% The Lie Hasse package. % % Version 1.0 % % % This package draws Hasse diagrams of root system posets of simple % Lie algebraS, in LaTeX documents, using the TikZ package. % % Benjamin McKay % b.mckay@ucc.ie % % Released under the LaTeX Project Public License v1.3c or later, see % http://www.latex-project.org/lppl.txt % % % % \NeedsTeXFormat{LaTeX2e}[1994/06/01] \ProvidesPackage{lie-hasse}[2020/02/02 Lie Hasse] \RequirePackage[rgb]{xcolor} \RequirePackage{dynkin-diagrams} \usetikzlibrary{ positioning, fadings, quotes} %%% %%% Application programming interface: %%% \newif\iflie@hasse@threeD \lie@hasse@threeDtrue \xdef\lie@hasse@three@D@shift{.3}% \newif\iflie@hasse@show@ordering \lie@hasse@show@orderingfalse \newif\iflie@hasse@show@name \lie@hasse@show@namefalse \newif\iflie@hasse@show@widths \lie@hasse@show@widthsfalse \newif\iflie@hasse@show@height \lie@hasse@show@heightfalse \newif\iflie@hasse@attach@dynkin@diagram \lie@hasse@attach@dynkin@diagramfalse \pgfkeys{ /Lie Hasse diagram/.is family, /Lie Hasse diagram, attach Dynkin diagram/.is if = lie@hasse@attach@dynkin@diagram, attach Dynkin diagram/.default = false, edge/.style={ultra thick}, edge quotes/.style={/Dynkin diagram/text style,auto,inner sep=2pt}, compact edge/.style={}, noncompact edge/.style={opacity=0}, compact root/.code={\dynkinDrawSolidRootMark{#1}}, noncompact root/.code={\dynkinDrawCrossRootMark{#1}}, embedded Dynkin diagram/.style={ edge length=.5cm, root radius=.05cm }, for all roots/.code 2 args={}, three D/.is if = lie@hasse@threeD, three D/.default = true, show ordering/.is if = lie@hasse@show@ordering, show ordering/.default = false, show name/.is if = lie@hasse@show@name, show name/.default = false, show widths/.is if = lie@hasse@show@widths, show widths/.default = false, show height/.is if = lie@hasse@show@height, show height/.default = false, z shift/.estore in = \lie@hasse@three@D@shift, z shift/.default = .3, top/.style={black!20,opacity=.4}, left/.style={black!20,opacity=.9}, right/.style={black!20,opacity=.6}, .search also={/Dynkin diagram}, } \NewDocumentCommand\studyHasseDiagramOfRootSystem{O{}mm}% {% \xdef\dynkin@parabolic{0}% \pgfkeys{/Lie Hasse diagram,#1}% \xdef\dynkin@user@series{#2}% \xdef\dynkin@user@string{#3}% \global\dynkin@ply=\dynkin@ply@value\relax% \xdef\dynkin@series{#2}% \dynkin@grok@series% \IfSubStr{ABCDEFGHI}{\dynkin@series}{}{\dynkin@error@series}% \xdef\dynkin@string{#3}% \IfInteger{\dynkin@string}% {% \dynkin@integer@rank% }% {% % Turn Satake codes into Dynkin diagram expressions in \dynkin@string. \dynkin@grok@Satake@codes% }% % Expand out any digits in \dynkin@string into multiples of the various root marks. \expand@Dynkin@Roots@Digits% % Assign to \dynkin@roots the input string \dynkin@string with all . symbols removed, % so we only get the symbols representing the marks for the various roots. \StrDel{\dynkin@string}{.}[\temp]% \xdef\dynkin@roots{\temp}% \StrLen{\dynkin@roots}[\temp]% \global\dynkin@nodes=\temp\relax% \dynkin@grok@indefinite@edges% \dynkin@find@rank{}% \dynkin@cross@out@parabolics{}% \dynkin@set@default@label@directions{}% \check@Dynkin@diagram{}% \xdef\lie@hasse@compact@roots{}% \initialize@roots@as@sums@table{}% }% \newcount\lie@hasse@height% \newcount\lie@hasse@width% \NewDocumentCommand\doRootThing{mm}% {% \pgfkeys{/Lie Hasse diagram/for all roots={#1}{#2}}% }% \NewDocumentCommand\hasse{O{}mm}% {% \ifdefined\filldraw% \IfStrEq{#1}{}% {% \studyHasseDiagramOfRootSystem{#2}{#3}% }% {% \studyHasseDiagramOfRootSystem[#1]{#2}{#3}% }% \label@item@count 0\relax% \expandafter\forcsvlist% \expandafter\add@label% \expandafter{\dynkin@label@list}% \ifnum\dynkin@rank=1% \dynkin{A}{1}% \else% \add@compact@simple@roots@to@compact@roots{}% \IfStrEqCase{#2}% {%% {A}{\hasse@A}% {B}{\hasse@BC}% {C}{\hasse@BC*}% {D}{\hasse@D}% {E}% {% \ifnum\dynkin@rank=6% \hasse@E@six% \else% \ifnum\dynkin@rank=7% \hasse@E@seven% \else% \ifnum\dynkin@rank=8% \hasse@E@eight% \else \ifnum\dynkin@rank>8% \ClassError% {Lie Hasse}% {Rank of E series Hasse diagram: #3{}% exceeds 8; not defined yet}% {}% \fi% \ifnum\dynkin@rank<6% \ClassError% {Lie Hasse}% {Rank of E series Hasse diagram: #3{} less than 6}% {}% \fi% \fi% \fi% \fi% }% {F}% {% \ifnum\dynkin@rank=4% \hasse@F% \else% \ClassError% {Lie Hasse}% {Rank of F series Hasse diagram: #3{} not 4}% {}% \fi% }% {G}% {% \ifnum\dynkin@rank=2% \hasse@G% \else% \ClassError% {Lie Hasse}% {Rank of G series Hasse diagram: #3{} not 2}% {}% \fi% }% }%% [\ClassError% {Lie Hasse}% {Lie algebra series #2{} not A,B,C,D,E,F or G}% {}]% \draw@simple@root@labels% \fi% \lie@hasse@write@widths{}% \forAllPositiveRootsInHasseDiagram{\doRootThing}% \iflie@hasse@attach@dynkin@diagram% \attachDynkin% \fi% \else% \begin{tikzpicture}\hasse[#1]{#2}{#3}\end{tikzpicture}% \fi% }% \NewDocumentCommand\drawRootAsDynkinSum{mm}% {% \rootSum{#1}{#2}{\rs}% \node at (#1;#2) {% \dynkin[% labels=\rs,% /Lie Hasse diagram/embedded Dynkin diagram]% {\dynkin@series}{\the\dynkin@rank}% };% }% \NewDocumentCommand\drawRootDecomp{mm}% {% \rootSum{#1}{#2}{\rs}% \StrSubstitute{\rs}{,}{}[\rss]% \node[above,/Dynkin diagram/text style] at (#1;#2) {$\rss$};% }% \NewDocumentCommand\lie@hasse@write@widths{}% {% \iflie@hasse@show@widths% \rootSystemHeight{\lie@hasse@height}% \foreach \lie@hasse@i in {1,...,\the\lie@hasse@height}% {% \rootSystemWidthAtGrade{\lie@hasse@i}{\lie@hasse@width}% \path let \p1 = (\lie@hasse@i;1) in node[left] at (0,\y1) {\(\the\lie@hasse@width\)};% }% \fi% }% \NewDocumentCommand\backwardsGtwo{}% {% \dynkin[label,backwards,x shift in edge lengths=-1]{G}{\dynkin@roots}% }% \newcount\lie@hasse@E@shift% \NewDocumentCommand\attachDynkin{}% {% \IfStrEqCase{\dynkin@series}{% {A}{\dynkin{A}{\dynkin@roots}}% {B}{\dynkin{B}{\dynkin@roots}}% {C}{\dynkin{C}{\dynkin@roots}}% {D}{\dynkin[y shift in edge lengths=-1.5,label]% {D}{\dynkin@roots}}% {E}% {%% \ifnum\dynkin@rank=6% \dynkin[% upsidedown,% labels*={1,...,6},% y shift in edge lengths=.5]% {E}{\dynkin@roots}% \else% \lie@hasse@E@shift\the\dynkin@rank\relax% \advance\lie@hasse@E@shift by -2\relax% \dynkin[% backwards,% upsidedown,% label,% x