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On the Dimension of an Irreducible Tensor Representation of the General Linear Group GL(d)

Published online by Cambridge University Press:  20 November 2018

J. A. J. Matthews  and
G. de B. Robinson
J. A. J. Matthews
Affiliation:
University of Toronto, Toronto, Ontario
G. de B. Robinson
Affiliation:
University of Toronto, Toronto, Ontario
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As has long been known, the irreducible tensor representations of GL(d) of rank n may be labeled by means of the irreducible representations of Sn, i.e., by means of the Young diagrams [λ], where λ1 + λ2 + … λr = n. We denote such a tensor representation by 〈λ〉. Using Young's raising operator Rij we can write [1, p. 42]

1.1

where the dot denotes the inducing process. For example, [3] . [2] is that representation of S5 induced by the identity representation of its subgroup S3 × S2.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 3 , September 1970 , pp. 389 - 390
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. de B. Robinson, G., Representation theory of the symmetric group, Toronto, 1961.Google Scholar
2. Whitworth, W. A., Choice and Chance, Cambridge, 1896.Google Scholar