I have a little problem in representation and/or invariant theory which I need help with.
Let's assume $G \leq \mathbb{C}^{n\times n}$ is a finite complex matrix group which operates linearly via $gp\mapsto p \circ g^{-1}$ on the vectorspace of polynomials over $\mathbb{C}$ in $n$ variables and I am given a natural number $k$. How do I compute the character of the corresponding representation if I restrict the operation to the finitely generated vectorspace of homogeneous polynomials of degree $k$? I already know the character of the given $n$-dimensional representation.
Any help would be greatly appreciated.