2

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.

shuhalo
  • 8,084

1 Answers1

5

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.

Qiaochu Yuan
  • 468,795
  • 1
    Also, if your representation of $G = S_n$ happens to be $\mathbb{C}^{n-1}$, then all of its exterior powers are irreducible. This is a nice exercise. – Qiaochu Yuan Dec 25 '15 at 21:47
  • Exactly that the case in the application. Thank you for pointing out. – shuhalo Dec 26 '15 at 01:36
  • @QiaochuYuan Is there an easy way to see that $\bigwedge^k V$ is irreducible as a representation of $GL(V)$? Thanks. I know this is classical, but google search didn't help. – Asaf Shachar Jan 02 '19 at 12:51
  • 1
    @Asaf: I don't know a proof "from the Book" off the top of my head. I would decompose the result into two lemmas: 1) every nonzero subrepresentation contains a pure tensor, and 2) $GL(V)$ acts transitively on pure tensors up to scale. The second lemma is straightforward so it's the first lemma that's interesting. If you wanted me to give a more detailed proof you could ask a separate question. – Qiaochu Yuan Jan 02 '19 at 19:10
  • 1
    Thanks, that was my approach too. I even asked here a more naive question about whether or not every subspace (of dimension $>1$) contains a pure tensor. Anyway, I also asked here for a proof of the second lemma. I am very interested to know how can we "produce" a pure tensor for any subrepresentation... – Asaf Shachar Jan 03 '19 at 08:58