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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 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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It seems we get one from any symmetry of the diagram; are these all of them?

No and no. The $A_n$ diagrams have a diagram symmetry of order $2$ for every $n$, but the induced automorphism of $S_{n+1}$ is inner for every $n$ (it's conjugation by the permutation $k \mapsto n - k$ acting on $\{ 0, 1, 2, \dots n \}$, say). On the other hand, $S_6$ has an exceptional outer automorphism.

Edit: The outer automorphism group of the Coxeter group $E_8$ appears to have order $2$ and is generated by $w \mapsto w_0^{\ell(w)} w$ where $w_0$ is the longest element, which is central. I found this result in this thesis by William Franszen (which has much more information about this problem), which was the second search result I got for "outer automorphisms of coxeter groups."

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  • $\begingroup$ Thanks! If we think of the Coxeter group concretely as transformations of $\mathbb{R}^8$, the only central element is $x \mapsto -x$. Then this outer automorphism must be $w \mapsto \det(w) w$: i.e., it does nothing to even products of reflections through roots, and multiplies odd products by $-1$. Sound right? $\endgroup$
    – John Baez
    Commented Dec 6, 2014 at 2:56
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It's useful to look at a short paper by Tits Sur le groupe des automorphismes de certains groupes de Coxeter, J. Algebra 113 (1988), no. 2, 346-357, available online here. While he doesn't treat arbitrary Coxeter groups he does include the ones of interest to you. However, the results for $E_8$ need to be made more explicit. Probably there is more recent literature coming from the study of the isomorphism problem for Coxeter groups.

In any case, the main point is to look at "commuting" subsets of the simple generators.

The article by Tits is partly based on an earlier one that year in the same journal by L.D. James here.

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