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authorKarl Berry <karl@freefriends.org>2017-09-13 22:04:42 +0000
committerKarl Berry <karl@freefriends.org>2017-09-13 22:04:42 +0000
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dynkin-diagrams (14sep17)
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+___________________________________
+
+ Dynkin diagrams
+
+ v1.0
+
+ 8 September 2017
+___________________________________
+
+Authors : Ben McKay
+Maintainer: Ben McKay
+E-mail : b.mckay@ucc.ie
+Licence : Released under the LaTeX Project Public License v1.3c or
+ later, see http://www.latex-project.org/lppl.txt
+
+----------------------------------------------------------------------
+
+Provides Dynkin diagrams drawn in TikZ.
+
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+\documentclass{amsart}
+
+\title{The Dynkin diagrams package}
+\author{Ben McKay}
+\date{\today}
+
+\usepackage{dynkin-diagrams}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{array}
+\usepackage{xstring}
+\usepackage{etoolbox}
+\usetikzlibrary{backgrounds}
+\usetikzlibrary{decorations.markings}
+\usepackage{longtable}
+\usepackage{showexpl}
+
+\newcommand{\C}[1]{\mathbb{C}^{#1}}
+
+
+\renewcommand*{\arraystretch}{1.5}
+
+\begin{document}
+\maketitle
+\tableofcontents
+
+\section{Quick introduction}
+
+This is a test of the Dynkin diagram package.
+Load the package via
+\begin{verbatim}
+\usepackage{dynkin-diagrams}
+\end{verbatim}
+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}.
+\end{LTXexample}
+or use the long form inside a \verb!\tikz! statement or environment:
+\begin{LTXexample}
+\tikz \dynkin[parabolic=3]{A}{3};
+\end{LTXexample}
+With labels for the roots:
+\begin{LTXexample}
+\tikz \dynkin[parabolic=3,label=true]{A}{3};
+\end{LTXexample}
+
+\bigskip
+
+Inside an environment:
+\begin{LTXexample}
+\begin{tikzpicture}
+\dynkin[parabolic=3,label=true]{A}{3}
+\end{tikzpicture}
+\end{LTXexample}
+
+\bigskip
+
+Make up your own labels for the roots:
+
+\begin{LTXexample}
+\begin{tikzpicture}
+\dynkin[parabolic=3]{A}{3};
+\node at (root label 2) {\scalebox{.7}{\(\alpha_2\)}};
+\end{tikzpicture}
+\end{LTXexample}
+
+\newpage
+
+Drawing curves between the roots:
+
+\begin{LTXexample}
+\begin{tikzpicture}
+\dynkin[parabolic=429]{E}{8}
+\draw[brown,-latex]
+ (root 3.south)
+ to [out=-90, in=-90]
+ (root 6.south);
+\end{tikzpicture}
+\end{LTXexample}
+
+Various options:
+
+\begin{LTXexample}
+\tikz \dynkin[color=brown]{G}{2};
+\end{LTXexample}
+
+\begin{LTXexample}
+\tikz \dynkin[edgelength=1.2,parabolic=3]{A}{3};
+\end{LTXexample}
+
+\begin{LTXexample}
+\tikz \dynkin[crosssize=.1cm,parabolic=3]{A}{3};
+\end{LTXexample}
+
+\begin{LTXexample}
+\tikz \dynkin[dotradius=.08cm,parabolic=3]{A}{3};
+\end{LTXexample}
+
+\begin{LTXexample}
+\begin{tikzpicture}[
+ show background rectangle,
+ background rectangle/.style={fill=lightgray}]
+\dynkin[parabolic=1,background color=lightgray]{G}{2}
+\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:
+
+\begin{LTXexample}
+\begin{tikzpicture}
+\dynkin[parabolic=5]{D}{*}
+\end{tikzpicture}
+\end{LTXexample}
+
+Outside a \verb!\tikz! environment, use \verb!\dynk! instead of \verb!\dynkin!.
+
+
+\bigskip
+
+\newcommand*{\typ}[1]{\(\left<\texttt{#1}\right>\)}
+\newcommand*{\optionLabel}[3]{%%
+\multicolumn{2}{l}{\(\texttt{#1}=\texttt{#2}, \texttt{default}=\texttt{#3}\)} \\
+}%%
+
+\section{Options}
+\par\noindent%
+\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}
+& 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{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{label}{true or false}{false}
+& whether to label the roots by their root numbers. \\
+\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.
+
+
+
+\section{Example: some parabolic subgroups}
+
+\newcommand{\drawparabolic}[3]%%
+{#1_{#2,#3} & \tikz \dynkin[parabolic=#3]{#1}{#2}; \\}
+
+\begin{center}
+\begin{longtable}{@{}>{$}r<{$}m{2cm}m{2cm}@{}}
+\endfirsthead
+\endhead
+\endfoot
+\endlastfoot
+\drawparabolic{A}{1}{0}
+\drawparabolic{A}{1}{1}
+\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{G}{2}{2}
+\end{longtable}
+\end{center}
+
+
+
+\section{Example: the Hermitian symmetric spaces}
+
+ \renewcommand*{\arraystretch}{1.5}
+\begin{center}
+\begin{longtable}{@{}>{$}r<{$}m{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
+\end{longtable}
+\end{center}
+\end{document}