summaryrefslogtreecommitdiff
path: root/graphics/pgf/contrib/rank-2-roots/rank-2-roots.tex
diff options
context:
space:
mode:
Diffstat (limited to 'graphics/pgf/contrib/rank-2-roots/rank-2-roots.tex')
-rw-r--r--graphics/pgf/contrib/rank-2-roots/rank-2-roots.tex306
1 files changed, 178 insertions, 128 deletions
diff --git a/graphics/pgf/contrib/rank-2-roots/rank-2-roots.tex b/graphics/pgf/contrib/rank-2-roots/rank-2-roots.tex
index 25b9b585f0..69fe728949 100644
--- a/graphics/pgf/contrib/rank-2-roots/rank-2-roots.tex
+++ b/graphics/pgf/contrib/rank-2-roots/rank-2-roots.tex
@@ -2,7 +2,7 @@
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenx}
-\title{The Rank 2 Roots Package \\ Version 1.1}
+\title{The Rank 2 Roots Package \\ Version 1.2}
\makeatletter
\DeclareRobustCommand{\scotsMc}{\scotsMcx{c}}
@@ -22,7 +22,7 @@
\author{Benjamin \scotsMc{}Kay}
\address{School of Mathematical Sciences, University College Cork, Cork, Ireland}
\email{b.mckay@ucc.ie}
-\date{31 August 2023}
+\date{3 September 2023}
\usepackage{etoolbox}
\usepackage{lmodern}
@@ -71,12 +71,10 @@ before upper={\widowpenalties=3 10000 10000 150}}
\begin{document}
\maketitle
\tableofcontents
-
\section{Introduction}
This package concerns mathematical drawings arising in representation theory.
The purpose of this package is to ease drawing of rank 2 root systems, with Weyl chambers, weight lattices, and parabolic subgroups, mostly imitating the drawings of Fulton and Harris \cite{Fulton.Harris:1991}.
We use definitions of root systems and weight lattices as in Carter \cite{Carter:2005} p. 540--609.
-
\begin{tcolorbox}[title={Load the \texttt{rank-2-roots} package}]
\begin{verbatim}
\documentclass{amsart}
@@ -91,9 +89,6 @@ The root system \(G_2\):
\end{document}
\end{verbatim}
\end{tcolorbox}
-
-
-
\section{Root systems}
\NewDocumentCommand\drawroots{m}%
{%
@@ -103,11 +98,8 @@ The root system \(G_2\):
\end{rootSystem}
\end{tikzpicture}
}%
-
\NewDocumentCommand\csdrawroots{m}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -115,10 +107,7 @@ The root system \(G_2\):
\texttt{\detokenize{\roots}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
-
\newcommand*\mytablecontents{}
\foreach \i in {A,B,C,G}{
\xappto\mytablecontents{$\i_2$ & \drawroots{\i} & \csdrawroots{\i}
@@ -149,8 +138,6 @@ The root system \(G_2\):
\RenewDocumentCommand\csdrawroots{m}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -158,8 +145,6 @@ The root system \(G_2\):
\texttt{\detokenize{\simpleroots}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\begin{longtable}{rcm{8cm}}
@@ -173,7 +158,21 @@ The root system \(G_2\):
\mytablecontents
\end{longtable}
-
+\begingroup
+To change the style of the simple roots:
+\[
+\verb!\pgfkeys{/root system/simple root/.style=black}!
+\]
+\pgfkeys{/root system/simple root/.style=black}
+\[
+\begin{tikzpicture}[baseline=-.5]
+\begin{rootSystem}{G}
+\roots
+\simpleroots
+\end{rootSystem}
+\end{tikzpicture}
+\]
+\endgroup
\section{Weights}
Type \verb!\wt{x}{y}! to get a weight at position \((x,y)\) (as measured in a basis of \emph{fundamental weights}).