shift in edge lengths=-\the\lie@hasse@E@shift,% y shift in edge lengths=1]% {E}{\dynkin@roots}% \fi% }%% {F}{\dynkin{F}{\dynkin@roots}}% {G}{\backwardsGtwo{}}% }% [\ClassError% {Lie Hasse}% {Lie algebra series \dynkin@series{} not A,B,C,D,E,F or G when attaching Dynkin diagram}% {}]% }% \newif\iflie@hasse@multiple@diagrams \newcommand\doHasseDiagram[1]%% {% \iflie@hasse@multiple@diagrams% \ % \fi% \begin{tikzpicture}% [show background rectangle,% baseline=(current bounding box.east)]]% \hasse#1% \iflie@hasse@show@height% \rootSystemHeight{\lie@hasse@height}% \fi% \iflie@hasse@show@name% \xdef\label@text{$\dynkinName{\dynkin@series}{\the\dynkin@rank}$}% \iflie@hasse@show@ordering% \xdef\label@text{\label@text{} in \currentDynkinOrdering{} ordering}% \fi% \iflie@hasse@show@height% \xdef\label@text{\label@text: height$=\the\lie@hasse@height$}% \fi% \node[anchor=north,yshift=1.5em] at (current bounding box.north) {\label@text};% \fi% \end{tikzpicture}% \iflie@hasse@multiple@diagrams% \ % \fi% }% \DeclareListParser*{\forscsvlist}{;}% \newcommand\hasseDiagrams[1]% {% \IfSubStr{#1}{;}% {% \global\lie@hasse@multiple@diagramstrue% }% {% \global\lie@hasse@multiple@diagramsfalse% }% \begin{center}% \expandafter\forscsvlist% \expandafter\doHasseDiagram% \expandafter{#1}% \end{center}% }% %%% %%% Package internals: %%% \pgfdeclarelayer{Dynkin middle} \pgfdeclarelayer{Dynkin above} \pgfsetlayers{background,Dynkin behind,Dynkin middle,Dynkin above,main} \newcommand*\label@list@at[1]{\csname FOOBAR-#1\endcsname} \newcommand*\defn@label@list@entry[1]{\expandafter\def\csname FOOBAR-#1\endcsname} \def\add@to@label@list#1#2{\global\defn@label@list@entry{#1}{#2}}% \newcount\label@item@count% \label@item@count 0\relax% \newcommand\add@label[1]% {% \advance\label@item@count by 1\relax% \global\add@to@label@list{\the\label@item@count}{#1}% }% \xdef\lie@hasse@compact@roots{}% \NewDocumentCommand\if@compact@root{mmm}% {% \IfSubStr{\lie@hasse@compact@roots}{(#1)}{#2}{#3}% }% \newcount\current@simple@root% \NewDocumentCommand\if@compact@simple@root{mmm}% {% % #1 is the index of a root, in the internal root ordering of % this Lie Hasse package, which is the Carter root ordering. % So first we need to convert it to the current root ordering % of the Dynkin diagram we are working on. \dynkinOrder{\dynkin@series}{\the\dynkin@rank}.Carter::#1->\dynkin@ordering.% {\current@simple@root}% \StrChar{\dynkin@roots}{\current@simple@root}[\lie@hasse@root@marker]% \IfStrEq{*}{\lie@hasse@root@marker}{#2}{#3}% }% \NewDocumentCommand\add@compact@root{m}% {% \if@compact@root{#1}% {%do nothing } {%else \edef\temp@lie@hasse@compact@roots{\lie@hasse@compact@roots}% \xdef\lie@hasse@compact@roots{\temp@lie@hasse@compact@roots{ }(#1)}% }% }% \newcount\lie@hasse@sr% \newcount\drpo% \NewDocumentCommand\add@compact@simple@roots@to@compact@roots{}% {% \lie@hasse@sr 1\relax% \drpo\the\dynkin@rank\relax% \advance\drpo by 1\relax% \loop% \if@compact@simple@root{\the\lie@hasse@sr}% {%% \add@compact@root{1;\the\lie@hasse@sr}% }%% {%% }%% \advance\lie@hasse@sr by 1\relax% \ifnum\lie@hasse@sr<\the\drpo% \repeat% }% \newcount\drmo% \newcount\lie@hasse@root@count% \newcount\lie@hasse@root@count@b% \newcount\max@lie@hasse@root@count% \newcount\lie@hasse@reorder \NewDocumentCommand\lie@hasse@fix@order{mm}% {% \dynkinOrder{\dynkin@series}{\dynkin@rank}.Carter::{#1}->\dynkin@ordering.{#2}% }% \NewDocumentCommand\draw@one@simple@root@label{mm}% %% \draw@one@simple@root@label{r}{s}, draw at Carter number r %% the root with text s. {% \node[below,% label=% {% [below,% lie@hasse@color,% /Lie Hasse diagram/edge quotes]% \(\pgfkeys{/Dynkin diagram/label macro=#2}\)% }% ]% at% (1;#1)% {};% }% \NewDocumentCommand\draw@simple@root@labels{}% {% \lie@hasse@root@count 1\relax% \max@lie@hasse@root@count \the\dynkin@rank\relax% \advance\max@lie@hasse@root@count 1\relax% \IfStrEq{\dynkin@label@list\dynkin@label@list@star}{}% {% \loop% \set@lie@hasse@color{\the\lie@hasse@root@count}% \lie@hasse@fix@order% {\the\lie@hasse@root@count}% {\lie@hasse@root@count@b}% \edef\label@text{\the\lie@hasse@root@count@b}% \draw@one@simple@root@label% {\the\lie@hasse@root@count}% {\label@text}% \global\advance\lie@hasse@root@count by 1\relax% \ifnum\lie@hasse@root@count<\the\max@lie@hasse@root@count% \repeat% }% {% \edef\lie@hasse@XXX{\dynkin@label@list}% \foreach \lie@hasse@i in \lie@hasse@XXX% {% \set@lie@hasse@color{\the\lie@hasse@root@count}% \lie@hasse@fix@order% {\the\lie@hasse@root@count}% {\lie@hasse@root@count@b}% \draw@one@simple@root@label% {\the\lie@hasse@root@count@b}% {\lie@hasse@i}% \global\advance\lie@hasse@root@count by 1\relax% }% }% }% \definecolor{lie@hasse@color}{hsb}{0,0,0} \NewDocumentCommand\set@lie@hasse@color{m}% {% \pgfmathsetmacro{\lie@hasse@huenum}{#1/\the\dynkin@rank}% \global\definecolor{lie@hasse@color}{hsb}{\lie@hasse@huenum,1,.6}% }% \NewDocumentCommand\draw@lie@hasse@root{m}% {% \if@compact@root{#1}% {%% \pgfkeys{/Lie Hasse diagram/compact root=#1}% }%% {%% \pgfkeys{/Lie Hasse diagram/noncompact root=#1}% }%% }% \xdef\lie@hasse@grade@a{} \xdef\lie@hasse@grade@b{} \xdef\lie@hasse@index@a{} \xdef\lie@hasse@index@b{} \newcount\root@gpo% \xdef\label@text{}% \NewDocumentCommand\draw@lie@hasse@edge{O{}mmmm}% %[edge label options]{from grade}{from index}{to index}{simple root} {% \root@gpo #2\relax% \advance\root@gpo by 1\relax% \set@lie@hasse@color{#5}% \lie@hasse@fix@order{#5}{\lie@hasse@root@count@b}% \IfStrEq{\dynkin@label@list}{}% {% \xdef\label@text{\the\lie@hasse@root@count@b}% }% {% \xdef\label@text% {% \label@list@at{\the\lie@hasse@root@count@b}% }% }% \if@compact@simple@root{#5}% {%%% Compact \draw[% lie@hasse@color,% /Lie Hasse diagram/edge,% /Lie Hasse diagram/compact edge]% (#2;#3) --% node[lie@hasse@color,/Lie Hasse diagram/edge quotes,#1]% {\(\pgfkeys{/Dynkin diagram/label macro=\label@text}\)}% (\the\root@gpo;#4);% \IfSubStr{\lie@hasse@compact@roots}{#2;#3}% {%% adding to a compact \add@compact@root{\the\root@gpo;#4}% }%% {%% }%% }%%% {%%% Noncompact \draw[% lie@hasse@color,% /Lie Hasse diagram/edge,% /Lie Hasse diagram/noncompact edge]% (#2;#3) --% node[lie@hasse@color,/Lie