}{ \global\dynkinrank=6}% + {II}% + {% + \global\dynkinfoldedtrue% + \global\dynkinrank=6% + }% + {III}% + {% + \global\dynkinfoldedtrue% + \global\dynkinrank=6% + }% + {IV}% + {% + \global\dynkinrank=6% + }% + {V}% + {% + \global\dynkinrank=7% + }% + {VI}% + {% + \global\dynkinrank=7% + }% + {VII}% + {% + \global\dynkinrank=7% + }% + {VIII}% + {% + \global\dynkinrank=8% + }% + {XI}% + {% + \global\dynkinrank=8% + }% + }% + [\errorRank]% + }% + {F}% + {% + \global\dynkinrank=4% + \IfStrEqCase{#3}% + {% + {I}{ }% + {II} {}% + }% + [\errorRank]% + }% + {G}% + {% + \global\dynkinrank=2% + \IfStrEqCase{#3}% + {% + {I}{ }% + }% + [\errorRank]% + }% + {H}% + {% + \IfStrEqCase{#3}% + {% + {*}% + {% + }% + }% + [\errorRank]% + }% + {I}% + {% + \IfStrEqCase{#3}% + {% + {*}% + {% + }% + }% + [\errorRank]% + }% + }% + [\errorSeries]% + }% + \checkDynkinDiagram% + \ifisaffine% + \csname affine#2dynkin\endcsname% + \else% + \csname#2dynkin\endcsname% + \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 { -\newcount\rmo -\rmo=\dynkinrank -\advance\rmo by -1 -\foreach \i in {0,...,\the\rmo} +\str_item:fn { #1 } { #2 } +} +\cs_generate_variant:Nn \str_item:nn {f} +\ExplSyntaxOff + +\NewDocumentCommand\errorRootOrdering{} +{% + \ClassWarning{Unrecognized root ordering: ``\dynkinordering'' in Dynkin diagram}% +}% + +\NewDocumentCommand\errorRank{}% +{% + \ClassWarning{Unrecognized \dynkinseries{} series rank: ``\the\dynkinrank'' in Dynkin diagram}% +}% + +\NewDocumentCommand\errorSeries{}% +{% + \ClassWarning{Unrecognized series ``\dynkinseries{}'' in Dynkin diagram}% +}% + +%% \checkDynkinDiagram +%% -> +%% Raises error messages for erroneous inputs. +\NewDocumentCommand\checkDynkinDiagram{}% +{% + \IfStrEqCase{\dynkinordering}% + {% + {Adams}{}% + {Bourbaki}{}% + {Carter}{}% + {Dynkin}{}% + {Kac}{}% + }% + [\ClassWarning{Unrecognized label ordering: ``\dynkinordering'' in Dynkin diagram}]% + \IfStrEqCase{\dynkinseries}% + {% + {A}{}% + {B}{}% + {C}{}% + {D}{}% + {E}% + {% + \ifnum\dynkinrank=6% + \else% + \ifnum\dynkinrank=7% + \else% + \ifnum\dynkinrank=8% + \else% + \errorRank% + \fi% + \fi% + \fi% + }% + {F}% + {% + \ifnum\dynkinrank=4% + \else% + \errorRank% + \fi% + }% + {G}% + {% + \ifnum\dynkinrank=2% + \else% + \errorRank% + \fi% + }% + {H}{}% + {I}{}% + }% + [\errorSeries]% +}% + +% 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\@fromRoot +\newcount\@toRoot + +%% \convertRootPair{<p>}{<q>} +%% -> +%% Stores conversions in \@fromRoot and \@toRoot. +\NewDocumentCommand\convertRootPair{mm} +{% + \convertRootNumber{#1}% + \@fromRoot=\RootNumber% + \convertRootNumber{#2}% + \@toRoot=\RootNumber% +}% + +%% \testbit{<n>}{<b>}{<f>}{<g>} +%% If bit number <b> of <n> is 1 then expand <f> else expand <g>. +\newcommand*{\testbit}[4]% +{% + \pgfmathparse{int(mod(div(#1,2^(#2)),2))}% + \let\tf\pgfmathresult% + \IfStrEq{\tf}{1}{#3}{#4}% +}% + +%% \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}% +{% + \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}) {};% + }% + {% + \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}) {};% + }% +}% + +%% \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}% +{% + \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}) {};% + }% + {% + \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}) {};% + }% +}% + +%% \Adynkinnodes +%% -> +%% Tell TikZ where to place the nodes for an A series Dynkin diagram. Draws nothing. +\newcommand*{\Adynkinnodes}% +{% + \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% + }% + \fi% + \else% + \foreach \b in {1,...,\the\dynkinrank}% + {% + \placeRoot{\b}{\b}{0}% + }% + \fi% +}% + +%% \Adynkin +%% -> +%% Draws an A series Dynkin diagram. +\newcommand*{\Adynkin} { -\node at (root label \i) {\scalebox{0.5}{\(\i\)}}; + \newif\ifwasfolded + \ifdynkinfolded + \global\wasfoldedtrue + \else + \global\wasfoldedfalse + \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}% + \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% + \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 + \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} + \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 } +%% \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 +} -\newcommand{\dynkincross}[2]{ -\dynkindot{#1}{#2} -\draw[thick,\dynkincolor] ({#1*\dykinedgelength},{#2*\dykinedgelength}) -- ++ ({(1/sqrt(2))*\dynkinXsize},{(1/sqrt(2))*\dynkinXsize}); -\draw[thick,\dynkincolor] ({#1*\dykinedgelength},{#2*\dykinedgelength}) -- ++ ({-(1/sqrt(2))*\dynkinXsize},{(1/sqrt(2))*\dynkinXsize}); -\draw[thick,\dynkincolor] ({#1*\dykinedgelength},{#2*\dykinedgelength}) -- ++ ({(1/sqrt(2))*\dynkinXsize},{-(1/sqrt(2))*\dynkinXsize}); -\draw[thick,\dynkincolor] ({#1*\dykinedgelength},{#2*\dykinedgelength}) -- ++ ({-(1/sqrt(2))*\dynkinXsize},{-(1/sqrt(2))*\dynkinXsize}); +%% \Ddynkinnodes +%% -> +%% Tell TikZ where to place the nodes for a D series Dynkin diagram. Draws nothing. +\newcommand*{\Ddynkinnodes} +{ + \newcount\rmo + \rmo=\dynkinrank + \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 } -\newcommand{\dynkindot}[2]{% -\fill[\dynkincolor] (\dykinedgelength*#1,\dykinedgelength*#2) circle (\dynkinradius);% +%% \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 } -% Line between nodes. -\newcommand{\dynkinline}[4]{\draw[thin,\dynkincolor] (\dykinedgelength*#1,\dykinedgelength*#2) -- (\dykinedgelength*#3,\dykinedgelength*#4);} +%% \Edynkinunfolded +%% -> +%% Draws an E series Dynkin diagram not folded. +\newcommand*{\Edynkinunfolded}% +{ + % Create the nodes. + \placeRoot{1}{1}{0} + \ifisaffine + \ifnum\dynkinrank=6 + \placeRootHorizontalLabels{2}{3}{1} + \else + \placeRoot*{2}{3}{1} + \fi + \else + \placeRoot*{2}{3}{1} + \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} +} -% Dotted line between nodes. -\newcommand{\dynkindots}[4]{\draw[densely dotted,\dynkincolor] (\dykinedgelength*#1,\dykinedgelength*#2) -- (\dykinedgelength*#3,\dykinedgelength*#4);} -% Double line between nodes. -\newcommand{\dynkindoubleline}[4]{\draw[double,postaction={decorate},\dynkincolor] (\dykinedgelength*#1,\dykinedgelength*#2) -- (\dykinedgelength*#3,\dykinedgelength*#4);} +%% \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} +} -% Triple