Combinatorics in tensor-integral reduction

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 d = 4 - 2 epsilon space-time dimensions. The main derivation is given in the ndimensional Euclidean space. The generalisation of the result to the Minkowski space is also discussed in order to provide graduate students and researchers with techniques of tensor-integral reduction for particle physics problems.