Symmetrizer and antisymmetrizer elements

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

Let $Γ 𝜇, Γ 𝜈$ be irreps of $𝑆 𝑛$ , and $Γ 𝜇 \otimes 𝜈 = Γ 𝜇 \otimes Γ 𝜈$ be their tensor product. Then the decomposition of $Γ 𝜇 \otimes 𝜈$ contains #m/thm/rep/sym

$𝜒 𝔰$ exactly once iff $Γ 𝜇 ≅ Γ 𝜈$ are equivalent representations, otherwise not at all
$𝜒 𝔞$ exactly once iff $Γ 𝜇 = 𝜒 𝔞 \otimes Γ 𝜈$ 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

(𝜒 𝔰 | 𝜒 𝜇 \otimes 𝜈) = 1 𝑛! \sum 𝑝 \in 𝑆 𝑛 ―――― 𝜒 𝜇 (𝑝) 𝜒 𝜈 (𝑝) = (𝜒 𝜇 | 𝜒 𝜈) (𝜒 𝔞 | 𝜒 𝜇 \otimes 𝜈) = 1 𝑛! \sum 𝑝 \in 𝑆 𝑛 ―――――― 𝜒 𝔞 (𝑝) 𝜒 𝜇 (𝑝) 𝜒 𝜈 (𝑝) = (𝜒 𝔞 \otimes 𝜇 | 𝜒 𝜈)

Since the right hand inner products only involve irreps, the first is one iff $Γ 𝜇 ≅ Γ 𝜈$ and zero otherwise, while the second is one iff $Γ 𝔞 \otimes 𝜇 ≅ Γ 𝜈$ , i.e. they are associate representations, and zero otherwise.

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