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author | Karl Berry <karl@freefriends.org> | 2020-02-03 22:30:53 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2020-02-03 22:30:53 +0000 |
commit | c968dc6a6e3b376bbba491ffad313ee8e82dab04 (patch) | |
tree | 11eb84dac11ad9b601a5c77745af1ebcb85dd2af /Master/texmf-dist/doc/latex/lie-hasse | |
parent | 80eeb8a04335539fde455e00d5cfde4e2c7ed5c6 (diff) |
lie-hasse (3feb20)
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diff --git a/Master/texmf-dist/doc/latex/lie-hasse/README b/Master/texmf-dist/doc/latex/lie-hasse/README new file mode 100644 index 00000000000..e63a82cc605 --- /dev/null +++ b/Master/texmf-dist/doc/latex/lie-hasse/README @@ -0,0 +1,21 @@ +___________________________________ + + Lie Hasse + + v1.0 + + 3 February 2020 +___________________________________ + +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 + +---------------------------------------------------------------------- + +This package draws Hasse diagrams of the posets +of the simple roots of any complex simple Lie algebra. +It uses the Dynkin diagrams package dynkin-diagrams. + diff --git a/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.bib b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.bib new file mode 100644 index 00000000000..d39af91eae2 --- /dev/null +++ b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.bib @@ -0,0 +1,495 @@ +% 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 {Langlands:1967, + AUTHOR = {Langlands, Robert P.}, + TITLE = {Euler products}, + NOTE = {A James K. Whittemore Lecture in Mathematics given at Yale + University, 1967, + Yale Mathematical Monographs, 1}, + PUBLISHER = {Yale University Press, New Haven, Conn.-London}, + YEAR = {1971}, + PAGES = {v+53}, + MRCLASS = {10D20 (22E55)}, + MRNUMBER = {0419366}, +MRREVIEWER = {Stephen Gelbart}, +} +@MISC {MathOverflow:123801, +TITLE = {Invariant subbundles of tangent bundle of flag variety (question)}, +AUTHOR = {Benjamin McKay}, +HOWPUBLISHED = {MathOverflow}, +DATE = {2013-03-06}, +NOTE = {URL:\url{http://mathoverflow.net/a/22350} (visited on 2020-01-29)}, +URL = {http://mathoverflow.net/a/123801}, +URLDATE = {2020-01-29}, +} +@phdthesis{Nutma:2010, + author = {Nutma, Teake Aant}, + title = {Kac-Moody symmetries and gauged supergravity}, + school = {Rijksuniversiteit Groningen}, + year = 2010, + address = {Groningen}, + month = 9, + note = {URL:\url{http://inspirehep.net/record/1283406/files/Thesis-2010-Nutma.pdf} (visited on 2020-01-29)} +} +@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} +} +@ARTICLE{Ringel:2013, + author = {Ringel, Claus Michael}, + title = {The root posets and their rich antichains}, + journal = {arXiv e-prints}, + keywords = {Mathematics - Combinatorics, Mathematics - Representation Theory}, + year = "2013", + month = "Jun", + eid = {arXiv:1306.1593}, + pages = {arXiv:1306.1593}, +archivePrefix = {arXiv}, + eprint = {1306.1593}, + primaryClass = {math.CO}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2013arXiv1306.1593R}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} +@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/lie-hasse/lie-hasse.pdf b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.pdf Binary files differnew file mode 100644 index 00000000000..5be5791e7ff --- /dev/null +++ b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.pdf diff --git a/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.tex b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.tex new