I was following Humphrey's Lie algebra for study, and came to study of Weyl groups of root systems. The book has stated orders of Weyl groups of exceptional Lie algebras, and there were no comments or exercises on their structures. I searched other books and came to Knapp's Lie algebra some exercises, and finally came to a confusion.

$E_8$ contains roots of $\mathbb{R}^8$ of the form $\pm (e_i \pm e_j)$ and $\frac{1}{2} (c_1e_1 + \cdots + c_8e_8)$ with $c_i\in \{1,-1\}$ for $i=1,2,\cdots,8$ and $c_i=1$ for even number of $i$'s.

$E_6$ contains those roots of $E_8$ which are orthogonal to $e_8+e_7$ and $e_8+e_6$. These are $$\pm \frac{1}{2} (e_8-e_7-e_6 + c_1e_1 +\cdots + c_5e_5), c_i\in \{1,-1\}, c_i=1 \mbox{ for odd no. times}.$$ and $\pm (e_i\pm e_j)$ for $i,j$ distinct from $1,2,...,5$.

Then Knapp asks for following:

Consider roots of $E_6$ orthogonal to $\frac{1}{2}(e_8-e_7-e_6+e_5+e_4+e_3+e_2+e_1).$ Show that they form a root system of type $A_5$.

Q.1 My simple question is whether it should be $A_5$ or $A_4$? Because, the answer (I think) to question gives roots $$\pm (e_i-e_j) , \hskip5mm i,j=1,2,3,4,5, \mbox{ and } i\neq j.$$

I confused, whether answer is incorrect or it is notational difference of $A_n$ in Humphrey's and Kanpps books? (According to Humphrey's $A_n$ contains roots of $\mathbb{R}^{n+1}$ of form $e_i-e_j$.)

Q.2 Where can I see structure description of Weyl groups of exceptional root systems? I was following Bourbaki's Algebra 4-6, in that some order formula for Weyl group is given. But I want to study structures also. Can you suggest some reference for it?


2 Answers 2


[Answer restated for clarity:]

Though it's a little awkward to combine two separate questions in one posting, the answer to Q2 is somewhat scattered in the literature due to the fact that exceptional Weyl groups tend to show up in various different places. A short survey, with references, is given in Section 2.12 of my 1990 Cambridge book Reflection Groups and Coxeter Groups. For example, the Atlas of Finite Groups has some relevant entries cited.

To answer Q1, Knapp's conclusion is correct, though I don't have his book at hand: the subsystem has type $A_5$. The confusion comes from the fact that the standard description of the root system $A_n$ starts with a euclidean space of dimension $n+1$ and then constructs the desired root system on a hyperplane. So the usual notation for roots is not directly comparable to that used for $E_8$ or its subsystem $E_6$.

Note that the given root for $E_6$ in the shaded box is just the highest root relative to the usual system of simple roots; this is called $\tilde{\alpha}$ by Bourbaki. Then it's clear from a look at the extended Dynkin diagram how to characterize the roots orthogonal to this one (or its negative) as a subsystem of type $A_5$.

Concerning Victor's answer to Q1, it would have prevented some confusion if he had tried to give a reference for his "general principles" remark.

[Small linguistic comment: When an English name like Jones or Humphreys ends in the letter 's', it's always tricky to follow the usual rule about forming a possessive. Probably the best solution is just to add a single mark such as Jones' (but sometimes people write Jones's instead). It's all the fault of our ancestors, who were mostly illiterate anyway.]


${\bf Q1}$ From the general principles [the following claim is false, as pointed out by Dave Witte Morris in the comments], the subsystem $R_{\alpha}^{\perp}=\{\gamma\in R: (\gamma, \alpha)=0\}$ of a rank $d$ root system $R$ formed by the roots orthogonal to $\alpha\in R$ is a root system of rank $d-1$. Thus if $R=E_6$, which has rank 6, the corresponding subsystem has rank 5, so it could not be $A_4$, which has rank 4. You must be missing some roots.

${\bf Q2}$ Exceptional root systems and Weyl groups of type $E_n, n=6,7,8$ play an important role in the theory of del Pezzo surfaces. One classic source is Manin's book "Cubic forms" (nonsingular cubic surfaces in ${\Bbb P}^3$ are del Pezzo surfaces of degree 3, with 27 lines corresponding to certain weights of $E_6$, the corresponding Weyl group action, etc). You should have no problem finding plenty of other resources.

  • 1
    $\begingroup$ I don't agree with (or misunderstand) your claim that $R_\alpha^\perp$ is always a root system of rank $d - 1$. If $R$ is of type $A_2$, then $R_\alpha^\perp$ is the empty set, which is not a root system of rank one. $\endgroup$ Jan 28, 2017 at 17:37
  • $\begingroup$ And you are right! Thank you for pointing it out, I'll amend the answer. $\endgroup$ Jan 28, 2017 at 18:50
  • $\begingroup$ It isn't hard to see that $R^\perp_\alpha$ is of rank $d-1$ except in type $A_d$ ($d\ge 2$), in which case it has rank $d-2$. $\endgroup$ Jan 30, 2017 at 1:54
  • $\begingroup$ In fact, Jim Humphreys' argument with affine Dynkin diagrams shows it for the long roots. It still requires some case-by-case analysis, though. $\endgroup$ Jan 30, 2017 at 2:53

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