Group character

Orthonormality of irreducible characters

Let be irreducible characters for each . Then form an Orthonormal basis of the Centre of the group ring (i.e. class functions into ) under a certain inner product.¹ rep In particular,

Proof of orthonormality and completeness

That each is central follows from Properties, since

Orthonormality follows easily from Orthonormality of irreps:

Completeness follows from that of irreps too, by first noting

and therefore for any , from completness of irreps for some , thus

thus .

Alternate proof of completeness via Schur's lemma and matrix algebra isomorphism

Let . Then by the Isomorphism between the complex group ring and direct sum of matrix algebras on carriers of irreducible representations, each commutes in its matrix algebra, which includes with the concrete reälization of

and therefore by Schur’s lemma so

as required.

Since are class functions, the orthonormality may be rewritten for the character table.

Corollaries

• The number of conjugacy classes equals the number of non-equivalent irreps of a group
• The decomposition of a character into irreducible characters is always possible reveals the composition of the characterised representation.
• Character irreducibility criterion

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Footnotes

1. 1996, Representations of finite and compact groups, §III.1 ↩