@@ -185,6 +184,7 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\begin{tikzpicture}[baseline=-.5]
\begin{rootSystem}{#1}
\roots
+\simpleroots
\wt[brown]{1}{0}
\wt[red]{0}{1}
\wt[multiplicity=4,blue]{1}{3}
@@ -196,12 +196,12 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\RenewDocumentCommand\csdrawroots{m}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
\par\noindent%
+\texttt{\detokenize{\simpleroots}}%
+\par\noindent%
\texttt{\detokenize{\wt[brown]{1}{0}}}%
\par\noindent%
\texttt{\detokenize{\wt[red]{0}{1}}}%
@@ -213,8 +213,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\texttt{\detokenize{\wt[blue]{-1}{3}}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -235,7 +233,117 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\mytablecontents
\end{longtable}
+{
+\pgfkeys{/root system/weight length=.75cm}
+\begin{longtable}{rcm{10cm}}
+\caption{The fundamental weights and the simple roots}\\
+\endfirsthead
+\caption{\dots continued}\\
+\endhead
+\multicolumn{3}{c}{continued \dots}\\
+\endfoot
+\endlastfoot
+$A_2$&
+\begin{tikzpicture}[baseline=-.5]
+\begin{rootSystem}{A}
+\roots
+\simpleroots
+\node [above] at \Root {1}{0} {\(\alpha_1\)};
+\node [right] at \Root {0}{1} {\(\alpha_2\)};
+\fundamentalweights
+\node [right] at \weight {1}{0} {\(\omega_1\)};
+\node [right] at \weight {0}{1} {\(\omega_2\)};
+\end{rootSystem}
+\end{tikzpicture}
+&
+\begin{verbatim}
+\begin{rootSystem}{A}
+\roots
+\simpleroots
+\node [above] at \Root {1}{0} {\(\alpha_1\)};
+\node [right] at \Root {0}{1} {\(\alpha_2\)};
+\fundamentalweights
+\node [right] at \weight {1}{0} {\(\omega_1\)};
+\node [right] at \weight {0}{1} {\(\omega_2\)};
+\end{rootSystem}
+\end{verbatim}
+\\
+$B_2$&
+\begin{tikzpicture}[baseline=-.5]
+\begin{rootSystem}{B}
+\roots
+\simpleroots
+\node [below] at \Root {1}{0} {\(\alpha_1\)};
+\node [above] at \Root {0}{1} {\(\alpha_2\)};
+\fundamentalweights
+\node [right] at \weight {1}{0} {\(\omega_1\)};
+\node [right] at \weight {0}{1} {\(\omega_2\)};
+\end{rootSystem}{B}
+\end{tikzpicture}
+&
+\begin{verbatim}
+\begin{rootSystem}{B}
+\roots
+\simpleroots
+\node [below] at \Root {1}{0} {\(\alpha_1\)};
+\node [above] at \Root {0}{1} {\(\alpha_2\)};
+\fundamentalweights
+\node [right] at \weight {1}{0} {\(\omega_1\)};
+\node [right] at \weight {0}{1} {\(\omega_2\)};
+\end{rootSystem}{B}
+\end{verbatim}
+\\
+$C_2$&
+\begin{tikzpicture}[baseline=-.5]
+\begin {rootSystem}{C}
+\roots
+\simpleroots
+\node [left] at \Root {1}{0} {\(\alpha_1\)};
+\node [right] at \Root {0}{1} {\(\alpha_2\)};
+\fundamentalweights
+\node [right] at \weight {1}{0} {\(\omega_1\)};
+\node [above] at \weight {0}{1} {\(\omega_2\)};
+\end{rootSystem}
+\end{tikzpicture}
+&
+\begin{verbatim}
+\begin{rootSystem}{C}
+\roots
+\simpleroots
+\node [left] at \Root {1}{0} {\(\alpha_1\)};
+\node [right] at \Root {0}{1} {\(\alpha_2\)};
+\fundamentalweights
+\node [right] at \weight {1}{0} {\(\omega_1\)};
+\node [above] at \weight {0}{1} {\(\omega_2\)};
+\end{rootSystem}
+\end{verbatim}
+\\
+$G_2$&
+\begin{tikzpicture}[baseline=-.5]
+\begin{rootSystem}{G}
+\roots
+\simpleroots
+\node [above] at \Root {1}{0} {\(\alpha_1\)};
+\node [below right] at \Root {0}{1} {\(\alpha_2\)};
+\fundamentalweights
+\node [right] at \weight {1}{0} {\(\omega_1\)};
+\node [right] at \weight {0}{1} {\(\omega_2\)};
+\end{rootSystem}
+\end{tikzpicture}
+&
+\begin{verbatim}
+\begin{rootSystem}{G}
+\roots
+\simpleroots
+\node [above] at \Root {1}{0} {\(\alpha_1\)};
+\node [below right] at \Root {0}{1} {\(\alpha_2\)};
+\fundamentalweights
+\node [right] at \weight {1}{0} {\(\omega_1\)};
+\node [right] at \weight {0}{1} {\(\omega_2\)};
+\end{rootSystem}
+\end{verbatim}
+\end{longtable}
\RenewDocumentCommand\drawroots{m}%
{%
@@ -247,10 +355,10 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\end{tikzpicture}
