# How to descent to smaller groups "by chopping off a node of the Dynkin diagram"?

I read in section 2 of this paper :

"There is a well-defined chain to descent from $E_8$ to smaller groups by chopping off a node of the Dynkin diagram."

What exactly is here referring to here? What is this process called to descent from a group to smaller groups and how does it work?

If you delete a node from a Dynkin diagram associated to a group $G$, you get a group which is a subgroup of $G$. In fact, it is a parabolic subgroup. (I can't come up with a reference for this, but I'm pretty sure that standard texts will state this fact. For example see Carter Simple groups of Lie type or Wilson The finite simple groups.) I don't know if this process has a name, it is a well known fact in the theory of (finite and algebraic) groups of Lie type, though.

For example you have $$A_1<A_2<...<A_7<E_8$$

• Thanks for your answer! One thing that is confusing me is that if we remove a node from the, say $SU(5)$, Dynkin diagram is the result $SU(3)\times SU(2)$ or $SU(3)\times SU(2) \times U(1)$, because so far I have read both and I thought maybe there is some systematic approach that clarifies such issues. The Dynkin diagrams are on the level of Lie algebras, and maybe there is a reason why we always get an extra $U(1)$ if we remove a node on the level of groups?