Char $p$ representations of $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$ 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 $Symm^k(V) \otimes Symm^l(V)$? 
 A: 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. 
