Characters and reality types

The character

The character χR(g) is the trace of the representing matrix. It is constant on conjugacy classes. For finite groups over ℂ, irreducible characters distinguish irreducible representations up to equivalence.

Use the inner product

The class-function inner product is the group average of the product of one character with the complex conjugate of the other. Orthogonality lets it extract multiplicities in tensor products and restricted representations.

Read the identity column

The value at the identity is the representation dimension. The sum of the squares of all irreducible dimensions equals the group order. This is one useful consistency check, although it does not determine the character table.

Reality needs more than real values

The Frobenius–Schur indicator distinguishes real type (+1), complex type (0), and quaternionic type (−1). The Q₈ doublet has a real-valued character but quaternionic type. Compare it with the real-type doublet of the dihedral group of order eight.

Use a concrete table

The A₄ page supplies the conjugacy classes, their sizes, and its four ordinary irreducible characters. The invariant tool applies these data to tensor and symmetric powers of the triplet.

Derive and practice

The finite-character formulas state the conjugation, class-size, and power-map conventions explicitly. Work through the S₃ and Q₈ exercise to derive complete decompositions and see why a quaternionic doublet has no identical-scalar quadratic singlet. The finite-symmetry path connects these calculations to a physical model.

Continue in the atlas

a4 · q8 · d8

Sources

• GAP · Characters and symmetrizations
• GAP · Character tables and Frobenius–Schur indicators
• GroupNames · A₄

Source curation and automated validation are separate from independent expert review.