Methods of representation theory. With applications to by CW Curtis, Irving Reiner

By CW Curtis, Irving Reiner

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Although a detailed analysis of the subjective, visual factors of symmetry is here omitted, mainly because of the complexity of the problems of visual perception, this work offers a potential approach to such problems. Continuous symmetry groups with continuous non-polar elements of symmetry allow an immediate visual interpretation, while for representing groups with polar or bipolar continuous elements of symmetry we can apply textures—an equal, average density of the asymmetric figures arranged along the invariant polar or bipolar line, in accordance with the given continuous symmetry group (Shubnikov, Belov et al, 1964).

Consequently, the same holds for all other geometric properties of such figures, so that the equiangularity (the congruence of the angles of homologous figures) and their equiformity (the same form of homologous figures) are the direct consequences of isometrism. The next class of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of the space En is a transformation which to each line segment of length AB assigns a line segment of the length kAB whereby k is a real positive number, the coefficient of similarity.

The colored symmetry group 1 1 / 1 1 . Therefore, we can conclude that the index of the subgroup 11 in the group 11 is any natural number N and that its minimal index is two. The results of computing the (minimal) indexes of subgroups in groups of symmetry, where the subgroups belong to the same category of symmetry groups as the groups discussed, based on the works of Coxeter and Moser (Coxeter and Moser, 1980; Coxeter, 1985, 1987) are completed with the results obtained by using antisymmetry and colored symmetry.

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