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-rw-r--r-- | Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.bib | 451 | ||||
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diff --git a/Master/texmf-dist/doc/latex/rank-2-roots/README b/Master/texmf-dist/doc/latex/rank-2-roots/README new file mode 100644 index 00000000000..66d307c0010 --- /dev/null +++ b/Master/texmf-dist/doc/latex/rank-2-roots/README @@ -0,0 +1,18 @@ +___________________________________ + + Rank 2 roots + + v1.0 + + 30 August 2018 +___________________________________ + +Authors : Ben McKay +Maintainer: Ben McKay +E-mail : b.mckay@ucc.ie +Licence : Released under the LaTeX Project Public License v1.3c or + later, see http://www.latex-project.org/lppl.txt + +---------------------------------------------------------------------- + +For mathematicians. Draws rank 2 root systems, with Weyl chambers, weight lattices, and parabolic subgroups. diff --git a/Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.bib b/Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.bib new file mode 100644 index 00000000000..38a25a6466e --- /dev/null +++ b/Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.bib @@ -0,0 +1,451 @@ +% Encoding: ISO-8859-1 + + +@Book{Adams:1996, + Title = {Lectures on exceptional {L}ie groups}, + Author = {Adams, J. F.}, + Publisher = {University of Chicago Press, Chicago, IL}, + Year = {1996}, + Note = {With a foreword by J. Peter May, + Edited by Zafer Mahmud and Mamoru Mimura}, + Series = {Chicago Lectures in Mathematics}, + + ISBN = {0-226-00526-7; 0-226-00527-5}, + Mrclass = {22-01 (22E10)}, + Mrnumber = {1428422}, + Mrreviewer = {William M. McGovern}, + Owner = {user}, + Pages = {xiv+122}, + Timestamp = {2018.07.22} +} + +@Article{Baba:2009, + Title = {Satake diagrams and restricted root systems of semisimple pseudo-{R}iemannian symmetric spaces}, + Author = {Baba, Kurando}, + Journal = {Tokyo J. Math.}, + Year = {2009}, + Number = {1}, + Pages = {127--158}, + Volume = {32}, + + Fjournal = {Tokyo Journal of Mathematics}, + ISSN = {0387-3870}, + Mrclass = {17B20 (17B22 53C35)}, + Mrnumber = {2541161}, + Mrreviewer = {Oksana S. Yakimova}, + Owner = {user}, + Timestamp = {2017.12.04}, + Url = {https://doi.org/10.3836/tjm/1249648414} +} + +@Book{Bourbaki:2002, + Title = {Lie groups and {L}ie algebras. {C}hapters 4--6}, + Author = {Bourbaki, Nicolas}, + Publisher = {Springer-Verlag, Berlin}, + Year = {2002}, + Note = {Translated from the 1968 French original by Andrew Pressley}, + Series = {Elements of Mathematics (Berlin)}, + + ISBN = {3-540-42650-7}, + Mrclass = {17-01 (00A05 20E42 20F55 22-01)}, + Mrnumber = {1890629}, + Owner = {user}, + Pages = {xii+300}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1007/978-3-540-89394-3} +} + +@Book{Carter:2005, + Title = {Lie algebras of finite and affine type}, + Author = {Carter, R. W.}, + Publisher = {Cambridge University Press, Cambridge}, + Year = {2005}, + Series = {Cambridge Studies in Advanced Mathematics}, + Volume = {96}, + + ISBN = {978-0-521-85138-1; 0-521-85138-6}, + Mrclass = {17-02 (17B67)}, + Mrnumber = {2188930}, + Mrreviewer = {Stephen Slebarski}, + Owner = {user}, + Pages = {xviii+632}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1017/CBO9780511614910} +} + +@InCollection{Carter:1995, + Title = {On the representation theory of the finite groups of {L}ie + type over an algebraically closed field of characteristic 0 [ + {MR}1170353 (93j:20034)]}, + Author = {Carter, R. W.}, + Booktitle = {Algebra, {IX}}, + Publisher = {Springer, Berlin}, + Year = {1995}, + Pages = {1--120, 235--239}, + Series = {Encyclopaedia Math. Sci.}, + Volume = {77}, + + Doi = {10.1007/978-3-662-03235-0_1}, + Mrclass = {20C33 (20-02 20G05)}, + Mrnumber = {1392478}, + Owner = {user}, + Timestamp = {2018.05.19}, + Url = {https://doi.org/10.1007/978-3-662-03235-0_1} +} + +@Article{Chuah:2013, + Title = {Cartan automorphisms and {V}ogan superdiagrams}, + Author = {Chuah, Meng-Kiat}, + Journal = {Math. Z.