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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}