I am new to representation theory.
Suppose that $G$ is a finite group with an irreducible representation over a (real or complex) vector space $V$. In my application, $G$ is a symmetric group and the representation is faithful.
What can be said about the representation of $G$ over the $k$-th exterior power $\Lambda^k V$ of $V$? I am particularly interested how to decompose the exterior power of the representation into irreducible representations in a canonical manner.
I presume that this is a standard topic, so perhaps a reference to an exposition would be helpful, or a brief outline what to expect in this situation.
Best Answer
I don't know what kind of answer you're expecting at this level of generality. $\wedge^k V$ is irreducible as a representation of $GL(V)$, so in some sense there is no additional decomposition that can be done knowing nothing about the group $G$.
Given the character of $V$, you can compute the character of $\wedge^k V$. For example,
$$\chi_{\wedge^2 V}(g) = \frac{\chi_V(g)^2 - \chi_V(g^2)}{2}.$$
Hence if you know all of the irreducible characters of $G$, you can compute the irreducible decomposition of $\wedge^k V$ using character theory.