Timeline for Easy proof that reflections generate $N(T)/T$ for connected compact group?
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14 events
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| Sep 28, 2017 at 3:20 | answer | added | Chris McDaniel | timeline score: 1 | |
| Sep 26, 2017 at 18:31 | comment | added | nfdc23 | A separate concern is that your UC1-UC4 are not sufficiently precise at distinguishing $\mathfrak{g}$ and $\mathfrak{g}_{\mathbf{C}}$ (e.g., the root lines only exist in the latter, as you know), due to which $c$ is really in $\mathbf{C}^{\times}$, so the triple $(I,J,K)$ doesn't seem to have any visible contact with the original $\mathbf{R}$-structure $\mathfrak{g}$ on $\mathfrak{g}_{\mathbf{C}}$. Hence, in UC4 the target is a-priori the complexification of $G$ rather than the original $G$ (and explaining "complexification" intrinsically without more black boxes will be another can of worms). | |
| Sep 26, 2017 at 14:25 | answer | added | Jim Humphreys | timeline score: 2 | |
| Sep 26, 2017 at 13:53 | comment | added | nfdc23 | Without having the SU(2) in hand, how do you know roots occur in opposite pairs with root spaces that are moreover 1-dimensional (presumably you aren't assuming knowledge of semisimple Lie algebras; that would be just replacing the machine of connected compact Lie groups with a different machine)? It remains unclear what exactly you are taking as known to the students, and whether it is worth all of these contortions instead of just telling them some general facts and illustrating those with examples, since it's just for motivation and not logical necessity anyway. | |
| Sep 26, 2017 at 13:21 | comment | added | David E Speyer | UC 1: The Lie algebra $\mathfrak{t}$ of $T$ is a maximal abelian subalgebra. Let $\pm \alpha$ be a root pair and $\mathfrak{g}_{\pm \alpha}$ the corresponding 2-dimensional subspace of $\mathfrak{g}$. Let $J_0$, $K_0$ be an orthogonal basis of $\mathfrak{g}_{\alpha}$ and let $I_0 = [J_0, K_0]/2$. UC 2: $I_0 \neq 0$. Then $[\mathfrak{t}, I_0]=0$ so $I_0 \in \mathfrak{t}$ so $[I_0,J_0] = c K_0$ and $[I_0, K_0] = - c J_0$ for some $c$. UC $3$: $c>0$. Put $(I,J,K) = \sqrt{\tfrac{2}{c}} (I_0, J_0, K_0)$, then $\mathrm{Span}(I,J,K) \cong \mathfrak{su}(2)$. UC 4: This exponentiates to $SU(2) \to G$. | |
| Sep 26, 2017 at 13:11 | comment | added | David E Speyer | I do know that $N(T)/T$ is finite -- $N(T)$ is a compact group and (since $T$ is maximal and $\mathrm{Aut}(T)$ is discrete) $T$ is the connected component of the identity in $N(T)$. I agree that it isn't clear that $N(T)/T \to N(T)/Z(T)$ is an isomorphism, though. I definitely agree that I need to work to get the map $\phi$. I thought I could bring it down to four plausible unproved claims: | |
| Sep 26, 2017 at 13:08 | comment | added | nfdc23 | I think you're sweeping some genuine issues under the rug in connection with compact Lie groups, since the combinatorial setup you're giving yourself is what has to be shown to exist from the connected compact Lie group, etc. Without knowing the equality of $Z_G(T)$ and $T$ it is not clear that $N(T)/T$ is finite, and without some structure theory to make the map $\phi$ one doesn't have the coroots you're giving yourself in the combinatorial setup, etc. And without knowing all such $T$ are $G$-conjugate, crucial motivation is missing. | |
| Sep 26, 2017 at 12:54 | answer | added | David E Speyer | timeline score: 1 | |
| Sep 26, 2017 at 12:41 | comment | added | David E Speyer | I've already classified finite orthogonal reflection groups, so I have the vast bulk of this. (One needs to point out that any finite group preserves an inner product, so once one knows the group is finite those results apply.) | |
| Sep 26, 2017 at 12:40 | comment | added | David E Speyer | @nfdc23 But that's all structure theory of Coxeter groups, I think! All you need is: Let $V$ and $V^{\vee}$ be dual finite dimensional real vector spaces. For $\alpha \in V$ and $\alpha^{\vee} \in V^{\vee}$ with $\langle \alpha, \alpha^{\vee} \rangle = 2$, define a reflection $s_{\alpha, \alpha^{\vee}}$ in $V$ by the usual formulas. Let $\Phi \subset V$ and $\Phi^{\vee} \subset V^{\vee}$ be finite sets equipped with a bijection $\alpha \mapsto \alpha^{\vee}$ such that the $s_{\alpha, \alpha^{\vee}}$ take $\Phi$ to itself. Then $W$ is a finite Coxeter group and $\Phi$ a root system for it. | |
| Sep 26, 2017 at 12:27 | comment | added | nfdc23 | For this definition of $W$ to be appropriate one has to know $Z_G(T)=T$ (and that all $T$ are $G$--conjugate). To make $\phi$ (and show roots come in pairs) uses more serious work with compact Lie groups. You also need that the centralizer in $G$ of any torus (such as the image of a cocharacter of $T$ corresponding to a hypothetical point in an open chamber of $\mathfrak{t}={\rm{X}}_*(T)_{\mathbf{R}}$ fixed by some product of reflections) is connected. It is unclear how much about connected compact Lie groups you take as known. It may be sufficiently instructive to work out some examples. | |
| Sep 26, 2017 at 12:10 | comment | added | nfdc23 | To know that the non-trivial $T$-weights on $\mathfrak{g}$ constitute a root system also requires quite a lot of structure theory, though this is not strictly what you are asking for. | |
| Sep 26, 2017 at 11:12 | comment | added | Robert Bryant | I think that most treatments of root systems define the Weyl group $W$ of a root system $R$ to be the subgroup of $\mathrm{Aut}(R)$ generated by the reflections in the roots. (For example, see Helgason's or Humphreys' treatment of root systems.) The result that $W=N(T)/T$ depends on the connectedness of $G$, so it sounds as though you will at least have to go into enough detail to show how connectedness of $G$ enters the picture. | |
| Sep 26, 2017 at 10:16 | history | asked | David E Speyer | CC BY-SA 3.0 |