}, + Year = {2013}, + Number = {3-4}, + Pages = {793--800}, + Volume = {273}, + + Fjournal = {Mathematische Zeitschrift}, + ISSN = {0025-5874}, + Mrclass = {17B20 (17B40)}, + Mrnumber = {3030677}, + Mrreviewer = {Zi-Xin Hou}, + Owner = {user}, + Timestamp = {2017.12.04}, + Url = {https://doi.org/10.1007/s00209-012-1030-z} +} + +@InCollection{Draper/Guido:2016, + Title = {On the real forms of the exceptional {L}ie algebra {$\mathfrak + e_6$} and their {S}atake diagrams}, + Author = {Draper Fontanals, Cristina and Guido, Valerio}, + Booktitle = {Non-associative and non-commutative algebra and operator + theory}, + Publisher = {Springer, Cham}, + Year = {2016}, + Pages = {211--226}, + Series = {Springer Proc. Math. Stat.}, + Volume = {160}, + + Mrclass = {17B20 (17A75 17B25 17B60)}, + Mrnumber = {3613831}, + Mrreviewer = {Alberto Elduque}, + Owner = {user}, + Timestamp = {2018.04.30} +} + +@Book{Dynkin:2000, + Title = {Selected papers of {E}. {B}. {D}ynkin with commentary}, + Author = {Dynkin, E. B.}, + Publisher = {American Mathematical Society, Providence, RI; International Press, Cambridge, MA}, + Year = {2000}, + Note = {Edited by A. A. Yushkevich, G. M. Seitz and A. L. Onishchik}, + + ISBN = {0-8218-1065-0}, + Mrclass = {01A75 (60Jxx)}, + Mrnumber = {1757976}, + Mrreviewer = {William M. McGovern}, + Owner = {user}, + Pages = {xxviii+796}, + Timestamp = {2017.11.15} +} + +@Article{Dynkin:1952, + Title = {Semisimple subalgebras of semisimple {L}ie algebras}, + Author = {Dynkin, E. B.}, + Journal = {Mat. Sbornik N.S.}, + Year = {1952}, + Note = {Reprinted in English translation in \cite{Dynkin:2000}.}, + Pages = {349--462 (3 plates)}, + Volume = {30(72)}, + + Mrclass = {09.1X}, + Mrnumber = {0047629}, + Mrreviewer = {I. Kaplansky}, + Owner = {user}, + Timestamp = {2017.11.15} +} + +@Article{Frappat/Sciarrino/Sorba:1989, + Title = {Structure of basic {L}ie superalgebras and of their affine extensions}, + Author = {Frappat, L. and Sciarrino, A. and Sorba, P.}, + Journal = {Comm. Math. Phys.}, + Year = {1989}, + Number = {3}, + Pages = {457--500}, + Volume = {121}, + + Fjournal = {Communications in Mathematical Physics}, + ISSN = {0010-3616}, + Mrclass = {17B70 (17A70 17B40)}, + Mrnumber = {990776}, + Mrreviewer = {A. Pianzola}, + Owner = {user}, + Timestamp = {2017.12.18}, + Url = {http://0-projecteuclid.org.library.ucc.ie/euclid.cmp/1104178142} +} + +@Book{Grove/Benson:1985, + Title = {Finite reflection groups}, + Author = {Grove, L. C. and Benson, C. T.}, + Publisher = {Springer-Verlag, New York}, + Year = {1985}, + Edition = {Second}, + Series = {Graduate Texts in Mathematics}, + Volume = {99}, + + ISBN = {0-387-96082-1}, + Mrclass = {20-01 (20B25 20H15)}, + Mrnumber = {777684}, + Owner = {user}, + Pages = {x+133}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1007/978-1-4757-1869-0} +} + +@Book{Helgason:2001, + Title = {Differential geometry, {L}ie groups, and symmetric spaces}, + Author = {Helgason, Sigurdur}, + Publisher = {American Mathematical Society, Providence, RI}, + Year = {2001}, + Note = {Corrected reprint of the 1978 original}, + Series = {Graduate Studies in Mathematics}, + Volume = {34}, + + ISBN = {0-8218-2848-7}, + Mrclass = {53C35 (22E10 22E46 22E60)}, + Mrnumber = {1834454}, + Owner = {user}, + Pages = {xxvi+641}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1090/gsm/034} +} + +@Book{Humphreys:1990, + Title = {Reflection groups and {C}oxeter groups}, + Author = {Humphreys, James E.}, + Publisher = {Cambridge University Press, Cambridge}, + Year = {1990}, + Series = {Cambridge Studies in Advanced Mathematics}, + Volume = {29}, + + ISBN = {0-521-37510-X}, + Mrclass = {20-02 (20F32 20F55 20G15 20H15)}, + Mrnumber = {1066460}, + Mrreviewer = {Louis Solomon}, + Owner = {user}, + Pages = {xii+204}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1017/CBO9780511623646} +} + +@Book{Kac:1990, + Title = {Infinite-dimensional {L}ie algebras}, + Author = {Kac, Victor G.}, + Publisher = {Cambridge University Press, Cambridge}, + Year = {1990}, + Edition = {Third}, + + ISBN = {0-521-37215-1; 0-521-46693-8}, + Mrclass = {17B65 (17B67 17B68 58F07)}, + Mrnumber = {1104219}, + Owner = {user}, + Pages = {xxii+400}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1017/CBO9780511626234} +} + +@Article{Khastgir/Sasaki:1996, + Title = {Non-canonical folding of {D}ynkin diagrams and reduction of affine {T}oda theories}, + Author = {Khastgir, S. Pratik and Sasaki, Ryu}, + Journal = {Progr. Theoret. Phys.