1-dimensional representation

Alternating character

The alternating character or signature is a 1-dimensional irrep (i.e. linear character) and group homomorphism of a Symmetric group. group It is given by

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

Proof of well-defined

proof

Properties

1. is the Alternating group
2. where is the crossing number of the Birdtrack notation of .

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