Character inner products and orthogonality

A character is constant on conjugacy classes because conjugating a group element conjugates its representing matrix, and trace is unchanged by similarity. Its value at the identity is the dimension. Direct sums add characters. For a finite-order complex matrix, every eigenvalue is a root of unity, so replacing the element by its inverse conjugates the trace. This is consistent with choosing an invariant Hermitian form and making all representing matrices unitary, although traces remain the same in the original basis.

Define the inner product on complex class functions to be linear in its first argument and conjugate linear in its second, with the group average as normalization. If a sum is organized by conjugacy classes, each representative must carry the number of elements in that class. Omitting these weights changes the inner product for nonabelian groups. Omitting conjugation breaks positivity for complex-valued characters, a failure already visible for a nontrivial character of the cyclic group of order three.

The group acts on linear maps from W to V by sending a map to the action on V, then that map, then the inverse action on W. The fixed maps are exactly the intertwiners. Averaging these operators gives a projection onto that fixed subspace, and the trace of a projection is its rank. The trace of the operator on the space of maps is the character of V times the conjugate character of W. Thus the inner product of the two characters equals the dimension of the intertwiner space.

For inequivalent irreducibles this dimension is zero, and for equivalent irreducibles it is one by the complex form of Schur’s lemma. Orthogonality follows without choosing decomposition vectors. If a general representation contains an irreducible with some multiplicity, taking its character inner product with that irreducible character returns the multiplicity. The result must be a nonnegative integer, providing a strong diagnostic when a proposed character or class-size list has been entered incorrectly. Rounding a substantially nonintegral answer would conceal an error.

The permutation representation of a group acting on a finite set has character equal to the number of fixed points, since each fixed basis vector contributes one diagonal entry. For the natural action of the symmetric group on three letters, the three class values are three, one, and zero: the identity fixes all points, a transposition fixes one, and a three-cycle fixes none. The fixed constant vector gives a trivial summand, and subtracting its character leaves the standard two-dimensional action.

Its remaining class values are two, zero, and minus one. With class sizes one, three, and two, its squared norm is one, confirming irreducibility. The original permutation action has squared norm two despite dimension three, because it contains two distinct irreducible summands once each. A representation containing two copies of a single irreducible instead has squared character norm four. These distinctions make character norms useful for finding repeated modes and symmetry-allowed linear maps before constructing any preferred basis.