Hasse diagram/edge quotes,#1]% {\(\pgfkeys{/Dynkin diagram/label macro=\label@text}\)}% (\the\root@gpo;#4);% }%%% }% \newcount\A@series@grade% \newcount\max@A@series@grade% \newcount\A@series@x% \newcount\A@series@y% \newcount\max@A@series@j% \newcount\A@series@j% \NewDocumentCommand\place@A@roots{}% {% \A@series@grade 1\relax% \max@A@series@grade \the\dynkin@rank\relax% \advance\max@A@series@grade by 1\relax% \A@series@x 0\relax% \A@series@y 0\relax% \loop% \max@A@series@j\the\dynkin@rank\relax% \advance\max@A@series@j by -\the\A@series@grade\relax% \advance\max@A@series@j by 2\relax% \A@series@j 1\relax% {% \loop% \coordinate (\the\A@series@grade;\the\A@series@j) at ({.5*\the\A@series@x*\dynkin@edge@length}, {.25*\the\A@series@y*\dynkin@edge@length}) {};% \advance\A@series@j by 1\relax% \advance\A@series@x by 2\relax% \ifnum\A@series@j<\the\max@A@series@j% \repeat% }% \advance\A@series@grade by 1\relax% \advance\A@series@x by 1\relax% \advance\A@series@y by 2\relax% \ifnum\A@series@grade<\the\max@A@series@grade% \repeat% }% \newcount\A@series@grade% \newcount\max@A@series@grade% \NewDocumentCommand\draw@A@roots{}% {% \A@series@grade 1\relax% \max@A@series@grade \the\dynkin@rank\relax% \advance\max@A@series@grade by 1\relax% \loop% \max@A@series@j\the\dynkin@rank\relax% \advance\max@A@series@j by -\the\A@series@grade\relax% \advance\max@A@series@j by 2\relax% \A@series@j 1\relax% {% \loop% \draw@lie@hasse@root% {\the\A@series@grade;\the\A@series@j}% \advance\A@series@j by 1\relax% \ifnum\A@series@j<\the\max@A@series@j% \repeat% }% \advance\A@series@grade by 1\relax% \ifnum\A@series@grade<\the\max@A@series@grade% \repeat% }% \newcount\maxA@i% \newcount\maxA@j% \newcount\A@lbl% \newcount\A@i% \newcount\A@j% \newcount\A@ipo% \newcount\A@rj% \newcount\A@rjmo% \NewDocumentCommand\draw@A@edges{}% {% \maxA@i\the\dynkin@rank\relax% \A@i 1\relax% \loop% \maxA@j \the\dynkin@rank\relax% \advance\maxA@j by -\the\A@i\relax% \advance\maxA@j by 1\relax% \ifnum\maxA@j>-1% \A@j 1\relax% {% \loop% \A@lbl\the\A@i\relax% \advance\A@lbl by \the\A@j\relax% \A@ipo \A@i\relax% \advance\A@ipo by 1\relax% \draw@lie@hasse@edge[left]% {\the\A@i}{\the\A@j}% {\the\A@j}{\the\A@lbl}% \A@rj\the\maxA@j\relax% \advance\A@rj by -\the\A@j\relax% \advance\A@rj by 1\relax% \A@rjmo\the\A@rj\relax% \advance\A@rjmo by -1\relax% \A@lbl-\A@lbl\relax% \advance\A@lbl by \the\dynkin@rank% \advance\A@lbl by 1% \draw@lie@hasse@edge[right]% {\the\A@i}{\the\A@rj}% {\the\A@rjmo}{\the\A@lbl}% \advance\A@j by 1\relax% \ifnum\A@j<\the\maxA@j% \repeat% }% \fi% \advance\A@i by 1\relax% \ifnum\A@i<\the\maxA@i% \repeat% }% %% \hasse@A@width@at@grade{g}{\w} sets the counter \w to %% the value of the width (number of roots) at grade g of %% the A root system with rank \the\dynkin@rank \NewDocumentCommand\hasse@A@width@at@grade{mm}% {% \global#2\the\dynkin@rank\relax% \global\advance#2 by -#1\relax% \global\advance#2 by 1\relax% }% \NewDocumentCommand\hasse@A{}% {% \place@A@roots% \draw@A@edges% \draw@A@roots% }% %% \hasse@BC@width@at@grade{g}{\w} sets the counter \w to %% the value of the width (number of roots) at grade g of %% the B or C root system with rank \the\dynkin@rank \NewDocumentCommand\hasse@BC@width@at@grade{mm}% {% \global#2\the\dynkin@rank\relax% \global\advance#2 by \the\dynkin@rank\relax% \global\advance#2 by -#1\relax% \global\advance#2 by 1\relax% \global\divide#2 by 2\relax% }% \newcount\tdrmt% \newcount\tdrmo% \newcount\max@BC@r% \newcount\BC@r% \newcount\BC@series@x% \newcount\BC@series@y% \NewDocumentCommand\place@BC@roots{}% {% \drmo \the\dynkin@rank\relax% \advance\drmo by -1\relax% \tdrmt \the\drmo\relax% \advance\tdrmt by \the\drmo\relax% \tdrmo \the\tdrmt\relax% \advance\tdrmo 1\relax% \max@BC@r \the\dynkin@rank\relax% \advance\max@BC@r by 1\relax% \BC@r 1\relax% \loop% \BC@series@x \the\BC@r\relax% \advance\BC@series@x by -1\relax% \advance\BC@series@x by \the\BC@series@x\relax% \BC@series@y 1\relax% {%% \loop% \coordinate (\the\BC@series@y;\the\BC@r) at ({.5*\the\BC@series@x*\dynkin@edge@length}, {.5*(\the\BC@series@y-1)*\dynkin@edge@length}) {};% \advance\BC@series@y by 1\relax% \advance\BC@series@x by 1\relax% \ifnum\BC@series@x<\the\tdrmo\repeat% }%% \advance\BC@r by 1\relax% \ifnum\BC@r<\the\max@BC@r% \repeat% }% \newcount\max@BC@i% \newcount\BC@i% \newcount\BC@series@x% \newcount\BC@series@y% \newcount\BC@ii% \newcount\BC@series@yy% \newcount\BC@series@yy% \newcount\lbl% \newcount\temp@lbl% \NewDocumentCommand\draw@BC@edges{s}% {% \tdrmo \the\dynkin@rank\relax% \advance\tdrmo by \the\dynkin@rank\relax% \advance\tdrmo by -1\relax% \max@BC@i \the\dynkin@rank\relax% \advance\max@BC@i by 1\relax% \BC@i 1\relax% \loop% \BC@series@x \the\BC@i\relax% \advance\BC@series@x by \the\BC@i\relax% \advance\BC@series@x by -2\relax% \BC@series@y 1\relax% {%% \loop% \ifnum\BC@i>1% % draw an edge upward to the left \BC@ii \the\BC@i\relax% \advance\BC@ii by -1\relax% \BC@series@yy \the\BC@series@y\relax% \advance\BC@series@yy by 1\relax% \draw@lie@hasse@edge[right]% {\the\BC@series@y}{\the\BC@i}% {\the\BC@ii}% {\the\BC@ii}% \fi% \ifnum\BC@series@y>1% % draw an edge downward to the left \BC@series@yy \the\BC@series@y\relax% \advance\BC@series@yy by -1\relax% \lbl \the\BC@i\relax% \advance\lbl by \the\BC@series@y\relax% \advance\lbl by -1\relax% \ifnum\dynkin@rank<\the\lbl% \temp@lbl \the\lbl\relax% \lbl \the\dynkin@rank\relax% \advance\lbl by \the\dynkin@rank% \advance\lbl by -\the\temp@lbl% %% For the B series: \IfBooleanF{#1}{\global\advance\lbl by 1\relax}% \draw@lie@hasse@edge[left]% {\the\BC@series@yy}{\the\BC@i}% {\the\BC@i}% {\the\lbl}% \else% \draw@lie@hasse@edge[left]% {\the\BC@series@yy}{\the\BC@i}% {\the\BC@i}% {\the\lbl}% \fi% \fi% \advance\BC@series@y by 1\relax% \advance\BC@series@x by 1\relax% \ifnum\BC@series@x<\the\tdrmo\repeat% }%% \advance\BC@i by 1\relax% \ifnum\BC@i<\the\max@BC@i% \repeat% }% \newcount\tdrpt% \newcount\BC@r% \newcount\BC@series@y% \newcount\maxy% \NewDocumentCommand\draw@BC@roots{}% {% \tdrpt \the\dynkin@rank\relax% \advance\tdrpt by \the\dynkin@rank\relax% \advance\tdrpt by 