On the theory of matrices with elements in the Clebsch Aronhold symbolic calculus
Donald Livingstone · ERA · 1949
In the theory of invariant matrices and in the classical invariant theory there arise a considerable number of rather surprising isonorphisms relating to symmetric functions of the latent roots and to the theory of matric representations of the symmetric group. Although these relations appear sooner or later in the development of either theory it would seem, on account of their fundamental simplicity, at least desirable that they should be brought into evidence by an analysis of the algebraic nature of the systems involved. Since many of the relations referred to are merely extensions of familiar determinantal theorems, and since the totality of reducing matrices for the general invariant matrix should give a complete basis for determinantal relations, the development of these results should embody an extension of determinant theory besides all results relating to determinants with unrestricted elements in the ground field. With this end in view, the discussion given below proceeds first from an analysis of the nature of various expressions involving determinants and permanents, to a construction for the orthogonal representations of the symmetric group. The reductions given here are related to the corresponding reductions for the central cores of invariant matrices by means of an isomorphism which can be expressed in terms of certain symbolic quantities obtained by writing each element of the fundamental matrix [a i j] as a symbolic product ai αj and forming direct products of compounds and Sohläflians of [a i j] by the use of equivalent symbols. A symbolisation of matrices in this way has been used by Professor Turnbull in his paper The Invariant Theory of a General Bi- linear Form" (Proc. L.:.S. Series II, Vol. 33, Part I). The process of deduction followed in this paper is, however, essentially non -symbolic. It could have been framed equally well in the terminology of invariant matrices, in which it forms an extension of some theorems developed by Professor Aitken in his Research Lectures. The actual representations obtained are essentially the same as the orthogonal forms developed by Young. The discussion is restricted to the orthogonal case for the sake of symmetry, although rational forms can be developed in a similar manner. In the paper quoted above, Professor Turnbull uses his symbolic forms to obtain expressions representing symmetric functions of the latent roots of the matrix [a i j]. His results are easily extended to the complete homogeneous symmetric functions and to bi- alternants, and suggest the manner in which the reducing matrices for the central core might be extended to reduce the full invariant matrix. A homo-morphism is developed for this extension