Character for kth symmetric and exterior power I am trying to compute the character of $\Lambda^k V$ and Sym$^k V$ for V an arbitrary representation of a group G. I already know how the characters look for $k=2$, but I cannot find a way to generalize that. My attempt would be to find a way to decompose $\Lambda^k V$ and Sym$^k v$ into powers of 2 and 1, but I am not sure that is possible. Does someone have an idea to solve that? 
 A: A partial answer: here is a a formula for $\chi_{\vee^kV}(g)$.
We note that $\chi_{\vee^kV}(g) = e_k(\lambda(g))$ and $\chi(g^k) = p_k(\lambda(g))$, where $e_k$ denotes the $k$th elementary symmetric polynomial, $p_i$ denotes the $i$th moment polynomial $p_i(x_1,\dots,x_n) = x_i^k + \cdots + x_n^k$, and $\lambda(g)$ denotes the vector of eigenvalues of $g$.  Newton's identities state that
$$
e_k = \frac 1k \sum_{i=1}^k e_{k-i} \cdot p_i.
$$
Plugging in the vector of eigenvalues into both sides of the equation yields
$$
\chi_{\vee^kV}(g) = \frac 1k \sum_{i=1}^k \chi_{\vee^{k-i}V}(g) \cdot \chi(g^i).
$$
This gives you a recursive formula for $\chi_{\vee^kV}(g)$ with arbitrary $k$.

To complete this answer, the corresponding formula for the alternating character (according to the post linked above) is 
$$
\chi_{\Lambda^k V}(g)=\frac{1}{k}\sum_{m=1}^k(-1)^{m-1}\chi_{\Lambda^{k-m}V}(g) \cdot \chi(g^m).
$$

If you look further along the Newton's identities page, there are further expressions for the elementary polynomials that could be applied here.  For instance,
$$
e_n = \frac1{n!}\begin{vmatrix}
    p_1     & 1       & 0      & \cdots       \\
    p_2     & p_1     & 2      & 0      & \cdots \\
    \vdots  &         & \ddots & \ddots       \\
    p_{n-1} & p_{n-2} & \cdots & p_1    & n-1 \\
    p_n     & p_{n-1} & \cdots & p_2    & p_1
  \end{vmatrix} \implies\\
\chi_{\vee^n}(g) = \frac1{n!}\begin{vmatrix}
    \chi(g)     & 1       & 0      & \cdots&0       \\
    \chi(g^2)     & \chi(g)     & 2      & \ddots      & \vdots \\
    \vdots  &         & \ddots & \ddots &0      \\
    \chi(g^{n-1}) & \chi(g^{n-2}) & \cdots & \chi(g)    & n-1 \\
    \chi(g^n)     & \chi(g^{n-1}) & \cdots & \chi(g^2)    & \chi(g)
  \end{vmatrix}
$$
