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Let $\Phi$ be an irreducible root system. A root subsystem of $\Phi$ is a subset $\Psi \subseteq \Phi$ which is a root system. One can find the possible types of root subsystems of $\Phi$ by deleting nodes in the extended Dynkin diagram of $\Phi$.

Now suppose that we fix a possible type $T$ (e.g. $A_2$) of root subsystems of $\Phi$. What I'm interested in is the following:

  1. How many root subsystems of type $T$ are there in $\Phi$? One could, of course, brute force this using a computer but I guess there is some kind of formula or more elaborate method?
  2. How do I find all root subsystems of type $T$ in $R$? Again, one could just generate all possible root subsystems and check their type but is there something smarter one could do?

Is there any reference for these types of questions? Has anyone read Carter's Conjugacy classes in the Weyl group and know it well enough to tell if I will find my answers there (I don't know this area well enough to quickly tell by browsing his paper)?

By the way, I'm especially interested in the case of root subsystems of type $D_2$ in $D_5$ and the case of root subsystems of type $A_2 \times A_2 \times A_2$ in $E_6$.

Fat Lip
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  • I have gazed at Carter's article a lot, and it has extensive tables at the end which should answer all your questions. I highly recommend looking through them. There are lists of all possible root subsystems in given systems, size of their conjugacy classes and much more.I was actually hoping somebody would have double checked those tables since the advent of computers. What I know is that every time I though I spotted a mistake in there, it turned out the mistake was on me. – Torsten Schoeneberg Jan 23 '22 at 20:30
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    By the way, I assume it is clear to you that such a subsystem can almost never be unique, because of course if you let any automorphism of the root system act on it, it will send it to a conjugate, isomorphic system, which very rarely is the one you started with, and you know root systems have many automorphisms. – Torsten Schoeneberg Jan 23 '22 at 20:32
  • Also, related: https://math.stackexchange.com/q/3864501/96384 which contains a general method to find certain subsystems (not all by far, but plenty already!) – Torsten Schoeneberg Jan 28 '22 at 18:28
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    Thank you @TorstenSchoeneberg! I have now managed to read the paper enough to understand that (at least) most of what I want is there and I have managed to check that the results agree with my intuition from moduli theory.

    I tried to convince some bachelor students to verify Carter's results with a computer as a thesis project for this spring but no takers so far. I still think it's a great idea for a project so I'll keep this in mind also for the future!

     – Fat Lip Feb 04 '22 at 07:12

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