What does the coefficient $a_j$ mean in $\bigoplus_j a_j \Gamma^j$?

Conditions

• The decomposition has been relabeled so that each $j$ corresponds to one inequivalent irreducible representation.

Reasoning, step by step

1. Identify the relabeled direct sum $\Gamma = \bigoplus_j a_j \Gamma^j$.
2. Understand that the index $j$ labels an inequivalent irreducible representation type.
3. Interpret the coefficient $a_j$ as the number of occurrences of $\Gamma^j$ in $\Gamma$.
4. Recognize that if $a_j > 1$, the same irreducible representation appears multiple times.

Example

The lecturer says: 'and then we have a multiplicity number here if a representation appears several times.' The slide states: 'i.e. in $\Gamma$ the irreducible representation $\Gamma^j$ is contained $a_j$ times.'

Common misconceptions

• Thinking that $a_j$ represents the dimension of the irreducible representation $\Gamma^j$.
• Assuming that $a_j$ can be zero for any label $j$ in the sum.

Watch the explanation

Explore next

• Why relabel the blocks by inequivalent irreducible representations instead of by order of appearance?
• What does it mean here that a representation is completely reducible?