**Direct Product Vector Space**: Let and be inner product spaces and and be orthonormal basis in the two spaces respectively. Then the direct product space consists of all linear combinations of the orthonormal basis vector , which can be regarded as , such that

(i) Inner product: ;

(ii) where the complex number are the components of ;

(iii) .

**Direct Product Representation**: Let be a symmetry group of a physical system, and be the direct product space of physical solutions consisting of two sets of degrees of freedom . Suppose and are the representations of on and respectively. The the operators on forms the direct product representation of (on ) and (on ).

The group characters of the direct product representation are equal to the product of the characters of the two representation and : i.e.

(1)

**Clebsch-Gordan Coefficients**: The vector space can be decomposed into a direct sum of invariant subspaces with basis , which are related to basis with a unitary transformation,

(2)

the matrix elements are called Clebsch-Gordan Coefficients.