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Some Enumerative Results in the Theory of Forms

Published online by Cambridge University Press:  24 October 2008

W. V. D. Hodge
W. V. D. Hodge
Affiliation:
Pembroke College, Cambridge
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 39 , Issue 1 , March 1943 , pp. 22 - 30
Copyright
Copyright © Cambridge Philosophical Society 1943

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References

* Hodge, W. V. D., Proc. Cambridge Phil. Soc. 38 (1942), 129.CrossRefGoogle Scholar

† Littlewood, D. E.. Proc. Cambridge Phil. Soc. 38 (1942), 394.CrossRefGoogle Scholar

* The idea of standard tableaux and of standard power products is due to Young, A., Proc. London Math. Soc. (2), 28 (1927), 255–292.Google Scholar

* These identities should. be compared with those given by Turnbull, H. W., Determinants, Matrices and Invariants (Blackie, 1928), pp. 45 et seq.Google Scholar

* See, for example, Weitzenbock, R., Proc. Akad. Wet. Amsterdam, 39 (1936), 503.Google Scholar

† Severi, F., Rend. Circ. Mat. Palermo, 28 (1909), 33.CrossRefGoogle Scholar

* Severi, F., Ann. Mat. (3), 24 (1915).Google Scholar