What does it mean here that a representation is completely reducible?

Conditions

• $\Gamma : G \to GL(V)$ is a completely reducible representation.
• One chooses a suitable basis of $V$.

Reasoning, step by step

1. Start with a completely reducible representation $\Gamma : G \to GL(V)$.
2. Choose a suitable basis of the vector space $V$.
3. Express the representation matrix $\Gamma(g)$ in this new basis.
4. Observe that the matrix takes a block diagonal form.
5. Identify the diagonal blocks as irreducible representations.

Example

The lecturer states: 'So first we have a completely reducible representation. This means by a suitable choice of basis we can write the representation matrices in this block diagonal form.' The slide displays $\Gamma(g)$ as a block diagonal matrix with $\Gamma^1(g), \Gamma^2(g), \Gamma^3(g), \ldots$ on the diagonal and zeros off the diagonal.

Common misconceptions

• Assuming that the representation is already in block diagonal form without changing the basis.
• Believing that the diagonal blocks can be reducible representations.

Watch the explanation

Explore next

• Why relabel the blocks by inequivalent irreducible representations instead of by order of appearance?
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