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This is to ask about references of Hasse diagrams of irreducible root systems. I found here and there nice pictures of root systems of type $E$. I would like to ask for Hasse diagrams of classical root systems ($A_n, B_n, C_n, D_n$).

Thanks in advance.

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    $\begingroup$ For type $A_{n-1}$ just take the strictly upper-triangular part of an $n\times n$ matrix and turn it on its side so that the first upper-diagonal is horizontal and the top right-hand corner of the matrix is now at the top. Then draw edges between position $(i,j)$ and positions $(i+1,j)$ and $(i,j+1)$. For the other classical types you can do something very similar. $\endgroup$
    – Paul Levy
    Commented Dec 1, 2015 at 16:03
  • $\begingroup$ Sorry, between position $(i,j)$ and positions $(i-1,j)$ and $(i,j+1)$ (whenever these make sense). $\endgroup$
    – Paul Levy
    Commented Dec 1, 2015 at 19:20
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    $\begingroup$ This paper arxiv.org/abs/1306.1593 by Ringel describes the posets for all the classical types, and has nice pictures for the exceptionals. $\endgroup$
    – Hugh Thomas
    Commented Dec 31, 2015 at 14:56
  • $\begingroup$ @HughThomas Thanks. The paper contains pictures of up to rank-6 classical root systems. $\endgroup$
    – user
    Commented Jan 1, 2016 at 15:19

3 Answers 3

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I put a LaTeX package up on CTAN today to draw the Hasse diagrams of all root systems, following Ringel's pictures. Code is as simple as

\documentclass{amsart}
\usepackage{lie-hasse}
\begin{document}
\hasseDiagrams{A4;B5}
\hasseDiagrams{F4;G2}
\end{document}

to display Four examples of Hasse diagrams of root systems

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  • $\begingroup$ That's awesome! And you also made dynkin-diagram package! That's gonna make my life so much easier! Thanks! $\endgroup$
    – Vít Tuček
    Commented Feb 5, 2020 at 11:07
  • $\begingroup$ Sorry but it seems that your code above doesn't work for me. The console says Argument of \lie@hasse@place@root has an extra }. <inserted tex> \par l.5 \hasseDiagrams{F4;G2} $\endgroup$
    – user
    Commented Mar 30, 2020 at 13:52
  • $\begingroup$ @user: I tried running that code again to check that it works. I don't have a problem with it, but that might have to do with the particular TeX configuration on my machine. I am using pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Debian), LaTeX2e <2018-12-01>. If you try to update your system to the latest version of TeXLive or MikTex, please let me know if you still have trouble. $\endgroup$
    – Ben McKay
    Commented Mar 30, 2020 at 14:42
  • $\begingroup$ I am using TeXShop. First it said the dynkin-diagrams.sty file is missing. I found that such a file can be downloaded here ctan.org/tex-archive/graphics/pgf/contrib/…. I simply put dynkin-diagrams.sty in the same folder with lie-hasse, create a new tex file, copy paste your code and compile. However, still it doesn't produce the desired figures. $\endgroup$
    – user
    Commented Mar 30, 2020 at 15:17
  • $\begingroup$ @user:I don't have a Mac, so I can't try out my package on TeXshop. It seems that TeXshop runs with a basic TeX installation, which you can extend to a full TeXLive installation, using the latest TeXLive version, by use of some Mac installer. If that doesn't help, please let me know. $\endgroup$
    – Ben McKay
    Commented Mar 30, 2020 at 15:43
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Hasse diagrams for root systems can be extracted from the following paper, which can be found on E. Plotkin's publication page:

  • E.Plotkin, A.Semenov, N.Vavilov, Visual basic representations: An atlas, Int.Journal of Algebra and Computation, 8 (1998), no. 1, 61-95.

The paper discusses more generally Hasse diagrams for the weights of various representations and various applications. The pictures for the adjoint representations give the Hasse diagram of the root system (but using all roots, not just the positive ones).

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  • $\begingroup$ I am aware of that paper. It's an excellent reference but containing only Hasse diagrams of weights. I was actually looking for Hasse diagrams which draw positive roots. That helps me easier to read from. $\endgroup$
    – user
    Commented Jan 31, 2016 at 10:58
  • $\begingroup$ Well, I mentioned the paper because the difference between all roots and the positive ones should merely be cutting off "in the middle". $\endgroup$
    – Matthias Wendt
    Commented Feb 1, 2016 at 7:34
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I did some research relating to this area for:

$5$ dimensional space over $\mathbb{R}$

There isn't much to be found on the Internet other than at Wikipedia, except for:

A FAMILY OF MARKED CUBIC SURFACES AND THE ROOT SYSTEM $D_5$, by Elisabetta Colombo and Bert Van Geemen

which is detailed but has no pictures.

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  • $\begingroup$ Thanks Jon. I can find Hasse diagram of $D_5$ in the first reference that I gave at the OP. Type $A_n$ is probably is an easiest object to understand. I need bigger/general pictures for other types. $\endgroup$
    – user
    Commented Dec 1, 2015 at 3:44

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