Algebraic investigations in landau model of structural phase transitions

Abstract

First we introduce the basic notions of the theory of permutation representations: stabilizers, orbits, stable subsets and strata. Then we consider the relation between permutation and linear representations which lead to some formulae connecting subduction coefficients, Clebsch-Gordan coefficients, and dimensions of stability spaces. This relation also leads to the concept of suborbits. Epikernels, the subgroups which are stabilizers of vectors of irreducible subspaces (either on the complex or on the real field) — are, studied and several theorems about them are proved. Further we consider the relation between epikernels, stability spaces and strata for subspaces irreducible on the real field as compared with subspaces irreducible on the complex field. Finally, the exomorphism is defined with use of permutation representations. The vectors of irreducible subspaces and corresponding epikernels (their stabilizers) for real ireps (representations irreducible on the real) of the classical crystal point groups are given in the appendix.