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

Conditions

• The original block diagonal decomposition is available.
• Equivalent irreducible summands are grouped under the same label.

Reasoning, step by step

1. Start with the block diagonal form where blocks are numbered by order of appearance.
2. Recognize that some blocks may represent the same irreducible representation.
3. Relabel the blocks so that the index $j$ refers to an inequivalent irreducible representation type.
4. Group all copies of the same irreducible type together.
5. Introduce the multiplicity number $a_j$ to count the occurrences of $\Gamma^j$.
6. Write the compact expression $\Gamma = \bigoplus_j a_j \Gamma^j$.

Example

The lecturer says: 'And now it may make sense to just use these labels 1, 2, 3 and so on as labels for inequivalent irreducible representations.' The slide introduces '(relabel)' and rewrites the decomposition as $\Gamma = \Gamma^1 \oplus \cdots \oplus \Gamma^1 \oplus \Gamma^2 \oplus \cdots \oplus \Gamma^2 \oplus \cdots = \bigoplus_j a_j \Gamma^j$.

Common misconceptions

• Thinking that the indices 1, 2, 3, ... always label distinct irreducible types simply because they mark different diagonal blocks.
• Believing that relabeling changes the mathematical content of the decomposition.

Watch the explanation

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