Suppose that V is a group representation of G, and W is a group representation of H. Then the vector space tensor product V⊗W is a group representation of the group direct product G×H. An element (g, h) of G×H acts on a basis element v⊗w by (g, h)(v⊗w) = g v⊗h w. To distinguish from the representation tensor product, the external tensor product is denoted V□x W, although the only possible confusion would occur when G = H.

group | group representation | irreducible representation | representation tensor product | vector space | vector space tensor product