Tensor product with a 1-dimensional representation

Let $𝜒 : 𝐺 \to 𝕂$ be an arbitrary 1-dimensional representation and $Γ 𝜇 : 𝐺 \to G L (𝑉)$ be an irrep over $𝕂$ . Then $𝜒 \otimes Γ 𝜇 : 𝑔 \mapsto 𝜒 (𝑔) \cdot Γ 𝜇 (𝑔)$ is an irrep. rep

Proof

For any $𝑣 \in 𝑈 \subseteq 𝑉$ where $𝑈$ is a subspace, $𝜒 (𝑔) 𝑣 \in 𝑈$ , Hence the invariant subspaces of $Γ$ are the same as those of $𝜒 \cdot Γ$ , and thus if $Γ$ is irreducible so is $𝜒 \cdot Γ$ .

As a result, 1-dimensional irreps form a group.

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Tensor product with a 1-dimensional representation

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