The vector space tensor product V⊗W of two group representations of a group G is also a representation of G. An element g of G acts on a basis element v⊗w by g(v⊗w) = g v⊗g w. If G is a finite group and V is a faithful representation, then any representation is contained in ⊗^n V for some n. If V_1 is a representation of G_1 and V_2 is a representation of G_2, then V_1 ⊗V_2 is a representation of G_1×G_2, called the external tensor product. The regular tensor product is a special case, with the diagonal embedding of G in G×G.

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