diff options
author | Karl Berry <karl@freefriends.org> | 2023-09-03 20:10:43 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2023-09-03 20:10:43 +0000 |
commit | fdea69177d193184118e3ced933bb1c1b4c21789 (patch) | |
tree | acb4066d7a974358d5f1e64926ee9faf4c00dfdb /Master/texmf-dist/doc | |
parent | f4b6afc9aecd7ce8e4dcbdd9080e1eaf09b10c15 (diff) |
rank-2-roots (3sep23)
git-svn-id: svn://tug.org/texlive/trunk@68161 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc')
-rw-r--r-- | Master/texmf-dist/doc/latex/rank-2-roots/README | 4 | ||||
-rw-r--r-- | Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.pdf | bin | 448699 -> 448000 bytes | |||
-rw-r--r-- | Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.tex | 306 |
3 files changed, 180 insertions, 130 deletions
diff --git a/Master/texmf-dist/doc/latex/rank-2-roots/README b/Master/texmf-dist/doc/latex/rank-2-roots/README index 91a7c6417b5..116783e3b33 100644 --- a/Master/texmf-dist/doc/latex/rank-2-roots/README +++ b/Master/texmf-dist/doc/latex/rank-2-roots/README @@ -2,9 +2,9 @@ ___________________________________ Rank 2 roots - v1.1 + v1.2 - 31 August 2023 + 03 September 2023 ___________________________________ Authors : Ben McKay diff --git a/Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.pdf b/Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.pdf Binary files differindex c9049e354f5..3a50f63c765 100644 --- a/Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.pdf +++ b/Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.pdf diff --git a/Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.tex b/Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.tex index 25b9b585f0e..69fe728949b 100644 --- a/Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.tex +++ b/Master/texmf-dist/doc/latex/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} |