2\relax% \max@BC@r \the\dynkin@rank\relax% \advance\max@BC@r by 1\relax% \BC@r 1\relax% \loop% \BC@series@y 1\relax% \maxy \tdrpt\relax% \advance\maxy -\the\BC@r\relax% \advance\maxy -\the\BC@r\relax% {%% \loop% \draw@lie@hasse@root{\the\BC@series@y;\the\BC@r}% \advance\BC@series@y by 1\relax% \ifnum\BC@series@y<\the\maxy\repeat% }%% \advance\BC@r by 1\relax% \ifnum\BC@r<\the\max@BC@r% \repeat% }% \NewDocumentCommand\hasse@BC{s}% star for C: {% \place@BC@roots% \IfBooleanTF{#1}% {% \draw@BC@edges*% }% {% \draw@BC@edges% }% \draw@BC@roots% }% \NewDocumentCommand\hasse@D{}% {% \place@D@roots% \draw@D@edges% \iflie@hasse@threeD% \draw@D@hasse@three@D% \fi% \draw@D@roots% }% \newcount\drmt% \NewDocumentCommand\draw@D@hasse@three@D{}% {% \drmo \the\dynkin@rank\relax% \advance\drmo by -1\relax% \drmt \the\dynkin@rank\relax% \advance\drmt by -2\relax% \begin{pgfonlayer}{Dynkin middle}% \fill[/Lie Hasse diagram/left] (\the\drmt;1) -- (\the\drmo;1) -- (\the\dynkin@rank;1) -- (\the\drmo;2) -- cycle;% \fill[/Lie Hasse diagram/top] (\the\drmt;1) -- (1;\the\drmt) -- (2;\the\drmo) -- (\the\drmo;2) -- cycle;% \fill[/Lie Hasse diagram/top] (\the\drmo;2) -- (\the\dynkin@rank;1) -- (3;\the\drmo) -- (2;\the\drmo) -- cycle;% \fill[/Lie Hasse diagram/top] (\the\drmo;1) -- (\the\dynkin@rank;1) -- (3;\the\drmo) -- (2;\the\drmt) -- cycle;% \fill[/Lie Hasse diagram/top] (\the\drmt;1) -- (\the\drmo;1) -- (2;\the\drmt) -- (1;\the\drmt) -- cycle;% \fill[/Lie Hasse diagram/left] (1;\the\drmt) -- (2;\the\drmt) -- (3;\the\drmo) -- (2;\the\drmo) -- cycle;% \end{pgfonlayer}% }% \NewDocumentCommand\hasse@D@width@at@grade{mm}% {% \global#2\the\dynkin@rank\relax% \global\advance#2 by\the\dynkin@rank\relax% \global\advance#2 by -#1\relax% \ifnum#1<\the\dynkin@rank% \global\advance#2 by 1\relax% \else% \global\advance#2 by -1\relax% \fi% \global\divide#2 by 2\relax% }% \newcount\max@D@grades% \newcount\max@D@grades@plus@one% \newcount\D@g% \newcount\maxD@grd% \newcount\D@grd% \newcount\maxxD% \newcount\iD% \newcount\xxD% \newcount\sum@D% \NewDocumentCommand\place@D@roots{}% {% \global\max@D@grades \the\dynkin@rank\relax% \global\advance\max@D@grades \the\dynkin@rank\relax% \global\advance\max@D@grades -3\relax% \max@D@grades@plus@one \the\max@D@grades\relax% \advance\max@D@grades@plus@one 1\relax% \D@g 1\relax% {% \loop% \place@D@roots@at@grade{\the\D@g}% \advance\D@g by 1\relax% \ifnum\D@g<\the\max@D@grades@plus@one% \repeat% }% }% \newcount\D@r% \NewDocumentCommand\place@D@roots@at@grade{m}% {% \hasse@D@width@at@grade{#1}{\max@D@roots}% \max@D@roots@plus@one \the\max@D@roots\relax% \advance\max@D@roots@plus@one 1\relax% \D@r 1\relax% {% \loop% \place@D@root@at{#1}{\the\D@r}% \advance\D@r by 1\relax% \ifnum\D@r<\the\max@D@roots@plus@one% \repeat% }% }% \newcount\drpo% \NewDocumentCommand\place@D@root@at{mm}% {% \xxD #1\relax% \advance\xxD by #2\relax% \advance\xxD by #2\relax% \advance\xxD by -3\relax% \drpo \the\dynkin@rank\relax% \advance\drpo by 1\relax% \sum@D #1\relax% \advance\sum@D by #2\relax% \ifnum\sum@D=\the\dynkin@rank% \global\def\D@root@x% {\the\xxD-\lie@hasse@three@D@shift}% \else% \ifnum\sum@D=\the\drpo% \ifnum#1<\the\dynkin@rank% \global\def\D@root@x% {\the\xxD-2+\lie@hasse@three@D@shift}% \else% \global\def\D@root@x{\the\xxD}% \fi% \else% \ifnum\sum@D<\the\dynkin@rank% \global\def\D@root@x{\the\xxD}% \else% % if \sum@D exceeds rank+1 \ifnum#1<\the\dynkin@rank% \global\def\D@root@x{\the\xxD-2}% \else% \global\def\D@root@x{\the\xxD}% \fi% \fi% \fi% \fi% \coordinate (#1;#2) at ({(\D@root@x)*\dynkin@edge@length/2},{(#1-1)*\dynkin@edge@length/2}) {};% }% \NewDocumentCommand\draw@D@edges{}% {% \global\max@D@grades \the\dynkin@rank\relax% \global\advance\max@D@grades \the\dynkin@rank\relax% \global\advance\max@D@grades -3\relax% \D@g 1\relax% {% \loop% \draw@D@edges@at@grade{\the\D@g}% \advance\D@g by 1\relax% \ifnum\D@g<\the\max@D@grades% \repeat% }% }% \newcount\max@D@roots% \newcount\max@D@roots@plus@one% \NewDocumentCommand\draw@D@edges@at@grade{m}% {% \hasse@D@width@at@grade{#1}{\max@D@roots}% \max@D@roots@plus@one \the\max@D@roots\relax% \advance\max@D@roots@plus@one 1\relax% \D@r 1\relax% {% \loop% \draw@D@edges@at{#1}{\the\D@r}{\the\max@D@roots}% \advance\D@r by 1\relax% \ifnum\D@r<\the\max@D@roots@plus@one% \repeat% }% }% \newcount\hasse@D@gpo% \newcount\hasse@D@imo% \newcount\hasse@D@w% \newcount\hasse@D@gpi% \newcount\hasse@D@target% \newif\ifhasse@D% \xdef\hasse@D@layer{}% \xdef\hasse@D@LR{}% \newcount\hasse@D@label% \NewDocumentCommand\draw@D@edges@at{mm}% {% \drmo\the\dynkin@rank\relax% \advance\drmo by -1\relax% \drpo \the\dynkin@rank\relax% \advance\drpo by 1\relax% \hasse@D@gpo#1\relax% \advance\hasse@D@gpo by 1\relax% \hasse@D@gpi#1\relax% \advance\hasse@D@gpi #2\relax% \hasse@D@imo#2\relax% \advance\hasse@D@imo by -1\relax% \hasse@Dfalse% \xdef\hasse@D@layer{Dynkin middle}% \xdef\hasse@D@LR{left}% \hasse@D@width@at@grade{#1}{\hasse@D@w}% \ifnum#2>1% % Not at the left edge, so draw an edge going up to the left \global\hasse@D@label#2\relax% \advance\hasse@D@label -1\relax% \ifnum\hasse@D@gpo=\the\dynkin@rank% \ifnum\hasse@D@imo=1% % (g;i)=(r-1;2) doesn't go anywhere up to the left. \else% % connect g;i to g+1;i-2 \global\hasse@Dtrue% \global\hasse@D@target \hasse@D@imo\relax \global\advance\hasse@D@target by -1\relax% \fi% \else% % connect g;i to g+1;i-1 \global\hasse@Dtrue% \global\hasse@D@target \hasse@D@imo\relax% \ifnum\hasse@D@gpi=\the\dynkin@rank% \xdef\hasse@D@layer{Dynkin above}% \else% % below or middle. \ifnum\hasse@D@gpi=\the\drpo% \xdef\hasse@D@layer{Dynkin behind}% \fi% \fi% \fi% \ifnum\hasse@D@w=#2% \xdef\hasse@D@LR{right}% \fi% \fi% \ifnum\dynkin@rank>#1% \ifnum\hasse@D@gpi<\the\drpo% \else% \global\advance\hasse@D@label -1\relax% \fi% \fi% \ifhasse@D% \begin{pgfonlayer}{\hasse@D@layer} \draw@lie@hasse@edge% [\hasse@D@LR]% {#1}{#2}% {\the\hasse@D@target}{\the\hasse@D@label}% \end{pgfonlayer}% \fi% \hasse@Dtrue% \xdef\hasse@D@LR{left}% \ifnum\hasse@D@w=#2% \ifodd#1% % not enough room to draw an edge going