irreducible representation

Definition

A group representation $G\to V$ is called irreducible if the only subrepresentations are $\{0\}$ and $V$.

Decomposition into irreducible representations

Every compact group representation can be decomposed into a direct sum of irreducible sub-representations.

$$V=\underset{i}{⨁}{V}_i{V}_i\text{irreducible}$$

Proof

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