multiplicity of representation

Definition

Given a compact group , with representation and . Let be irreducible.

Then is called the multiplicity of in .

Intuition

Why is this somewhat esoteric seeming number called the multiplicity?

can be decomposed into irreducible subrepresentations

Therefore,

We know from Schur’s Lemma that

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