Maschke: averaging over the group
Suppose W ⊆ V is a subrepresentation. Can we find a complementary subrepresentation W' so that V = W ⊕ W' and *both* are G-invariant? A naive vector-space complement exists but need not be G-invariant. Maschke's theorem says: if G is finite and char k does not divide |G| (e.g. k = C), then yes — an invariant complement always exists. Consequently every finite-degree representation is completely reducible, a direct sum of irreducibles.
The proof is the averaging trick, the single most important move in the subject. Take any projection π from V onto W (just linear, not yet G-invariant). Average its conjugates over the group: π₀ = (1/|G|) Σ_g ρ(g) π ρ(g)⁻¹. Dividing by |G| is exactly where we need char k ∤ |G|. The averaged map π₀ is a G-equivariant projection onto W, and its kernel is the invariant complement.
Schur: maps between irreducibles
Maschke gives the pieces; Schur's lemma controls how they fit. A G-equivariant (or intertwining) map φ: V → V' is a linear map with φ ρ(g) = ρ'(g) φ for all g. Schur: if V, V' are irreducible then any such φ is either 0 or an isomorphism. Reason: ker φ and im φ are subrepresentations, so each is 0 or everything; the only way to avoid forcing φ = 0 is for φ to be a bijection.
Over an algebraically closed field like C there is a sharper second half: any equivariant φ: V → V from an irreducible to itself is a scalar, φ = λ·id. Why? φ has an eigenvalue λ; then φ − λ·id is equivariant with nonzero kernel, so by the first half it is 0. This is the engine behind every orthogonality relation in guides 3 and 4.
Schur in action: which matrices commute with an irreducible rep?
Take G = Z/4Z acting on C^2 by powers of J = [0,-1; 1,0] (i acts as J).
This 2-dim rep is NOT irreducible over C: J has eigenvalues +i, -i,
so C^2 = (eigenline for +i) (+) (eigenline for -i), two 1-dim subreps.
Matrices commuting with all rho(g): a full 2-dim space {diag in that basis}.
Now an honestly irreducible example: G = Q8 (quaternion group) on C^2
via i -> [i,0; 0,-i], j -> [0,1; -1,0].
The only matrices commuting with BOTH are scalars lambda*I.
That lone scalar-matrix algebra is exactly Schur's lemma over C.Why this is the whole structure theory
Put them together. Maschke says V = m₁V₁ ⊕ m₂V₂ ⊕ ··· decomposes into irreducibles. Schur says the multiplicities mᵢ are unique: the number of copies of Vᵢ equals dim Hom_G(Vᵢ, V), which is basis-independent. So a finite-group representation over C is determined up to isomorphism by a list of nonnegative integers — one per irreducible. This is exactly the Artin–Wedderburn decomposition of k[G] read through the semisimple dictionary.