Group representation

1-dimensional irrep

A 1-dimensional-irrep or linear character¹ is a homomorphism from a group to the multiplicative group of a field. It is both a representation and the corresponding Group character. Every 1-dimensional representation is clearly irreducible.

Irrep group

Given a finite group , the set of 1-dimensional irreps forms a group under multiplication (Tensor product with a 1-dimensional representation), where the inverse of is the complex conjugate . is isomorphic to the dual group of the Abelianization of , since there is a one-to-one correspondance between 1-dimensional irreps of and the irreps of . In particular

since The number of irreps of an abelian group equals its order.

Proof

Since Universal property, every Abelian representation factors via , and since An abelian representation is a sum of 1-dimensional irreps, the 1-dimensional irreps of are precisely the irreps of .

Properties

• Tensor product with a 1-dimensional representation
• Irreps of abelian groups are 1-dimensional
• An abelian representation is a sum of 1-dimensional irreps

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Footnotes

1. See Linear character ↩