Timeline for Suzuki and Ree groups, from the algebraic group standpoint
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 7, 2018 at 20:38 | vote | accept | Marty | ||
| Jul 30, 2010 at 5:47 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Added an example
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| Jul 29, 2010 at 23:19 | comment | added | Marty | I am amused! I'll check out the P-Red book shortly. | |
| Jul 29, 2010 at 23:09 | comment | added | BCnrd | Marty, you may be amused to hear that these purely inseparable special isogenies for types B, C, and F in characteristic 2 and type G_2 in characteristic 3 (i.e., types with a bond of multiplicity $p$ in characteristic $p$), which are bijective on rational points over a perfect field but not otherwise, underlies the "exceptional" pseudo-reductive groups. See Chapter 7 (especially section 7.1) of the book "Pseudo-reductive groups". | |
| Jul 29, 2010 at 21:13 | comment | added | Marty | Thanks Jim! I'll look back at Deligne-Lusztig for more. I'm always a bit slow going between generalities (defining groups via descent data) and the special cases (using Frobenius morphisms over finite fields, and Cartan involutions over the reals). | |
| Jul 29, 2010 at 20:56 | comment | added | Jim Humphreys | Marty: The "Frobenius endomorphism" here incorporates the special isogeny, so its fixed points give the Suzuki or Ree group in question as Steinberg explained in uniform fashion. This broader notion of Frobenius morphism is now standard in the work on characters and such coming out of the Deligne-Lusztig construction in 1976. | |
| Jul 29, 2010 at 20:42 | comment | added | Marty | Ah - so the Suzuki-Ree groups are not $F$-points of an algebraic group over $F$ after all then, I guess. I guess that the endomorphism $\phi$ cannot be used to define descent data as required. So weird... | |
| Jul 29, 2010 at 20:35 | history | answered | Torsten Ekedahl | CC BY-SA 2.5 |