Typical integrity bases of abelian crystal point groups

Abstract

It is shown that an integrity basis of absolute invariants for any representation of a finite abelian group follows from one integrity basis universal of this group-the typical integrity basis(TIB). An algorithm based on a subsequent multiplication of monomial bases to irreducible representations is found for deriving the TIB. The concept of integrity basis is extended to relative invariants that transform by non-identical representations. It is shown that there exists a finite set of relative invariants such that any other relative invariant can be expressed linearly to this set and polynomially to the integrity basis. As an application, integrity bases in polarization vector and strain tensor components are found for all abelian crystal point groups. Together with invariants that are pure in polarization or in strain tensor and were previously derived by other authors, invariants composed from both quantities, which describe the piezoelectric coupling are presented.