Invariant Theory, Tensors and Group Characters

D. E. Littlewood

Abstract

In Part I is developed the theory of the tensor as a device for the construction of concomitants. This part includes the specific separation of a complete cogredient tensor of rank r into simple tensors, with a formula indicating the corresponding separation of a mixed tensor; also the corresponding theory in tensors to the Clebsch theory of algebraic forms, and a compact proof of the fundamental theorem that all concomitants under the full linear group can be obtained by the multiplication and contraction of tensors. The general equivalence is demonstrated, so far as elementary applications are concerned, of the method of tensors with the classical symbolic method of invariant theory. The first part forms a foundation for the principal theory of the paper which is developed in Part II. This primarily consists of an analysis of the properties of S-functions which provides methods for predicting the exact number of linearly independent concomitants of each type, of a given set of ground forms. Complementary to this, a method of substitutional analysis based on the tableaux which must be constructed in obtaining a product of S-functions, enables the specific concomitants of each type to be written down. Part III consists of applications to the classical problems of invariant theory. For ternary perpetuants a generating function is obtained which is not only simpler than that given by Young, but is also more general, in so far as it indicates, as well as the covariants, also the mixed concomitants. Extension is made to any number of variables. The complete sets of concomitants, up to degree 5 or 6 in the coefficients, are obtained for the cubic, quartic and quadratic complex in any number of variables. Alternating concomitant types are described and enumerated. A theorem of conjugates is proved which associates the concomitants of one ground form with the concomitants of a ground form of a different type, namely, that which corresponds to the conjugate partition. Some indication is made of the extension of this theory to invariants under restricted groups of transformations, e.g. the orthogonal group, but the full development of this extended theory is to be the subject of another paper.