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This is my first post here :)

I have the following two related questions. While looking in Conway's ATLAS (the 1985 one) for $SL_3(Z/pZ)$, where he does the cases $p=2,3,5,7$, I saw that the dimension of the biggest irreducible representation is ~$p^3$. More precise, as the order of the group is $p^3(p^3-1)(p^2-1)$, it looks like the dimension of the biggest irreducible representation is a polynomial of degree 3 in p, that "obviously" dividers the order.

For p=2, the biggest one has dimension $8=2^3$

For p=3, the biggest one has dimension $39 = 3(3^2+3+1)$

For p=5, the biggest one has dimension $186=(5+1)(5^2+5+1)$

For p=7, the biggest one has dimension $456=(7+1)(7^2+7+1)$

Is this true in general? Are there infinitely many primes for which the biggest irr rep of $SL_3(Z/pZ)$ has dimension $O(p^3)$ (or even weaker bounds, like $o(p^4)$).

The second question is about the number of irreducible representations. While looking in the ATLAS, looks like there are quite few of them, relative to the dimension of the group. Is there a result that computes the growth rate of the number of conjugacy classes/number of irr rep, as $p\to\infty$?

I am interested in a result that somehow combines both. Something in the spirit "small", but "few" representations.

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I cannot follow from the answers - does dimension really equals p(p^2+p+1) ? –  Alexander Chervov Jan 2 at 16:20

3 Answers 3

up vote 9 down vote accepted

To add another viewpoint to what Paul has already said, it's important to realize that a lot of general theory exists by now for these particular groups and for others of Lie type (especially the work inspired by Lusztig). The Atlas authors relied on only some of this work, but started with older literature on special cases and some direct computer work for large groups like the Monster.

But for finite groups of Lie type there is an extensive theory to tap into, beyond the few special cases treated in earlier literature. Paul has indicated some of the ideas coming from the 1955 paper by J.A. Green in Trans. Amer. Math. Soc. Green treated the somewhat easier case of finite general linear groups, using combinatorial methods. But special linear groups are harder to work with, due to the finite center in the ambient algebraic group. Here the explicit character values have only recently been made fairly explicit for cases beyond $SL_3(\mathbb{F}_q)$.

The limited case you start with involves (as do other groups of Lie type) some exceptions for small primes, but when $p$ is large enough the picture becomes rather uniform. Two specific references deal with the questions you raise:

1) Before the "modern" theoretical era, W.A. Simpson and J.S. Frame worked out character tables for some rank 2 groups including $SL_3(\mathbb{F}_q)$ for a power $q$ of $p$. Their results are mostly reliable, with a little bit of adjustment, and published in Canad. J. Math. 25 (1973): PDF here.

In particular, the largest degrees occur for principal series characters (as Paul indicates). Here as in other families of Lie-type groups there are standard degree polynomials in $q$ whose highest term is always $q$ raised to the number of positive roots (in your example 3). But for small primes all of these families of characters may be degenerate, preventing the generic degrees from occurring in the character table. The details do get complicated, but keep in mind that as $p$ grows the character tables become more uniform and predictable. There are lots of references, but the bottom line is that much is known. Not everything, of course.

2) Quite a few people have worked out the total number of classes (hence of complex irreducible characters) for families of Lie type groups. I think the joint paper by Deriziotis and Holt in Comm. Algebra 21 (1993) might contain the most explicit results for the next-larger family $SL_4$. In your case $SL_3$ it may be easiest to compile the number of characters from the Simpson-Frame paper linked above. Roughly speaking, the total number is given by a polynomial in $q$ of degree equal to the Lie rank (in your example 2).

ADDED: If my arithmetic is correct, the number of classes (or characters) for $SL_3(\mathbb{F}_q)$ with $q$ a power of $p$ is $q^2+q$ when $d=1$ and $q^2 + q + 8$ when $d=3$. Here $d = \gcd(3,q-1)$, the order of the center.

One other remark: the Atlas gives little help with questions about asymptotic behavior as $p \rightarrow \infty$, but a comparison with $p$-modular representations may help to "explain" the significance of the asymptotic values given by highest powers in polynomials above. Here $p^2$ is the number of modular irreducibles, while $p^3$ is the dimension of the largest of these (the Steinberg representation). See for instance J. Algebra 72 (1981), 8--16.

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The linked PDF is unfortunately behind a UMass login wall. –  Kevin Ventullo Sep 9 '13 at 1:29
    
Dear Paul Garrett and Jim Humphreys, thank you very much for your clean and nice answers. –  Alin Galatan Sep 9 '13 at 1:51
    
@Kevin: I've replaced with a direct link (only the most recent years require library access). –  Jim Humphreys Sep 9 '13 at 10:28
    
@Jim: Thanks! $\phantom{}$ –  Kevin Ventullo Sep 10 '13 at 2:24

A partial answer, but one which may help in some refinement of the original question:

For every $GL_n(\mathbb F_q)$ (and for reductive Lie groups and reductive $p$-adic groups) there is a substantial family of mostly-irreducible repns which "tend to" be the largest among irreducibles, namely, "principal series" repns, induced from a one-dimensional repn of a minimal parabolic (here $P=$ upper-triangular), with, therefore, easily predictable dimension: dimension of the whole group divided by dimension of the parabolic, which is $(q^3-1)(q^3-q)(q^3-q^2)/(q-1)^3q^3$ for $GL_3$. These numbers are the numbers you mention. I do not have a proof that these are largest among all irreducibles, but based on the behavior for reductive Lie groups, I'd expect that these are largest for finite groups.

The number of irreducibles for $GL_3(\mathbb F_q)$ is pretty easy: it is the number of Jordan forms, which is the number of chains of elementary divisors, which is one of monic cubic, monic linear dividing monic quadratic, and three copies of the same monic linear. That is, $q^3+q^2+q$.

Already for $GL_2(\mathbb F_q)$ there are subtler repns that are also smaller than the principal series. Even without constructing them, we can see that they are smaller by looking at the isotypes for characters of the unipotent radical of the parabolic in that case: the isotypes are multiplicity one, except for the trivial character, which must not appear or by Frobenius reciprocity these things would map to principal series. For $GL_3(\mathbb F_q)$ there is surely a similar argument, but it has to be somewhat more complicated.

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The question of the maximal dimension of irreducible representations of simple finite groups (including those of Lie type) is considered in "The largest irreducible representations of simple groups" by Larsen, Malle and Tiep (arXiv:1010.5628); for finite groups of Lie type (not necessarily simple) over large enough field, it is a theorem of Seitz ("Cross-characteristic embeddings of finite groups of Lie type", Proc. LMS 1990, 166-200, section 2) that the maximal dimension is at most $|G:T|_{p'}$, where $T$ is a maximail torus of smallest possible order (i.e., maximally split). In particular, if there is a split maximal torus, this shows that the principal series representations are indeed those with maximal dimension.

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