The alternating group $G=A_4$ acts by permutations on a set of four elements. I am considering its permutation representation on $\mathbb{R^4}$ (or $\mathbb{C^4}$). Clearly $(1,1,1,1)$ is invariant so generates a $1$-dimensional $\mathbb{C}$G-submodule. How do I find a complementary $\mathbb{C}$G-submodule to this?
My thinking is that it would obviously have to be a three dimensional space, but I don't know how to find it - I think this comes from my lack of understanding of a general strategy for going about finding complementary submodules. I would imagine that a solution arises from basic results of character theory, but I'm very unsure.
Any insight would be appreciated!
You should trust your geometric intuition! Why not take the three-dimensional subspace in $V = \mathbb C^4$ that is orthogonal to $ (1,1,1,1)$? You can very easily check that this three-dimensional orthogonal complement is preserved under the action of $G = A_4$, and hence, is the complementary $\mathbb CG$-submodule that you are looking for.
The reason why this orthogonality idea works so well in this example is that the natural inner product on our $\mathbb C G$-module $V $ is invariant under the action of $G$: that is, for every $v_1, v_2 \in V$ and for every $g \in G$, we have $\langle g(v_1), g(v_2) \rangle = \langle v_1,v_2 \rangle $. It is this observation that enables us to prove that, whenever $W$ is a $\mathbb C G$-submodule of our $\mathbb C G$-module $V$, its orthogonal complement $W^\perp$ is also a $\mathbb C G$-submodule of $V$.
But what are we to do if we're faced with a horrible example where the natural inner product on $V$ is not invariant under the action of $G$? In this situation, we can engineer a new inner product that is invariant under the action of $G$. We do this by taking a group average over the old inner product, i.e. we define $\langle v_1, v_2\rangle _{\rm new} = \frac{1}{|G|} \langle g(v_1), g(v_2) \rangle_{\rm old}$. The complementary submodule to $W$ is then the orthogonal complementof $W$ with respect to our new inner product.