Complete Reducibility

Let $V$ be a representation of $G$. Let be an inner product on$V$. Let $W\subset V$ be a sub-representation. The inner product

$(v,w)=\frac{1}{|G|}{\sum }_{g\in G}⟨\rho (g)v,\rho (g)w⟩$

on $V$ is $G$-invariant.

Remark: Every representation of a finite group is unitary: by changing the inner product to claim 1.7, we get that all operators $\rho (x),x\in G$ are unitary.

Proposition(Complete Reducibility):
Every representation is a direct sum of irreducible ones.

Lemma 1.11 (Schur’s Lemma):
Let $G$ be a finite group and $V,W$ irreducible representations. Let $T:V\to W$ be a $G$-map. Then

• Either $T$ is zero or an isomorphism.

• If $V=W$, then $T=\lambda {\text{Id}}_V$ for some $\lambda \in \mathbb{C}$.