orthogonality relations

The orthogonality relations say that the irreducible characters of a finite group, viewed as vectors of numbers, are mutually perpendicular and of unit length under a natural inner product. This single geometric fact turns the messy bookkeeping of representations into clean linear algebra: to test irreducibility, decompose a representation, or count copies of an irreducible, you just take inner products.

Define on class functions of G over C the Hermitian inner product (a, b) = (1/|G|) sum over g in G of a(g) times the complex conjugate of b(g). The first orthogonality relation (row orthogonality) states that the irreducible characters chi_i satisfy (chi_i, chi_j) = 1 if i = j and 0 otherwise — they form an orthonormal system, in fact an orthonormal basis of the class functions. The second orthogonality relation (column orthogonality) is the dual statement on columns of the character table: summing chi_i(g) times the conjugate of chi_i(h) over all irreducibles gives |C_G(g)| if g and h are conjugate, and 0 otherwise.

These relations descend from Schur's lemma applied to averaged intertwiners between irreducibles, then take their cleanest form as character identities. Their consequences are pervasive: the multiplicity of an irreducible chi_i in a representation with character psi is exactly (psi, chi_i); a class function is a character of an actual representation iff it is a nonnegative-integer combination of the chi_i; and (psi, psi) = 1 characterizes irreducibility.