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Dear community, I'm looking for a table that lists the full commutation relations for the Chevalley group $G_2$. I could do this myself (although it would take quite a while), but maybe there is already such a table out there? Sincerely, and thank you, Moshe Adrian |
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It would help to clarify the context of "Chevalley group" here, but over any field one gets uniform results by starting with a Chevalley basis for the associated complex Lie algebra. The commutation relations for rank 2 groups including ADDED: There are two cases not involving a pair of positive roots. 1) If the two roots are independent, they both lie in some positive system. Using the standard picture of the 12 roots, it's easy to visualize the Weyl group transformation involved. Then just relabel the two roots by the corresponding two in the fixed positive system and apply the formulas worked out there. 2) If the roots are negatives of each other, life gets a little more complicated. But here the commutation takes place in a rank 1 group. Explicit commutation formulas here are usually avoided in the literature but take place just in a group like Maybe it would also help to add two relevant online references. Both are based ultimately on a Chevalley basis (keeping in mind that such a basis is unique only up to certain sign choices) as well as the ideas in Chevalley's 1956-58 classification seminar. SGA3 is now online here. See in particular Expose XXIII. Steinberg's 1967-68 Yale lectures on Chevalley groups are available at his homepage here. See Section 6, page 66. |
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At first: sorry, I could't find the source you were looking for, but I'm so free to sketch two ways how you in my opinion get very fast to the informations you need (especially the latter, generic, might be the reason rarely somebody wants to work out all relations?) Maybe you've a more specific question than just knowing the entire table, which schould be quite huge? By the way: With Chevalley-group you mean the finite simple group? But not the twisted one? I assume yes on both, as this is what you wrote ;-) 1) The probably easiest way to directly and elementary get ahold of $G_2$ is as folding of $D_4$ by the cyclic interchange of the three outer nodes. I.e. take $D_4=so_8$ and take the sub-algebra, sub-root-system, etc. fixed by the outer order-3-automorphism often named triality. I'm not 100% sure and don't have a proper source, where this is done for Chevalley groups, but I'm pretty positive, you just do it ;-) ....do NOT confuse this with the construction of the Steinberg group $^3D_4(q^3)$, where one takes (analogously to the construction of unitary groups) the elements, where the action of the automorphisms matches the order 3 galois field automorphisms forced to exist by taking $\mathbb{F}_{q^3}$. This way you're commutator relations of $G_2(q)$ are just a subset of those in $SO_8(\mathbb{F}_q)$, where calculating is obvious (but tedious?) 2) The more uniform, direct way would be starting with a Chevalley basis of the Lie algebra $G_2$ (which I'm sure is not hard and/or can be found in literature - I'll look into it, if I find time...) - the relations are then just exponentiated to the Chevalley group. This way you can read off the commutator relations just from the structure constants of the Lie algebra $G_2$, that I'm sure are worked out at many places |
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