I am now reading the book 'Invariant theory' by Neusel. It seems that the logic is like this: You have a representation of a group $G$ on some vector space $V$, this might be a reducible representation. Actually, in the book by Neusel, she always takes a reducible representation. Then you try to find invariant polynomials.
It seems that the irreducible representation idea is not so relevant here.
Is it so?