Lemma 13.2.1 (Existence of -invariant inner product). If is a -representation of a finite group , there exists a positive-definite Hermitian inner product on that is -invariant: for all ,. In other words, all operators are unitary with respect to .
Proof. Start with any Hermitian inner product on , and average over : Then is still Hermitian and positive-definite (a positive combination of positive-definite forms), and -invariant: (reindexing ).∎
Theorem 13.2.1 (Maschke’s Theorem). Let be a representation of a finite group over (or any field with ), and let be a subrepresentation. Then there exists a complementary subrepresentation with .
Proof 1 (via invariant inner product). Equip with a -invariant Hermitian inner product (by the lemma). Since each is unitary, (if and , then since ). So is a -invariant complement. ∎
Proof 2 (via averaging projection). Choose any complementary subspace with (not necessarily -invariant). Let be the projection onto with kernel (,). Average: Then is -equivariant (), and (since for , as ). So is a -equivariant projection onto , and is a -invariant complement. ∎
Note. Maschke’s theorem fails when divides (modular representation theory), and also fails for infinite groups in general: the representation of (or ) on via has the invariant subspace but no complementary invariant subspace.
Corollary 13.2.1 (Complete reducibility). Any finite-dimensional representation of a finite group (over ) decomposes as a direct sum of irreducible representations.