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When finding representations of finite groups over $\mathbb{F}_p$, (i.e. homomorphism from $G$ to $GL(n,\mathbb{F}_p$), it requires many times presentation of $GL(n,p)$. What is presentation of GL(n,p)?

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Have a look at any introductory group theory book. This is not the right place to ask such questions. MO ist for reasearch level questions! –  Johannes Hahn Jan 25 '11 at 11:18
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@Johannes : I don't know what introductory books you have on group theory, but the ones on my shelf (including books by Rotman, Robinson, Marshall Hall, and Alperin) do not include presentations for $GL(n,p)$. It's an old and venerable topic, but I think that's its a fine question for MO. However, it's worth mentioning that I don't think such presentations have ever proven useful in representation theory... –  Andy Putman Jan 25 '11 at 17:40
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I'd second Andy's comment but add that the "question" asked isn't yet a real question. I can't think offhand of any significant representation-theoretic problem that requires a "presentation" of the finite general linear group. So "requires many times" is serious overkill here. The question needs much more specific detail. What literature is relevant, for instance? –  Jim Humphreys Jan 25 '11 at 18:21
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P.S. If a presentation is really wanted here in the context of finite groups of Lie type, probably the most natural one would be based on the BN-pair (or Tits system) and resulting Bruhat decomposition. Finite group theorists often recognize a new group as being of Lie type by finding such a BN-pair structure in it. –  Jim Humphreys Jan 25 '11 at 21:17
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Possible duplicate: mathoverflow.net/questions/10123/presentation-for-gln-k –  Guntram Jan 25 '11 at 22:39

6 Answers 6

up vote 3 down vote accepted

To focus just on the question of giving a presentation of a finite general linear group over the prime field (or other finite field), leaving aside the fuzzy connection with representations of finite groups: This can be looked at profitably in the broader setting of finite groups of Lie type as studied by Steinberg in the early 1960s, even though there is a little distance between general linear and special linear groups. For a "universal" Chevalley group such as a special linear group over an arbitrary field, Steinberg gives a nice presentation in section 6 of his 1967-68 Yale lecture notes, building on his earlier Brussels conference note (published in French). None of this material seems to be readily available online, unfortunately, though Steinberg's papers are collected in an affordable AMS volume. Attempts over the years to publish the Yale lectures more formally fell through.

Steinberg's presentation is based on the BN-pair structure, but with some refinements. Treating the finite general linear group doesn't require too much modification, but perhaps isn't written down explicitly (?) In any case, this type of presentation is transparent and has the added merit of leading (over commutative rings) to fundamental ideas in algebraic K-theory involving Steinberg symbols, etc. Meanwhile finite group theorists involved with the classification of simple groups have made their own good use of BN-pair ideas following Tits and Steinberg.

ADDED: Notice at the end of Steinberg's section 6 (page 72) the simple explicit presentation of $SL_n(\mathbb{F}_q)$ given by his method. You could get this by taking shortcuts in his proof, but keep in mind that the case $n=2$ requires extra care. To get instead the general linear group you just have to add some generators corresponding to scalar matrices along with suitable relations. Nothing is gained in this approach by working only over the prime field, which anyway would be inadequate for most applications.

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His Yale notes are available online: math.wisc.edu/%7Eram/YaleNotes.pdf –  fherzig Feb 8 '11 at 16:22
    
@Florian: Thanks for pointing this out. The Yale department used to sell bulky mimeographed copies, but there was hope for a while that a typeset version could be published (with an index and directory of theorems). –  Jim Humphreys Feb 8 '11 at 17:13
    
@Jim Humphreys: You are welcome. Yes, it's a shame they weren't properly published. –  fherzig Feb 9 '11 at 3:03

It is straightforward to read off a presentation of $GL_n(p)$, and for that matter any group of Lie type, by representing it as a BN-pair. For $GL_n$, $B$ is the group of upper triangular matrices, and $N$ is the group of matrices with one non-zero entry in each row and column. Each of these two subgroups has an easy presentation, and the axioms for a BN pair combine these into a presentation for $GL_n$.

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I think, this BN pair is just giving generators, but not relations. –  Martin David Jan 26 '11 at 3:37
    
@Martin: Note my added answer to your question. –  Jim Humphreys Feb 8 '11 at 15:10

In line with commenters, I'm not convinced that your motivation for the question is well founded, but it's an interesting question in its own right. My impresssion is that finding reasonably nice presentations for finite simple groups in general is challenging, and there is probably much still to be learned. Perhaps the geometric "inside" view of finite groups still has room to develop, to catch up a little bit with the far more mature "outside" view based on representation theory.

One place to start concerning presentations of finite quasisimple groups (from which you can work out $GL(n,p)$) is the work of Guralnick, Kanotor, Kassabov and Lubotzky, link text. They're focused here on profinite presentations, which only distinguish the given group from other finite and residually finite groups, but these may be what you actually want, and it seems that it's easier to establish simpler profinite presentations than ordinary presentations. I think chasing references and citations should lead you to the relevant material.

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This isn't a presentation, but this file gives generating sets (of size 2) for $GL_n(F_q)$ and other finite matrix groups. I think it's kind of awesome: http://talus.maths.usyd.edu.au:8000/u/don/papers/genAC.pdf

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Its fine. But it would be better, if found presentation... –  Martin David Jan 26 '11 at 3:38

Generators and relations for $GL(n, Z)$ (which answer the question over $\mathbb{F}_p$, as well) are given in Morris Newman's "Integral Matrices" (which is a wonderful book for many other reasons as well). Newman derives them in a completely elementary way, as well (no BN pairs), which, I am guessing, should appeal to OP.

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Can you give a specific page reference (and especially explain a little more about the parenthetic comment)? Presentations using elementary matrices are the most natural, translated into the language of BN-pairs and Bruhat decomposition in general, but relations over rings and fields then lead to complications. I'm not sure what is the most elegant answer for finite general linear groups over arbitrary fields. –  Jim Humphreys Feb 8 '11 at 17:17
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See my other answer, which makes this answer obsolete. –  Igor Rivin Feb 9 '11 at 21:02

See

MR1871620 (2002i:20068) Chiaselotti, Giampiero(I-CLBR) Some presentations for the special linear groups on finite fields. (English summary) Ann. Mat. Pura Appl. (4) 180 (2001), no. 3, 359–372.

http://dl.dropbox.com/u/5188175/slnpresentations.pdf

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