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I am starting to learn reductive groups, and I don't have good knowledge of the root subgroups. For a root $\alpha \in \Phi$, I do know they are define as the (unique?) one-parameter subgroup $x: G_a \to U_\alpha$ such that $$tx(a)t^{-1} = x(\alpha(t) a)$$

If I take two different root subgroups $U_\alpha$ and $U_\beta$, how can I think of the product $U_\alpha U_\beta$?

Consider matrix groups, and I did my examples in $GL(n)$. Does $U_\alpha U_\beta$ correspond to the matrices with stars (nonzero elements) at the "places" indicated by $\alpha$ and $\beta$ (e.g. for $\alpha = \epsilon_1 - \epsilon_2$ and $\beta = \epsilon_1 - \epsilon_3$) I would see that $U_\alpha$ are the matrices with $1$ on the diagonal and nonzero elements only at $(1,2)$ and $(1,3)$. In that case, they commute.

Is it always the case, or is there other recipes to think about them/construct them?

Wirdspan
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  • What is $t$? You can only talk of matrices and entries if your group is a matrix group have you even tried to compute an example of what the product be of two U's looks like in an example, like GL_n or some other group in which you can actually compute? – Mariano Suárez-Álvarez Jan 25 '23 at 08:03
  • @MarianoSuárez-Álvarez Oh sorry I indeed think of $G$ as a matrix group, and did my examples in $GL(n)$. I edit my OP – Wirdspan Jan 25 '23 at 08:08

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