Talk:Weyl group

inconsistent definition
There is something that I do not understand. It looks to me, from the picture, that if I choose the hyperplane v^ to be above the root gamma, then the system of positive roots will change because it will now include -gamma and not gamma anymore. Therefore, the positive roots may change even if v^ stays in the same hyperplane. Isn't then better to say the weyl chambers are those portion of the plane delimited by the half lines generated by two roots?

Alberto

The article is correct. Roots and Weyl chambers/v live in spaces *dual* one to each other. Think of what happens in higher dimension to vectors, roots, hyperplanes...

Weyl group of A2 has only order 6
Root system claims that the Weyl group associated to A_n hat (n+1)! elements. I haven't checked this myself, but at least for n=2 they seem to be right: The hexagon has 3 diagonals meeting at its center. The reflections on these diags genearate the dihedral group of the equilateral triangle. Rgds --Boobarkee (talk) 13:56, 7 July 2009 (UTC)
 * The Weyl group associated to A_n is the symmetric group on n+1 points and has (n+1)! elements. I have no idea what the intro to this article was talking about, but I deleted the incorrect statement. JackSchmidt (talk) 14:10, 7 July 2009 (UTC)

Two different definitions
I am under the impression that there is a definition of the Weyl group of a root system and a definition of a Weyl group of a split reductive algebraic group with a given maximal torus, and the fact that these two coincide is a theorem. I think it would be useful to state both definitions. As it stands, it's a bit strange that the section on Bruhat Decomposition gives a completely different definition of the Weyl group.

Owen Jones (talk) 14:53, 10 November 2009 (UTC)

there is no definition
In this article of wikipedia there is no definition of a weyl group. only different perspectives how to see it. what about N_G(T)/T as definition? —Preceding unsigned comment added by 132.230.30.96 (talk) 14:49, 13 January 2010 (UTC)

Relation to Conformal Group?
There seems to be a relation between Weyl symmetry and Conformal symmetry. Is there also a relation between a Weyl group and a Conformal group? My hunch would be yes. Can't find it in the article, though. 70.247.162.34 (talk) 04:16, 30 September 2014 (UTC)

Is this right?
This definition appears in the section Weyl chambers:

"If $$\Phi\subset V$$ is a root system, we may consider the hyperplane perpendicular to each root $$\alpha$$. Recall that $$\sigma_\alpha$$ denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of $$V$$ generated by all the $$\sigma_\alpha$$'s. The complement of the set of hyperplanes is disconnected, and each connected component is called a Weyl chamber. If we have fixed a particular set Δ of simple roots, we may define the fundamental Weyl chamber associated to Δ as the set of points $$v\in V$$ such that $$(\alpha,v)>0$$ for all $$\alpha\in\Delta$$."

It's the last sentence that seems possibly wrong to me. It does not seem consistent with the illustration that has as caption "The Weyl group of the A2 root system is the symmetry group of an equilateral triangle" (but the illustration seems correct).

Am I missing something here?216.161.117.162 (talk) 16:33, 29 September 2020 (UTC)


 * Probably, but since you don't explain what you think is inconsistent, I don't know how you expect anyone to help you. (The illustration does not illustrate the fundamental chamber.)  The Math Reference Desk would be a better place for questions like this. --JBL (talk) 16:05, 3 October 2020 (UTC)