file mode 100644 index 00000000000..82643fee141 --- /dev/null +++ b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.tex @@ -0,0 +1,555 @@ +\documentclass{amsart} +\title[The Lie Hasse package]{The Lie Hasse package \\ Version 1.0} +%% My name: +\makeatletter +\DeclareRobustCommand{\scotsMc}{\scotsMcx{c}} +\DeclareRobustCommand{\scotsMC}{\scotsMcx{\textsc{c}}} +\DeclareRobustCommand{\scotsMcx}[1]{% + M% + \raisebox{\dimexpr\fontcharht\font`M-\height}{% + \check@mathfonts\fontsize{\sf@size}{0}\selectfont + \kern.3ex\underline{\kern-.3ex #1\kern-.3ex}\kern.3ex + }% +} +\expandafter\def\expandafter\@uclclist\expandafter{% + \@uclclist\scotsMc\scotsMC +} +\makeatother +\newcommand{\authorsname}{\texorpdfstring{Benjamin \scotsMc{}Kay}{Benjamin McKay}} +\author{\authorsname} +\address{School of Mathematical Sciences, University College Cork, Cork, Ireland} +\email{b.mckay@ucc.ie} +\date{3 February 2020} +\usepackage{etex} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenx} +\usepackage{etoolbox} +\usepackage{lmodern} +\RequirePackage[tt=lining]{cfr-lm} +\usepackage[kerning=true,tracking=true]{microtype} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{mathtools} +\usepackage{mathtext} +\usepackage[english]{babel} +\usepackage[pagebackref]{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{lie-hasse} +\usetikzlibrary{positioning} +\usepackage{fancyvrb}\fvset{obeytabs,tabsize=2,fontsize=\small} +\usepackage[listings]{tcolorbox} +\tcbuselibrary{breakable} +\tcbuselibrary{skins} +\usepackage{varwidth} +\usepackage{xspace} +\newcommand{\TikZ}{Ti\textit{k}Z\xspace} +\definecolor{example-color}{gray}{1} +\definecolor{example-border-color}{gray}{.8} +\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, + before upper={\widowpenalties=3 10000 10000 150} +} +\tikzset{ + /Dynkin diagram, + edge length=1cm, + ordering=Carter, + vertical shift=0} +\tikzset{ + background rectangle/.style={ + shade, + top color=olive!20, + bottom color=white, + draw=olive!15, + very thick, + rounded corners}, +} +\begin{document} +\maketitle +\begin{center}% + \begin{tikzpicture}[show background rectangle] + \hasse[ + edge/.style={}, + root radius=.02cm, + edge length=.5cm, + edge quotes/.style={opacity=0}% + ]{E}{8}% + \end{tikzpicture}% +\end{center}% +\begin{center} +\begin{varwidth}{\textwidth} +\tableofcontents +\end{varwidth} +\end{center} +\setlength{\arrayrulewidth}{1.5pt} + +\section{Quick introduction} +This package draws the Hasse diagram of the poset of the positive simple roots of each complex simple Lie group, as drawn by Ringel \cite{Ringel:2013}. +\begin{tcolorbox}[title={Load the package}] +\begin{Verbatim} +\documentclass{article} +\usepackage{lie-hasse} +\begin{document} +The Hasse diagram of \(F_4\) is +\begin{center} +\hasse[edge length=1cm]{F}{4} +\end{center} +\end{document} +\end{Verbatim} +\end{tcolorbox} +\par\noindent{}The Hasse diagram of \(F_4\) is +\begin{center} +\hasse[edge length=1cm]{F}{4} +\end{center} +Each edge is labelled with the simple root by which vertices differ. +\begin{tcblisting}{title={Inside a \TikZ statement}} +\(B_4\) has Dynkin diagram \tikz \dynkin[edge length=.35cm]{B}{4};, Hasse diagram +\begin{center} +\hasse[edge length=1cm]{B}{4} +\end{center} +\end{tcblisting} +\begin{tcblisting}{title={Inside a Dynkin diagram environment, diagrams fit together}} +The Hasse diagram of \(B_4\) is +\begin{dynkinDiagram}[vertical shift=0,edge length=1cm]{B}{4} +\hasse{B}{4} +\end{dynkinDiagram} +\end{tcblisting} +We shut off the default vertical shift of the Dynkin diagram, so that it starts at the origin. +There is an option to \verb!