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John Baez
John Baez
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I want to know the outer automorphisms of the Weyl group of $\mathrm{E}_8$, if any.

But I might as well ask the question more generally. Suppose we have a Coxeter diagram. This gives a Coxeter group. What are the outer automorphisms of this group? It seems we get one from any symmetry of the diagram; are these all of them?

(If we were forming the simply connected compact Lie group $G$ from a Dynkin diagram, we'd know every outer automorphismsautomorphism of $G$ comes from a symmetry of the diagram. So, I'm hoping this analogous result is true. But maybe it's too bold a generalization; I'll settle for Coxeter diagrams that come from Dynkin diagrams. The Dynkin diagram of $\mathrm{E}_8$ has no symmetries, of course.)

I want to know the outer automorphisms of the Weyl group of $\mathrm{E}_8$, if any.

But I might as well ask the question more generally. Suppose we have a Coxeter diagram. This gives a Coxeter group. What are the outer automorphisms of this group? It seems we get one from any symmetry of the diagram; are these all of them?

(If we were forming the simply connected compact Lie group $G$ from a Dynkin diagram, we'd know every outer automorphisms of $G$ comes from a symmetry of the diagram. So, I'm hoping this analogous result is true. But maybe it's too bold a generalization; I'll settle for Coxeter diagrams that come from Dynkin diagrams.)

I want to know the outer automorphisms of the Weyl group of $\mathrm{E}_8$, if any.

But I might as well ask the question more generally. Suppose we have a Coxeter diagram. This gives a Coxeter group. What are the outer automorphisms of this group? It seems we get one from any symmetry of the diagram; are these all of them?

(If we were forming the simply connected compact Lie group $G$ from a Dynkin diagram, we'd know every outer automorphism of $G$ comes from a symmetry of the diagram. So, I'm hoping this analogous result is true. But maybe it's too bold a generalization; I'll settle for Coxeter diagrams that come from Dynkin diagrams. The Dynkin diagram of $\mathrm{E}_8$ has no symmetries, of course.)

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John Baez
John Baez
  • 22.3k
  • 3
  • 85
  • 170

What are the outer automorphisms of a Coxeter group?

I want to know the outer automorphisms of the Weyl group of $\mathrm{E}_8$, if any.

But I might as well ask the question more generally. Suppose we have a Coxeter diagram. This gives a Coxeter group. What are the outer automorphisms of this group? It seems we get one from any symmetry of the diagram; are these all of them?

(If we were forming the simply connected compact Lie group $G$ from a Dynkin diagram, we'd know every outer automorphisms of $G$ comes from a symmetry of the diagram. So, I'm hoping this analogous result is true. But maybe it's too bold a generalization; I'll settle for Coxeter diagrams that come from Dynkin diagrams.)