1-dimensional irrep

A 1-dimensional-irrep or linear character¹ $𝜒 : 𝐺 \to 𝕂$ 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 $̂ 𝐺 1$ of 1-dimensional irreps forms a group under multiplication (Tensor product with a 1-dimensional representation), where the inverse of $𝜒 𝜇$ is the complex conjugate $――― 𝜒 𝜇$ . $̂ 𝐺 1$ 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

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See Linear character ↩

1-dimensional irrep

Irrep group

Properties

Footnotes