}, + Year = {1996}, + Number = {3}, + Pages = {503--518}, + Volume = {95}, + + Fjournal = {Progress of Theoretical Physics}, + ISSN = {0033-068X}, + Mrclass = {81T10 (17B81 58F07 81R10)}, + Mrnumber = {1388245}, + Mrreviewer = {Mehmet Koca}, + Owner = {user}, + Timestamp = {2017.12.18}, + Url = {https://doi.org/10.1143/PTP.95.503} +} + +@Book{OnishchikVinberg:1990, + Title = {Lie groups and algebraic groups}, + Author = {Onishchik, A. L. and Vinberg, {\`E}. B.}, + Publisher = {Springer-Verlag}, + Year = {1990}, + + Address = {Berlin}, + Note = {Translated from the Russian and with a preface by D. A. Leites}, + Series = {Springer Series in Soviet Mathematics}, + + ISBN = {3-540-50614-4}, + Mrclass = {22-01 (17B20 20G20 22E10 22E15)}, + Mrnumber = {91g:22001}, + Mrreviewer = {James E. Humphreys}, + Owner = {user}, + Pages = {xx+328}, + Timestamp = {2017.11.15} +} + +@Book{Onishchik/Vinberg:1990, + Title = {Lie groups and algebraic groups}, + Author = {Onishchik, A. L. and Vinberg, \`E. B.}, + Publisher = {Springer-Verlag, Berlin}, + Year = {1990}, + Note = {Translated from the Russian and with a preface by D. A. Leites}, + Series = {Springer Series in Soviet Mathematics}, + + ISBN = {3-540-50614-4}, + Mrclass = {22-01 (17B20 20G20 22E10 22E15)}, + Mrnumber = {1064110}, + Mrreviewer = {James E. Humphreys}, + Owner = {user}, + Pages = {xx+328}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1007/978-3-642-74334-4} +} + +@Article{Ransingh:2013, + Title = {Vogan diagrams of untwisted affine {K}ac-{M}oody superalgebras}, + Author = {Ransingh, Biswajit}, + Journal = {Asian-Eur. J. Math.}, + Year = {2013}, + Number = {4}, + Pages = {1350062, 10}, + Volume = {6}, + + Fjournal = {Asian-European Journal of Mathematics}, + ISSN = {1793-5571}, + Mrclass = {17B67 (17B05 17B22 17B40)}, + Mrnumber = {3149279}, + Mrreviewer = {Xiangqian Guo}, + Owner = {user}, + Timestamp = {2018.01.11} +} + +@Article{Ransingh:unpub, + Title = {{Vogan diagrams of affine twisted Lie superalgebras}}, + Author = {Ransingh, B.}, + Journal = {ArXiv e-prints}, + Year = {2013}, + + Month = mar, + Pages = {1--9}, + + Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, + Adsurl = {http://adsabs.harvard.edu/abs/2013arXiv1303.0092R}, + Archiveprefix = {arXiv}, + Eprint = {1303.0092}, + Keywords = {Mathematical Physics, Mathematics - Representation Theory}, + Owner = {user}, + Primaryclass = {math-ph}, + Timestamp = {2018.01.11} +} + +@Article{Regelskis/Vlaar:2016, + Title = {{Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type}}, + Author = {{Regelskis}, V. and {Vlaar}, B.}, + Journal = {ArXiv e-prints}, + Year = {2016}, + + Month = feb, + Pages = {1--118}, + + Adsnote = {Provided by the SAO/NASA Astrophysics Data System}, + Adsurl = {http://adsabs.harvard.edu/abs/2016arXiv160208471R}, + Archiveprefix = {arXiv}, + Eprint = {1602.08471}, + Keywords = {Mathematical Physics, Mathematics - Quantum Algebra, Mathematics - Representation Theory, Nonlinear Sciences - Exactly Solvable and Integrable Systems}, + Owner = {user}, + Primaryclass = {math-ph}, + Timestamp = {2017.12.04} +} + +@Book{Satake:1980, + Title = {Algebraic structures of symmetric domains}, + Author = {Satake, Ichir\^o}, + Publisher = {Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J.}, + Year = {1980}, + Series = {Kan\^o Memorial Lectures}, + Volume = {4}, + + Mrclass = {32-02 (17C35 32Mxx 53C35)}, + Mrnumber = {591460}, + Mrreviewer = {S. Murakami}, + Owner = {user}, + Pages = {xvi+321}, + Timestamp = {2017.11.15} +} + +@Book{Springer:2009, + Title = {Linear algebraic groups}, + Author = {Springer, T. A.}, + Publisher = {Birkh\"auser Boston, Inc., Boston, MA}, + Year = {2009}, + Edition = {second}, + Series = {Modern Birkh\"auser Classics}, + + ISBN = {978-0-8176-4839-8}, + Mrclass = {20G15 (14L10)}, + Mrnumber = {2458469}, + Owner = {user}, + Pages = {xvi+334}, + Timestamp = {2018.03.31} +} + +@InCollection{Zuber:1998, + Title = {Generalized {D}ynkin diagrams and root systems and their folding}, + Author = {Zuber, Jean-Bernard}, + Booktitle = {Topological field theory, primitive forms and related topics ({K}yoto, 1996)}, + Publisher = {Birkh\"auser Boston, Boston, MA}, + Year = {1998}, + Pages = {453--493}, + Series = {Progr. Math.