Timeline for Order of finite unitary group
Current License: CC BY-SA 4.0
21 events
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| Jan 25, 2019 at 0:31 | history | edited | Jim Humphreys | CC BY-SA 4.0 |
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| Dec 16, 2014 at 12:21 | vote | accept | Pooja Singla | ||
| Nov 1, 2014 at 14:57 | comment | added | Olórin | @BCnrd Brian, would you have a reference for Boyarchenko's calculation using étale cohomology ? | |
| Dec 28, 2011 at 15:36 | answer | added | Jim Humphreys | timeline score: 8 | |
| Aug 17, 2010 at 17:32 | history | edited | darij grinberg | CC BY-SA 2.5 |
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| Aug 4, 2010 at 4:37 | comment | added | Pooja Singla | @Robin Thanks a lot for the reference. @Jim Thanks a lot for amazing advice. It is precious. | |
| Aug 3, 2010 at 22:10 | comment | added | Jim Humphreys | Advice: These further comments on unified proofs of the most general order formula are not directly an answer about finite unitary groups, but eventually it's the direction to go. In the short run, if access is a problem, try one of the more classical sources mentioned already or if necessary online notes. Don't spend a lot of time rediscovering proofs of such old theorems. There are many things left to do in mathematics. | |
| Aug 3, 2010 at 20:22 | comment | added | Jim Humphreys | (cont) To be specific about references, at the end of a section of SGA 4-1/2 written by Deligne (Lecture Notes 569, p. 230), a geometric approach to Steinberg's uniform order formula is sketched. The ideas are worked out in more detail in MR685938 (84g:20086), Springer, T. A., The order of a finite group of Lie type. Algebraists’ homage: papers in ring theory and related topics (New Haven, Conn., 1981), pp. 81–89, Contemp. Math., 13, Amer. Math. Soc., Providence, R.I., 1982. | |
| Aug 3, 2010 at 16:51 | comment | added | Jim Humphreys | @BCnrd: I think Steinberg's 1968 paper was the first to streamline the older Chevalley approach adapted separately by him and others for twisted ADE, Suzuki, Ree groups (Bruhat decomposition, Weyl group identities). Lusztig brought more topological background to his 1976 Annals paper with Deligne which emphasizes Lefschetz fixed point methods in etale cohomology. But I don't recall now what got written down about computing the group orders in that setting. Lusztig started in that direction while still at Warwick. | |
| Aug 3, 2010 at 16:10 | comment | added | Robin Chapman | See Peter Cameron's notes on classical groups: maths.qmul.ac.uk/~pjc/class_gps . | |
| Aug 3, 2010 at 15:20 | comment | added | BCnrd | Jim, do you know of a published proof of Steinberg's general formula other than his 1968 book? More specifically, M. Boyarchenko showed me a proof using etale cohomology, and I wonder if an argument along those lines already appears in the literature (such as in some of the various papers which discuss $\ell$-adic cohomology of connected semisimple groups). | |
| Aug 3, 2010 at 15:08 | comment | added | Jim Humphreys | The question is really just a reference-request (for an old, standard result). The unified Lie-theoretic viewpoint is best, but there are nice older treatments such as Emil Artin's "The orders of the classical simple groups", Comm. Pure Appl. Math. 8 (1955). That was the year of Chevalley's famous Tohoku paper, to which Artin alludes. Calculating such orders of finite linear groups is not a trivial exercise, since it requires a strategy, but is by now fairly elementary. | |
| Aug 3, 2010 at 15:02 | comment | added | Pooja Singla | @Anweshi Thanks a lot @BCnrd, @Stanley Thanks for your references. But I don't have access to both neither Steinberg's paper nor to Wan Zhexian's book :( | |
| Aug 3, 2010 at 14:56 | comment | added | Richard Stanley | A good reference for elementary proofs of such results is Wan Zhexian, Geometry of Classical Groups over Finite Fields, second ed., Science Press, Beijing/New York, 2002. | |
| Aug 3, 2010 at 14:39 | comment | added | Pooja Singla | I am getting n=1 case. But not n=2. I think that will help me to see general induction proof. | |
| Aug 3, 2010 at 14:33 | history | edited | Pooja Singla | CC BY-SA 2.5 |
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| Aug 3, 2010 at 14:30 | comment | added | BCnrd | There's a "classical" formula of Steinberg for order of $G(k)$ for any conn'd ss group $G$ over finite field $k$: see 11.16 in his "Endomorphisms of linear alg. gps". Steinberg imposes extra hypotheses on $G$, but unnecessary, as explained in 1.6 in Oesterle's paper "Nombres de Tamagawa" (which gives an elegant description of the formula). The inputs in formula are dimension of $G$, size of $k$, action of geometric Weyl group on geometric character gp of max. $k$-torus $T$, and Frob. action on this char. gp. There's a shorter cohomological proof. | |
| Aug 3, 2010 at 14:27 | comment | added | Charles Matthews | Well, have you tried induction? | |
| Aug 3, 2010 at 14:13 | comment | added | Anweshi | @Pooja: I have replaced your link to the cached pdf file with the more stable link ams.org/mathscinet-getitem?mr=150210 instead. You can get such links by clicking on "Make Link" in the Math Review page. | |
| Aug 3, 2010 at 14:11 | history | edited | Anweshi | CC BY-SA 2.5 |
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| Aug 3, 2010 at 14:09 | history | asked | Pooja Singla | CC BY-SA 2.5 |