Maschke's theorem tells us that any representation of a finite group $G$ can be decomposed into a direct sum of irreducible representations. The proof does make intuitive sense to me (Intuition behind Maschke's theorem), but my question is really about why we should expect it to be true in the first place.
Sure, we can do some explicit examples (perhaps a most obvious starting point is the standard permutation representation of $S_n$, which has an obvious invariant subspace $\{(x,x,\ldots, x) \in \mathbb C^n: x\in \mathbb C\}$, and almost as obvious complement subspace $\{(x_1,\ldots, x_n) \in \mathbb C^n: \sum x_i = 0\}$), and perhaps one has already seen the result for abelian groups over $\mathbb C$ in the guise of linear algebra: Matrices commute if and only if they share a common basis of eigenvectors?.
However I wonder if there is any other point of view that makes Maschke's theorem feel "inevitable"…since right now, it just seems absurdly powerful/magical.
Best Answer
If $\rho \colon G \to \operatorname{GL}(V)$ is a representation of a finite group $G$ on a finite-dimensional $\mathbb{C}$-vector space $V$, then $V$ can be equipped with an inner product $\langle -, - \rangle \colon V \times V \to \mathbb{C}$ such that every element of $G$ acts by a unitary matrix. This inner product is not mysterious either: take any old inner product $(-, -) \colon V \times V \to \mathbb{C}$ and average it over the group: $$ \langle u, v \rangle := \frac{1}{|G|} \sum_{g \in G} (\rho(g) u, \rho(g) v).$$ (The same is true over $\mathbb{R}$ if we replace "unitary" with "orthogonal"). The slogan is: we can always find a basis such that every $\rho(g)$ is a unitary matrix.
This makes Maschke's theorem completely inevitable: if a subspace $W \subseteq V$ is invariant under a unitary operator $\rho(g)$, then its orthogonal complement $W^\perp$ is also invariant under $\rho(g)$. So every subrepresentation has a complement, and moreover, doing standard orthogonal projection stuff is an effective way to decompose a vector into its components in subrepresentations.
Of course, Maschke's theorem is true in more generality than just $\mathbb{R}$ and $\mathbb{C}$, and this inner product argument really only works over them, but I think it is illustrative of why it is completely inevitable in certain contexts, and many of the other settings end up being related to these settings anyway.