Group representation theory MOC

Irreducible orthonormal basis

Let be a unitary representation of a finite group with decomposition

and let be an orthonormal basis transforming under in¹ a unitary irrep for each , i.e. for all

then .² rep

Proof

Applying orthogonality or irreps on the third line:

as required.

are thus called irreducible basis vectors transforming under irrep . Every may then be expressed as such, with

and the application of gives

which motivates the Generalized projection operator of a representation.

Explanation

Irreducible basis functions have special symmetry properties under , and the above theorem basically states these functions are orthogonal to each other.

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Footnotes

1. the invariant subspace of corresponding to ↩

2. 2023, Groups and representations, p. 44 (§4.1 lemma 8) ↩