This is related to this question. Suppose I have an $n$dimensional representation of a finitely generated group, and I want to know whether it is absolutely irreducible. This can, of course, be done if you are willing to compute the eigenspaces of the generators, but that involves working over the splitting field of the characteristic polynomial (it should, presumably, be the same for all the generators if the representation is reducible), but that is not what I would describe as pleasant. Is there anything more efficient? (note that I can, of course, diagonalize the generators numerically, but I am not sure what this tells me).

There are very complicated groups $G$ such that $G = \langle x,y : x^{2} = y^{3} = 1 \rangle,$ possibly with further relations. In a sense, the most complicated such group is ${\rm PSL}(2,\mathbb{Z}),$ which is the free product of a group of order $2$ and a group of order $3$. For any given representation of such a group, it is easy to calculate the invariant subspaces of the generators, but calculating the invariant subspaces for the whole group seems hard to me. 

