Proposition 13.5.1 (Averaging projection). If is a representation of , the map is a projection onto , the space of -invariant vectors. In particular, .
Note. For representations :, so . If and are irreducible, Schur’s lemma gives .
Definition 13.5.1 (Hermitian inner product on class functions). Define by
Theorem 13.5.1 (Character orthogonality). The characters of the irreducible representations of are orthonormal with respect to . That is, if are the distinct irreducible representations of , then In particular, the characters of distinct irreducible representations are linearly independent as class functions.
Corollary 13.5.1 (Number of irreducibles). The number of irreducible representations of is at most the number of conjugacy classes of (since the irreducible characters are linearly independent class functions, and the space of class functions has dimension = number of conjugacy classes). In fact, equality holds: the number of irreducible representations equals the number of conjugacy classes.
Corollary 13.5.2 (Character determines representation). Every representation of is completely determined (up to isomorphism) by its character. If , then , and .
Corollary 13.5.3 (Irreducibility criterion). A representation is irreducible if and only if . More generally, where , so the inner product counts the “number of irreducible summands” (with multiplicity squared).
13.5.2 The Regular Representation
Definition 13.5.2 (Regular representation). The regular representation of is the vector space with basis , on which acts by left multiplication: .
Proposition 13.5.2 (Decomposition of the regular representation). In the regular representation, each irreducible appears with multiplicity :.
Proof. (permutation representation where only has fixed points). Then .∎
Corollary 13.5.4 (Dimension formula)., where the sum is over all irreducible representations.
Proof..∎
13.5.3 Column Orthogonality
The row orthogonality relations (character orthonormality) have a dual version: orthogonality of the columns of the character table.
Proposition 13.5.3 (Column orthogonality relations). Let be the conjugacy classes of , with representatives and sizes . Then
Proof. Consider the character table matrix with entries . Row orthonormality states . In matrix form, where . Since is square and invertible, . Taking the transpose: , so , giving . Entry-by-entry, this is the column orthogonality relation. ∎
Note. Column orthogonality provides an independent check on computed character tables. It also has a group-theoretic interpretation: two elements are conjugate if and only if for every representation . Characters separate conjugacy classes.