Some Enumerative Results in the Theory of Forms

Source abstract

In a recent note I attempted to obtain the postulation formula for the Grassmannian of k -spaces in [ n ] by the consideration of forms of a certain type in k + 1 sets of r + 1 homogeneous variables, which I called k -connexes. My attempt was not entirely successful; I obtained a formula for k -connexes which suggested what the required postulation formula should be, but was unable to prove it. D. E. Littlewood has now written a paper to show that my problem is intimately connected with the theory of invariant matrices, and has thereby established the truth of the postulation formula which I had conjectured. Littlewood's proof requires a considerable knowledge of the theory of invariant matrices, and this paper results from an attempt to re-write his proof in a form which is intelligible to a student not having this specialized knowledge. Prof. H. W. Turnbull has pointed out to me the importance of the so-called k -connexes in the theory of forms, particularly in connexion with the Gordan-Capelli series, and for this reason I am taking the k -connexes as the principal topic of this paper, leaving the deduction of certain postulation formulae which are the more immediate concern of a geometer to the end.