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I am looking for a detailed mapping of the root subgroups and elements (and their height) of the simply connected Chevalley groups of type other than $A_n$, and their generators into $\operatorname{GL}_n(\mathbb{C})$ (relative to a maximal torus of the Borel subgroup).

For example, for $A_n$ we have $G = \operatorname{SL}_n(\mathbb{C})$ and the roots are $\phi_{i,j}: \operatorname{diag}(a_1, \ldots, a_n) \rightarrow a_i - a_j$ for $1 \leq i \neq j \leq n$ (the positive ones are $1 \leq i < j \leq n$), and the matching root subgroups are $U_{i ,j} = \{e_{i, j}(t) | t \in \mathbb{C}\}$ (where $e_{i, j}(t)$ is the matrix with ones on the diagonal and $t$ in the $(i, j)$ spot).

Any good sources?

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    $\begingroup$ Did you read the LECTURES ON CHEVALLEY GROUPS by Robert Steinberg? $\endgroup$
    – Wilberd van der Kallen
    Commented Sep 17, 2019 at 8:17
  • $\begingroup$ Yes, he did explained about the root subgroups in general but did not go into specifics except for the case $A_n$. $\endgroup$
    – Ami
    Commented Sep 17, 2019 at 9:08
  • $\begingroup$ Since you ask for "their generators into $\operatorname{GL}_n(\mathbb C)$", is part of this question a faithful representation of each type? $\endgroup$
    – LSpice
    Commented Jun 14, 2020 at 14:51
  • $\begingroup$ @SamuelLelièvre, since they bump questions to the front page, purely cosmetic edits to old questions are usually discouraged. $\endgroup$
    – LSpice
    Commented Oct 12, 2020 at 16:28

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I don't have these references in front of me, but for root systems Humphreys gives explicit coordinates for all simple systems in his Lie algebras book. On the level of the group, Dan Bump's Lie groups book has several examples for classical groups.

As for the root groups, I would recommend writing down the corresponding root spaces in the Lie algebra and working out the root groups via the exponential as an exercise. There are some explicit examples in Bump's Lie groups book though.

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  • $\begingroup$ I think, though it's not entirely clear to me (so I've asked), that @Ami may be looking not just for a description of the roots in an abstract Euclidean space, but even for a faithful representation of each type. (Of course, something like this is necessary even for your answer: I'm not sure what it means to look for the root spaces in the Lie algebra unless you have some handle on the Lie algebra, e.g., via a faithful representation.) $\endgroup$
    – LSpice
    Commented Jun 14, 2020 at 14:52
  • $\begingroup$ @LSpice Does not the choice of a Borel subgroup determine all the necessary data? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 13, 2020 at 1:37
  • $\begingroup$ @მამუკაჯიბლაძე, maybe; I'm not really sure what the question is. I don't know what to make of "their [the root subgroups'] generators into $\operatorname{GL}_n(\mathbb C)$" unless there is a map (from the entire group) into $\operatorname{GL}_n$ in the first place, and presumably that map isn't most interesting unless it's faithful. (Of course it is meaningful to talk about root groups and root spaces even without a representation, and even without a choice of Borel; but presumably the most satisfactory description of them for this question would be in terms of a faithful representation.) $\endgroup$
    – LSpice
    Commented Oct 13, 2020 at 14:07

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