2

If W is the Weyl group of some ADE-type Lie algebra, and w is an element corresponding to a reflection (not just an involution), does it necessarily correspond to a root?

Matt Samuel
  • 58,164
woody
  • 23
  • for further reading you may be interested in "Introduction to Lie algebras and representation theory" or "Reflection groups and Coxeter groups" by Humphreys. – Matt Samuel Dec 29 '14 at 18:51

1 Answers1

3

Choose a base of simple roots $x_1,x_2,\ldots,x_n$ corresponding to simple reflections $s_1,s_2,\ldots,s_n$. If $t$ is a reflection, then there exists a $w$ in the Weyl group and an $i$ such that $t=ws_iw^{-1}$ . Then the root corresponding to this reflection is $w(x_i)$. This construction may give either a positive or a negative root, but if $w$ is if minimal length then the root will be positive. The correspondence goes both ways, as there is a bijection between reflections and positive roots.

Matt Samuel
  • 58,164
  • Hi Matt, thanks for the answer, but this leads me to ask: why should any two reflections be conjugate? (I guess this is some Coxeter basics?) – woody Dec 29 '14 at 18:24
  • @woody I figured from the question that you were defining reflections as the elements conjugate to simple reflections. For simply laced irreducible root systems it's true that any two reflections are conjugate, but that isn't true in general. In general we can say that reflections are conjugate to simple reflections. It's standard theory that the only reflections are the reflections coming from roots. – Matt Samuel Dec 29 '14 at 18:38
  • @RosaAmbre Not sure I understand your question. Irreducible just means the graph is connected, there's no reason the edges have to be labeled with odd numbers. – Matt Samuel Apr 07 '21 at 20:23
  • @MattSamuel , sorry to be more specific : https://math.stackexchange.com/questions/3411156/what-sort-of-groups-are-generated-by-a-single-conjugacy-class –  Apr 08 '21 at 22:02
  • @Rosa In that question it says the edge labels are odd. That's not required for it to be irreducible. – Matt Samuel Apr 09 '21 at 00:49
  • @MattSamuel can you explain (or share a source for) the "standard theory" fact that the only reflections are reflections corresponding to roots? I don't see how to prove this from sections 9-10 on Humphreys' book on Lie algebras, for example. – Tamar May 04 '22 at 15:34
  • @MattSamuel By reflection I'm thinking of an element of the Weyl group that is also a reflection of Euclidean space—so, something that fixes a hyperplane and sends any vector that's perpendicular to the hyperplane to its negative. Is that what you're referring to in your comment? – Tamar May 04 '22 at 15:43
  • I think I now have a source for this, which I'll leave here in case anyone else is looking in the future: see Theorem 1.14 of "Reflection groups and Coxeter groups" by Humphreys. – Tamar May 04 '22 at 16:34