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

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## Statement

Suppose and are groups and is a field. Let denote the set of irreducible representation of the group over . Then, there is a natural bijection:

.

The bijection is given using the outer tensor product of linear representations, as follows. For irreducible representations of and , to vector spaces and , is defined as a linear representation on the tensor product , with:

.