Reducibility of representations

Character irreducibility criterion

Let be a complex Group representation and be the corresponding Group character. Then is reducible iff #m/thm/rep

and otherwise the sum is .

Proof

Let be a (in general reducible) representation with

i.e. each irrep occurs times. Then it follows from the definition of a character as a trace that

and then since by Orthonormality of irreducible characters

it follows that

as required.

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