}, + Volume = {160}, + + Mrclass = {17B20 (05C25 20F55)}, + Mrnumber = {1653035}, + Mrreviewer = {Saeid Azam}, + Owner = {user}, + Timestamp = {2017.12.18} +} + +@Book{Vinberg:1994, + Title = {Lie groups and {L}ie algebras, {III}}, + Editor = {Vinberg, \`E. B.}, + Publisher = {Springer-Verlag, Berlin}, + Year = {1994}, + Note = {Structure of Lie groups and Lie algebras, A translation of {{\i}t Current problems in mathematics. Fundamental directions. Vol. 41} (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990 [ MR1056485 (91b:22001)], Translation by V. Minachin [V. V. Minakhin], Translation edited by A. L. Onishchik and \`E. B. Vinberg}, + Series = {Encyclopaedia of Mathematical Sciences}, + Volume = {41}, + + ISBN = {3-540-54683-9}, + Mrclass = {22-06 (17-06 22Exx)}, + Mrnumber = {1349140}, + Owner = {user}, + Pages = {iv+248}, + Timestamp = {2017.11.15}, + Url = {https://doi.org/10.1007/978-3-662-03066-0} +} + +@Book{Fulton.Harris:1991, + title = {Representation theory}, + publisher = {Springer-Verlag, New York}, + year = {1991}, + author = {Fulton, William and Harris, Joe}, + volume = {129}, + series = {Graduate Texts in Mathematics}, + isbn = {0-387-97527-6; 0-387-97495-4}, + note = {A first course, Readings in Mathematics}, + doi = {10.1007/978-1-4612-0979-9}, + mrclass = {20G05 (17B10 20G20 22E46)}, + mrnumber = {1153249}, + mrreviewer = {James E. Humphreys}, + pages = {xvi+551}, + url = {https://doi.org/10.1007/978-1-4612-0979-9}, +} + +@Comment{jabref-meta: databaseType:bibtex;} 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 differnew file mode 100644 index 00000000000..b152c8360e3 --- /dev/null +++ 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 new file mode 100644 index 00000000000..742682aecef --- /dev/null +++ b/Master/texmf-dist/doc/latex/rank-2-roots/rank-2-roots.tex @@ -0,0 +1,1243 @@ +\documentclass{amsart} +\usepackage{etex} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenx} + +\title{The Rank 2 Roots Package \\ Version 1.0} +\author{Ben McKay} +\date{30 August 2018} + +\usepackage{etoolbox} +\usepackage{lmodern} +\usepackage[kerning=true,tracking=true]{microtype} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{array} +\usepackage{xparse} +\usepackage{xstring} +\usepackage{longtable} +\usepackage{rank-2-roots} +\usepackage{tikz} +\usepackage[listings]{tcolorbox} +\tcbuselibrary{breakable} +\tcbuselibrary{skins} +\definecolor{example-color}{gray}{.85} +\definecolor{example-border-color}{gray}{.7} +\tcbset{coltitle=black,colback=white,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, +before upper={\widowpenalties=3 10000 10000 150}} +\usepackage[pdftex]{hyperref} +\hypersetup{ + colorlinks = true, %Colours links instead of ugly boxes + urlcolor = black, %Colour for external hyperlinks + linkcolor = black, %Colour of internal links + citecolor = black %Colour of citations +} +\usepackage{booktabs} +\usepackage{colortbl} +\usepackage{varwidth} +\usepackage{dynkin-diagrams} +\usepackage{fancyvrb} +\usepackage{xspace} +\newcommand{\TikZ}{Ti\textit{k}Z\xspace} +\usepackage{filecontents} +\usetikzlibrary{decorations.markings} +\usetikzlibrary{arrows,decorations.pathmorphing,backgrounds,positioning,fit} +\arrayrulecolor{white} +\makeatletter + \def\rulecolor#1#{\CT@arc{#1}} + \def\CT@arc#1#2{% + \ifdim\baselineskip=\z@\noalign\fi + {\gdef\CT@arc@{\color#1{#2}}}} + \let\CT@arc@\relax +\rulecolor{white} +\makeatother + + + + + +\NewDocumentCommand\todo{m}% +{% +\textcolor{blue}{\textit{#1}} +}% + +\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. + + +\section{Root systems} +\NewDocumentCommand\drawroots{m}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\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}}% +\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} +} + \gappto\mytablecontents{\\ \\} +} + +\begin{longtable}{rcm{8cm}} +\caption{The root systems}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + +\section{Weights} +Type \verb!\wt{x}{y}! to get a weight at position \((x,y)\) (as measured in a basis of \emph{fundamental weights}). +Type \verb!\wt[multiplicity=n]{x}{y}! to get multiplicity \(m\). +Add an option: \verb!