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 #m/thm/rep

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.

#state/tidy | #lang/en | #SemBr