[[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]],
$p \sim p^{-1}$ are [[Conjugation by an element|conjugate]] for all $p \in S_{n}$.
Thus since [[Every finite complex representation of a compact group is equivalent to a unitary representation]] $\chi(p) = \overline{\chi(p)}$ for all $p \in S_{n}$, 
i.e. the characters of a finite symmetric group are real. #m/thm/rep/sym 

#
---
#state/tidy | #lang/en | #SemBr