\wt[Z]{x}{y}{m}! to get \verb!Z! passed to TikZ. + + +\RenewDocumentCommand\drawroots{m}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\wt[brown]{1}{0} +\wt[red]{0}{1} +\wt[multiplicity=4,blue]{1}{3} +\wt[blue,multiplicity=2]{2}{2} +\wt[blue]{-1}{3} +\end{rootSystem} +\end{tikzpicture} +}% + +\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{\wt[brown]{1}{0}}}% +\par\noindent% +\texttt{\detokenize{\wt[red]{0}{1}}}% +\par\noindent% +\texttt{\detokenize{\wt[multiplicity=4,blue]{1}{3}}}% +\par\noindent% +\texttt{\detokenize{\wt[blue,multiplicity=2]{2}{2}}}% +\par\noindent% +\texttt{\detokenize{\wt[blue]{-1}{3}}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \xappto\mytablecontents{$\i_2$ & \drawroots{\i} & \csdrawroots{\i} +} + \gappto\mytablecontents{\\ \\} +} + +\begin{longtable}{rcm{8cm}} +\caption{Some weights drawn with multiplicities}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + +\RenewDocumentCommand\drawroots{m}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\wt[multiplicity=2,root]{0}{0} +\end{rootSystem} +\end{tikzpicture} +}% + +\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{\wt[multiplicity=2,root]{0}{0}}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \xappto\mytablecontents{$\i_2$ & \drawroots{\i} & \csdrawroots{\i} +} + \gappto\mytablecontents{\\ \\} +} + +\begin{longtable}{rcm{8cm}} +\caption{The root systems with all multiplicities of the adjoint representation, like Fulton and Harris}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + + + + + + + + + + + + + + + + + + +\RenewDocumentCommand\drawroots{m}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\WeylChamber +\end{rootSystem} +\end{tikzpicture} +}% + +\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{\WeylChamber}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \xappto\mytablecontents{$\i_2$ & \drawroots{\i} & \csdrawroots{\i} +} + \gappto\mytablecontents{\\ \\} +} + +\begin{longtable}{rcm{8cm}} +\caption{Weyl chambers}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + + + + + + + + + + + + + + + + + +\section{Parabolic subgroups} + +\RenewDocumentCommand\drawroots{m}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\positiveRootHyperplane +\end{rootSystem} +\end{tikzpicture} +}% + +\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{\positiveRootHyperplane}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \xappto\mytablecontents{$\i_2$ & \drawroots{\i} & \csdrawroots{\i} +} + \gappto\mytablecontents{\\ \\} +} + +\begin{longtable}{rcm{8cm}} +\caption{The positive root hyperplane}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + + + + + + + + + + + + + + + + + + + + + +\RenewDocumentCommand\drawroots{mm}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\parabolic{#2} +\end{rootSystem} +\end{tikzpicture} +}% + +\RenewDocumentCommand\csdrawroots{mm}% +{% +\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}% +\par\noindent% +\texttt{\detokenize{\begin{rootSystem}}\{#1\}}% +\par\noindent% +\texttt{\detokenize{\roots}}% +\par\noindent% +\texttt{\detokenize{\parabolic}\{#2\}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \foreach \j in {1,2,3}{ + \xappto\mytablecontents{$\i_{2,\j}$ & \drawroots{\i}{\j} & \csdrawroots{\i}{\j} + } + \gappto\mytablecontents{\\ \\} +} +} + +\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. }\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + + + + + + + + + + + + + + + + + + + + + + + + +\RenewDocumentCommand\drawroots{mm}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\parabolic{#2} +\parabolicgrading +\end{rootSystem} +\end{tikzpicture} +}% + +\RenewDocumentCommand\csdrawroots{mm}% +{% +\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}% +\par\noindent% +\texttt{\detokenize{\begin{rootSystem}}\{#1\}}% +\par\noindent% +\texttt{\detokenize{\roots}}% +\par\noindent% +\texttt{\detokenize{\parabolic}\{#2\}}% +\par\noindent% +\texttt{\detokenize{\parabolicgrading}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \foreach \j in {1,2,3}{ + \xappto\mytablecontents{$\i_{2,\j}$ & \drawroots{\i}{\j} & \csdrawroots{\i}{\j} + } + \gappto\mytablecontents{\\ \\} +} +} + +\begin{longtable}{rcm{8cm}} +\caption{Parabolic subgroups with grading of the positive roots}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + +\NewDocumentCommand{\labelWt}{mmmm}% +{% +\node[#1,black] at \weight{#2}{#3} {\(#4\)}; +}% + + +{ +\NewDocumentCommand\labelRoots{}% +{% +\labelWt{above