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In this clip, a completely reducible representation $\Gamma : G \to GL(V)$ means that by choosing a suitable basis of $V$, every representation matrix $\Gamma(g)$ can be written in block diagonal form, with irreducible representations placed along the diagonal.

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

In this clip, a completely reducible representation $\Gamma : G \to GL(V)$ means that by choosing a suitable basis of $V$, every representation matrix $\Gamma(g)$ can be written in block diagonal form, with irreducible representations placed along the diagonal.

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

The coefficient $a_j$ is the multiplicity number, which records how many copies of the irreducible representation $\Gamma^j$ occur inside $\Gamma$. If $a_j > 1$, the same irreducible type appears on several different subspaces.

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

The coefficient $a_j$ is the multiplicity number, which records how many copies of the irreducible representation $\Gamma^j$ occur inside $\Gamma$. If $a_j > 1$, the same irreducible type appears on several different subspaces.

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