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Diffstat (limited to 'Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.tex')
-rw-r--r-- | Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.tex | 228 |
1 files changed, 213 insertions, 15 deletions
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 97a121f1439..5557830a45c 100644 --- a/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.tex +++ b/Master/texmf-dist/doc/latex/dynkin-diagrams/dynkin-diagrams.tex @@ -1,8 +1,8 @@ \documentclass{amsart} -\title{The Dynkin diagrams package \\ Version 3.11} +\title{The Dynkin diagrams package \\ Version 3.12} \author{Ben McKay} -\date{28 February 2018} +\date{30 April 2018} \usepackage{etex} \usepackage[T1]{fontenc} @@ -33,7 +33,6 @@ \usepackage{xspace} \newcommand{\TikZ}{Ti\textit{k}Z\xspace} \usepackage{filecontents} -\usetikzlibrary{backgrounds} \usetikzlibrary{decorations.markings} \arrayrulecolor{white} \makeatletter @@ -49,9 +48,11 @@ \renewcommand*{\arraystretch}{1.5} \NewDocumentCommand\wdtA{}{.7cm} \NewDocumentCommand\wdtD{}{3cm} +\NewDocumentCommand\wdtE{}{6cm} \NewDocumentCommand\wdtL{}{3cm} \newcolumntype{A}{@{}>{\columncolor[gray]{.9}$}m{\wdtA}<{$}} \newcolumntype{D}{>{\columncolor[gray]{.9}}m{\wdtD}} +\newcolumntype{E}{>{\columncolor[gray]{.9}}m{\wdtE}} \newcolumntype{L}{>{\columncolor[gray]{.9}}p{\wdtL}} \newcolumntype{P}{>{\columncolor[gray]{.9}}p{10cm}} \NewDocumentCommand\textleftcurly{}{\texttt{\char'173}}% @@ -78,9 +79,10 @@ \NewDocumentCommand\dyn{omom}% {% - {#2}_{\typesetSubseries{#4}}^{\IfInteger{#3}{#3}{}} & \dynk[#1]{#2}[#3]{#4}% + {#2}_{\typesetSubseries{#4}}^{\IfInteger{#3}{#3}{\IfStrEq{#1}{extended}{1}{}}} & \dynk[#1]{#2}[#3]{#4}% }% + \NewDocumentEnvironment{dynkinTable}{mmm}% {% \RenewDocumentCommand\wdtD{}{#2} @@ -101,7 +103,7 @@ \definecolor{example-color}{gray}{1} -\definecolor{example-border-color}{gray}{.7} +\definecolor{example-border-color}{gray}{.7} \tcbset{coltitle=black,colback=example-color,colframe=example-border-color,enhanced,breakable,pad at break*=1mm, toprule=1.2mm,bottomrule=1.2mm,leftrule=1mm,rightrule=1mm,toprule at break=-1mm,bottomrule at break=-1mm, @@ -127,6 +129,11 @@ before upper={\widowpenalties=3 10000 10000 150}} \begin{document} + + + + + \maketitle \begin{center} \begin{varwidth}{\textwidth} @@ -138,23 +145,26 @@ before upper={\widowpenalties=3 10000 10000 150}} \setlength{\arrayrulewidth}{1.5pt} - \section{Quick introduction} - - \begin{tcolorbox}[title={Load the Dynkin diagram package (see options below)}] \begin{verbatim} +\documentclass{amsart} \usepackage{dynkin-diagrams} +\begin{document} +The Dynkin diagram of \(B_3\) is \dynkin{B}{3}. +\end{document} \end{verbatim} \end{tcolorbox} \begin{tcblisting}{title={Invoke it}} The Dynkin diagram of \(B_3\) is \dynkin{B}{3}. \end{tcblisting} \begin{tcblisting}{title={Inside a \TikZ statement}} -\tikz \dynkin{B}{3}; +The Dynkin diagram of \(B_3\) is +\tikz[baseline=-0.5ex] \dynkin{B}{3}; \end{tcblisting} \begin{tcblisting}{title={Inside a \TikZ environment}} -\begin{tikzpicture} +The Dynkin diagram of \(B_3\) is +\begin{tikzpicture}[baseline=-0.5ex] \dynkin{B}{3} \end{tikzpicture} \end{tcblisting} @@ -441,6 +451,116 @@ For such rare situations, there is an option: \end{tcblisting} +\begingroup +\RenewDocumentCommand\wdtA{}{.35cm} +\RenewDocumentCommand\wdtE{}{6.55cm} +\begin{dynkinTable}{Springer's table of indices \cite{Springer:2009}, pp. 320-321, with one form of \(E_7\) corrected}{2.5cm}{3.7cm} +% 1 +A_n & +\multicolumn{2}{E}{ +\begin{tikzpicture}[baseline=0pt] +\dynkin{A}{o.o*o.o*o.o} +\dynkinLabelRoot{3}{d} +\dynkinLabelRoot{6}{n-d} +\end{tikzpicture} +} +\\ +% 2 +A_n & +\multicolumn{2}{E}{ +\begin{tikzpicture}[baseline=0pt] +\dynkin{A}{o.o*o.o*o.o*o.o*o.o} +\dynkinLabelRoot{3}{d} +\dynkinLabelRoot{6}{rd} +\dynkinLabelRoot{9}{n-rd} +\dynkinLabelRoot{12}{n-d} +\end{tikzpicture} +} +\\ +% 3 +B_n & +\multicolumn{2}{E}{ +\begin{tikzpicture}[baseline=0pt] +\dynkin{B}{**.*.o.oo} +\dynkinLabelRoot{3}{r} +\end{tikzpicture} +} +\\ +% 4 +C_n & +\multicolumn{2}{E}{ +\begin{tikzpicture}[baseline=0pt] +\dynkin{C}{o.o*o.o*o.oo} +\dynkinLabelRoot{3}{d} +\dynkinLabelRoot{6}{rd} +\end{tikzpicture} +} +\\ +% 5 +D_n & +\multicolumn{2}{E}{ +\begin{tikzpicture}[baseline=0pt] +\dynkin{D}{o.o*o.o*o.ooo} +\dynkinLabelRoot{3}{d} +\dynkinLabelRoot{6}{rd} +\end{tikzpicture} +} +\\ +% 6 +E_6 & +\dynk{E}{*oooo*} +% 7 +E_6 & +\dynk{E}{o*o*oo} +% 8 +E_6 & +\dynk{E}{o*oooo} +% 9 +E_6 & +\dynk{E}{**ooo*} +% 10 +E_7 & +\dynk{E}{*oooooo} +% 11 +E_7 & +\dynk{E}{ooooo*o} +% 12 +E_7 & +\dynk{E}{oooooo*} +% 13 +E_7 & +\dynk{E}{*oooo*o} +% 14 - corrected from Springer. +E_7 & +\dynk{E}{*oooo**} +% 15 +E_7 & +\dynk{E}{*o**o*o} +% 16 +E_8 & +\dynk{E}{*ooooooo} +% 17 +E_8 & +\dynk{E}{ooooooo*} +% 18 +E_8 & +\dynk{E}{*oooooo*} +% 19 +E_8 & +\dynk{E}{oooooo**} +% 20 +E_8 & +\dynk{E}{*oooo***} +% 21 +F_4 & +\dynk{F}{ooo*} +% 22 +D_4 & +\dynk{D}{o*oo} +\end{longtable} +\endgroup + + \section{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: @@ -641,7 +761,7 @@ The Dynkin diagrams package has limited support for folding Dynkin diagrams. \end{tcblisting} Some Dynkin diagrams have multiple foldings, which we attempt to distinguish (not entirely successfully) by their \emph{ply}: the maximum number of roots folded together. -Most diagrams can only allow a 2-ply folding, so \verb!fold! is a synonym form \verb!ply=2!. +Most diagrams can only allow a 2-ply folding, so \verb!fold! is a synonym for \verb!ply=2!. \begin{tcblisting}{title={3-ply}} \dynkin[ply=3]{D}{4} @@ -714,7 +834,7 @@ We have to be careful about the 4-ply foldings of \(D^{(1)}_{2\ell}\), for which }% \begin{filecontents*}{DoneTwoElBendy.tex} -\begin{tikzpicture} +\begin{tikzpicture}[baseline=0pt] \dynkin[ply=4]{D}[1]{****.*****.