Timeline for How to check whether a given matrix is in the image of a representation?
Current License: CC BY-SA 4.0
11 events
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| Feb 21, 2020 at 23:23 | comment | added | YCor | @DavidESpeyer very nice!. Its 27-dim rep is absolutely irreducible, so has a centralizer reduced to scalars, hence has trivial centralizer in $\mathrm{SO}(27)$. Moreover $G_2$ (in its compact form or any other) has a trivial Out (and trivial center), so its normalizer in any overgroup is the direct product with its centralizer. Hence $G_2$ equals its own normalizer. (This also works with the split form, in $\mathrm{SL}_{27}(\mathbf{R})$). | |
| Feb 21, 2020 at 22:20 | answer | added | Moishe Kohan | timeline score: 4 | |
| Feb 21, 2020 at 21:25 | comment | added | David E Speyer | Is $G_2$ self-normalizing in $SO(27)$? Because it would be easy to test if $g G_2 g^{-1} = G_2$ -- it's enough to check the corresponding equality of Lie algebras, which is just a Linear Algebra computation. | |
| Feb 21, 2020 at 20:40 | comment | added | Robert Bryant | @BenMcKay: Actually, checking the Euler angles would only show you that the matrix is conjugate in $\mathrm{SO}(27)$ to an element of $\rho(\mathrm{G}_2)$, wouldn't it? That's not enough to show that it actually lies in $\rho(\mathrm{G}_2)$ because $\rho(\mathrm{G}_2)$ is not a normal subgroup of $\mathrm{SO}(27)$. | |
| Feb 21, 2020 at 17:24 | comment | added | Ben McKay | The paper I was thinking of: arxiv.org/abs/hep-th/0503106 | |
| Feb 21, 2020 at 16:50 | comment | added | Ben McKay | The group G2, in its compact real form, has explicit Euler angles, worked out a few years ago by some physicists. You can immediately see using trigonometry if a matrix belongs. | |
| Feb 21, 2020 at 16:47 | answer | added | David A. Craven | timeline score: 1 | |
| Feb 10, 2020 at 0:15 | comment | added | YCor | The image is Zariski closed, and probably after conjugation can be chosen defined over a number field ($\mathbf{Q}$ itself?). The issue is then to determine an explicit family of polynomials whose zero set equals $\rho(G)$. Possibly some computational commutative algebra machinery does this in a somewhat complicated way, and maybe there are clever ways to do so too. | |
| Feb 10, 2020 at 0:11 | comment | added | user6976 | If the entries are real numbers what does "given matrix" mean? | |
| Feb 10, 2020 at 0:05 | comment | added | AccidentalFourierTransform | FWIW: I posted this on math.SE (cf. math.stackexchange.com/q/3531073/289977). After a week with no activity whatsoever I decided to cross-post here. Hopefully the question is not as easy as to be off-topic here, but if so please let me know and I'll delete. Cheers! | |
| Feb 10, 2020 at 0:03 | history | asked | AccidentalFourierTransform | CC BY-SA 4.0 |