right}{0}{0}{0}% +\labelWt{right}{1}{1}{e_1-e_3}% +\labelWt{right}{2}{-1}{e_1-e_2}% +\labelWt{below}{1}{-2}{e_3-e_2}% +\labelWt{left}{-1}{-1}{e_3-e_1}% +\labelWt{left}{-2}{1}{e_2-e_1}% +\labelWt{above}{-1}{2}{e_2-e_3}% +}% +\setlength{\weightLength}{1cm} +\begin{tikzpicture} +\begin{rootSystem}{A} +\roots +\wt{0}{0} +\labelRoots +\end{rootSystem} +\end{tikzpicture} +} + + +\tikzstyle{weight arrow}=[black,-stealth,shorten <=.25cm,shorten >=.25cm] + +{ +\NewDocumentCommand\wa{O{}mm}% +{% +\IfStrEq{#1}{0}% +{% +\draw[weight arrow] \weight{#2}{#3} -- \weight{#2+1}{#3+1} node[right=-4pt]{\(0\)};% +}% +{% +\draw[weight arrow] \weight{#2}{#3} -- \weight{#2+1}{#3+1};% +}% +}% +\setlength{\weightLength}{.75cm} +\begin{tikzpicture} +\begin{rootSystem}{A} +\setlength{\weightRadius}{1.5pt} +\roots +\wt{0}{0} +\labelWt{above left}{0}{0}{0} +\labelWt{right}{1}{1}{e_1-e_3} +\labelWt{right}{2}{-1}{e_1-e_2} +\labelWt{below}{1}{-2}{e_3-e_2} +\labelWt{left}{-1}{-1}{e_3-e_1} +\labelWt{left}{-2}{1}{e_2-e_1} +\labelWt{above left}{-1}{2}{e_2-e_3} +\wa{0}{0} +\wa[0]{1}{1} +\wa[0]{2}{-1} +\wa[0]{-1}{2} +\wa{1}{-2} +\wa{-1}{-1} +\wa{-2}{1} +\end{rootSystem} +\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} +\roots +\wt[root]{0}{0} +\end{rootSystem} +\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} +\roots +\wt[root]{0}{0} +\positiveRootHyperplane +\end{rootSystem} +\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. +We can also use weight coordinates like +\[ +\verb!\draw \weight{0}{1} -- \weight{1}{0};! +\] +The square system, used like \verb!\draw (square cs:x=1,y=2) circle (2pt);!, is simply the standard Cartesian coordinate system measured so that the minimum distance between weights is one unit. +The hex coordinate system has basis precisely the fundamental weights of the \(A_2\) lattice. +We can use the hex system in drawing on the \(A_2\) or \(G_2\) weight lattices, as below, as they are the same lattices. + +\begin{tcblisting}{title={Automatic sizing of the weight lattice (the default) \dots}} +\begin{tikzpicture} +\begin{rootSystem}{A} +\wt{0}{0} +\fill[gray!50,opacity=.2] (hex cs:x=5,y=-7) -- (hex cs:x=1,y=1) -- (hex cs:x=-7,y=5) arc (150:270:{7*\weightLength}); +\draw[black,very thick] (hex cs:x=5,y=-7) -- (hex cs:x=1,y=1) -- (hex cs:x=-7,y=5); +\node[above right=-2pt] at (hex cs:x=1,y=1) {\small\(\alpha\)}; +\end{rootSystem} +\end{tikzpicture} +\end{tcblisting} + +\begin{tcblisting}{title={\dots and here with manual sizing, setting the weight lattice to include 3 steps to the right of the origin}} +\begin{tikzpicture} +\AutoSizeWeightLatticefalse +\begin{rootSystem}{A} +\wt{0}{0} +\weightLattice{3} +\fill[gray!50,opacity=.2] (hex cs:x=5,y=-7) -- (hex cs:x=1,y=1) -- (hex cs:x=-7,y=5) arc (150:270:{7*\weightLength}); +\draw[black,very thick] (hex cs:x=5,y=-7) -- (hex cs:x=1,y=1) -- (hex cs:x=-7,y=5); +\node[above right=-2pt] at (hex cs:x=1,y=1) {\small\(\alpha\)}; +\end{rootSystem} +\end{tikzpicture} +\end{tcblisting} + +\begin{tcblisting}{title={Fulton and Harris p. 170}} +\begin{tikzpicture} +\begin{rootSystem}{A} +\draw \weight{3}{1} -- \weight{-4}{4.5}; +\foreach \i in {1,...,4}{\wt{5-2*\i}{\i}} +\node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\alpha\)}; +\end{rootSystem} +\end{tikzpicture} +\end{tcblisting} + + + + + +\begin{tcblisting}{title={Automatic sizing of the weight lattice (the default) \dots}} +\begin{tikzpicture} +\begin{rootSystem}{A} +\setlength{\weightRadius}{2pt} +\draw \weight{3}{1} -- \weight{-3}{4}; +\draw \weight{3}{1} -- \weight{4}{-1}; +\wt{4}{-1} +\foreach \i in {1,...,4}{\wt{5-2*\i}{\i}} +\node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\alpha\)}; +\draw[very thick] \weight{0}{-4} -- \weight{0}{4.5} node[above]{\small\(\left<H_{12},L\right>=0\)}; +\draw[very thick] \weight{-4}{0} -- \weight{4.5}{0} node[right]{\small\(\left<H_{23},L\right>=0\)}; +\end{rootSystem} +\end{tikzpicture} +\end{tcblisting} + + +\begin{tcblisting}{title={\dots and manual sizing}} +\begin{tikzpicture} +\AutoSizeWeightLatticefalse +\begin{rootSystem}{A} +\setlength{\weightRadius}{2pt} +\weightLattice{4} +\draw \weight{3}{1} -- \weight{-3}{4}; +\draw \weight{3}{1} -- \weight{4}{-1}; +\wt{4}{-1} +\foreach \i in {1,...,4}{\wt{5-2*\i}{\i}} +\node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\alpha\)}; +\draw[very