Suppose I have a matrix group $G \leq GL_n$, acting on $V = k^n$ in the usual way. Is there some way to calculate the $G$-invariant subspaces of $V$? Failing that, is there an easy way to ask if $V$ is irreducible as a $G$-module?
Suppose I have a [EDIT: finitely-generated] matrix group $G \leq GL_n$, acting on $V = k^n$ in the usual way. Is there some way to calculate the $G$-invariant subspaces of $V$? Failing that, is there an easy way to ask if $V$ is irreducible as a $G$-module?