Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-09T01:07:16.072Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Disjoint Product of Irreducible Representations of the Symmetric Group

Published online by Cambridge University Press:  20 November 2018

G. de B. Robinson
G. de B. Robinson*
Affiliation:
The University of Toronto
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The results of the present paper can be interpreted (a) in terms of the theory of the representations of the symmetric group, or (b) in terms of the corresponding theory of the full linear group. In the latter connection they give a solution to the problem of the expression of an invariant matrix of an invariant matrix as a sum of invariant matrices, in the sense of Schur's Dissertation. D. E. Littlewood has pointed out the significance of this problem for invariant theory and has attacked it via Schur functions, i.e. characters of the irreducible representations of the full linear group. We shall confine our attention here to the interpretation (a).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 2 , 01 April 1949 , pp. 166 - 175
Copyright
Copyright © Canadian Mathematical Society 1949

References

[1] Burnside, W., The Theory of Groups, 2nd ed. (Cambridge, 1911).Google Scholar
[2] Littlewood, D. E., “Polynomial Concomitants and Invariant Matrices,” J. London Math. Soc., vol. 11 (1936), 49-55.Google Scholar
[3] Littlewood, D. E., “Invariant Theory, Tensors and Group Characters,” Philos. Trans. Roy. Soc. (A), vol. 239 (1944), 305–365.Google Scholar
[4] Littlewood, D. E., “Invariants of Systems of Quadrics,” Proc. London Math. Soc, vol. 49 (1947), 282-306.Google Scholar
[5] Murnaghan, F. D., “The Analysis of the Direct Product Representation,” Amer. J, Math., vol. 60 (1938), 44–65.Google Scholar
[6] Robinson, G. de B., “On the Representations of the Symmetric Group,” Amer. J.Math., vol. 60 (1938), 745-760.Google Scholar
[7] Robinson, G. de B. (second paper), ibid., vol. 69 (1947), 286-298.Google Scholar
[8] Robinson, G. de B. (third paper), ibid., vol. 70 (1948), 277-294.Google Scholar
[9] Schur, I., “Ueber eine Klasse von Matrizen … ,” Dissertation, Berlin, 1901.Google Scholar
[10] Schur, I., “Neue Begrundung der Théorie der Gruppencharaktere,” Preuss. Akad. Wiss. Sitzungsber., Berlin, 1905, 406–432.Google Scholar
[11] Schur, I., “Ueber die rationalen Darstellungen der allgemeinen linearen Gruppe,” ibid. 1927, 58-75.Google Scholar
[12] A. Speiser, , Théorie der Gruppen, 3rd ed. (Berlin, 1937).Google Scholar
[13] Weyl, H., Gruppentheorie und Quantenmechanik, 2nd ed. (Leipzig, 1931).Google Scholar