thick] \weight{0}{-4} -- \weight{0}{4.5} node[above]{\small\(\left<H_{12},L\right>=0\)}; +\draw[very thick] \weight{-4}{0} -- \weight{4.5}{0} node[right]{\small\(\left<H_{23},L\right>=0\)}; +\end{rootSystem} +\end{tikzpicture} +\end{tcblisting} + +\begin{tcblisting}{} +\begin{tikzpicture} +\AutoSizeWeightLatticefalse +\begin{rootSystem}{A} +\setlength{\weightRadius}{2pt} +\weightLattice{4} +\draw \weight{3}{1} -- \weight{-3}{4}; +\draw \weight{3}{1} -- \weight{4}{-1}; +\draw \weight{-3}{4} -- \weight{-4}{3}; +\wt{4}{-1} +\wt{-4}{3} +\foreach \i in {1,...,4}{\wt{5-2*\i}{\i}} +\node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\alpha\)}; +\draw[very thick] \weight{0}{-4} -- \weight{0}{4.5} node[above]{\small\(\left<H_{12},L\right>=0\)}; +\draw[very thick] \weight{-4}{0} -- \weight{4.5}{0} node[right]{\small\(\left<H_{23},L\right>=0\)}; +\draw[very thick] \weight{4}{-4} -- \weight{-4.5}{4.5} node[above]{\small\(\left<H_{13},L\right>=0\)}; +\end{rootSystem} +\end{tikzpicture} +\end{tcblisting} + + +\begin{tcblisting}{} +\setlength{\weightRadius}{2pt} +\setlength\weightLength{.75cm} +\begin{tikzpicture} +\begin{rootSystem}{A} +\foreach \x/\y in {1/0, -1/1, 0/-1, -2/0, 0/2, 2/-2}{\wt{\x}{\y}} +\node[above] at \weight{1}{0} {\small\(L_1\)}; +\node[above] at \weight{-1}{1} {\small\(L_2\)}; +\node[above] at \weight{0}{-1} {\small\(L_3\)}; +\end{rootSystem} +\end{tikzpicture} +\end{tcblisting} + +\begin{tcblisting}{title={Changing the weight length rescales}} +\begin{tikzpicture} +\setlength\weightLength{.3cm} +\begin{rootSystem}{A} +\wt[multiplicity=2]{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} +\roots +\draw[green!50!black,very thick] \weight{0}{1} -- \weight{1}{0}; +\weightLattice{3} +\wt[blue]{1}{0}{1} +\wt[red]{0}{1}{1} +\end{rootSystem} +\end{tikzpicture} +\end{tcblisting} + + + + + + +Without automatic stretching of the weight lattice to fit the picture, you won't see the weight lattice at all unless you ask for it. + +\AutoSizeWeightLatticefalse + + + + + + +\RenewDocumentCommand\drawroots{m}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\end{rootSystem} +\end{tikzpicture} +}% + +\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{} +\foreach \i in {A,B,C,G}{ + \xappto\mytablecontents{$\i_2$ & \drawroots{\i} & \csdrawroots{\i} +} + \gappto\mytablecontents{\\ \\} +} + +\begin{longtable}{rcm{8cm}} +\caption{The root systems}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + +Type \verb!\wt{x}{y}{m}! to get a weight at position \((x,y)\) (as measured in a basis of \emph{fundamental weights}) with multiplicity \(m\). +Add an option: \verb!\wt[Z]{x}{y}{m}! to get \verb!Z! passed to TikZ. + + +\RenewDocumentCommand\drawroots{m}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\wt[brown]{1}{0}{1} +\wt[red]{0}{1}{1} +\wt[blue]{1}{3}{4} +\wt[blue]{2}{2}{2} +\wt[blue]{-1}{3}{1} +\end{rootSystem} +\end{tikzpicture} +}% + +\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{\wt[brown]{1}{0}{1}}}% +\par\noindent% +\texttt{\detokenize{\wt[red]{0}{1}{1}}}% +\par\noindent% +\texttt{\detokenize{\wt[blue]{1}{3}{4}}}% +\par\noindent% +\texttt{\detokenize{\wt[blue]{2}{2}{2}}}% +\par\noindent% +\texttt{\detokenize{\wt[blue]{-1}{3}{1}}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \xappto\mytablecontents{$\i_2$ & \drawroots{\i} & \csdrawroots{\i} +} + \gappto\mytablecontents{\\ \\} +} + +\begin{longtable}{rcm{8cm}} +\caption{Some weights drawn with multiplicities}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + +\RenewDocumentCommand\drawroots{m}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\wt[multiplicity=2]{0}{0} +\end{rootSystem} +\end{tikzpicture} +}% + +\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{\wt[multiplicity=2]{0}{0}}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \xappto\mytablecontents{$\i_2$ & \drawroots{\i} & \csdrawroots{\i} +} + \gappto\mytablecontents{\\ \\} +} + +\begin{longtable}{rcm{8cm}} +\caption{The root systems with all multiplicities of the adjoint representation, like Fulton and Harris}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + + + + + + + + + + + + + + + + + + +\RenewDocumentCommand\drawroots{m}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\WeylChamber +\end{rootSystem} +\end{tikzpicture} +}% + +\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{\WeylChamber}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \xappto\mytablecontents{$\i_2$ & \drawroots{\i} & \csdrawroots{\i} +} + \gappto\mytablecontents{\\ \\} +} + +\begin{longtable}{rcm{8cm}} +\caption{Weyl chambers}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + + + + + + + + + + + + + + + + + + + + + + + + +\RenewDocumentCommand\drawroots{m}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\positiveRootHyperplane +\end{rootSystem} +\end{tikzpicture} +}% + +\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{\positiveRootHyperplane}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \xappto\mytablecontents{$\i_2$ & \drawroots{\i} & \csdrawroots{\i} +} + \gappto\mytablecontents{\\ \\} +} + +\begin{longtable}{rcm{8cm}} +\caption{The positive root hyperplane}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + + + + + + + + + + + + + + + + + + + + + +\RenewDocumentCommand\drawroots{mm}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\parabolic{#2} +\end{rootSystem} +\end{tikzpicture} +}% + +\RenewDocumentCommand\csdrawroots{mm}% +{% +\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}% +\par\noindent% +\texttt{\detokenize{\begin{rootSystem}}\{#1\}}% +\par\noindent% +\texttt{\detokenize{\roots}}% +\par\noindent% +\texttt{\detokenize{\parabolic}\{#2\}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \foreach \j in {1,2,3}{ + \xappto\mytablecontents{$\i_{2,\j}$ & \drawroots{\i}{\j} & \csdrawroots{\i}{\j} + } + \gappto\mytablecontents{\\ \\} +} +} + +\begin{longtable}{rcm{8cm}} +\caption{Parabolic subgroups}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + + + + + + + + + + + + + + + + + + + + + + + + +\RenewDocumentCommand\drawroots{mm}% +{% +\begin{tikzpicture}[baseline=-.5] +\begin{rootSystem}{#1} +\roots +\parabolic{#2} +\parabolicgrading +\end{rootSystem} +\end{tikzpicture} +}% + +\RenewDocumentCommand\csdrawroots{mm}% +{% +\texttt{\detokenize{\begin{tikzpicture}[baseline=-.5]}}% +\par\noindent% +\texttt{\detokenize{\begin{rootSystem}}\{#1\}}% +\par\noindent% +\texttt{\detokenize{\roots}}% +\par\noindent% +\texttt{\detokenize{\parabolic}\{#2\}}% +\par\noindent% +\texttt{\detokenize{\parabolicgrading}}% +\par\noindent% +\texttt{\detokenize{\end{rootSystem}}}% +\par\noindent% +\texttt{\detokenize{\end{tikzpicture}}}% +}% + +\renewcommand*\mytablecontents{} +\foreach \i in {A,B,C,G}{ + \foreach \j in {1,2,3}{ + \xappto\mytablecontents{$\i_{2,\j}$ & \drawroots{\i}{\j} & \csdrawroots{\i}{\j} + } + \gappto\mytablecontents{\\ \\} +} +} + +\begin{longtable}{rcm{8cm}} +\caption{Parabolic subgroups with grading of the positive roots}\\ +\endfirsthead +\caption{\dots continued}\\ +\endhead +\multicolumn{3}{c}{continued \dots}\\ +\endfoot +\endlastfoot +\mytablecontents +\end{longtable} + + + + + +\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} +\foreach \x/\y/\m in +{0/ 1/5, -1/0/5, 1/-1/5, 2/ 0/4, -2/ 2/4, 0/-2/4, + 1/ 2/2, -1/3/2, 3/-2/2, 2/-3/2, -2/-1/2, -3/ 1/2, + 4/-1/1, 3/1/1, -3/ 4/1, -4/ 3/1, -1/-3/1, 1/-4/1} +{\wt[multiplicity=\m]{\x}{\y}} +\end{rootSystem} +\end{tikzpicture} +\end{tcblisting} + + +\begin{tcblisting}{title={A representation of \(G_2\)}} +\setlength\weightLength{1cm} +\begin{tikzpicture} +\begin{rootSystem}{G} +\roots +\foreach \m/\x/\y in { +1/1/1, 1/4/-1, 1/-1/2, 2/2/0, 1/5/-2, +2/0/1, 2/3/-1, 2/-2/2, 4/1/0, 1/-4/3, +2/4/-2, 4/-1/1, 4/2/-1, 2/-3/2, 1/5/-3, +4/0/0, 1/-5/3, 2/3/-2, 4/-2/1, 4/1/-1, +2/-4/2, 1/4/-3, 4/-1/0, 2/2/-2, 2/-3/1, +2/0/-1, 1/-5/2, 2/-2/0, 1/1/-2, 1/-4/1, +1/-1/-1}{\wt[multiplicity=\m]{\x}{\y}} +\positiveRootHyperplane +\WeylChamber +\end{rootSystem} +\end{tikzpicture} +\end{tcblisting} + + +\begin{tcblisting}{title={Dimensions of representations of \(G_2\), parameterized by highest weight}} +\setlength\weightLength{1cm} +\begin{tikzpicture} +\begin{rootSystem}{G} +\roots +\foreach \x/\y/\d in { +0/1/14, 0/2/77, 0/3/273, 1/0/7, 1/1/64, +1/2/286, 2/0/27, 2/1/189, 2/2/729, 3/0/77, +4/0/182, 5/0/318, 6/0/714, 3/1/448, 4/1/924} +{\wt{\x}{\y}\node[black,above] at \weight{\x}{\y} {\(\d\)};} +\positiveRootHyperplane +\WeylChamber +\end{rootSystem} +\end{tikzpicture} +\end{tcblisting} + + +\bibliographystyle{amsplain} +\bibliography{rank-2-roots} + +\end{document} |