A simplified calculation and tabulation of tensorial covariants for magnetic point groups belonging to the same Laue class

Four types of static tensors can be distinguished according to their parity with respect to space inversion and to time reversal. However, all magnetic point groups belonging to the same (oriented) Laue class consist, apart from inversions, of the same proper rotations. Tensors differing only by parities transform identically under the same proper rotations; their transformation properties under different groups of the same Laue class may therefore differ only by an additional change of sign, which depends on the tensor parity and on the way in which inversions are combined with proper rotations in a given group. It is shown that, for a certain natural choice of typical representations of magnetic point groups of the same Laue class, it is sufficient to calculate tensorial covariants (symmetry-adapted tensorial bases) of even parity with respect to both space inversion and time reversal for the group of proper rotations. Tensorial covariants of other parities and for other magnetic point groups of the same Laue class can then be obtained by the use of a simple conversion table and of parity arguments. The scheme is illustrated by an example from the Laue class D₄.