Suppose that is a group representation of , and is a group representation of . Then the vector space tensor product is a group representation of the group direct product . An element of acts on a basis element by

To distinguish from the representation tensor product, the external tensor product is denoted , although the only possible confusion would occur when .

When and are irreducible representations of and respectively, then so is the external tensor product. In fact, all irreducible representations of arise as external direct products of irreducible representations.