}%
+}
+
\RenewDocumentCommand\csdrawroots{m}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -258,8 +366,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\texttt{\detokenize{\wt[multiplicity=2,root]{0}{0}}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -280,8 +386,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\mytablecontents
\end{longtable}
-
-
\RenewDocumentCommand\drawroots{m}%
{%
\begin{tikzpicture}[baseline=-.5]
@@ -294,8 +398,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\RenewDocumentCommand\csdrawroots{m}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -303,8 +405,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\texttt{\detokenize{\WeylChamber}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -340,8 +440,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\RenewDocumentCommand\csdrawroots{m}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -349,8 +447,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\texttt{\detokenize{\positiveRootHyperplane}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -384,8 +480,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\RenewDocumentCommand\csdrawroots{mm}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -393,8 +487,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\texttt{\detokenize{\parabolic}\{#2\}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -407,7 +499,7 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
}
\begin{longtable}{rcm{8cm}}
-\caption{Parabolic subgroups. Each set of roots is assigned a number, with each binary digit zero or one to say whether the corresponding root is crossed or not: \(A_{5,37}\) means the parabolic subgroup of \(A_5\) so that the binary digits of \(37=2^5+2^2+2^0\) give us roots \(0,2,5\) in Bourbaki ordering being compact roots, i.e. having the root vectors of both that root and its negative inside the parabolic subgroup. }\\
+\caption{Parabolic subgroups. Each set of roots is assigned a number, with each binary digit zero or one to say whether the corresponding root is crossed or not: \(A_{5,37}\) means the parabolic subgroup of \(A_5\) so that the binary digits of \(37=2^5+2^2+2^0\) give us roots \(0,2,5\) in Bourbaki ordering being crossed roots, i.e. noncompact roots, i.e. having the root vectors of that root but not of its negative inside the parabolic subgroup. }\\
\endfirsthead
\caption{\dots continued}\\
\endhead
@@ -430,8 +522,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\RenewDocumentCommand\csdrawroots{mm}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -441,8 +531,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\texttt{\detokenize{\parabolicgrading}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -493,7 +581,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\end{tikzpicture}
}
-
\tikzstyle{weight arrow}=[black,-stealth,shorten <=.25cm,shorten >=.25cm]
{
@@ -531,8 +618,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\end{tikzpicture}
}
-
-
\begin{tcblisting}{title={Drawing the \(A_2\) root system and a weight at the origin. The option \texttt{root} indicates that this weight is to be coloured like a root.}}
\begin{tikzpicture}
\begin{rootSystem}{A}
@@ -542,7 +627,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\end{tikzpicture}
\end{tcblisting}
-
\begin{tcblisting}{title={Drawing the \(A_2\) root system and a weight at the origin and the positive root hyperplane}}
\begin{tikzpicture}
\begin{rootSystem}{A}
@@ -553,9 +637,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ.
\end{tikzpicture}
\end{tcblisting}
-
-
-
\section{Coordinate systems}
The package provides three coordinate systems: hex, square and weight.
Above we have seen the weight coordinates: a basis of fundamental weights.
