orbit of a Dynkin diagram automorphism

Question

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.

Answer 1

Assuming $\Delta$ is an irreducible root system, the set of orbits actually can be parametrised by a (not necessarily reduced) root system, which in a way is a quotient ("folding") of the one you started with. The cases for $f \neq id$ are:

$A_{n\ge 2}$, $n$ even: folded root system of type $BC_{\frac{n}{2}}$.

$A_{n\ge 3}$, $n$ odd: folded root system of type $C_{\frac{n+1}{2}}$.

$D_{n\ge 3}$ (and $f^2=id$ for $n=4$): folded root system of type $B_{n-1}$.

$D_4$ and $f$ of order $3$ or $6$: folded root system of type $G_2$.

$E_6$: folded root system of type $F_4$.

To make clear what I mean, e.g. for $\Delta$ of type $A_2$, you have four orbits: $\beta_1:=\{\alpha_1, \alpha_2\}$, $\beta_2=2\beta_1:=\{\alpha_1+\alpha_2\}$, and "their negatives". For $\Delta$ of type $A_3$, the orbits can be given the structure of a root system of type $C_2$ with $\beta_1:=\{\alpha_1, \alpha_3\}$ the short root and $\beta_2:=\{\alpha_2\}$ the long root of a basis, the two other "positive" orbits being $\beta_2+\beta_1:=\{\alpha_1+\alpha_2, \alpha_2+\alpha_3\}$ and $\beta_2+2\beta_1:=\{\alpha_1+\alpha_2+\alpha_3\}$ etc.

For "folding", cf. the comments to https://mathoverflow.net/a/244895/27465 with further links, as well as e.g. https://mathoverflow.net/q/138083/27465 and https://mathoverflow.net/q/111469/27465. An easy method to find the folded root systems as above is explained in section 3.4.1 of my thesis, but I got it from (whom else but) J. Tits who sketched in section 2.5 of his article in the Boulder proceedings, and explained it a bit more thoruoghly in section 42.3.5 of his book (with R.M.Weiss) on Moufang Polygons.

I would be interested to know what is the background of this question.