Integrity Bases for Vectors — The Crystal Classes

Abstract

In a previous paper (Smith, Smith & Rivlin (1963)), irreducible integrity bases for a symmetric three-dimensional tensor and absolute vector under the transformation groups describing each of the 32 crystal symmetries have been obtained and their minimality demonstrated. In the present paper, rather similar methods are used to determine irreducible integrity bases for an arbitrary number of absolute vectors under the transformation groups describing 31 of the 32 crystal symmetries. The remaining crystal class, which is the gyroidal class in the cubic system, has so far proven intractable for technical reasons although the methods used for the four classes of the cubic system, for which irreducible integrity bases have been found, are in principle applicable.