Maschke's theorem ↩

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created: 2021-10-19 14:22:42
modified: 2022-01-10 04:13:04

Very shortly: every finite dimensional representation of a finite group is fully reducible.

Every finite dimensional representation of every finite order (or compact) group is equivalent to a unitary representation, and hence fully reducible.

Proof steps

  1. Define a new scalar product (for compact groups the sum is an integral instead)
  2. Define a new basis in the space equipped with the new scalar product. The new basis can be expressed in the old basis using the Gram–Schmidt ortogonalization. Then there is a basis transformation between these bases.
  3. With this new scalar product, our representation is unitary and we can find a similarity transformation (the basis transformation) that brings into that's unitary for the old scalar product
  4. We see that so it is unitary-equivalent fully reducible according to Every reducible unitary representation is fully reducible