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I have two questions:

  1. Let $G$ be a finite group. Because complex representations of $G$ are completely reducible to know all representations is same as knowing irreducible ones. In case of modules over group algebra ${\overline{\mathbb F_p}}[G]$ when $p|o(G),$ we know there are possibly indecomposable modules of arbitrary large degree. In such cases when do we say that we know "all" representations of $G$ with appropriate sense the word "all"?
  2. Is there any way or reference to compute order of group GL$_2(\mathbb Z/p^n\mathbb Z)$? I am trying to work with modules over $\overline{\mathbb F_p}[\mbox{GL}_2(\mathbb Z/p^n\mathbb Z)].$ Is it something very trivial or very difficult?

I request all to give answer in elementary language if possible.

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    $\begingroup$ The order of $GL_2(\mathbb{Z}/p^n)$ is $(p^2-1)(p^2-p)p^{4(n-1)}$.The surjection $GL_2(\mathbb{Z}/p^n) \to GL_2(\mathbb{Z}/p)$, has a kernel which is clearly of size $p^{4(n-1)}$. The order of $GL_2(\mathbb{Z}/p)$ is $(p^2-1)(p^2-p)$. Your other questions will be more difficult. $\endgroup$
    – David E Speyer
    Commented Jul 13, 2010 at 16:30
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    $\begingroup$ (By the way, your two questions are sufficiently different that they would be, IMO, better split in two) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jul 13, 2010 at 16:42
  • $\begingroup$ @Jim, you have been able for a while now to edit other people's questions! :) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jul 13, 2010 at 18:10
  • $\begingroup$ Added the tags. $\endgroup$
    – José Figueroa-O'Farrill
    Commented Jul 13, 2010 at 18:13
  • $\begingroup$ @Mariano, Jose: I've generally confined my edits to misspelled names or the like, not wanting to impose my own choice of tags on other people's questions. Anyway, this really should be posted as two questions. which I can't edit. $\endgroup$
    – Jim Humphreys
    Commented Jul 13, 2010 at 20:11

2 Answers 2

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For (1), in a technical sense, for sufficiently complicated groups $G$ we never say we know all modular representations of $G$, because the classification problem is in that case wild.

A theorem of Higman tells us exactly when there are finitely many indecomposable modules in terms of the structure of Sylow subgroups: a necessary and sufficient condition is that the $p$-Sylow subgroups be cyclic—and if I recall correctly in that case one can in principle construct them all.

There is a remaining case, that of groups of tame representation type, where there is a lot of technology available, and I would say that in that case we "know" the representation theory when you can make a picture of of Auslander-Reiten quiver of the group.

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  • $\begingroup$ Concerning the actual construction of indecomposables in the cyclic Sylow subgroup case, I'm not sure how explicitly this can be done. It was done for the groups PSL$(2,p)$ with $p$ by G.J. Janusz, Trans. AMS 125 (1966). Here the projective indecomposables are easy to describe and other indecomposables are submodules of these or close relatives. But his main result is a precise count of the number of nonisomorphic indecomposables whenever the Sylow $p$-subgroup is cyclic. $\endgroup$
    – Jim Humphreys
    Commented Jul 15, 2010 at 16:13
  • $\begingroup$ Well, if there is an effective bound on the dimension of the indecomposable modules or on their number when there are finitely many of them, then their determination is, in principle, reduced a finite computation. Of course, this will end up in a rather unsatisfactory 'description'! Without looking, I guess Janusz's construction is much more satisfying that this :) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jul 15, 2010 at 16:36
  • $\begingroup$ Yes, I was overlooking the more comprehensive 1969 Annals paper by Janusz in which he uses the Brauer tree for any cyclic $p$-block to construct the indecomposables (counted in his 1966 paper). Sorry for relying too much on memory when I wrote my comment. $\endgroup$
    – Jim Humphreys
    Commented Jul 15, 2010 at 16:50
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Mariano has addressed question (1), but let me add that finite representation type is extremely rare especially for interesting classes of groups like the simple nonabelian ones: in characteristic $p$, the Sylow $p$-subgroups must be cyclic. Also, the initial work by Higman was refined by looking at individual $p$-blocks of a finite group and their representation type (Brauer, Dade, Janusz). To get a block of tame representation type, you need $p=2$ while the block must have defect group of a very special type: dihedral, semidihedral, or generalized quaternion. A general reference is:

MR1064107 (91c:20016) 20C20 (16G60 16G70) Erdmann, Karin (4-OX), Blocks of tame representation type and related algebras. Lecture Notes inMathematics, 1428. Springer-Verlag, Berlin, 1990. xvi+312 pp.

Concerning question (2), it really goes off in another direction and has been studied mainly by people interested in $p$-adic representation theory. It would be useful for them to know more about representations of various linear groups over rings of $p$-adic integers or over finite residue rings other than the residue field such as $\mathbb{Z}/p^n \mathbb{Z}$ when $n>1$. This is a tough problem to attack using methods of finite group theory, whether you start by working over a ground field like $\mathbb{C}$ or else look at $p$-modular representations of these finite matrix groups. As far as I know, results in this direction have been rather few and far between.

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    $\begingroup$ Comment for those not in the know: "$p$-adic representation theory" mentioned by Jim studies (smooth or unitary; not algebraic) representations of $p$-adic reductive groups, such as $GL_k(\mathbb{Q}_p).$ One of the most successful approaches to constructing them is via induction from compact open subgroups, such as $GL_k(\mathbb{Z}_p),$ hence the need to understand finite-dimensional representations of the latter group that factor through $GL_k(\mathbb{Z}/p^n\mathbb{Z})$ and the corresponding Hecke algebras. $\endgroup$
    – Victor Protsak
    Commented Jul 14, 2010 at 1:29
  • $\begingroup$ Victor's version is what I had in mind here: the study of representations of the analogues of real Lie groups over other local fields, motivated especially by the Langlands program. $\endgroup$
    – Jim Humphreys
    Commented Jul 14, 2010 at 11:36

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