Timeline for What is the minimum possible k-rank of a quasi-split reductive group over a field?
Current License: CC BY-SA 4.0
14 events
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| Oct 18, 2023 at 20:57 | history | edited | LSpice | CC BY-SA 4.0 |
Links to comments; `\operatorname`
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| Oct 18, 2023 at 15:17 | comment | added | Mikhail Borovoi | @DavidESpeyer: Corrected! Thank you! | |
| Oct 18, 2023 at 15:16 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
Mistake noticed by David E Speyer corrected.
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| Oct 18, 2023 at 14:57 | comment | added | David E Speyer | Possible typo: I think all the $2n$'s when you discuss type $D$ should just be $n$. (Or, rather, what you wrote is true, but it is also true for $D_{\text{odd}}$.) Good answer! | |
| Oct 18, 2023 at 14:24 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
added 1018 characters in body
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| Oct 18, 2023 at 13:16 | comment | added | YCor | @MikhailBorovoi OK that's a clear-cut conclusion, you might edit to add it. | |
| Oct 18, 2023 at 12:59 | comment | added | Mikhail Borovoi | @YCor: On the other hand, when $G$ is absolutely simple, we obtain the positive answer, see the first comment of David. | |
| Oct 18, 2023 at 12:56 | comment | added | Mikhail Borovoi | @YCor: I think that yes, it does answer this question in the negative. We can take $G=R_{K/k}{\rm SL}_{2,K}$, where $K/k$ is a finite separable extension of degree $n$. Then the absolute rank of $G$ is $n$, whereas the $k$-rank of $G$ is 1. | |
| Oct 18, 2023 at 12:54 | comment | added | David E Speyer | I'm not sure about the semisimple case, though. The Dynkin diagram of $\text{SL}_2^n$ is $n$ disjoint copies of the $A_1$ diagram, and you could have a group that permutes them transitively. I don't know if this can occur. | |
| Oct 18, 2023 at 12:53 | comment | added | David E Speyer | @YCor In the simple case, yes. I don't know all the vocab in this question, so I can't check the correctness of the argument, but in the end it came down to counting orbits of a diagram automorphism group acting on a Dynkin diagram. The only connected Dynkin diagram with $>2$ automorphisms is $D_4$, and even in that case, there are $2$ orbits. | |
| Oct 18, 2023 at 8:05 | comment | added | YCor | Does this answer the question whether the $k$-rank is $\ge$ rank$/2$? | |
| Oct 18, 2023 at 5:22 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
Details added
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| Oct 17, 2023 at 14:39 | vote | accept | C.D. | ||
| Oct 17, 2023 at 14:16 | history | answered | Mikhail Borovoi | CC BY-SA 4.0 |