rightward. \global\hasse@Dfalse\relax% \else% \xdef\hasse@D@LR{right}% \fi% \fi% \ifnum#1=1% \ifnum\drmo=#2% \global\hasse@Dfalse\relax% \else% \xdef\hasse@D@LR{right}% \fi% \fi% \ifhasse@D% \global\hasse@D@label \hasse@D@gpi\relax% \ifnum\hasse@D@label>\the\dynkin@rank% \hasse@D@label -\hasse@D@label\relax% \advance\hasse@D@label \the\dynkin@rank\relax% \advance\hasse@D@label \the\drmo\relax% \fi% \ifnum\drmo<#1% \else% \ifnum#2=1% \else% \ifnum\hasse@D@gpi<\the\drpo% \else% \advance\hasse@D@label 1\relax% \fi% \fi% \fi% \global\hasse@D@target #2\relax% \ifnum\hasse@D@gpi=\the\dynkin@rank% \ifnum#2=1% \else% \global\advance\hasse@D@target by 1\relax% \fi% \else% \ifnum\drmo=#1% \global\advance\hasse@D@target by -1\relax% \fi% \fi% \ifnum\hasse@D@gpi=\the\drpo% \ifnum#2>1% \xdef\hasse@D@layer{Dynkin behind}% \xdef\hasse@D@LR{right}% \fi% \else% \ifnum\hasse@D@gpi=\the\dynkin@rank% \xdef\hasse@D@layer{Dynkin above}% \else% \ifnum\hasse@D@gpi=\the\drmo% \xdef\hasse@D@layer{Dynkin above}% \else% \xdef\hasse@D@layer{Dynkin middle}% \fi% \fi% \fi% \fi% \ifhasse@D% \begin{pgfonlayer}{\hasse@D@layer} \draw@lie@hasse@edge% [\hasse@D@LR]% {#1}{#2}% {\the\hasse@D@target}{\the\hasse@D@label}% \end{pgfonlayer}% \fi% % % Draw any other edges. \ifnum\hasse@D@gpi=\the\drmo% \hasse@D@target#2\relax% \advance\hasse@D@target by 1\relax% \begin{pgfonlayer}{Dynkin behind} \draw@lie@hasse@edge% [right,near start]% {#1}{#2}% {\the\hasse@D@target}{\the\dynkin@rank}% \end{pgfonlayer}% \fi% }% \newcount\maxD@grd% \newcount\D@grd% \newcount\maxxD% \newcount\sum@D% \NewDocumentCommand\draw@D@roots{}% {% \drmo \the\dynkin@rank\relax% \advance\drmo by -1\relax% \maxD@grd \the\dynkin@rank\relax% \advance\maxD@grd by \the\dynkin@rank\relax% \advance\maxD@grd by -2\relax% \D@grd 1\relax% \loop% % Construct the roots at grade \D@grd. \draw@D@roots@at@grade{\D@grd}% \advance\D@grd by 1\relax% \ifnum\D@grd<\the\maxD@grd% \repeat% }% \NewDocumentCommand\draw@D@roots@at@grade{m}% {% \hasse@D@width@at@grade{#1}{\max@D@roots}% \max@D@roots@plus@one \the\max@D@roots\relax% \advance\max@D@roots@plus@one 1\relax% \D@r 1\relax% {% \loop% \draw@lie@hasse@root{\the\D@grd;\the\D@r}% \advance\D@r by 1\relax% \ifnum\D@r<\the\max@D@roots@plus@one% \repeat% }% }% %}% \newcount\lie@hasse@grade@no \newcount\lie@hasse@root@no \def\lie@hasse@XX{} \def\lie@hasse@ZZ{} \NewDocumentCommand\lie@hasse@read@root@pair{m}% {% \IfSubStr{#1}{:}% {% \StrBefore{#1}{:}[\lie@hasse@XX]% \StrBehind{#1}{:}[\lie@hasse@ZZ]% }% {% \xdef\lie@hasse@XX{#1}% \xdef\lie@hasse@ZZ{0}% }% \IfStrEq{\dynkin@series}{G}% {% \pgfmathsetmacro\lie@hasse@XX% {\lie@hasse@XX}% }% {% else \pgfmathsetmacro\lie@hasse@XX% {\lie@hasse@XX+\lie@hasse@three@D@shift*(\the\lie@hasse@grade@no-\lie@hasse@ZZ-1)}% }% }% \NewDocumentCommand\lie@hasse@place@root{m}% {% \lie@hasse@read@root@pair{#1}% \coordinate (\the\lie@hasse@grade@no;\the\lie@hasse@root@no) at ({\lie@hasse@XX*\dynkin@edge@length/2}, {(\the\lie@hasse@grade@no-1)*\dynkin@edge@length/2}) {};% \global\advance\lie@hasse@root@no by 1% }% \NewDocumentCommand\lie@hasse@place@root@list{m}% {% \global\lie@hasse@root@no 1\relax% \expandafter\forcsvlist\expandafter{\lie@hasse@place@root}{#1}% \global\advance\lie@hasse@grade@no by 1% }% \NewDocumentCommand\lie@hasse@place@roots{m}% {% \global\lie@hasse@grade@no 1\relax% \renewcommand*{\do}[1]% {% \lie@hasse@place@root@list{##1}% }% \expandafter\docsvlist\expandafter{#1}% }% \NewDocumentCommand\lie@hasse@draw@root{m}% {% \draw@lie@hasse@root{\the\lie@hasse@grade@no;\the\lie@hasse@root@no}% \global\advance\lie@hasse@root@no by 1% }% \NewDocumentCommand\lie@hasse@root@draw@list{m}% {% \global\lie@hasse@root@no 1\relax% \expandafter\forcsvlist\expandafter{\lie@hasse@draw@root}{#1}% \global\advance\lie@hasse@grade@no by 1% }% \NewDocumentCommand\lie@hasse@draw@roots{m}% {% \global\lie@hasse@grade@no 1\relax% \renewcommand*{\do}[1]% {% \lie@hasse@root@draw@list{##1}% }% \expandafter\docsvlist\expandafter{#1}% }% \newcount\lie@hasse@grade@no@plus@one% \newcount\lie@hasse@edge@count% \def\lie@hasse@edge@go{% \@ifstar\lie@hasse@process@edge@above\lie@hasse@process@edge@behind}% \NewDocumentCommand\lie@hasse@process@edge{m}% {% \lie@hasse@edge@go#1!% }% \def\lie@hasse@process@edge@above#1!% {% \global\advance\lie@hasse@edge@count by 1\relax% \begin{pgfonlayer}{Dynkin above}% \lie@hasse@finish@edge#1!% \end{pgfonlayer}% }% \def\lie@hasse@process@edge@behind#1!% {% \global\advance\lie@hasse@edge@count by 1\relax% \begin{pgfonlayer}{Dynkin behind}% \lie@hasse@finish@edge#1\relax% \end{pgfonlayer}% }% \def\lie@hasse@finish@edge#1+#2#3#4% {% \ifx#3<\relax% \draw@lie@hasse@edge[left]% {\the\lie@hasse@grade@no}{#1}% {#4}% {#2}% \else% \draw@lie@hasse@edge[right]% {\the\lie@hasse@grade@no}{#1}% {#4}% {#2}% \fi% }% \NewDocumentCommand\lie@hasse@process@edge@list{m}% {% \global\lie@hasse@edge@count 0\relax% \expandafter\forcsvlist\expandafter\lie@hasse@process@edge{#1}% \global\advance\lie@hasse@grade@no by 1\relax% }% \NewDocumentCommand\lie@hasse@process@edge@lists{m}% {% \global\lie@hasse@grade@no 1\relax% \expandafter\forcsvlist\expandafter{\lie@hasse@process@edge@list}{#1}% }% \NewDocumentCommand\hasse@X{mm}% {% \lie@hasse@place@roots{#1}% \lie@hasse@process@edge@lists{#2}% \lie@hasse@draw@roots{#1}% }% \NewDocumentCommand\hasse@E@six@width@at@grade{mm}% {% \ifnum#1=1% #2 6% \else\ifnum#1=2% #2 5% \else\ifnum#1=3% #2 5% \else\ifnum#1=4% #2 5% \else\ifnum#1=5% #2 4% \else\ifnum#1=6% #2 3% \else\ifnum#1=7% #2 3% \else\ifnum#1=8% #2 2% \else\ifnum#1<1% #2 0% \else\ifnum#1>11% #2 0% \else% #2 1% \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi% }% % In the following code, edges are described by grade: in grade g, the expression i+r instead of < to get the right. Use a star in front to get this edge on the higher layer. \NewDocumentCommand\hasse@E@six{}% {% \hasse@X% {{0,2,4,5:1,6,8},{1,3,4:1,5,7},{2,3:1,4,5:1,6},{2:1,3,4:1,5,6:1},% {3:1,4:2,4,5:1},{3:2,4:1,5:2},{2:2,4:2,6:2},{3:2,5:2},{4:2},{4:3},{4:4}}% {%%% grades: {% 1 1+2<1,*2+3>2,*3+4<3,*5+6>5,*2+1<1,*3+2<2,*3+5>4,*4+3>3,*5+3<4,6+5>5}, {% 2 *1+3<1,*2+4<2,*3+5<4,4+2<3,*4+4<4,*4+6<5,*2+1<1,2+5>3,*3+2>2,*5+3>5}, {% 3 *1+4<1,3+1<2,*2+5<3,*2+1<1,3+4<3,3+6<4,*4+2<3,*4+6<5,1+5>2,5+2>4,*5+4>5}, {% 4 *1+5<1,2+4<1,2+6<3,*3+1<1,*3+3>2,4+1<3,4+4<4,*3+6<4,*5+2>4}, {% 5 *1+3<1,1+6<2,3+4<2,4+1>2,*2+1<1,*2+6>3,*4+3>3}, {% 6 *1+2<1,*1+6<2,2+3<2,*3+1<2,*3+5>3}, {% 7 *1+6<1,*2+5<2,*2+2<1,*3+1>2}, {% 8 *1+5<1,*2+2>1}, {% 9 1+3<1}, {% 10 1+4<1} }%%% \iflie@hasse@threeD% \begin{pgfonlayer}{Dynkin middle}% \fill[/Lie Hasse diagram/top] (2;1) -- (5;3) -- (2;5) -- (1;5) -- (2;4) -- (1;3) -- (2;2) -- (1;2) -- cycle;% \fill[/Lie Hasse diagram/top] (2;3) -- (4;1) -- (6;2) -- (4;5) -- cycle;% \fill[/Lie Hasse diagram/right] (1;3) -- (2;3) -- (4;5) -- (3;5) -- cycle;% \fill[/Lie Hasse diagram/left] (3;1) -- (4;1) -- (2;3) -- (1;3) -- cycle;% \fill[/Lie Hasse diagram/left] (4;3) -- (5;1) -- (6;1) -- (5;2) -- cycle;% \fill[/Lie Hasse diagram/right] (4;3) -- (5;2) -- (6;3) -- (5;4) -- cycle;% \fill[/Lie Hasse diagram/top] (5;2) -- (7;1) -- (9;1) -- (7;3) -- cycle;% \end{pgfonlayer}% \fi% }% \NewDocumentCommand\hasse@E@seven@width@at@grade{mm}% {% \ifnum#1=1% #2 7% \else\ifnum#1=2% #2 6% \else\ifnum#1=3% #2 6% \else\ifnum#1=4% #2 6% \else\ifnum#1=5% #2 6% \else\ifnum#1=6% #2 5% \else\ifnum#1=7% #2 5% \else\ifnum#1=8% #2 4% \else\ifnum#1=9% #2 4% \else\ifnum#1=10% #2 3% \else\ifnum#1=11% #2 3% \else\ifnum#1=12% #2 2% \else\ifnum#1=13% #2 2% \else\ifnum#1<1% #2 0% \else\ifnum#1>17% #2 0% \else% #2 1% \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi% }% \NewDocumentCommand\hasse@E@seven{}% {% \hasse@X% {{0,2,4,6,7:1,8,10},%1 {1,3,5,6:1,7,9},%2 {2,4,5:1,6,7:1,8},%3 {3,4:1,5,6:1,7,8:1},%4 {3:1,4,5:1,6:2,6,7:1},%5 {4:1,5:2,5,6:1,7:2},%6 {4:2,5:3,5:1,6:2,8:2},%7 {4:3,5:2,6:3,7:2},%8 {3:3,5:3,6:2,7:3},%9 {4:3,6:3,8:3},%10 {5:3,7:3,9:3},%11 {6:3,8:3},%12 {5:3,7:3},%13 {6:3},%14 {6:4},%15 {6:5},%16 {6:6}%17 } {%%% grades: {% 1 *1+2<1,*2+1<1,*2+3>2,*3+2<2,*3+4>3,*4+3<3,*4+5<4,4+6>5,*5+4<4,*6+4<5,*6+7>6,*7+6>6}, {% 2 *1+3<1,*2+1<1,*2+4<2,*3+2<2,*3+5<3,3+6<4,*4+3<3,*4+6>5,5+3>4,*5+5>5,*5+7>6,*6+4>6}, {% 3 *1+4<1,*2+1<1,*2+5<2,2+6<3,*3+2<2,*3+6<4,4+2<3,4+5<4,4+7<5,*5+3<4,*5+7>6,6+3>5,*6+5>6}, {% 4 *1+5<1,1+6<2,*2+1<1,*2+6<3,3+1<2,3+5<3,3+7<5,*4+2<3,*4+4>4,5+2<5,*4+7<6,5+5<6,*6+3>6}, {% 5 *1+6<1,2+5<1,2+7>3,*3+1<1,*3+4<2,5+1<3,3+7<4,*4+2<2,*4+7>5,5+5<4,6+2>4,*6+4>5}, {% 6 *1+4<1,1+7<3,*2+1<1,*2+3>2,*2+7>4,3+5<3,4+1>3,4+4>4,*5+2>4,*5+6>5}, {% 7 *1+3<1,1+7<2,*2+1<1,3+4<2,4+1>2,*4+3>3,*2+7>3,*4+6>4,*5+2>4}, {% 8 *1+2<1,*1+7<2,2+3<2,2+6<3,*3+1<2,*3+6<4,4+1>3,*4+3>4}, {% 9 *1+7<1,*2+2<1,*2+6<2,3+3<2,*4+1>2,*4+4>3}, {% 10 *1+6<1,*2+2<1,*2+4<2,*3+1<2,*3+5>3}, {% 11 *1+4<1,*2+2<1,*2+5<2,*3+1<2}, {% 12 *1+3<1,*1+5<2,*2+2<2}, {% 13 *1+5<1,*2+3<1}, {% 14 1+4<1}, {% 15 1+6<1}, {% 16 1+7<1}, }%%% \iflie@hasse@threeD% \begin{pgfonlayer}{Dynkin middle}% \fill[/Lie Hasse diagram/top] (2;1) -- (6;3) -- (2;6) -- (1;6) -- (2;5) -- (1;4) -- (2;3) -- (1;3) -- (2;2) -- (1;2) -- cycle;% \fill[/Lie Hasse diagram/top] (5;1) -- (7;3) -- (4;6) -- (2;4) -- cycle;% \fill[/Lie Hasse diagram/top] (7;1) -- (9;3) -- (7;5) -- (5;4) -- cycle;% \fill[/Lie Hasse diagram/top] (9;1) -- (12;1) -- (13;1) -- (14;1) -- (11;3) -- (7;2) -- cycle;% \fill[/Lie Hasse diagram/right] (1;4) -- (2;4) -- (4;6) -- (3;6) -- cycle;% \fill[/Lie Hasse diagram/left] (4;1) -- (5;1) -- (2;4) -- (1;4) -- cycle;% \fill[/Lie Hasse diagram/left] (4;4) -- (6;1) -- (8;1) -- (7;2) -- (6;2) -- (5;4) -- cycle;% \fill[/Lie Hasse diagram/right] (4;4) -- (5;4) -- (6;5) -- (5;6) -- cycle;% \fill[/Lie Hasse diagram/right] (6;2) -- (7;2) -- (9;4) -- (8;4) -- cycle;% \end{pgfonlayer}% \fi% }% \NewDocumentCommand\hasse@E@eight@width@at@grade{mm}% {% \ifnum#1=1% #2 8% \else\ifnum#1=2% #2 7% \else\ifnum#1=3% #2 7% \else\ifnum#1=4% #2 7% \else\ifnum#1=5% #2 7% \else\ifnum#1=6% #2 7% \else\ifnum#1=7% #2 7% \else\ifnum#1=8% #2 6% \else\ifnum#1=9% #2 6% \else\ifnum#1=10% #2 6% \else\ifnum#1=11% #2 6% \else\ifnum#1=12% #2 5% \else\ifnum#1=13% #2 5% \else\ifnum#1=14% #2 4% \else\ifnum#1=15% #2 4% \else\ifnum#1=16% #2 4% \else\ifnum#1=17% #2 4% \else\ifnum#1=18% #2 3% \else\ifnum#1=19% #2 3% \else\ifnum#1=20% #2 2% \else\ifnum#1=21% #2 2% \else\ifnum#1=22% #2 2% \else\ifnum#1=23% #2 2% \else\ifnum#1<1% #2 0% \else\ifnum#1>29% #2 0% \else% #2 1% \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi% }% \NewDocumentCommand\hasse@E@eight{}% {% \hasse@X{% {0,2,4,6,8,9:1,10,12},%1 {1,3,5,7,8:1,9,11},%2 {2,4,6,7:1,8,9:1,10},%3 {3,5,6:1,7,8:1,9,10:1},%4 {4,5:1,6,7:1,8:2,8,9:1},%5 {4:1,5,6:1,7:2,7,8:1,9:2},%6 {5:1,6:2,6,7:3,7:1,8:2,10:2},%7 {5:2,6:3,6:1,7:2,8:3,9:2},%8 {5:3,6:4,6:2,7:3,8:2,9:3},%9 {5:4,6:3,7:4,7:2,8:3,10:3},%10 {4:4,6:4,7:3,8:4,9:3,11:3},%11 {5:4,7:4,8:3,9:4,10:3},%12 {6:4,8:4,9:5,9:3,10:4},%13 {7:4,8:5,9:4,10:5},%14 {7:5,8:4,9:5,11:5},%15 {6:5,8:5,10:5,12:5},%16 {7:5,9:5,11:5,13:5},%17 {8:5,10:5,12:5},%18 {7:5,9:5,11:5},%19 {8:5,10:5},%20 {7:5,9:5},%21 {6:5,8:5},%22 {7:5,9:5},%23 {8:5},%24 {8:6},%25 {8:7},%26 {8:8},%27 {8:9},%28 {8:10}%29 }% {%%% grades: {% 1 *1+2<1,*2+1<1,*2+3>2,*3+2<2,*3+4>3,*4+3<3,*4+5>4,*5+4<4,*5+6<5,5+7>6,*6+5>5,*7+5<6,*7+8>7,8+7>7}, {% 2 *1+3<1,*2+1<1,*2+4<2,*3+2<2,*3+5<3,*4+3<3,*4+6<4,4+7>5,*5+4<4,*5+7<6,6+4<5,*6+6>6,*6+8>7,*7+5>7}, {% 3 *1+4<1,*2+1<1,*2+5<2,*3+2<2,*3+6<3,3+7>4,*4+3<3,*4+7<5,5+3<4,5+6>5,5+8>6,*6+4>5,*6+8<7,7+4>6,*7+6>7}, {% 4 *1+5<1,*2+1<1,*2+6<2,2+7>3,*3+2<2,*3+7<4,4+2<3,4+6<4,4+8<6,*5+3<4,*5+5>5,*5+8<7,6+3>6,6+6>7,*7+4>7}, {% 5 6+2<5,*1+6<1,1+7<2,*2+1<1,*2+7<3,3+1<2,3+6<3,3+8<5,*4+2<3,*4+5<4,4+8<6,*5+3<4,*5+8>7,6+6<6,7+3>6,*7+5>7}, {% 6 *1+7<1,2+6<1,2+8<3,*3+1<1,*3+5<2,*4+2<2,3+8<5,*4+4<4,*4+8>6,5+1<3,5+6<5,6+2>5,6+5>6,*7+3>6,*7+7>7}, {% 7 5+1<3,*1+5<1,1+8<3,*2+1<1,*2+4<2,2+8<4,3+6<3,*4+2<2,*4+8>5,5+5<4,6+2>4,*6+4>5,*6+7>6,*7+3>6}, {% 8 *1+4<1,1+8<3,*2+1<1,*2+3>2,*2+8>4,3+5<3,4+1<3,4+4<4,4+7<5,*5+2<4,*5+7<6,*6+4>6,6+2>5}, {% 9 *1+3<1,1+8<2,*2+1<1,*2+8<3,3+4<2,3+7<4,*4+3>3,4+1<2,*4+7<5,5+1>4,5+4>5,*6+2>5,*6+5>6}, {% 10 *1+2<1,*1+8<2,2+3<2,2+7<3,*3+1<2,*3+7>4,4+4<3,5+1>3,*5+3>4,*5+5>5,*6+2<5,*6+6>6}, {% 11 *1+8<1,*2+2<1,*2+7<2,3+3<2,3+5>3,*4+1<2,*4+5<4,*5+3>4,5+1>3,*5+6>5,*6+2>5}, {% 12 *1+7<1,*2+2<1,*2+5<2,3+3<2,3+6<4,*4+1<2,*4+4>3,*4+6<5,5+1>4,*5+3>5}, {% 13 *1+5<1,*2+2<1,*2+4<2,2+6<3,*3+1<2,*3+6>4,4+3<3,5+1>3,*5+4>4}, {% 