Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 12B / v / Solution

Contracting two antisymmetric tensors gives the nonzero isotropic rank-four tensorA nonzero isotropic rank-five tensor is, for example,

More generally, one obtains the rank-five isotropic building blocks by choosing which two of the five indices belong to the Kronecker delta and putting the remaining three into the Levi-Civita symbol. There aresuch placements:Thus the unrestricted isotropic rank-five ansatz requires ten scalar component coefficients, one for each delta-epsilon placement.