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On Determinants of Symmetric Functions

Published online by Cambridge University Press:  20 January 2009

A. C. Aitken
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The result of dividing the alternant |aαbβcγ…| by the simplest alternant |a0b1c2…| (the difference-product of a, b, c, …) is known to be a symmetric function expressible in two distinct ways, (1) as a determinant having for elements the elementary symmetric functions C, of a, b, c, …, (2) as a determinant having for elements the complete homogeneous symmetric functions Hr. For example

The formation of the (historically earlier) H-determinant is evident. The suffixes in the first row are the indices of the alternant; those of the other rows decrease by unit steps. This result is due to Jacobi.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 1 , Issue 1 , May 1927 , pp. 55 - 61
Copyright
Copyright © Edinburgh Mathematical Society 1927

References

page 55 note 1 De functionibus alternantibus. J. für Math., 22 (1841), pp. 370–371.Google Scholar

page 55 note 2 Theory of Determinants, vol. III, pp. 145–146.Google Scholar

page 55 note 3 Ueber eino Classe symmetrischen Functionen. Sch.-Programm, Zweibrücken, 1871.Google Scholar

page 56 note 1 Bemerkungen über symmetrischen Funktionen. J. für Math. 132 (1907), pp. 159, 161.Google Scholar

page 56 note 2 Combinatory Analysis, vol. I, p. 205.Google Scholar

page 60 note 1 On a determinantal theorem due to Jacobi. Messenger of Math. 21 (1892), pp. 148, 150.Google Scholar

page 60 note 2 On a theorem of Segar's. Messenger of Math. 36 (1906), pp. 77–78.Google Scholar

page 60 note 3 Note on a determinant whose elements are aleph functions. Messenger of Math. 46 (1916), pp. 108–110.Google Scholar

page 60 note 4 In this form the theorem had really been given by Naegelsbach, , op. cit., in 1871.Google Scholar CfMuir's, History, Vol. III, p. 147.Google Scholar