14 *1+4<1,1+6<2,*2+2<1,*2+6<3,3+2<2,3+4<3,*4+1<3,*4+5>4}, {% 15 *1+3<1,*1+6<2,2+4<2,*3+2>2,*3+5<3,*4+1<3,*4+7>4}, {% 16 *1+6<1,*2+3<1,*2+5<2,*3+2<2,*3+7<3,*4+1<3,*4+8>4}, {% 17 *1+5<1,*2+3<1,*2+7<2,*3+2<2,*3+8<3,*4+1>3}, {% 18 *1+4<1,*1+7<2,*2+3<2,*2+8<3,*3+2>3}, {% 19 *1+7<1,*2+4<1,*2+8<2,*3+3>2}, {% 20 *1+5<1,*1+8<2,*2+4>2}, {% 21 *1+6<1,*1+8<2,*2+5>2,}, {% 22 *1+8<1,*2+6<1,*2+7>2}, {% 23 *1+7<1,*2+6>1}, {% 24 1+5>1}, {% 25 1+4>1}, {% 26 1+3>1}, {% 27 1+2>1}, {% 28 1+1>1} }%%% \iflie@hasse@threeD% \begin{pgfonlayer}{Dynkin middle}% \fill[/Lie Hasse diagram/top] (2;1) -- (7;3) -- (2;7) -- (1;7) -- (2;6) -- (1;5) -- (2;4) -- (1;4) -- (2;3) -- (1;3) -- (2;2) -- (1;2) -- cycle;% \fill[/Lie Hasse diagram/left] (5;1) -- (6;1) -- (2;5) -- (1;5) -- cycle;% \fill[/Lie Hasse diagram/right] (2;5) -- (4;7) -- (3;7) -- (1;5) -- cycle;% \fill[/Lie Hasse diagram/left] (7;1) -- (10;1) -- (9;2) -- (8;2) -- (7;4) -- (6;4) -- (5;5) -- (4;5) -- cycle;% \fill[/Lie Hasse diagram/right] (4;5) -- (5;5) -- (6;7) -- (5;7) -- cycle;% \fill[/Lie Hasse diagram/right] (6;4) -- (7;4) -- (9;6) -- (8;6) -- cycle;% \fill[/Lie Hasse diagram/right] (8;2) -- (9;2) -- (12;4) -- (13;3) -- (14;4) -- (12;5) -- cycle;% \fill[/Lie Hasse diagram/top] (6;1) -- (8;3) -- (4;7) -- (2;5) -- cycle;% \fill[/Lie Hasse diagram/left] (12;4) -- (14;1) -- (15;1) -- (13;3) -- cycle;% \fill[/Lie Hasse diagram/top] (5;5) -- (8;1) -- (10;4) -- (7;7) -- cycle;% \fill[/Lie Hasse diagram/top] (7;4) -- (9;1) -- (13;4) -- (11;6) -- cycle;% \fill[/Lie Hasse diagram/top] (9;2) -- (11;1) -- (15;2) -- (13;5) -- cycle;% \fill[/Lie Hasse diagram/top] (13;3) -- (16;1) -- (18;1) -- (19;1) -- (20;1) -- (22;1) -- (24;1) -- (23;2) -- (22;2) -- (17;4) -- cycle;% \end{pgfonlayer}% \fi% }% \NewDocumentCommand\hasse@F@width@at@grade{mm}% {% \ifnum#1=1% #2 4% \else\ifnum#1=2% #2 3% \else\ifnum#1=3% #2 3% \else\ifnum#1=4% #2 3% \else\ifnum#1=5% #2 3% \else\ifnum#1=6% #2 2% \else\ifnum#1=7% #2 2% \else\ifnum#1<1% #2 0% \else\ifnum#1>11% #2 0% \else% #2 1% \fi\fi\fi\fi\fi\fi\fi\fi\fi% }% \NewDocumentCommand\hasse@F{}% {% \hasse@X% {{0,2,4,6},{1,3,5},{2,3:1,4},{2:1,3,4:1},{1:1,3:1,5:1},{2:1,4:1},{1:1,3:1},{2:1},{2:2},{2:3},{2:4}}% {% grades: {% 1 1+2<1,*2+3<2,*3+4<3,*2+1>1,*3+2>2,4+3>3}, {% 2 *1+3<1,*2+4<3,*2+3<2,*2+1<1,3+2>3}, {% 3 *1+3<1,1+4<2,*2+4<3,*2+1<1,3+1>2,*3+3>3}, {% 4 *1+2<1,*1+4<2,2+3<2,*3+1>2,*3+4>3}, {% 5 *1+4<1,*2+4<2,*2+2>1,*3+1>2}, {% 6 *1+3<1,*1+4<2,*2+2>2}, {% 7 *1+4<1,*2+3>1}, {% 8 1+3<1}, {% 9 1+2<1}, {% 10 1+1<1} }% \iflie@hasse@threeD% \begin{pgfonlayer}{Dynkin middle}% \fill[/Lie Hasse diagram/top] (2;1) -- (4;2) -- (2;3) -- (1;3) -- (2;2) -- (1;2) -- cycle;% \fill[/Lie Hasse diagram/left] (2;2) -- (3;1) -- (4;1) -- (3;2) -- cycle;% \fill[/Lie Hasse diagram/right] (2;2) -- (3;2) -- (4;3) -- (3;3) -- cycle;% \fill[/Lie Hasse diagram/top] (3;2) -- (5;1) -- (6;1) -- (7;1) -- (8;1) -- (5;3) -- cycle;% \end{pgfonlayer}% \fi% }% \NewDocumentCommand\hasse@G@width@at@grade{mm}% {% \ifnum#1=1% #2 2% \else\ifnum#1<1% #2 0% \else\ifnum#1>5% #2 0% \else% #2 1% \fi\fi\fi% }% \NewDocumentCommand\hasse@G{}% {% \hasse@X{{0,2},{1},{1},{1},{1}}{{1+2<1,2+1>1},{1+1<1},{1+1<1},{1+2<1}}% }% \ExplSyntaxOn \DeclareExpandableDocumentCommand{\lie@hasse@repeat@string}{O{}mm} { \int_compare:nT { #2 > 0 } { #3 \prg_replicate:nn { #2 - 1 } { #1#3 } } } \ExplSyntaxOff \newcommand*\exceptional@root@sum[1]{\csname FOO-#1\endcsname} \newcommand*\defn@root@sum[1]{\expandafter\def\csname FOO-#1\endcsname} \def\@root@sum#1=#2!{\global\defn@root@sum{#1}{#2}}% \newcount\left@part@length% \newcount\middle@part@length% \newcount\right@part@length% \NewDocumentCommand\hasse@A@root@sum{mmm}% {% \left@part@length#2\relax% \advance\left@part@length by -1\relax% \middle@part@length#1\relax% \right@part@length\the\dynkin@rank\relax% \advance\right@part@length by -#1\relax% \advance\right@part@length by -#2\relax% \advance\right@part@length by 1\relax% \ifnum\left@part@length=0% \ifnum\right@part@length=0% \xdef#3{\lie@hasse@repeat@string[.]{\the\middle@part@length}{1}}% \else% \xdef#3{\lie@hasse@repeat@string[.]{\the\middle@part@length}{1}.\lie@hasse@repeat@string[.]{\the\right@part@length}{0}}% \fi% \else% \ifnum\right@part@length=0% \xdef#3{\lie@hasse@repeat@string[.]{\the\left@part@length}{0}.\lie@hasse@repeat@string[.]{\the\middle@part@length}{1}}% \else% \xdef#3{\lie@hasse@repeat@string[.]{\the\left@part@length}{0}.\lie@hasse@repeat@string[.]{\the\middle@part@length}{1}.\lie@hasse@repeat@string[.]{\the\right@part@length}{0}}% \fi% \fi% }% \NewDocumentCommand\hasse@B@root@sum{mmm}% {% \right@part@length#1\relax% \advance\right@part@length by -\the\dynkin@rank\relax% \advance\right@part@length by #2\relax% \advance\right@part@length by -1\relax% \ifnum\right@part@length>0% \left@part@length#2\relax% \advance\left@part@length by -1\relax% \middle@part@length\the\dynkin@rank\relax% \advance\middle@part@length by\the\dynkin@rank\relax% \advance\middle@part@length by -#2\relax% \advance\middle@part@length by -#2\relax% \advance\middle@part@length by 2\relax% \advance\middle@part@length by -#1\relax% \ifnum\left@part@length=0% \ifnum\middle@part@length=0% \xdef#3{\lie@hasse@repeat@string[.]{\the\right@part@length}{2}}% \else% \xdef#3{\lie@hasse@repeat@string[.]{\the\middle@part@length}{1}.\lie@hasse@repeat@string[.]{\the\right@part@length}{2}}% \fi% \else% \ifnum\middle@part@length=0% \xdef#3{\lie@hasse@repeat@string[.]{\the\left@part@length}{0}.\lie@hasse@repeat@string[.]{\the\right@part@length}{2}}% \else% \xdef#3{\lie@hasse@repeat@string[.]{\the\left@part@length}{0}.\lie@hasse@repeat@string[.]{\the\middle@part@length}{1}.\lie@hasse@repeat@string[.]{\the\right@part@length}{2}}% \fi% \fi% \else% \hasse@A@root@sum{#1}{#2}{#3}% \fi% }% \newcount\hasse@C@len% \NewDocumentCommand\hasse@C@root@sum{mmm}% {% \hasse@B@root@sum{#1}{#2}{\hasse@C@root@tmp}% \StrRight{\hasse@C@root@tmp}{1}[\hasse@C@last@char]% \IfStrEq{\hasse@C@last@char}{2}% {% \StrGobbleRight{\hasse@C@root@tmp}{1}[\hasse@C@left@chunk]% \xdef#3{\hasse@C@left@chunk{}1}% }% {% }% }% \newcount\hasse@D@len% \NewDocumentCommand\hasse@D@root@sum{mmm}% {% \hasse@D@len#1\relax% 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\advance\middle@part@length by 2\relax% % g+i-r-2 twos \right@part@length#1\relax% \advance\right@part@length by #2\relax% \advance\right@part@length by -\the\dynkin@rank\relax% \advance\right@part@length by -2\relax% \ifnum\left@part@length>0% \ifnum\right@part@length>0% \xdef#3% {%% \lie@hasse@repeat@string[.]