Orthonormality of irreducible characters
Let
Proof of orthonormality and completeness
That each
is central follows from Properties, since π π π π ( π¦ π₯ π¦ β 1 ) = t r β‘ Ξ π ( π¦ π₯ π¦ β 1 ) = t r β‘ Ξ π ( π₯ π¦ β 1 π¦ ) = t r β‘ Ξ π ( π₯ ) = π π ( π₯ ) Orthonormality follows easily from Orthonormality of irreps:
( π πΌ | π π½ ) = π πΌ β π = 1 π π½ β π = 1 ( Ξ πΌ π π | Ξ π½ π π ) = π πΌ β π = 1 π π½ β π = 1 1 π πΌ πΏ πΌ π½ πΏ π π = πΏ πΌ π½ Completeness follows from that of irreps too, by first noting
1 | πΊ | β π¦ β πΊ Ξ π π π ( π¦ π₯ π¦ β 1 ) = 1 | πΊ | β π¦ β πΊ β π , π Ξ π π π ( π¦ ) Ξ π π π ( π₯ ) ββββ Ξ π π π ( π¦ ) = β π , π ( Ξ π π π | Ξ π π π ) Ξ π π π ( π₯ ) = β π , π 1 π π πΏ π π πΏ π π Ξ π π π ( π₯ ) = 1 π π πΏ π π π π ( π₯ ) and therefore for any
, from completness of irreps π β π ( β [ πΊ ] ) for some π = β π ; π , π π π π π Ξ π π π , thus π π π π π ( π₯ ) = 1 | πΊ | β π¦ β πΊ π ( π¦ π₯ π¦ β 1 ) = β π ; π , π π π π π 1 | πΊ | β π¦ β πΊ Ξ π π π ( π¦ π₯ π¦ β 1 ) = β π ; π , π π π π π 1 π π πΏ π π π π ( π¦ ) thus
. π β s p a n β‘ { π π }
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 π β π ( β [ πΊ ] ) commutes in its Μ π πΌ matrix algebra, which includes with the concrete reΓ€lization of π πΌ Γ π πΌ Ξ πΌ Μ π πΌ Ξ πΌ ( π ) = Ξ πΌ ( π ) Μ π πΌ and therefore by Schurβs lemma
so Μ π πΌ = π πΌ π πΌ π = 1 | πΊ | β πΌ ; π π π πΌ π πΌ πΏ π π Ξ πΌ π π = 1 | πΊ | β πΌ π πΌ π πΌ π πΌ as required.
Since
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
Footnotes
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1996, Representations of finite and compact groups, Β§III.1 β©