Alternating character

The alternating character or signature $𝜒 𝔞 = s g n : 𝑆 𝑛 \to 𝑆 2 \in ℂ$ 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

s g n 𝜎 = (- 1) 𝑚

where $𝑚$ is the number of 2-cycles in any decomposition of $𝜎$ .

Proof of well-defined

𝑚

#missing/proof

Properties

$k e r s g n$ is the Alternating group
$s g n 𝑝 = (- 1) 𝑘$ where $𝑘$ is the crossing number of the Birdtrack notation of $𝑝$ .

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