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Let $f$ be a Dynkin diagram automorphism. Extend $f$ linearly to the root system $\Delta$.

What is a set of representatives of the orbits of $\Delta$ under $f$ ?

Thanks!

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  • $\begingroup$ It's unclear to me what "orbit" means in this non-group context. Aside from that, the point of the question isn't obvious. For each relevant simple type, it's very easy to see what each such map does, using the explicit lists of roots in Bourbaki (for $E_6$ the table at the end of Springer's 1966 paper in IHES No. 30, available online through www.numdam.org, is convenient). $\endgroup$
    – Jim Humphreys
    Jan 10, 2011 at 17:15
  • $\begingroup$ @Jim: Thanks for the attention. My problem is exactly in understand the term "orbit" in this situation. What is your idea? I know that the explicit answer will not be obvious. But I am happy if someone give me an insight in what it can be! $\endgroup$
    – Binai
    Jan 10, 2011 at 18:17
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    $\begingroup$ "Orbit" should probably understood as the orbit of the group $\langle f\rangle$. $\endgroup$
    – Johannes Hahn
    Jan 10, 2011 at 19:15
  • $\begingroup$ @Chris: As Johannes points out, "orbit" refers to a group action. In any case, the explicit answer to your question requires some case-by-case enumeration and is rather easy, but see the further information given by Hugh Thomas for the folding process. $\endgroup$
    – Jim Humphreys
    Jan 10, 2011 at 23:06

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In a bunch of important cases, the orbits (of the group $\langle f \rangle$) acting on $\Delta$, correspond to roots in another root system. This is the procedure called "folding", which gives a way to reduce non simply-laced root systems to simply-laced ones. For example, $A_{2n-1}$ folds to $C_n$, $D_{n+1}$ folds to $B_n$, $E_6$ folds to $F_4$. $D_4$ folds to $G_2$ (if you take an automorphim of order 3).

If you start with $A_{2n}$, I guess the result is a non-reduced root system (i.e., one in which $\alpha$ and $2\alpha$ can both be roots for some $\alpha$), but I don't have a reference that includes non-reduced root systems handy, so I haven't confirmed that.

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    $\begingroup$ I'm still uncertain what the motivation behind the question is or what the point of giving "orbit" representatives is, but a useful online source giving concrete examples of the folding process is the classification article by Tits in the first part of the 1965 AMS conference volume posted on their online book page at e-math.ams.org. The non-reduced root system is usually labelled $BC_n$. $\endgroup$
    – Jim Humphreys
    Jan 10, 2011 at 23:02

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