\hasse! for this: +\begin{tcblisting}{title={Attaching the Dynkin diagram}} +The Hasse diagram of \(B_4\) is +\begin{center} +\hasse[attach Dynkin diagram=true]{B}{4} +\end{center} +\end{tcblisting} +Unfortunately, attaching a Dynkin diagram looks terrible for \(D\) or \(E\) series, so a Dynkin diagram appears below. +\begin{tcblisting}{title={Attaching the Dynkin diagram}} +The Hasse diagram of \(D_5\) is +\begin{center} +\hasse[attach Dynkin diagram=true]{D}{5} +\end{center} +\end{tcblisting} +\begin{tcblisting}{title={Inside a \TikZ environment}} +\begin{tikzpicture} +\hasse{A}{4} +\draw (4;1) circle (5pt); +\draw[red] (2;3) circle (5pt); +\end{tikzpicture} +\end{tcblisting} +In this example, we see that the roots of the Hasse diagram are \TikZ{} nodes labelled \(g;i\) for grade \(g\) (i.e. \(g\) units up the page) and index \(i\) (i.e. \(i^{\text{th}}\) root of grade \(g\) drawn on the page, starting from the left). + +\section{Inherited options} +The Lie Hasse package inherits options from the Dynkin diagrams package: the edge lengths are set with +\begin{Verbatim} +\tikzset{/Dynkin diagram/edge lengths=1.2cm} +\end{Verbatim} +and similarly the ordering of roots with +\begin{Verbatim} +\tikzset{/Dynkin diagram/ordering=Bourbaki} +\end{Verbatim} + +\section{Prettier} +The package includes a more elaborate \verb!\hasseDiagrams! command, taking a list of semicolon separated Dynkin diagram identfiers. +\begin{tcolorbox}[title={With some global options to make prettier diagrams}] +\begin{Verbatim} +\tikzset{ + background rectangle/.style={ + shade, + top color=olive!20, + bottom color=white, + draw=olive!15, + very thick, + rounded corners}, + /Lie Hasse diagram, + edge length=1.2cm, + show name=true, + vertical shift=0} +\hasseDiagrams{A4;B4;C4} +\end{Verbatim} +\end{tcolorbox} +\begingroup +\tikzset{ + background rectangle/.style={ + shade, + top color=olive!20, + bottom color=white, + draw=olive!15, + very thick, + rounded corners}, + /Lie Hasse diagram, + edge length=1.2cm, + show name=true, + vertical shift=0} +\hasseDiagrams{A4;B4;C4} +\endgroup +Global options: +\begin{verbatim} + edge/.style={ultra thick}, + edge quotes/.style={/Dynkin diagram/text style,auto,inner sep=2pt}, +\end{verbatim} +allow to change the edges, and to change the way that labels are printed, and how close labels are to the edges. + + + +\section{Root order} +We order the roots as in the Dynkin diagram package: with orderings Adams, Bourbaki, Carter, Dynkin and Kac. +\emph{Warning:} the default is Carter, \emph{not} Bourbaki; the default in the Dynkin diagram package is Bourbaki. +We can use this like: +\begin{Verbatim} +\tikzset{/Lie Hasse diagram,show name=true,show ordering=true} +\hasseDiagrams{[ordering=Adams]E6;[ordering=Bourbaki]E6} +\hasseDiagrams{[ordering=Carter]E6;[ordering=Dynkin]E6} +\hasseDiagrams{[ordering=Kac]E6} +\end{Verbatim} + +\begingroup +\tikzset{/Lie Hasse diagram,show name=true,show ordering=true} +\hasseDiagrams{[ordering=Adams]E6;[ordering=Bourbaki]E6} +\hasseDiagrams{[ordering=Carter]E6;[ordering=Dynkin]E6} +\hasseDiagrams{[ordering=Kac]E6} +\endgroup + +\section{Graph height and width} +The \emph{height} of a Hasse diagram is the number of grades. +The \emph{width} of each grade is the number of vertices on that grade. +We recover these with +\begin{Verbatim} +\newcount\h +\rootSystemHeight[G][2]{\h} +\end{Verbatim} +to store the height of \(G_2\) in a counter called \verb!