Exterior power representation of $S_n$

Question

Let $V = \left\{x \in \mathbb C^n \mid x_1 + x_2 + \cdots + x_n = 0 \right\} \subset\mathbb C^n$. We know that $V$ is an irreducible representation of $S_n$. Why is the representation $\Lambda^k(V)$ irreducible for all $k$ with $1\leq k\leq n-1$?

Answer 1

It is a general fact that if $W$ is a finite (complex) reflection group with irreducible reflection representation $V$ then the exterior powers $\Lambda^i(V)$ are all irreducible. If $W$ can be generated by $\mathrm{dim}(V)$ reflections, then this is the content of exercise 3. to §2 of Chapter 5 of Bourbaki's book Lie groups and Lie algebras, a fact which Bourbaki credits to Steinberg (this already answers your question, modulo the exercise and the fact that the symmetric group is such a reflection group). The general case is a corollary: any irreducible reflection group contains a subgroup that is generated by $\mathrm{dim}(V)$ reflections and which still acts irreducibly on $V$.