[[Representation theory of finite symmetric groups]]
# 1-dimensional irreps of a finite symmetric group

A symmetric group $S_{n}$ with $n\geq 2$ has exactly two[^two] non-equivalent [[1-dimensional irrep|1-dimensional irreps]] #m/thm/rep

- $\chi^\mathfrak{s}$ the trivial irrep
- $\chi^\mathfrak{a}$ the [[alternating character]]

[^two]: this is because the [[Alternating group]] $A_{n} \trianglelefteq S_{n}$ is the [[commutator subgroup]] and therefore $S_{2}$ is the [[Abelianization]].

In the [[Group ring]] $\mathbb{C}[S_{n}]$ these irreps are carried by left ideäls generated by the [[Symmetrizer and antisymmetrizer elements]].

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