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It is a well known fact that the irreducible representations of $SL_2(\mathbb{F}_p)$ over $\overline{\mathbb{F_p}}$ are given by the symmetric powers $Symm^k(V)$, where $V = \overline{\mathbb{F}_p}^2$, for $k$ ranging in $0,\ldots,p-1$. In the case of $GL_2(\mathbb{F}_p)$ irreducible representations are exhausted by:

$$\sigma_{k,n} = Symm^k(V) \otimes det^n$$

My question is then if there is a systematic way to know which of these irreducible representations appear in higher symmetric powers ($k>p-1$), either for $SL_2$ or $GL_2$?

And what about tensor products $Symm^k(V) \otimes Symm^l(V)$?

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    $\begingroup$ Steinberg's tensor product theorem? $\endgroup$
    – user30035
    Commented Mar 15, 2013 at 7:47
  • $\begingroup$ Section 5.4 of Kleidman & Liebeck should shed light on this question. They deal with all finite groups of Lie type, and also explain Steinberg's tensor product theorem, as per the comment of @wccanard. I don't know whether STPT quite answers your question as it gives a decomposition by tensors (as the name suggests), and I'm not sure if that's what you mean when you say "which of these appear". Still, it's a good first place to look. If you need an e-copy email me. $\endgroup$
    – Nick Gill
    Commented Mar 15, 2013 at 9:58
  • $\begingroup$ Off-topic: One hour later I just understood the pun in the moniker @wccanard. I like it! $\endgroup$
    – Nick Gill
    Commented Mar 15, 2013 at 10:25
  • $\begingroup$ @wccanard: Steinberg's tensor product theorem only becomes relevant for the groups over larger finite fields than the prime field (and in this rank 1 case actually goes back to much earlier work of Brauer-Nesbitt). $\endgroup$
    – Jim Humphreys
    Commented Mar 16, 2013 at 13:12

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The short answer to your basic question is no; but a lot is known (and written down in various places including a 1978 paper in J. Algebra by D.J. Glover based on his A.N.U. thesis). The long answer is that the bookkeeping involved even for this small case gets quite involved, a little more so for the finite general linear groups than for the special linear groups.

There is a detailed survey with references in Chapter 19 of my 2006 LMS Lecture Note Series 326 Modular Representations of Finite Groups of Lie Type (Cambridge U. Press). The emphasis throughout is on the "defining" characteristic $p$. In Chapter 19 the main theme is the decomposition of all symmetric powers of the natural representation for general or special linear groups of any rank, specialized to rank 1 in 19.7.

Even in the rank 1 case it's unreasonable to expect explicit closed formulas, but the fact that weight multiplicities are 1 already simplifies the problem a lot. A nontrivial test case involves the multiplicity of the trivial representation in each space of homogeneous polynomials in two indeterminates: this leads to the Dickson invariants (for the finite groups over arbitrary finite fields). But this already requires a lot of ingenuity. As far as I know, the approaches described in this chapter are the only ones so far attempted.

Concerning your added question about tensoring two symmetric powers, this must get a lot more complicated to work out in detail. Since representations in prime characteristic generally fail to be completely reducible, you start to run into a large assortment of indecomposable modules including the projective ones. But if you are only asking for composition factor multiplicities, there is a better chance of obtaining these (in principle) from study of the individual symmetric powers.

It's important here to take a close look at what has already been done, since the results go back all the way to older work of Dickson and involve quite a range of methods in the modular representation theory of finite groups.

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  • $\begingroup$ Thanks for the answer! The kind of description that appears in the paper of Glover seems to be enough for the cases I needed. $\endgroup$
    – mcampo
    Commented Mar 19, 2013 at 16:12

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