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!



orbit of a Dynkin diagram automorphism
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.