@@ -625,10 +706,6 @@ We can use the hex system in drawing on the \(A_2\) or \(G_2\) weight lattices,
\end{tikzpicture}
\end{tcblisting}
-
-
-
-
\begin{tcblisting}{title={Automatic sizing of the weight lattice (the default) \dots}}
\begin{tikzpicture}
\begin{rootSystem}{A}
@@ -693,31 +770,16 @@ We can use the hex system in drawing on the \(A_2\) or \(G_2\) weight lattices,
\end{rootSystem}
\end{tikzpicture}
\end{tcblisting}
+
\begin{tcblisting}{title={Changing the weight length rescales}}
\begin{tikzpicture}
-\setlength\weightLength{.3cm}
+\pgfkeys{/root system/weight length=0.3cm}
\begin{rootSystem}{A}
-\wt[multiplicity=2]{0}{0}
+\wt[multiplicity=2,draw=gray]{0}{0}
\foreach \x/\y in {1/1, 2/-1, 1/-2, -1/-1, -2/1, -1/2}{\wt{\x}{\y}}
\end{rootSystem}
\end{tikzpicture}
\end{tcblisting}
-\begin{tcblisting}{}
-\begin{tikzpicture}
-\setlength\weightLength{.3cm}
-\begin{rootSystem}{A}
-\foreach \x/\y in {0/0, 3/0, 2/-1, 1/-2, 0/-3, 1/1, -1/-1, -1/2, -2/1, -3/3}{\wt{\x}{\y}}
-\end{rootSystem}
-\end{tikzpicture}
-\end{tcblisting}
-\begin{tcblisting}{}
-\begin{tikzpicture}
-\setlength\weightLength{.3cm}
-\begin{rootSystem}{A}
-\foreach \x/\y in {0/0, -3/0, 2/-1, 1/-2, 3/-3, 1/1, -1/-1, -1/2, -2/1, 0/3}{\wt{\x}{\y}}
-\end{rootSystem}
-\end{tikzpicture}
-\end{tcblisting}
\begin{tcblisting}{title={We use a basis of fundamental weights, as given in Carter's book \cite{Carter:2005} p. 540--609}}
\begin{tikzpicture}
\begin{rootSystem}{B}
@@ -744,15 +806,11 @@ Without automatic stretching of the weight lattice to fit the picture, you won't
\RenewDocumentCommand\csdrawroots{m}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -774,8 +832,6 @@ Without automatic stretching of the weight lattice to fit the picture, you won't
\end{longtable}
-
-
Type \verb!\wt{x}{y}! to get a weight at position \((x,y)\) (as measured in a basis of \emph{fundamental weights}).
Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with option \verb!multiplicity=!\texttt{n} to get multiplicity $n$.
@@ -796,8 +852,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\RenewDocumentCommand\csdrawroots{m}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -813,8 +867,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\texttt{\detokenize{\wt[blue]{-1}{3}}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -847,8 +899,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\RenewDocumentCommand\csdrawroots{m}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -856,8 +906,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\texttt{\detokenize{\wt[multiplicity=2]{0}{0}}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -892,8 +940,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\RenewDocumentCommand\csdrawroots{m}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -901,8 +947,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\texttt{\detokenize{\WeylChamber}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -936,8 +980,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\RenewDocumentCommand\csdrawroots{m}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -945,8 +987,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\texttt{\detokenize{\positiveRootHyperplane}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -980,8 +1020,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\RenewDocumentCommand\csdrawroots{mm}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -989,8 +1027,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\texttt{\detokenize{\parabolic}\{#2\}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -1028,8 +1064,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\RenewDocumentCommand\csdrawroots{mm}%
{%
-\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}%
-\par\noindent%
\texttt{\detokenize{\begin{rootSystem}}\{#1\}}%
\par\noindent%
\texttt{\detokenize{\roots}}%
@@ -1039,8 +1073,6 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\texttt{\detokenize{\parabolicgrading}}%
\par\noindent%
\texttt{\detokenize{\end{rootSystem}}}%
-\par\noindent%
-\texttt{\detokenize{\end{tikzpicture}}}%
}%
\renewcommand*\mytablecontents{}
@@ -1068,12 +1100,10 @@ Add an option: \verb!\wt[Z]{x}{y}! to get \verb!Z! passed to TikZ, or with optio
\section{Examples of weights of various representations}
-
Henceforth assume \verb!\AutoSizeWeightLatticetrue! (the default).
\AutoSizeWeightLatticetrue
-
\begin{tcblisting}{title={Fulton and Harris, p. 186}}
\begin{tikzpicture}
\begin{rootSystem}{A}
@@ -1088,9 +1118,8 @@ Henceforth assume \verb!\AutoSizeWeightLatticetrue! (the default).
\begin{tcblisting}{title={A representation of \(G_2\)}}
-\setlength\weightLength{1cm}
\begin{tikzpicture}
-\begin{rootSystem}{G}
+\begin{rootSystem}[weight length=1cm,weight/.style={draw=gray,fill=none}]{G}
\roots
\foreach \m/\x/\y in {
1/1/1, 1/4/-1, 1/-1/2, 2/2/0, 1/5/-2,
@@ -1108,9 +1137,8 @@ Henceforth assume \verb!\AutoSizeWeightLatticetrue! (the default).
\begin{tcblisting}{title={Dimensions of representations of \(G_2\), parameterized by highest weight}}
-\setlength\weightLength{1cm}
\begin{tikzpicture}
-\begin{rootSystem}{G}
+\begin{rootSystem}[weight length=1cm]{G}
\roots
\foreach \x/\y/\d in {
0/1/14, 0/2/77, 0/3/273, 1/0/7, 1/1/64,
@@ -1128,6 +1156,37 @@ Henceforth assume \verb!\AutoSizeWeightLatticetrue! (the default).
