Author’s summary: Symmetry properties of \(r\)-times covariant tensors \(T\) can be described by certain linear subspaces \(W\) of the group ring \(\mathbb{K}[{\mathcal S}_r]\) of a symmetric group \({\mathcal S}_r\). If for a class of tensors \(T\) such a \(W\) is known, the elements of the orthogonal subspace \(W^\bot\) of \(W\) within the dual space \(\mathbb{K}[{\mathcal S}_r]^*\) of \(\mathbb{K}[{\mathcal S}_r]\) yield linear identities needed for a treatment of the term combination problem for the coordinates of the \(T\). We give the structure of these \(W\) for every situation which appears in symbolic tensor calculations by computer. Characterizing idempotents of such \(W\) can be determined by means of an ideal decomposition algorithm which works in every semisimple ring up to an isomorphism. Furthermore, we use tools such as the Littlewood-Richardson rule, plethysms and discrete Fourier transforms for \({\mathcal S}_r\) to increase the efficience of calculations. All described methods were implemented in a Mathematica package called PERMS.