Timeline for Gelfand's trick (Gelfand's lemma) in positive characteristic?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Jul 24, 2022 at 5:01 | comment | added | LSpice | @preamble, that link has rotted. Was it to Zhang - Modular Gelfand pairs and multiplicity-free representations? | |
| Sep 14, 2020 at 13:49 | comment | added | preamble | Googling around, I found this paper that looks like it does the problem, at least for finite groups. | |
| Sep 7, 2020 at 18:50 | comment | added | Bugs Bunny | Let me sit on it before I edit the answer. Filling the hole may not require any assumptions after all. | |
| Sep 7, 2020 at 18:47 | comment | added | Bugs Bunny | Your $P$, Doc, is not my $P$. It is not going to work for any module you can find on a street. Yet I agree there is a hole. Off the top of head, I can fill this hole only if $|H|$ is coprime to $p$: in this case $k_{triv}$ is projective, induction takes projectives to projectives, so $P$ is projective. This will tell you that the map from Nakayama Lemma $A/Rad(A) \rightarrow End(P/Rad(P))$ is surjective. Then it is clearly injective... | |
| Sep 5, 2020 at 17:29 | comment | added | preamble | See this thread for a finite-dimensional counterexample where $End(P/Rad(P))\not\cong A/Rad(A)$. There must be some other special circumstance or condition on $P$ for the isomorphism to hold? | |
| Sep 5, 2020 at 13:04 | comment | added | preamble | The isomorphism claim involving the Nakayama Lemma is not true for all $P$. It's still not clear how the isomorphism for $P = \mathrm{Ind}_H^G (k_{\mathrm{triv}})$ (or all finitely generated $P$?) in particular should follow from the Nakayama Lemma. | |
| May 27, 2020 at 6:11 | history | edited | Bugs Bunny | CC BY-SA 4.0 |
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| May 27, 2020 at 6:11 | comment | added | Bugs Bunny | sets? I will change to Func | |
| May 26, 2020 at 21:33 | comment | added | LSpice | In what category are we taking hom's for $\operatorname{Hom}(H\backslash G/H, k)$? (I would have expected just $\operatorname{Func}$.) | |
| Oct 19, 2019 at 5:31 | comment | added | preamble | @BugsBunny, could you explain how the isomorphism follows from Nakayama's Lemma? | |
| Jul 11, 2019 at 1:35 | vote | accept | ferrari | ||
| Jul 9, 2019 at 16:11 | comment | added | Bugs Bunny | The socle of P. It is the sum of all simple submodules, or the largest semisimple submodule. | |
| Jul 9, 2019 at 15:27 | comment | added | ferrari | Sorry, what is $\text{Soc}(P)$? | |
| Jul 9, 2019 at 15:27 | vote | accept | ferrari | ||
| Jul 9, 2019 at 15:27 | |||||
| Jun 27, 2019 at 9:17 | history | edited | Bugs Bunny | CC BY-SA 4.0 |
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| Jun 27, 2019 at 8:58 | history | answered | Bugs Bunny | CC BY-SA 4.0 |