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.

This is not quite an answer, but Fulton & Harris, §2.1 on page 13 gives a Formula for $k=2$:
$$\chi_{\bigwedge^2 V}(g) = \frac{1}{2}\cdot\left( \chi_V(g)^2 - \chi_V(g^2)\right)$$
as well as, in the Exercise below,
$$\chi_{\mathrm{Sym}^2(V)}(g) = \frac{1}{2}\cdot\left( \chi_V(g)^2 + \chi_V(g^2)\right)$$
Maybe you can look into the proof for the first equality and generalize.
