Timeline for Root system terminology
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Oct 4, 2023 at 12:51 | history | edited | RobPratt |
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| Oct 4, 2023 at 8:46 | history | edited | YCor |
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| Oct 4, 2023 at 0:09 | comment | added | LSpice | Re, excellent! If you're interested beyond the reductive case, it is discussed in Section 3.3 of Conrad, Gabber, and Prasad - Pseudo-reductive groups. Just to have the other link here, as you certainly know, it's Borel - Linear algebraic groups. | |
| Oct 3, 2023 at 23:57 | comment | added | Eric | @LSpice: Ah, that's nice! I had included a proof since I didn't know a reference, but that shortens my paper by a page. | |
| Oct 3, 2023 at 23:17 | comment | added | LSpice | Re, if you're just looking for such a theorem, then the condition with rational $c$ can be weakened just to integral $c$, and it is Proposition 21.9(ii) of Borel. | |
| Oct 3, 2023 at 21:14 | comment | added | Eric | @Lspice: Yes, that does sound like what these probably are. What's useful about them for me is that if $\Phi$ is the (relative) root system of a reductive algebraic group, then my condition ensures that the group generated by the root subgroups corresponding to elements of $\Phi'$ is a unipotent subgroup that can be decomposed as the set-theoretic product of the root subgroups associated to the non-divisible elements of $\Phi'$. Mostly I want to make sure that there isn't a standard name for these, since otherwise I'll look like an idiot if I make one up. | |
| Oct 3, 2023 at 20:35 | comment | added | LSpice | I suspect (still assuming $c$s are integers) that $\Phi'$ arises as follows: replace $\Phi$ by a closed subsystem, take a set of positive roots, and then take the complement of a further closed subsystem. But I'm not certain. (Oh, and I just saw your response. Then probably, instead of closed, take "Levi" (intersection of two opposite parabolic subsets).) | |
| Oct 3, 2023 at 20:27 | comment | added | Eric | @LSpice: They are arbitrary rationals. I'm also not assuming that my root system is reduced, so it's possible that for a root $\lambda$ the element $\frac{1}{2}\lambda$ is also a root. | |
| Oct 3, 2023 at 20:25 | comment | added | LSpice | @SamHopkins, as you doubtless know, if the $c$s are integers, then (1) is equivalent to being closed. | |
| Oct 3, 2023 at 20:24 | comment | added | Sam Hopkins | A subset $C \subseteq \Phi$ is called closed if $\alpha, \beta \in C$ and $\alpha+\beta \in \Phi$ imply $\alpha+\beta \in C$. Not exactly the same as your conditions, but this is an important notion which has been studied a lot. | |
| Oct 3, 2023 at 20:24 | comment | added | LSpice | This condition seems natural, though I don't know a term; but, to clarify, are your $c$s integers, or arbitrary rationals? | |
| Oct 3, 2023 at 20:22 | history | edited | Eric | CC BY-SA 4.0 |
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| S Oct 3, 2023 at 20:21 | review | First questions | |||
| Oct 3, 2023 at 20:56 | |||||
| S Oct 3, 2023 at 20:21 | history | asked | Eric | CC BY-SA 4.0 |