Group representation

1-dimensional irrep

A 1-dimensional-irrep or linear character¹ 𝜒 :𝐺 →𝕂 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 𝐺, the set ̂𝐺1 of 1-dimensional irreps forms a group under multiplication (Tensor product with a 1-dimensional representation), where the inverse of 𝜒𝜇 is the complex conjugate ―――𝜒𝜇. ̂𝐺1 is isomorphic to the dual group of the Abelianization 𝐴 =𝐺/[𝐺,𝐺] of 𝐺, since there is a one-to-one correspondance between 1-dimensional irreps of 𝐺 and the irreps ̂𝐴 of 𝐴. In particular

|̂𝐴|=|𝐺/[𝐺,𝐺]|=|𝐺||[𝐺,𝐺]|

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

Proof

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

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

---

develop | en | SemBr

Footnotes

1. See Linear character ↩