line between nodes. -\newcommand{\dynkintripleline}[4]{ -\draw[triple={[line width=.1mm,\dynkincolor] in - [line width=.6mm,\dynkinbackcolor] in - [line width=.8mm,\dynkincolor]}] (\dykinedgelength*#3,\dykinedgelength*#4) -- (\dykinedgelength*#1,\dykinedgelength*#2); -\draw[postaction={decorate},double,\dynkincolor] ({0.401*\dykinedgelength*#3+0.599*\dykinedgelength*#1},\dykinedgelength*#4) -- ({0.399*\dykinedgelength*#3+0.601*\dykinedgelength*#1},\dykinedgelength*#2); +%% \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}% + }% + }% + }% } -\tikzset{ - triple/.style args={[#1] in [#2] in [#3]}{ - #1,preaction={preaction={draw,#3},draw,#2} - } -} -\newcommand*{\testbit}[4]% -% if bit number #2 of #1 is 1 then expand #3 else expand #4. -{% -\pgfmathparse{mod(div(#1,2^(#2)),2)}% -\let\tf\pgfmathresult% -\IfStrEq{\tf}{1.0}{#3}{#4}% -}%% - - -\newcommand*{\Adynkin}[2][0]% -%\Adynkin[p]{n} gives the Dynkin diagram of An with parabolic subgroup p. -%\Adynkin[p]{*} allows for an indefinite choice of n, but then the possible parabolics are only from 0 to 2^7. -{%% -\IfStrEq{#2}{*}% -{%% - \dynkinrank=7 - \dynkinline{0}{0}{1}{0}; - \dynkindots{1}{0}{2}{0}; - \dynkinline{2}{0}{4}{0}; - \dynkindots{4}{0}{5}{0}; - \dynkinline{5}{0}{6}{0}; - \foreach \b in {0,...,6}%%% - {%%% - \testbit{#1}{\b}{\dynkincross{\b}{0}}{\dynkindot{\b}{0}} - \node (root \b) at ({\b*\dykinedgelength},0) {}; - \node[below] (root label \b) at ({\b*\dykinedgelength},0) {}; - \node[above] (root label swap \b) at ({\b*\dykinedgelength},0) {}; - }%%% -}%% -{%% -% \draw[\dykinbackcolor] (0,{-\dykinedgelength}) rectangle ({#2*\dykinedgelength},{\dykinedgelength}); - \newcount\rmo - \rmo=#2 - \advance\rmo by -1 - \dynkinline{0}{0}{\the\rmo}{0};% - \foreach \b in {0,...,\the\rmo}%%% - {%%% - \testbit{#1}{\b}{\dynkincross{\b}{0}}{\dynkindot{\b}{0}} - \node (root \b) at ({\b*\dykinedgelength},0) {}; - \node[below] (root label \b) at ({\b*\dykinedgelength},0) {}; - \node[above] (root label swap \b) at ({\b*\dykinedgelength},0) {}; - }%%% +%% \Fdynkin +%% -> +%% Draws an F series Dynkin diagram. +\newcommand*{\Fdynkin}% +{ + \Adynkinnodes + \ifdynkincoxeter + \dynkinline*{1}{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}}% + }% + \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}% + }% + }% + \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 +} -\newcommand*{\Bdynkin}[2][0]% -%\Bdynkin[p]{n} gives the Dynkin diagram of Bn with parabolic subgroup p. -%\Bdynkin[p]{*} allows for an indefinite choice of n, but then the possible parabolics are only from 0 to 2^5. -{% -\IfStrEq{#2}{*}% -{%% - \dynkinrank=5 - \dynkinline{0}{0}{1}{0}; - \dynkindots{1}{0}{2}{0}; - \dynkinline{2}{0}{3}{0}; - \dynkindoubleline{3}{0}{4}{0}; - \foreach \b in {0,...,4}%%% - {%%% - \testbit{#1}{\b}{\dynkincross{\b}{0}}{\dynkindot{\b}{0}} - \node (root \b) at ({\b*\dykinedgelength},0) {}; - \node[below] (root label \b) at ({\b*\dykinedgelength},0) {}; - \node[above] (root label swap \b) at ({\b*\dykinedgelength},0) {}; - }%%% -}%% -{%% -\pgfmathparse{subtract(#2,1)}% -\let\rmo\pgfmathresult% -\pgfmathparse{subtract(\rmo,1)}% -\let\rmt\pgfmathresult% -\dynkinline{0}{0}{\rmo}{0};% -\dynkindoubleline{\rmt}{0}{\rmo}{0}; -\foreach \b in {0,...,\rmo}%%% -{%%% -\testbit{#1}{\b}{\dynkincross{\b}{0}}{\dynkindot{\b}{0}} -\node (root \b) at ({\b*\dykinedgelength},0) {}; -\node[below] (root label \b) at ({\b*\dykinedgelength},0) {}; -\node[above] (root label swap \b) at ({\b*\dykinedgelength},0) {}; -}%%% -}%% -}% +%% \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}}; +} -\newcommand*{\Cdynkin}[2][0]% -%\Cdynkin[p]{n} gives the Dynkin diagram of Cn with parabolic subgroup p. -%\Cdynkin[p]{*} allows for an indefinite choice of n, but then the possible parabolics are only from 0 to 2^5. -{%% -\IfStrEq{#2}{*}% -{%% - \dynkinrank=5 - \dynkinline{0}{0}{1}{0}; - \dynkindots{1}{0}{2}{0}; - \dynkinline{2}{0}{3}{0}; - \dynkindoubleline{4}{0}{3}{0}; - \foreach \b in {0,...,4}%%% - {%%% - \testbit{#1}{\b}{\dynkincross{\b}{0}}{\dynkindot{\b}{0}} - \node (root \b) at ({\b*\dykinedgelength},0) {}; - \node[below] (root label \b) at ({\b*\dykinedgelength},0) {}; - \node[above] (root label swap \b) at ({\b*\dykinedgelength},0) {}; - }%%% -}%% -{%% -\pgfmathparse{subtract(#2,1)}% -\let\rmo\pgfmathresult% -\pgfmathparse{subtract(\rmo,1)}% -\let\rmt\pgfmathresult% -\dynkinline{0}{0}{\rmo}{0};% -\dynkindoubleline{\rmo}{0}{\rmt}{0}; -\foreach \b in {0,...,\rmo}%%% -{%%% -\testbit{#1}{\b}{\dynkincross{\b}{0}}{\dynkindot{\b}{0}} -\node (root \b) at ({\b*\dykinedgelength},0) {}; -\node[below] (root label \b) at ({\b*\dykinedgelength},0) {}; -\node[above] (root label swap \b) at ({\b*\dykinedgelength},0) {}; -}%%% -}%% -}% +%% \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}}; +} +\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 +} -\newcommand*{\Ddynkin}[2][0]% -%\Ddynkin[p]{n} gives the Dynkin diagram of Dn with parabolic subgroup p. -%\Ddynkin[p]{*} allows for an indefinite choice of n, but then the possible parabolics are only from 0 to 2^6. -{%% -\IfStrEq{#2}{*}% -{%% - \dynkinrank=6 - \foreach \x in {0,...,3} - { - \dynkindot{\x}{0} - } - \dynkinline{0}{0}{1}{0} - \dynkindots{1}{0}{2}{0} - \dynkinline{2}{0}{3}{0} - \dynkinline{3}{0}{3.5}{.9} - \dynkinline{3}{0}{3.5}{-.9} -\foreach \b in {0,...,3}%%% -{%%% -\testbit{#1}{\b}{\dynkincross{\b}{0}}{\dynkindot{\b}{0}} -\node (root \b) at ({\b*\dykinedgelength},0) {}; -\node[below] (root label \b) at ({\b*\dykinedgelength},0) {}; -}%%% -\testbit{#1}{4}{\dynkincross{3.5}{-.9}}{\dynkindot{3.5}{-.9}} -\node (root 4) at ({3.5*\dykinedgelength},{-.9*\dykinedgelength}) {}; -\node[below] (root label 4) at ({3.5*\dykinedgelength},{-.9*\dykinedgelength}) {}; -\testbit{#1}{5}{\dynkincross{3.5}{.9}}{\dynkindot{3.5}{.9}} -\node (root 5) at ({3.5*\dykinedgelength},{.9*\dykinedgelength}) {}; -\node[above] (root label 5) at ({3.5*\dykinedgelength},{.9*\dykinedgelength}) {}; -}%% -{%% -\newcount\rmo -\rmo=#2 -\advance\rmo by -1 -\newcount\rmt -\rmt=\rmo -\advance\rmt by -1 -\newcount\rmtt -\rmtt=\rmt -\advance\rmtt by -1 -\dynkinline{0}{0}{\the\rmtt}{0};% -\pgfmathparse{subtract(\the\rmt,.5)} -\let\rmh\pgfmathresult% -\dynkinline{\the\rmtt}{0}{\rmh}{.9} -\dynkinline{\the\rmtt}{0}{\rmh}{-.9} -\foreach \b in {0,...,\the\rmtt}%%% -{%%% -\testbit{#1}{\b}{\dynkincross{\b}{0}}{\dynkindot{\b}{0}} -\node (root \b) at ({\b*\dykinedgelength},0) {}; -\node[below] (root label \b) at ({\b*\dykinedgelength},0) {}; -}%%% -\testbit{#1}{\the\rmt}{\dynkincross{\rmh}{-.9}}{\dynkindot{\rmh}{-.9}} -\node (root \the\rmt) at ({\rmh*\dykinedgelength},{-.9*\dykinedgelength}) {}; -\node[below] (root label \the\rmt) at ({\rmh*\dykinedgelength},{-.9*\dykinedgelength}) {}; -\testbit{#1}{\the\rmo}{\dynkincross{\rmh}{.9}}{\dynkindot{\rmh}{.9}} -\node (root \the\rmo) at ({\rmh*\dykinedgelength},{.9*\dykinedgelength}) {}; -\node[above] (root label \the\rmo) at ({\rmh*\dykinedgelength},{.9*\dykinedgelength}) {}; -}%% -}% +\newcommand*{\affineBdynkin}% +{ + \placeRoot*{0}{2}{1} + \Bdynkin +} -\newcommand*{\Edynkin}[2][0]% -%\Edynkin[p]{n} gives the Dynkin diagram of En, n=6,7,8, with parabolic subgroup p. +\newcommand*{\affineCdynkin} { -\pgfmathparse{subtract(#2,1)}% -\let\rmo\pgfmathresult% -\pgfmathparse{subtract(\rmo,1)}% -\let\rmt\pgfmathresult% -\dynkinline{0}{0}{\rmt}{0};% -\dynkinline{2}{0}{2}{1} -\testbit{#1}{0}{\dynkincross{0}{0}}{\dynkindot{0}{0}} -\node (root 0) at (0,0) {}; -\node[below] (root label 0) at (0,0) {}; -\testbit{#1}{1}{\dynkincross{2}{1}}{\dynkindot{2}{1}} -\node (root 1) at ({2*\dykinedgelength},{1*\dykinedgelength}) {}; -\node[above] (root label 1) at ({2*\dykinedgelength},{1*\dykinedgelength}) {}; -\foreach \b in {2,...,\rmo}%%% -{%%% -\pgfmathparse{subtract(\b,1)}% -\let\bmo\pgfmathresult% -\testbit{#1}{\b}{\dynkincross{\bmo}{0}}{\dynkindot{\bmo}{0}} -\node (root \b) at ({\bmo*\dykinedgelength},0) {}; -\node[below] (root label \b) at ({\bmo*\dykinedgelength},0) {}; -}%%% + \placeRoot{0}{0}{0} + \Cdynkin } +\newcommand*{\affineDdynkin} +{ + \placeRoot*{0}{2}{1} + \Ddynkin +} -\newcommand*{\Ffourdynkin}[1][0]% -%\Fdynkin[p]{n} gives the Dynkin diagram of F4 with parabolic subgroup p. +\newcommand*{\affineEdynkin} { -\dynkinline{0}{0}{3}{0};% -\dynkindoubleline{1}{0}{2}{0} -\foreach \b in {0,...,3}%%% -{%%% -\testbit{#1}{\b}{\dynkincross{\b}{0}}{\dynkindot{\b}{0}} -\node (root \b) at ({\b*\dykinedgelength},0) {}; -\node[below] (root label \b) at ({\b*\dykinedgelength},0) {}; -\node[above] (root label swap \b) at ({\b*\dykinedgelength},0) {}; -}%%% + \ifnum\dynkinrank=6 + \placeRoot*{0}{3}{2} + \Edynkin + \else + \placeRoot{0}{0}{0} + \Edynkin + \fi } -\newcommand*{\Gtwodynkin}[1][0]% -%\Gtwodynkin[p] gives the Dynkin diagram of G2 with parabolic subgroup p. -{%% -\dynkintripleline{0}{0}{1}{0};% -\foreach \b in {0,...,1}%%% -{%%% -\testbit{#1}{\b}{\dynkincross{\b}{0}}{\dynkindot{\b}{0}} -\node (root \b) at ({\b*\dykinedgelength},0) {}; -\node[below] (root label \b) at ({\b*\dykinedgelength},0) {}; -\node[above] (root label swap \b) at ({\b*\dykinedgelength},0) {}; -}%%% -}%% +\newcommand*{\affineFdynkin} +{ + \placeRoot{0}{0}{0} + \Fdynkin +} +\newcommand*{\affineGdynkin} +{ + \placeRoot{0}{3}{0} + \Gdynkin +} \endinput |