Symmetric and exterior power of representation Does there exist some simple formulas for the characters
$$\chi_{\Lambda^{k}V}~~~~\text{and}~~~\chi_{\text{Sym}^{k}V},$$
where $V$ is a representation of some finite group?
Thanks.

I ran into this question when trying to find a reference for some formulas that I think should be true. I will leave them here, in case they are of some help to someone. Here are the formulas, and I comment on the derivation below:
$$
\sum_{n=0}^{\infty}t^n\chi_{S^n\alpha}(g)=\exp\left\{\sum_{p=1}^{\infty}t^p\frac{\chi_\alpha(g^p)}{p}\right\},\\
\sum_{n=0}^{\infty}t^n\chi_{\Lambda^n\alpha}(g)=\exp\left\{-\sum_{p=1}^{\infty}(-t)^p\frac{\chi_\alpha(g^p)}{p}\right\}.
$$
And in particular
$$
\sum_{n=0}^{\infty}t^n\chi_{S^n\alpha}=\frac{1}{\sum_{n=0}^{\infty}(-t)^n\chi_{\Lambda^n\alpha}}.
$$
The latter formula also can be seen as a direct consequence of the facts described in anon's answer and the relation between generating functions for complete homogeneous symmetric polynomials and for elementary symmetric polynomials.
As for the first two formulas, the key observation is given in Qiaochu's answer
$$
\chi_{\Lambda^n \alpha}(g)=\frac{1}{n!}\sum_{\pi\in S_n}\mathrm{sgn}{\pi} \,\mathrm{tr}_\pi \alpha(g),
$$
where $\mathrm{tr}_\pi(A)$ is defined in Qiaochu's answer and in anon's answer above. Slight modification gives
$$
\chi_{S^n \alpha}(g)=\frac{1}{n!}\sum_{\pi\in S_n}\mathrm{tr}_\pi \alpha(g).
$$
Now these can be more or less straightforwardly made into the generating functions above by the use of Polya enumeration (Qiaochu gives the relevant corollary, $Z_G$ is defined here). One just has to express the sign of the permutation in terms of the cycle type.
