Symmetrizer and antisymmetrizer elements

Trivial and alternating characters of a finite symmetric group in tensor product decomposition

Let be irreps of , and be their tensor product. Then the decomposition of contains #m/thm/rep/sym

exactly once iff are equivalent representations, otherwise not at all
exactly once iff are associate representations, otherwise not at all

Proof

Using Orthonormality of irreducible characters and the fact that Characters of a finite symmetric group are real to find multiplicities

Since the right hand inner products only involve irreps, the first is one iff and zero otherwise, while the second is one iff , i.e. they are associate representations, and zero otherwise.

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