Opechowski's magic relations

Abstract

Transformation properties of tensors under the action of the full magnetic group SO(3) × E_o are specified by the intrinsic symmetry which defines how the tensor transforms under the action of the proper rotation group SO(3) and by their parity which defines how the tensor transforms under the action of the group of inversions E_o = {e, i, e′, i′}, where i denotes the space inversion, e′ the magnetic inversion and i′ = i.e′ their combination. There exist therefore four types of tensors to each intrinsic symmetry whose transformation properties under the group SO(3) are identical while their transformation under the group E_o determines their parities with respect to the three inversions. As a result of this separation, transformation properties of tensors of the same intrinsic symmetry under the action of magnetic groups of the same oriented Laue class are strongly correlated by relations for which we propose the name Opechowski's magic relations in homage to late Prof. Opechowski. The origin of these relations is explained and tables which describe them are presented. It is found, in particular, that the number of different forms of tensor decompositions under the action of groups of oriented Laue class of magnetic point groups equals the number of one-dimensional real irreducible representations of the proper rotation group which generates the Laue class. Accordingly, there exists also the same number of allowed forms of tensors which do not vanish.