What would you like to understand?

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 $V$.

Relabeling the blocks by inequivalent irreducible representations groups equivalent irreducible summands under a single label. This makes it possible to introduce a multiplicity number $a_j$ that records how many times the irreducible representation $\Gamma^j$ occurs in $\Gamma$, leading to the compact expression $\Gamma = \bigoplus_j a_j \Gamma^j$.

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

Relabeling the blocks by inequivalent irreducible representations groups equivalent irreducible summands under a single label. This makes it possible to introduce a multiplicity number $a_j$ that records how many times the irreducible representation $\Gamma^j$ occurs in $\Gamma$, leading to the compact expression $\Gamma = \bigoplus_j a_j \Gamma^j$.

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

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.