On invariant theory under restricted groups

Abstract

A previous paper, ‘Invariant theory, tensors and group characters’, dealt with invariant theory under the full linear group. In this paper the methods are extended to restricted groups of transformation such as the orthogonal group. A knowledge of the characters of the group is shown to be an essential preliminary to any adequate study of invariants under the group. The characters of the orthogonal and symplectic groups, previously obtained by Schur and Weyl by transcendental methods involving group integration, are here obtained by methods entirely algebraic. Concerning transformation groups with a system of fundamental tensors, a fundamental theorem is proved that every concomitant may be obtained by multiplication and contraction of ground-form tensors, tensor variables, fundamental tensors and the alternating tensor. A characteristic analysis is developed, involving the operation denoted by ®, which enables the numbers and types of the concomitants of any given degree in any system of ground forms to be predicted. The determination of the actual concomitants is also discussed. Application is made for the orthogonal group to the quadratic, the ternary cubic and the quaternary quadratic complex; for the ternary symplectic group, to the quadratic, the linear complex and the quadratic complex. Various applications are also made for intransitive and imprimitive groups of transformation.