multiplicity of representation

Definition

Given a compact group $G$, with representation $V$ and $W$. Let $W$ be irreducible.

Then ${\dim }_{\mathbb{C}}{\text{Hom}}_G(W,V)$ is called the multiplicity of $W$ in $V$.

Intuition

Why is this somewhat esoteric seeming number called the multiplicity?

$V$ can be decomposed into irreducible subrepresentations

$$V=\underset{i}{⨁}{V}_i.$$

Therefore,

$${\text{Hom}}_G(W,V)={\text{Hom}}_G(W,{\oplus }_i{V}_i)=\underset{i}{⨁}{\text{Hom}}_G(W,V_i).$$

We know from Schur’s Lemma that

$${\dim }_{\mathbb{C}}{\text{Hom}}_G(W,V_i)=\{\begin{array}{ll}1& W\cong V_i\\ 0& W≆V_i\end{array}$$

Therefore, the multiplicity tells us how many times $W$ is isomorphic to one of the subrepresentations $V_i$, or in other words, how many times does $W$ “show up” in the decomposition of $V$.