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:


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



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> 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 $GL\_2$?
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And what about tensor products $Symm^k(V) \otimes Symm^l(V)$? 



Char $p$ representations of $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$
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. 


