Timeline for Chromatic representation theory of the symmetric groups?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Sep 20, 2023 at 13:54 | answer | added | Tyler Lawson | timeline score: 8 | |
| Sep 18, 2023 at 17:27 | comment | added | Tim Campion | @NicholasKuhn Maybe I'm misunderstanding something, but the thought is that these idempotents correspond naturally to functorial splittings of spectra with $\Sigma_n$ action. Tate vanishing tells us that there is a functorial splitting of the fixed points = orbits, so this should correspond to an idempotent in $\mathbb S_{T(h)}[\Sigma_n]$ which doesn't exist in $\mathbb Z_{(p)}[\Sigma_n]$ -- the one onto fixed points. | |
| Sep 18, 2023 at 3:36 | comment | added | Nicholas Kuhn | I don't understand your sentence "Certainly it restores ..." How does this work when h = 1 and n = 2? | |
| Sep 17, 2023 at 21:21 | comment | added | Maxime Ramzi | For representation theory purposes, you probably want central idempotents (in a strong homotopy coherent sense). For both $K(h)$ and $\mathbb S_{K(h)}$ it seems relevant to start by computing the ones for $E_h$ (in fact for $K(h)$ the answer might be the same), and there the answer will look like a computation of the idempotents in $\pi_0(E_n^{LB\Sigma_n})$ with some interesting ring structure (here $L = $ free loop space). I don't know if HKR character theory contains enough information to do this because it works away from the torsion | |
| Sep 17, 2023 at 20:57 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Sep 17, 2023 at 19:53 | history | asked | Tim Campion | CC BY-SA 4.0 |