From ba6f6ac5705667b8a0d7d3ddcb50a275d06278a1 Mon Sep 17 00:00:00 2001 From: Karl Berry Date: Sat, 18 Nov 2017 21:46:02 +0000 Subject: dynkin-diagrams (18nov17) git-svn-id: svn://tug.org/texlive/trunk@45846 c570f23f-e606-0410-a88d-b1316a301751 --- Master/texmf-dist/doc/latex/dynkin-diagrams/README | 4 +- .../doc/latex/dynkin-diagrams/dynkin-diagrams.bib | 212 +++++++ .../doc/latex/dynkin-diagrams/dynkin-diagrams.pdf | Bin 153870 -> 233754 bytes .../doc/latex/dynkin-diagrams/dynkin-diagrams.tex | 667 ++++++++++++++++++--- 4 files changed, 787 insertions(+), 96 deletions(-) create mode 100644 Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.bib (limited to 'Master/texmf-dist/doc') 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 index fa4ed5acbe9..851c6ae2200 100644 Binary files a/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.pdf and b/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.pdf differ 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[]{}{}! where \verb!! is \(A,B,C,D,E,F\) or \(G\), the family of root system for the Dynkin diagram, and \verb!! 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[]{}{}! where \verb!! is \(A,B,C,D,E,F\) or \(G\), the family of root system for the Dynkin diagram, and \verb!! 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} -- cgit v1.2.3