% {\the\left@part@length}{0}.% \lie@hasse@repeat@string[.]% {\the\middle@part@length}{1}.% \lie@hasse@repeat@string[.]% {\the\right@part@length}{2}.1.1% }%% \else% \xdef#3% {%% \lie@hasse@repeat@string[.]% {\the\left@part@length}{0}.% \lie@hasse@repeat@string[.]% {\the\middle@part@length}{1}.1.1% }%% \fi% \else% \ifnum\right@part@length>0% \xdef#3% {%% \lie@hasse@repeat@string[.]% {\the\middle@part@length}{1}.% \lie@hasse@repeat@string[.]% {\the\right@part@length}{2}.1.1% }%% \else% \xdef#3{\lie@hasse@repeat@string[.]{\the\middle@part@length}{1}.1.1}% \fi% \fi% \else% % i-1 zeroes \left@part@length#2\relax% \advance\left@part@length by -1\relax% % 2r-g-2i ones \middle@part@length\the\dynkin@rank\relax% \advance\middle@part@length by \the\dynkin@rank\relax% \advance\middle@part@length by-#1\relax% \advance\middle@part@length by-#2\relax% \advance\middle@part@length by-#2\relax% % g+i-r-1 twos \right@part@length#1\relax% \advance\right@part@length by #2\relax% \advance\right@part@length by -\the\dynkin@rank\relax% \advance\right@part@length by -1\relax% \ifnum\left@part@length>0% \xdef#3% {%% \lie@hasse@repeat@string[.]% {\the\left@part@length}{0}.% \lie@hasse@repeat@string[.]% {\the\middle@part@length}{1}.% \lie@hasse@repeat@string[.]% {\the\right@part@length}{2}.1.1% }%% \else% \xdef#3% {%% \lie@hasse@repeat@string[.]% {\the\middle@part@length}{1}.% \lie@hasse@repeat@string[.]% {\the\right@part@length}{2}.1.1% }%% \fi% \fi% \fi% \fi% }% \newif\ifroot@sum@reset% \root@sum@resettrue% \NewDocumentCommand\initialize@roots@as@sums@table{}% {% \IfStrEqCase{\dynkin@series}% {% {E}% {% \ifnum\dynkin@rank=6% \hasse@set@E@six@root@sums% \else% \ifnum\dynkin@rank=7% \hasse@set@E@seven@root@sums% \else% \hasse@set@E@eight@root@sums% \fi% \fi% }% {F}{\hasse@set@F@root@sums}% {G}{\hasse@set@G@root@sums}% }% \global\root@sum@resetfalse% }% \xdef\my@root@sum{}% \xdef\my@fixed@root@sum{}% \NewDocumentCommand\rootSum{oommm}% {% \IfValueT{#1}% {% \IfStrEq{\dynkin@series}{#1}% {% }% {% \global\root@sum@resettrue% }% }% \IfValueT{#2}% {% \ifnum\dynkin@rank=#2% \else% \global\root@sum@resettrue% \fi% }% \ifroot@sum@reset% \xdef\dynkin@series{#1}% \global\dynkin@rank #2\relax% \initialize@roots@as@sums@table% \fi% \IfStrEqCase{\dynkin@series}% {% {A}{\hasse@A@root@sum{#3}{#4}{\my@root@sum}}% {B}{\hasse@B@root@sum{#3}{#4}{\my@root@sum}}% {C}{\hasse@C@root@sum{#3}{#4}{\my@root@sum}}% {D}{\hasse@D@root@sum{#3}{#4}{\my@root@sum}}% {E}{\xdef\my@root@sum{\exceptional@root@sum{#3;#4}}}% {F}{\xdef\my@root@sum{\exceptional@root@sum{#3;#4}}}% {G}{\xdef\my@root@sum{\exceptional@root@sum{#3;#4}}}% }% \StrSubstitute{\my@root@sum}{.}{,}[#5]% }% %% \rootSystemHeight[G][2]{\h} sets the counter \h to the height of the poset %% of the root system G2. \rootSystemHeight{\h} uses the current root system. \NewDocumentCommand\rootSystemHeight{oom}% {% \global\root@sum@resetfalse% \IfValueT{#1}% {% \IfStrEq{\dynkin@series}{#1}% {% }% {% \global\root@sum@resettrue% }% }% \IfValueT{#2}% {% \ifnum\dynkin@rank=#2% \else% \global\root@sum@resettrue% \fi% }% \ifroot@sum@reset% \xdef\dynkin@series{#1}% \global\dynkin@rank #2\relax% \initialize@roots@as@sums@table% \fi% \IfStrEqCase{\dynkin@series}% {% {A}{\global#3\the\dynkin@rank}% {B}% {%% \global#3\the\dynkin@rank\relax% \global\advance#3 by \the\dynkin@rank\relax% \global\advance#3 by -1\relax% }%% {C}% {%% \global#3\the\dynkin@rank\relax% \global\advance#3 by \the\dynkin@rank\relax% \global\advance#3 by -1\relax% }%% {D}% {%% \global#3\the\dynkin@rank\relax% \global\advance#3 by \the\dynkin@rank\relax% \global\advance#3 by -3\relax% }%% {E}% {%% \ifnum\dynkin@rank=6% \global#3 11% \else% \ifnum\dynkin@rank=7% \global#3 17% \else% \global#3 29% \fi% \fi% }%% {F}{\global#3 11}% {G}{\global#3 5}% }% }% %% \rootSystemWidthAtGrade[G][2]{5}{\w} sets the counter \w to the width %% of the 5th grade of the poset of the root system G2. %% \rootSystemWidth{5}{\w} uses the current root system. \NewDocumentCommand\rootSystemWidthAtGrade{oomm}% {% \global\root@sum@resetfalse% \IfValueT{#1}% {% \IfStrEq{\dynkin@series}{#1}% {% }% {% \global\root@sum@resettrue% }% }% \IfValueT{#2}% {% \ifnum\dynkin@rank=#2% \else% \global\root@sum@resettrue% \fi% }% \ifroot@sum@reset% \xdef\dynkin@series{#1}% \global\dynkin@rank #2\relax% \initialize@roots@as@sums@table% \fi% \IfStrEqCase{\dynkin@series}% {% {A}{\hasse@A@width@at@grade{#3}{#4}}% {B}{\hasse@BC@width@at@grade{#3}{#4}}% {C}{\hasse@BC@width@at@grade{#3}{#4}}% {D}{\hasse@D@width@at@grade{#3}{#4}}% {E}% {%% \ifnum\dynkin@rank=6% \hasse@E@six@width@at@grade{#3}{#4}% \else% \ifnum\dynkin@rank=7% \hasse@E@seven@width@at@grade{#3}{#4}% \else% \hasse@E@eight@width@at@grade{#3}{#4}% \fi% \fi% }%% {F}% {% \hasse@F@width@at@grade{#3}{#4}% }% {G}% {% \hasse@G@width@at@grade{#3}{#4}% }% }% }% \NewDocumentCommand\do@hasse@list{m}{\expandafter\@root@sum#1!}% \NewDocumentCommand\hasse@set@E@six@root@sums{}% {% \forcsvlist{\do@hasse@list}{% 1;1=1.0.0.0.0.0,% 1;2=0.1.0.0.0.0,% 1;3=0.0.1.0.0.0,% 1;4=0.0.0.1.0.0,% 1;5=0.0.0.0.1.0,% 1;6=0.0.0.0.0.1,% 2;1=1.1.0.0.0.0,% 2;2=0.1.1.0.0.0,% 2;3=0.0.1.1.0.0,% 2;4=0.0.1.0.1.0,% 2;5=0.0.0.0.1.1,% 3;1=1.1.1.0.0.0,% 3;2=0.1.1.1.0.0,% 3;3=0.1.1.0.1.0,% 3;4=0.0.1.1.1.0,% 3;5=0.0.1.0.1.1,% 4;1=1.1.1.1.0.0,% 4;2=1.1.1.0.1.0,% 4;3=0.1.1.1.1.0,% 4;4=0.1.1.0.1.1,% 4;5=0.0.1.1.1.1,% 5;1=1.1.1.1.1.0,% 5;2=0.1.2.1.1.0,% 5;3=1.1.1.0.1.1,% 5;4=0.1.1.1.1.1,% 6;1=1.1.2.1.1.0,% 6;2=1.1.1.1.1.1,% 6;3=0.1.2.1.1.1,% 7;1=1.2.2.1.1.0,% 7;2=1.1.2.1.1.1,% 7;3=0.1.2.1.2.1,% 8;1=1.2.2.1.1.1,% 8;2=1.1.2.1.2.1,% 9;1=1.2.2.1.2.1,% 10;1=1.2.3.1.2.1,% 11;1=1.2.3.2.2.1}% }% \NewDocumentCommand\hasse@set@E@seven@root@sums{}% {% 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