\h!, and +\begin{Verbatim} +\newcount\w +\rootSystemWidthAtGrade[G][2]{3}{\w}% +\end{Verbatim} +to store the width of \(G_2\) at grade \(3\) in a counter called \verb!\w!. + +Once you use \verb!\dynkin{G}{2}! or \verb!\hasse{G}{2}! or the other commands, like +\begin{Verbatim} +\rootSystemHeight[G][2]{\h} +\end{Verbatim} +the system stores that your default root system is \(G_2\). +Subsequently calls to +\begin{Verbatim} +\rootSystemHeight{\h} +\end{Verbatim} +and +\begin{Verbatim} +\rootSystemWidthAtGrade{3}{\w} +\end{Verbatim} + do not need to specify the root system. + +\begingroup +The \verb!show height! option: +\begin{Verbatim} +\tikzset{/Lie Hasse diagram,show name=true,show height=true} +\hasseDiagrams{G2} +\end{Verbatim} +\tikzset{/Lie Hasse diagram,show name=true,show height=true} +\hasseDiagrams{G2} +The \verb!show widths! option: +\begin{Verbatim} +\tikzset{/Lie Hasse diagram/show widths=true} +\hasseDiagrams{G2} +\end{Verbatim} +\tikzset{/Lie Hasse diagram/show widths=true} +\hasseDiagrams{G2} +\tikzset{/Lie Hasse diagram/show height=false} +\tikzset{/Lie Hasse diagram/show widths=false} +\endgroup + +\section{Root decompositions} +Each positive root in a root system is a unique nonnegative integer linear combination of positive simple roots. +We can recover this expression as +\begin{Verbatim} +\rootSum[G][2]{5}{1}{\rs} +\end{Verbatim} +which, for the root system \(G_2\), and the root at position \(5;1\) in our Hasse diagram, stores in the variable \verb!\rs! a string which looks like \rootSum[G][2]{5}{1}{\rs}\texttt{\rs}. +This is a comma separated list of the integer coefficients. +\emph{Warning:} for the moment, this list of coefficients is in Carter ordering. +If we omit \verb![G][2]!, the current default root system is implied. + +Here is the Dynkin diagram of \(E_8\), indicating the order of the roots in Carter ordering. +\begin{Verbatim} +\dynkin[label,ordering=Carter,edge length=.35cm]{E}{8} +\end{Verbatim} +\begin{center} +\dynkin[label,ordering=Carter,edge length=.35cm]{E}{8} +\end{center} +Here is the same Dynkin diagram, except showing, at each simple root, the coefficient of that simple root in the highest root. +\begin{Verbatim} +\rootSum[E][8]{29}{1}{\rs} +\dynkin[labels=\rs,ordering=Carter,edge length=.35cm]{E}{8} +\end{Verbatim} +\rootSum[E][8]{29}{1}{\rs} +\begin{center} +\dynkin[labels=\rs,ordering=Carter,edge length=.35cm]{E}{8} +\end{center} + +The option \verb!for all roots! allows execution of code once on every root. +\begin{Verbatim} +\tikzset{/Lie Hasse diagram, + edge length=3.2cm, + compact root/.code={}, + noncompact root/.code={}, + edge quotes/.style={opacity=0}, + embedded Dynkin diagram/.style={ + edge length=.4cm, + root radius=.05cm + }, + for all roots/.code 2 args={\drawRootAsDynkinSum{#1}{#2}}} +\hasseDiagrams{D5} +\end{Verbatim} +\begingroup +\tikzset{/Lie Hasse diagram, + edge length=3.2cm, + compact root/.code={}, + noncompact root/.code={}, + edge quotes/.style={opacity=0}, + embedded Dynkin diagram/.style={ + edge length=.4cm, + root radius=.05cm + }, + for all roots/.code 2 args={\drawRootAsDynkinSum{#1}{#2}}} +\hasseDiagrams{D5} +\endgroup +See more below on compact versus noncompact roots; the code \verb!compact! is applied to draw all of the compact roots, and the code \verb!noncompact! to draw the noncompact roots. +Setting those codes to be empty, and setting \verb!edge quotes! to be transparent, we get a much simpler Hasse diagram, so that we can see the embedded Dynkin diagrams more clearly. + +\section{\texorpdfstring{For all roots \ldots}{For all roots ...