Representation theory of finite symmetric groups

Characters of a finite symmetric group are real

Since Conjugacy classes of a symmetric group are determined by cycle structure, 𝑝 ∼𝑝−1 are conjugate for all 𝑝 ∈𝑆𝑛. Thus since Every finite complex representation of a compact group is equivalent to a unitary representation 𝜒(𝑝) =―――𝜒(𝑝) for all 𝑝 ∈𝑆𝑛, i.e. the characters of a finite symmetric group are integers. #m/thm/rep/sym

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