How to visualize intuitively root systems and Weyl group?

Question

I am just learning about root systems for the first time and I am wondering how people visualize intuitively the notion of a root system and the Weyl group.

Answer 1

I like this question because I would really like to learn about better visualisations than I have ... Come on, people, give answers!

Honestly, most often I just visualise the examples of rank 2 (there are just four up to isomorphism) and see what happens there. For higher rank root systems, I sometimes look at this image of $A_3$, tell myself that it goes on like that (it's kind of a crystal in higher dimensions) and keep in mind that

a) it has incredibly many symmetries: you can reflect and rotate it around in many ways and still get your crystal back: and these rotations and reflections (or almost all of them -- here I remember how some symmetries of $A_2$ are outer) are exactly the Weyl group);

b) "very many" sub-structures of it are again root systems; so roughly, if I restrict my focus on parts of the high-dimensional crystal, these parts are again made of (possibly lower-dimensional) crystals;

c) as a particular instance of b), if I intersect the root system with any plane (through the origin), and look at what happens in that plane: Well, either it contains no roots (so I put it too "skew") or it just contains one root and its negative (so it's still a bit too "skew"), or -- it happens to be one of the rank-2-examples which I can visualise. In the mentioned example of $A_3$, note how in the "horizontal plane", we catch a system of type $A_1 \times A_1$ (four roots forming a cross), whereas there are four "diagonal" planes which each contain a copy of $A_2$ (six roots forming a hexagon). But some roots have to "play a part" in more than one of these copies. -- And now I tell myself that "more complicated variants of this happen in higher dimensions": each root can play parts in several sub-structures, which can look quite different from each other, but somehow they all are part of a big symmetric structure.