}} +You can make your own macros loop over all of the roots: you define a macro \verb!\foo{g}{i}!, which is fed the grade \(g\) of each root in the diagram, and the \emph{index} \(i\). +A simple example: +\begin{Verbatim} +\newcommand{\foo}[2]% +{% + \node[below,scale=.5] at (#1;#2) {\(#1,#2\)};% +}% +\end{Verbatim} +\newcommand{\foo}[2]% +{% + \node[below,scale=.75] at (#1;#2) {\(#1,#2\)};% +}% +Inside a \TikZ{} or \verb!dynkinDiagram! environment: +\begin{Verbatim} +\tikzset{/Lie Hasse diagram/edge quotes/.style={opacity=0}, + /Dynkin diagram/edge length=1.5cm} +\begin{tikzpicture} + \hasse{D}{6}% + \forAllPositiveRootsInHasseDiagram{\foo}% +\end{tikzpicture} +\end{Verbatim} +\begingroup +\tikzset{/Lie Hasse diagram/edge quotes/.style={opacity=0}, + /Dynkin diagram/edge length=1.5cm} +\begin{tikzpicture} + \hasse{D}{6}% + \forAllPositiveRootsInHasseDiagram{\foo}% +\end{tikzpicture} + +If you put this into the \verb!for all roots! option, it executes on its own: +\begin{Verbatim} +\tikzset{/Lie Hasse diagram/for all roots/.code 2 args={\foo{#1}{#2}}} +\hasseDiagrams{C4;D4} +\end{Verbatim} +\begingroup +\tikzset{/Lie Hasse diagram/for all roots/.code 2 args={\foo{#1}{#2}}} +\hasseDiagrams{C4;D4} +\endgroup +\endgroup + +\section{Three dimensional effect} +We draw the \(D,E,F\) Hasse diagrams, following Ringel \cite{Ringel:2013}, as an arrangement of cubes. +Nutma \cite{Nutma:2010} draws the Hasse diagrams using a more elementary approach, but including also the affine Kac--Moody algebras. +Opposite sides of any square have the same edge label, by commutativity of addition. +Hence we don't need to see every edge perfectly. +The three dimensional effect is the default: +\begin{Verbatim} +\hasse{D}{4}\hasse{E}{6} +\end{Verbatim} +\begin{center} +\hasse{D}{4}\hasse{E}{6} +\end{center} +We can turn it off: +\begin{Verbatim} +\hasse[three D=false]{D}{4} +\hasse[three D=false]{E}{6} +\end{Verbatim} +\begin{center} +\hasse[three D=false]{D}{4} +\hasse[three D=false]{E}{6} +\end{center} +or globally with \verb!\tikzset{/Lie Hasse diagram/three D=false}!. + +The astute reader will perhaps notice that the three dimensional effect is not realistic. +To be Hasse diagrams, the roots have to line up horizontally by grade. +This is inconsistent with three dimensional projection of our cubes. +We have also tried to use only a small number of layers in the three dimensional geometry, so the images are not perfect, but easy enough to read. + +We can change the \verb!z shift! to slant the three dimensional images to the right: +\begingroup +\begin{Verbatim} +\hasse[z shift=.1]F4\hasse[z shift=.2]F4\hasse[z shift=.3]F4\hasse[z shift=.4]F4 +\end{Verbatim} +\hasse[z shift=.1]F4\hasse[z shift=.2]F4\hasse[z shift=.3]F4\hasse[z shift=.4]F4 +\endgroup + +We only use three colours and opacities for the faces: +\begin{Verbatim} + top/.style={black!20,opacity=.4}, + left/.style={black!20,opacity=.9}, + right/.style={black!20,opacity=.6}, +\end{Verbatim} +You can change these: +\begin{Verbatim} +\hasse[ + top/.style={red,opacity=.1}, + right/.style={red,opacity=.2}, + left/.style={red,opacity=.4}]E6 +\end{Verbatim} +\begin{center} +\hasse[ + top/.style={red,opacity=.1}, + right/.style={red,opacity=.2}, + left/.style={red,opacity=.4}]E6 +\end{center} + +\section{Label the simple roots} +Ringel \cite{Ringel:2013} labels his edges like +\begin{Verbatim} +\hasseDiagrams{[labels={f,e,d,c,u,b,a}]E7} +\end{Verbatim} +\hasseDiagrams{[labels={f,e,d,c,u,b,a}]E7} + +\section{Parabolic subgroups} +This package offers nothing over Ringel's original pictures, except that the user can pick some simple roots whose associated edges are drawn differently. +The chosen simple roots are called \emph{compact}, following terminology from the theory of parabolic subgroups. +We let the reader explore the notation for parabolic subgroups in the Dynkin diagrams package, and use this to declare various roots compact. +\begin{Verbatim} +\tikzset{/Lie Hasse diagram,attach Dynkin diagram=true,three D=false} +\hasseDiagrams{D{**x*x*x*}} +\end{Verbatim} +\begingroup +\tikzset{/Lie Hasse diagram,attach Dynkin diagram=true,three D=false} +\hasseDiagrams{D{**x*x*x*}} +\endgroup +Our motivation comes from trying to identify the invariant vector subbundles of the tangent bundle of a rational homogeneous variety \cite{MathOverflow:123801}. +Such diagrams are often unreadable if we don't turn off the three dimensional graphics. +By default, noncompact root edges are not drawn. +\begingroup +\tikzset{/Lie Hasse diagram,attach Dynkin diagram=true,show name=false,three D=false} +\begin{Verbatim} +\hasseDiagrams{E{*xx*x*}} +\end{Verbatim} +\hasseDiagrams{E{*xx*x*}} +\begin{Verbatim} +\hasseDiagrams{A{x*x*}} +\end{Verbatim} +\hasseDiagrams{A{x*x*}} +\begin{Verbatim} +\hasseDiagrams{[parabolic=113]B8} +\end{Verbatim} +\hasseDiagrams{[parabolic=113]B8} +\begin{Verbatim} +\hasseDiagrams{C{**xx*x**}} +\end{Verbatim} +\hasseDiagrams{C{**xx*x**}} +\newpage +\begin{Verbatim} +\hasseDiagrams{E{*x*x*x**}} +\end{Verbatim} +\hasseDiagrams{E{*x*x*x**}} +\newpage +\begin{Verbatim} +\hasseDiagrams{F{**xx}} +\end{Verbatim} +\hasseDiagrams{F{**xx}} +\begin{Verbatim} +\hasseDiagrams{G{*x}} +\end{Verbatim} +\hasseDiagrams{G{*x}} +\endgroup + +\section{Examples} +\begingroup +\tikzset{/Lie Hasse diagram,attach Dynkin diagram=true,show name=true} +\begin{Verbatim} +\hasseDiagrams{A1;A2;A3;A4;A5;A6} +\hasseDiagrams{B3;B4;B5} +\hasseDiagrams{C2;C3;C4} +\hasseDiagrams{C5;C6} +\hasseDiagrams{E6;E7} +\hasseDiagrams{E8} +\hasseDiagrams{F4;G2} +\end{Verbatim} +\hasseDiagrams{A1;A2;A3;A4;A5;A6} +\hasseDiagrams{B3;B4;B5} +\hasseDiagrams{C2;C3;C4} +\hasseDiagrams{C5;C6} +\hasseDiagrams{E6;E7} +\hasseDiagrams{E8} +\hasseDiagrams{F4;G2} +\endgroup + +\section{Black and white} +Publishing in colour on paper can be expensive. +Simple global options: +\begin{Verbatim} +\tikzset{ + background rectangle/.style={ + shade, + top color=gray!15, + bottom color=white, + draw=gray!5, + very thick, + rounded corners}, + /Dynkin diagram/text style/.style={black,scale=.75}, + /Lie Hasse diagram, + edge length=1cm, + edge/.style={draw=black!50,ultra thick}, + edge quotes/.style={black,auto,inner sep=3pt,scale=.75}, + three D=true, + show name=true} +\end{Verbatim} +\begingroup +\tikzset{ + background rectangle/.style={ + shade, + top color=gray!15, + bottom color=white, + draw=gray!5, + very thick, + rounded corners}, + /Dynkin diagram/text style/.style={black,scale=.75}, + /Lie Hasse diagram, + edge length=1cm, + edge/.style={draw=black!50,ultra thick}, + edge quotes/.style={black,auto,inner sep=3pt,scale=.75}, + three D=true, + show name=true}% +change our examples to +\hasseDiagrams{A1;A2;A3;A4;A5;A6} +\hasseDiagrams{B3;B4;B5} +\hasseDiagrams{C2;C3;C4} +\hasseDiagrams{C5;C6} +\hasseDiagrams{E6;E7} +\hasseDiagrams{E8} +\hasseDiagrams{F4;G2} +\endgroup + +\bibliographystyle{amsplain} +\bibliography{lie-hasse} +\end{document} |