Let $V$ be a representation with character $\chi$. I would like to have a formula for the characters of the representations $\mathrm{Sym}^m[V]$ and $\wedge ^m[V]$ in terms of $\chi$. Fulton and Harris presents such a formula for $m=2$, but not for general $m$. Is there a "nice" such expression?
I found a paper by Zhou and Pulay, but this doesn't seem to be what I'm looking for (it doesn't seem to give a general formula in terms of $\chi$).
In case the link does not work, here is the reference:
X. Zhou, P. Pulay. Characters for Symmetric and Antisymmetric Higher Powers of Representations: Application to the Number of Anharmonic Force Constants in Symmetrical Molecules. Journal of Computational Chemistry, Vol 10, Issue 7, (1989).