Maschke's Theorem and Complete Reducibility
Prove that over a field of the right characteristic every representation splits into irreducible pieces, and see exactly where the proof needs that hypothesis.
What a learner can do afterwards
- Prove Maschke's theorem by averaging a projection over the group
- Point at the division by the group order and give a case where the theorem fails
- Apply Schur's lemma to constrain maps between irreducible representations
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Maschke's theorem says that over a field of the right characteristic, every representation of a finite group splits into irreducible pieces. The proof starts with any projection onto a subrepresentation and forces it to respect the group. You conjugate the projection by each group element and average the results: the averaged map is equivariant, and its kernel and image split the space into stable complements.
For the symmetric group on 3 letters, averaging runs over 6 elements, so the sum is divided by 6. A smaller cousin shows the arithmetic: with a group of 3 elements, a sum of conjugated projections with diagonal entries 6 and 6 becomes 2 and 2 after dividing each entry by 3. Division by the group order is what normalizes the sum into a genuine projection.
That division is exactly where the hypothesis lives: one over the group order must exist in the field. When the characteristic divides the group order, the step collapses. The classic failure is the cyclic group of order 2 in characteristic 2, where 2 equals 0 and division is impossible. Over the complex numbers the order is never zero, so every representation splits as a direct sum of irreducibles.
Once a representation splits, Schur's lemma constrains the maps between the pieces: any nonzero map between irreducible representations is an isomorphism. Use it to read off which pieces repeat and how they pair up. If a map between irreducibles is neither zero nor invertible, recheck your irreducibility claim first.
Average conjugates, divide by the order where it exists, and split every representation into irreducibles.
More to read
7.2 Permutations
Permutations illustrate averaging concretely. Conjugating a projection by each group element and averaging yields an equivariant map. Division by the group order normalizes the sum. When order vanishes in the field, the step collapses. Characteristic two with order two shows the breakdown.
3.1 Prime and Composite Numbers
Primes determine characteristics where averaging fails. If characteristic divides group order, one over order does not exist. Order two in characteristic two is the smallest case. Complex numbers avoid the issue entirely. This arithmetic controls complete reducibility.
5.9 Systems of Linear Equations in Two Variables
Systems of linear equations split when equivariant projections exist. Maschke supplies the projection by averaging. Schur lemma then constrains maps between the resulting irreducibles. Nonzero maps become isomorphisms. Endomorphisms reduce to scalars over algebraically closed fields.
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Where it sits
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.