Representation theory of finite symmetric groups

1-dimensional irreps of a finite symmetric group

A symmetric group 𝑆𝑛 with 𝑛 ≥2 has exactly two¹ non-equivalent 1-dimensional irreps rep

• 𝜒𝔰 the trivial irrep
• 𝜒𝔞 the alternating character

In the Group ring ℂ[𝑆𝑛] these irreps are carried by left ideäls generated by the Symmetrizer and antisymmetrizer elements.

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develop | en | SemBr

Footnotes

1. this is because the Alternating group 𝐴𝑛 ⊴𝑆𝑛 is the commutator subgroup and therefore 𝑆2 is the Abelianization. ↩