Outer tensor product establishes bijection between irreducible representations of direct factors and direct product

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Suppose G and H are groups and k is a field. Let \operatorname{Irr}(S) denote the set of irreducible representation of the group S over k. Then, there is a natural bijection:

\operatorname{Irr}(G) \times \operatorname{Irr}(H) \leftrightarrow \operatorname{Irr}(G \times H).

The bijection is given using the outer tensor product of linear representations, as follows. For irreducible representations \alpha,\beta of G and H, to vector spaces V and W, \alpha \otimes \beta is defined as a linear representation on the tensor product V \otimes W, with:

(\alpha \otimes \beta)(g,h) = \alpha(g) \otimes \beta(h).

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