The Kronecker product in terms of Hubbard operators and the Clebsch–Gordan decomposition of SU(2)×SU(2)SU(2)×SU(2)

• The Kronecker product is studied in terms of Hubbard operators.
• Complicated calculations involving large matrices are reduced to simple relations of subscripts.
• The algebraic properties of the quantum observables of multipartite systems are studied.
• The Clebsch–Gordan coefficients are given in terms of hypergeometric 3F23F2 functions.
• The results can be further developed in many different directions.

We review the properties of the Kronecker (direct, or tensor) product of square matrices A⊗B⊗C⋯A⊗B⊗C⋯ in terms of Hubbard operators. In its simplest form, a Hubbard operator Xni,j can be expressed as the nn-square matrix which has entry 1 in position (i,j)(i,j) and zero in all other entries. The algebra and group properties of the observables that define a multipartite quantum system are notably straightforward in such a framework. In particular, we use the Kronecker product in Hubbard notation to get the Clebsch–Gordan decomposition of the product group SU(2)×SU(2)SU(2)×SU(2). Finally, the nn-dimensional irreducible representations so obtained are used to derive closed forms of the Clebsch–Gordan coefficients that rule the addition of angular momenta. Our results can be further developed in many different directions.