  Formulating non-Abelian gauge theories as a tensor network is known to be
challenging due to the internal degrees of freedom that result in the
degeneracy in the singular value spectrum. In two dimensions, it is
straightforward to 'trace out' these degrees of freedom with the use of
character expansion, giving a reduced tensor network where the degeneracy
associated with the internal symmetry is eliminated. In this work, we show that
such an index loop also exists in higher dimensions in the form of a closed
tensor network we call the 'armillary sphere'. This allows us to completely
eliminate the matrix indices and reduce the overall size of the tensors in the
same way as is possible in two dimensions. This formulation allows us to
include significantly more representations with the same tensor size, thus
making it possible to reach a greater level of numerical accuracy in the tensor
renormalization group computations.