*****} \dynkinFold[bend right=65]{1}{13} \dynkinFold[bend right=65]{0}{14} @@ -723,7 +843,7 @@ We have to be careful about the 4-ply foldings of \(D^{(1)}_{2\ell}\), for which \begin{filecontents*}{DoneTwoElStraight.tex} -\begin{tikzpicture} +\begin{tikzpicture}[baseline=0pt] \dynkin[ply=4]{D}[1]{****.*****.*****} \dynkinFold{0}{1} \dynkinFold{1}{13} @@ -938,9 +1058,9 @@ Below we draw the Vogan diagrams of some affine Lie superalgebras \cite{Ransingh \fi% }% -\tikzset{/Dynkin diagram,labelMacro/.code=\labls{#1},label,radius=.06cm} - +\begingroup +\tikzset{/Dynkin diagram,labelMacro/.code=\labls{#1},label,radius=.06cm} \tcbset{text width=10cm} \RenewDocumentCommand\wdtA{}{2cm} @@ -1171,8 +1291,86 @@ Below we draw the Vogan diagrams of some affine Lie superalgebras \cite{Ransingh \end{tikzpicture}% \end{tcblisting} \end{Category} +\endgroup +\begingroup +\small +\newcolumntype{F}{>{\columncolor[gray]{.9}}>{$}m{\wdtD}<{$}} +\newcolumntype{G}{>{\columncolor[gray]{.9}}>{$}m{\wdtE}<{$}} +\newcolumntype{H}{>{\columncolor[gray]{.9}}>{$}m{\wdtL}<{$}} +\newcolumntype{I}{>{\columncolor[gray]{.9}}>{$}m{3cm}<{$}} +\RenewDocumentCommand\wdtA{}{.2cm} +\RenewDocumentCommand\wdtD{}{2.1cm} +\RenewDocumentCommand\wdtE{}{4.1cm} +\RenewDocumentCommand\wdtL{}{3.5cm} +\NewDocumentCommand\LieG{}{\ensuremath{\mathfrak{g}}} +\NewDocumentCommand\R{m}{\ensuremath{\mathbb{R}^{#1}}} +\renewcommand*{\arraystretch}{1.5} +\begin{longtable}{ADIGH} +\LieG & \text{diagram} & V & \Delta & \alpha_i \\ \midrule +\endfirsthead +\LieG & \text{diagram} & V & \Delta & \alpha_i \\ \midrule +\endhead +A_n & \dynkin{A}{} & +\R{n+1}/\left<\sum e_j\right> & e_i-e_j & e_i-e_{i+1} \\ +B_n & \dynkin{B}{} & +\R{n} & \pm e_i, \pm e_i \pm e_j, i\ne j & e_i-e_{i+1}, e_n \\ +C_n & \dynkin{C}{} & +\R{n} & \pm 2 e_i, \pm e_i \pm e_j, i\ne j & e_i-e_{i+1}, 2e_n \\ +D_n & \dynkin{D}{} & \R{n} & \pm e_i \pm e_j, i\ne j & \begin{cases}e_i-e_{i+1}, & i\le n-2 \\ e_{n-1}+e_n \end{cases} \\ +E_8 & \dynkin{E}{8} & \R{8} & +\begin{cases} +\pm 2 e_i \pm 2 e_j, & i \ne j, \\ +\sum_i (-1)^{m_i} e_i, & \sum m_i \text{ even} +\end{cases} +& +\begin{cases} +2e_1-2e_2, \\ +2e_2-2e_3, \\ +2e_3-2e_4, \\ +2e_4-2e_5, \\ +2e_5-2e_6, \\ +2e_6+2e_7, \\ +-\sum e_j, \\ +2e_6-2e_7 +\end{cases} +\\ +E_7 & \dynkin{E}{7} & +\R{8}/\left<e_1-e_2\right> +& \text{quotient of } E_8 & \text{quotient of } E_8 +\\ +E_6 & \dynkin{E}{6} & \R{8}/\left<e_1-e_2,e_2-e_3\right> & \text{quotient of } E_8 & \text{quotient of } E_8 +\\ +F_4 & \dynkin{F}{4} & \R{4} & +\begin{cases} +\pm 2e_i, \\ +\pm 2e_i \pm 2e_j, i \ne j, \\ +\pm e_1 \pm e_2 \pm e_3 \pm e_4 +\end{cases} +& +\begin{cases} +2e_2-2e_3, \\ +2e_3-2e_4, \\ +2e_4, \\ +e_1-e_2-e_3-e_4 +\end{cases} +\\ +G_2 & \dynkin{G}{2} & \R{3}/\left<\sum e_j\right> +& \begin{cases} +\pm(1,-1,0), & \\ +\pm(-1,0,1), & \\ +\pm(0,-1,1), & \\ +\pm(2,-1,-1), & \\ +\pm(1,-2,1), & \\ +\pm(-1,-1,2) +\end{cases} +& \begin{cases} +(-1,0,1), & \\ +(2,-1,-1) +\end{cases} +\end{longtable} +\endgroup |