\newpage
\section{More options}
+Options can be set using global TikZ keys
+\[
+\verb!\pgfkeys{/root system/simple root/.style=black}!
+\]
+{
+\pgfkeys{/root system/simple root/.style=black}
+\[
+\begin{tikzpicture}
+\begin{rootSystem}{G}
+\roots
+\end{rootSystem}
+\end{tikzpicture}
+\]
+}
+or, in each root system, using
+\begin{verbatim}
+\[
+\begin{tikzpicture}
+\begin{rootSystem}[weight length=.2cm]{G}
+\roots
+\end{rootSystem}
+\end{tikzpicture}
+\]
+\end{verbatim}
+\[
+\begin{tikzpicture}
+\begin{rootSystem}[weight length=.2cm]{G}
+\roots
+\end{rootSystem}
+\end{tikzpicture}
+\]
\newcommand*{\typ}[1]{#1}
\newcommand*{\optionLabel}[3]{%%
@@ -1148,16 +1207,22 @@ Henceforth assume \verb!\AutoSizeWeightLatticetrue! (the default).
\multicolumn{2}{c}{continued \dots}\\
\endfoot
\endlastfoot
-\commandLabel{weightRadius}{\typ{length}}{1.2pt}
+\optionLabel{weight radius}{\typ{length}}{1.2pt}
& Radius of dots used when marking specified weights. \\
-\commandLabel{weightLength}{\typ{length}}{.5cm}
+\optionLabel{weight length}{\typ{length}}{.5cm}
& Minimum distance between distinct weights. \\
-\commandLabel{gradingDot}{\typ{length}}{2pt}
+\optionLabel{grading dot radius}{\typ{length}}{2pt}
& Size of dot around a root using to indicate a grading of a parabolic subalgebra which only contains one root. \\
\optionLabel{weight lattice}{\typ{TikZ style data}}{gray!40}
& Style for drawing weight lattice lines. \\
\optionLabel{root}{\typ{TikZ style data}}{gray}
& Style for drawing roots. \\
+\optionLabel{simple root}{\typ{TikZ style data}}{fill=white,draw=gray}
+& Style for drawing simple roots. \\
+\optionLabel{weight}{\typ{TikZ style data}}{fill=gray,draw=white}
+& Style for drawing weights. \\
+\optionLabel{fundamental weight}{\typ{TikZ style data}}{fill=black,draw=gray}
+& Style for drawing fundamental weights. \\
\optionLabel{root polygon}{\typ{TikZ style data}}{gray!40,opacity=.5}
& Style for drawing a polygon which indicates the locations of the roots. \\
\optionLabel{hyperplane}{\typ{TikZ style data}}{gray!50,fill opacity=.5}
@@ -1168,21 +1233,6 @@ Henceforth assume \verb!\AutoSizeWeightLatticetrue! (the default).
& Style for drawing a thick line over top of some roots to indicate that they lie in the same grading associated to a parabolic subgroup. \\
\end{longtable}
-\begin{verbatim}
-\newlength\weightRadius
-\setlength\weightRadius{1.2pt}
-\newlength\weightLength
-\setlength\weightLength{.5cm}
-\newlength\gradingDot
-\setlength\gradingDot{2pt}
-\tikzstyle{weight lattice}=[gray!40]
-\tikzstyle{root}=[gray]
-\tikzstyle{root polygon}=[gray!40,opacity=.5]
-\tikzstyle{hyperplane}=[gray!50,fill opacity=.5]
-\tikzstyle{Weyl chamber}=[gray!60,fill opacity=.5]
-\tikzstyle{grading}=[line width=3pt,gray,opacity=0.5,line cap=round]
-\end{verbatim}
-
\bibliographystyle{amsplain}
\bibliography{rank-2-roots}