Relation between multidimensional representations of the Fedorov groups and the groups of color symmetry

Abstract

Relations have been traced between multidimensional irreducible representations (IR) of the Fedorov groups and the groups of color symmetry using as an example the phase transitions in crystals. Unlike the case of the one-dimensional real IR's, for which the Indenbom-Niggli theorem is valid, for the multidimensional IR's a one-to-one relationship between them and the color symmetry groups is disturbed. Namely, each multidimensional IR of the Fedorov group is associated not with one, but rather with a whole set of the color symmetry groups, and, conversely, each group of the color symmetry is associated not with one IR but with a certain set of these. Complete correspondence between the color symmetry groups and the IR's of the Fedorov group can be established on the basis of an analysis of the complete condensate of order parameters. It has been shown that on the set of different domains of a low-symmetry phase (macroscopic level), there are acting certain P-symmetry groups, when, however, the structure of one domain of such a phase (microscopic level) is described, the W p -symmetry groups have to be used. (The distinguishing feature of the W p -symmetry, as compared to the P-symmetry, is this: each element of the classical group is assigned not one color permutation, but different permutations, depending on which point of the space is subjected to the action of this element.) It has also been shown that to the W p -symmetry groups describing the structure of one domain, there corresponds, generally speaking, nonrectangular Young diagrams.