We illustrate a rigorous approach to express the totally symmetric isotropic tensors of arbitrary
rank in the n -dimensional Euclidean space as a linear combination of products of Kronecker deltas.
By making full use of the symmetries, one can greatly reduce the efforts to compute cumbersome
angular integrals into straightforward combinatoric counts. This method is generalised into the
cases in which such symmetries are present in subspaces. We further demonstrate the mechanism of the
tensor-integral reduction that is widely used in various physics problems such as perturbative
calculations of the gauge-field theory in which divergent integrals are regularised in ##IMG##
[http://ej.iop.org/images/0143-0807/38/2/025801/ejpaa54ceieqn1.gif] {$d=4-2\epsilon $} space–time
dimensions. The main derivation is given in the n -dimensional Euclidean space. The generalisation
of the result to the Minkowski space is also discussed in order to provide graduat…