[[1-dimensional irrep|1-dimensional representation]]
# Alternating character

The **alternating character** or **signature** $\chi^\mathfrak{a} = \sgn : S_{n} \to S_{2} \in \mathbb{C}$ is a [[1-dimensional irrep]] (i.e. linear character) and group homomorphism $G \to \{ -1,1 \}$ of a [[Symmetric group]]. #m/def/group 
It is given by
$$
\begin{align*}
\sgn \sigma = (-1)^m
\end{align*}
$$
where $m$ is the number of 2-cycles in any decomposition of $\sigma$.

> [!missing]- Proof of well-defined $m$
> #missing/proof

## Properties

1. $\ker \sgn$ is the [[Alternating group]]
2. $\sgn p = (-1)^k$ where $k$ is the crossing number of the [[Birdtrack notation]] of $p$.

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