  Tensor network decompositions offer an efficient description of certain
many-body states of a lattice system and are the basis of a wealth of numerical
simulation algorithms. We discuss how to incorporate a global symmetry, given
by a compact, completely reducible group G, in tensor network decompositions
and algorithms. This is achieved by considering tensors that are invariant
under the action of the group G. Each symmetric tensor decomposes into two
types of tensors: degeneracy tensors, containing all the degrees of freedom,
and structural tensors, which only depend on the symmetry group. In numerical
calculations, the use of symmetric tensors ensures the preservation of the
symmetry, allows selection of a specific symmetry sector, and significantly
reduces computational costs. On the other hand, the resulting tensor network
can be interpreted as a superposition of exponentially many spin networks. Spin
networks are used extensively in loop quantum gravity, where they represent
states of quantum geometry. Our work highlights their importance also in the
context of tensor network algorithms, thus setting the stage for
cross-fertilization between these two areas of research.
