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Let $\Phi$ be a root system. In a paper I'm writing, I need to work with subsets $\Phi' \subset \Phi$ satisfying the following two conditions:

  1. For all $\lambda_1,\lambda_2 \in \Phi'$ and $c_1,c_2 \geq 0$ such that $c_1 \lambda_1 + c_2 \lambda_2 \in \Phi$, we have $c_1 \lambda_1 + c_2 \lambda_2 \in \Phi'$.

  2. For all $\lambda \in \Phi'$, we have $-\lambda \notin \Phi'$.

One example would be a choice of positive roots.

Is there a term for such subsets? I'm not an expert in root systems or Lie theory, so I might be missing something obvious.

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  • $\begingroup$ This condition seems natural, though I don't know a term; but, to clarify, are your $c$s integers, or arbitrary rationals? $\endgroup$
    – LSpice
    Commented Oct 3, 2023 at 20:24
  • $\begingroup$ 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. $\endgroup$
    – Sam Hopkins
    Commented Oct 3, 2023 at 20:24
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    $\begingroup$ 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. $\endgroup$
    – LSpice
    Commented Oct 3, 2023 at 23:17
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    $\begingroup$ @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. $\endgroup$
    – Eric
    Commented Oct 3, 2023 at 23:57
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    $\begingroup$ 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. $\endgroup$
    – LSpice
    Commented Oct 4, 2023 at 0:09

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