Representation theory of finite symmetric groups

Symmetrizer and antisymmetrizer elements

The symmetrizer and antisymmetrizer are essential idempotents of the complex group ring $ℂ [𝑆 𝑛]$ of the symmetric group $𝑆 𝑛$ , defined as follows #m/def/rep/sym

𝔰 = \sum 𝑝 \in 𝑆 𝑛 𝛿 𝑝 𝔞 = \sum 𝑝 \in 𝑆 𝑛 s g n (𝑝) 𝛿 𝑝

The symmetrizer $𝔰$ generates the left ideäl carrying the trivial representation, whereas the antisymmetrizer $𝔞$ generates that carrying the alternating character.

Proof

For the symmetrizer see Trivial irrep carrying ideal of the group ring. For the antisymmetrizer note

𝔞 2 = \sum 𝑝, 𝑞 \in 𝑆 𝑛 s g n (𝑝) 𝛿 𝑝 * s g n (𝑞) 𝛿 𝑞 = \sum 𝑝, 𝑞 \in 𝑆 𝑛 s g n (𝑝 𝑞) 𝛿 𝑝 𝑞 = 𝑛! \sum 𝑝 \in 𝑆 𝑛 s g n (𝑝) 𝛿 𝑝 = 𝑛! 𝔞

and for any $𝑝 \in 𝑆 𝑛$ and $𝑓 \in ℂ [𝑆 𝑛]$

Λ (𝑝) 𝑓 * 𝔞 = \sum 𝑥, 𝑦 \in 𝑆 𝑛 𝑓 (𝑥) s g n (𝑦) 𝛿 𝑝 𝑥 𝑦 = \sum 𝑥, 𝑧 \in 𝑆 𝑛 𝑓 (𝑥) s g n (𝑥 - 1 𝑝 - 1 𝑥 𝑧) 𝛿 𝑥 𝑧 = s g n (𝑝) \sum 𝑥, 𝑧 \in 𝑆 𝑛 𝑓 (𝑥) s g n (𝑧) 𝛿 𝑥 𝑧 = s g n (𝑝) 𝑓 * 𝔞

where we used

s g n 𝑥 - 1 𝑝 - 1 𝑥 𝑧 = s g n 𝑥 - 1 s g n 𝑝 - 1 s g n 𝑥 s g n 𝑧 = s g n 𝑥 s g n 𝑝 s g n 𝑥 s g n 𝑧 = (s g n 𝑥 ⏟ \pm 1) 2 s g n 𝑝 s g n 𝑧 = s g n 𝑝 s g n 𝑧

Thus $𝔞$ generates the minimal ideäl carrying $𝜒 𝔞$ . The nonequivalence of these irreps may also be shown using the Equivalence of irreps on left ideals criterion: For any $𝑓 \in ℂ [𝐺]$ ,

𝔰 * 𝑓 * 𝔞 = 𝔰 * 𝔞 = \sum 𝑝, 𝑞 \in 𝑆 𝑛 s g n (𝑞) 𝛿 𝑝 𝑞 = \sum 𝑝 \in 𝑆 𝑛 s g n (𝑝) \sum 𝑞 \in 𝑆 𝑛 s g n (𝑝 𝑞) 𝛿 𝑝 𝑞 = \sum 𝑝 \in 𝑆 𝑛 s g n (𝑝) 𝔞 = 0

since there exist equal even and odd permutations.

The symmetrizer and antisymmetrizer elements fall into the more general category of Young operators, the former corresponding to the one-row diagram and the latter to the one-column diagram.

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