Timeline for Representations of $\mathrm{SL}(2)$ in characteristic 2
Current License: CC BY-SA 4.0
14 events
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| Mar 26, 2022 at 11:09 | history | edited | YCor | CC BY-SA 4.0 |
formatting
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| Mar 4, 2014 at 9:00 | comment | added | Tobias Kildetoft | BTW, another case where quite a bit can be said is when $d = p^r-1$ for some $r$. In this case, the module is tilting, and therefore a direct sum of indecomposable tilting modules. The highest weights of these can easily be found (and I think we also know the characters of the tilting modules for $SL_2$). | |
| Feb 28, 2014 at 15:34 | vote | accept | Lloyd Yu-West | ||
| Feb 26, 2014 at 9:16 | comment | added | Tobias Kildetoft | I have now added a proof that the module is indecomposable. It could probably be done in a much more elementary way, but this was the easiest way that came to mind. | |
| Feb 25, 2014 at 16:12 | answer | added | Jim Humphreys | timeline score: 4 | |
| Feb 25, 2014 at 15:28 | comment | added | Tobias Kildetoft | After some more thought I realized that this module is indecomposable. I am on a tablet now, so I will elaborate later. | |
| Feb 25, 2014 at 15:25 | comment | added | Lloyd Yu-West | @WilberdvanderKallen: Thank you for your comment. I am naive about invariant theory in positive characteristic. How much carries over from the classical case? I'd be grateful of any pointers/references. | |
| Feb 25, 2014 at 15:23 | history | edited | Lloyd Yu-West | CC BY-SA 3.0 |
Removed ambiguity in the question.
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| Feb 25, 2014 at 15:16 | comment | added | Lloyd Yu-West | @TobiasKildetoft: Sorry for the ambiguity (I have edited the question to remove it). And thank you for your answer. I want to be explicit as possible, so I want a composition series as you give below. I would also like to know whether $S^3V\otimes V\cong H^0(4)\oplus H^0(2)$. | |
| Feb 25, 2014 at 8:25 | answer | added | Tobias Kildetoft | timeline score: 7 | |
| Feb 25, 2014 at 8:21 | comment | added | Wilberd van der Kallen | And what do you mean by `the invariant theory'? Just the invariants in this module? (They are the same as predicted by Clebsch-Gordan.) | |
| Feb 25, 2014 at 8:00 | comment | added | Tobias Kildetoft | Could you be a bit more specific about what you mean by decompose? The modules will not decompose as a direct sum of simples in positive characteristic. Do you want the composition factors? Or a specific composition series? A good way to start is probably to do the usual decomposition to get a good filtration of the tensor product, and then use that we actually do know the characters of the modules in such a good filtration in terms of the characters of the simple modules, since this is $SL_2$. | |
| Feb 25, 2014 at 1:34 | history | edited | Lloyd Yu-West | CC BY-SA 3.0 |
added 51 characters in body
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| Feb 25, 2014 at 1:27 | history | asked | Lloyd Yu-West | CC BY-SA 3.0 |