We proved that the irreducible characters are orthonormal with respect to the Hermitian inner product on the space of class functions . We also showed that the number of irreducible representations is at most the number of conjugacy classes. We now prove that these characters in fact form a basis for the space of class functions, completing the proof that the number of irreducible representations equals the number of conjugacy classes.
The key tool is a generalization of the averaging projection.
Proposition 14.1.1 (Generalized averaging operator). Given any class function and any representation of , define Then is -equivariant (i.e. ).
Proof. For any , where we substituted (so ) and used that is a class function. ∎
Note. When (the constant function), is the projection onto that we used previously.
Theorem 14.1.1 (Characters form an orthonormal basis). The characters of the irreducible representations of form an orthonormal basis for the space of class functions with respect to . In particular, the number of irreducible representations equals the number of conjugacy classes.
Proof. We already proved orthonormality. It remains to show spanning: if for all , then .
Given any class function and any irreducible representation of dimension , the operator is -equivariant by the proposition above (since is also a class function). By Schur’s lemma, for some . Taking traces:
So if for all irreducible , then for all irreducible . By complete reducibility, for every representation of (since respects direct sums). In particular, taking the regular representation with basis :
Since the are linearly independent, for all , hence .∎
14.1.1 Projection onto Isotypic Components
The proof above also yields an explicit projection formula. For irreducible, consider the class function . Then
Proposition 14.1.2 (Isotypic projection formula). If is any representation of with decomposition , then the map where , is the projection onto the isotypic component (i.e. the identity on and zero on all other summands).
Note. The case (trivial representation, ,) recovers the averaging projection onto .