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The Dimensions of Irreducible Representations of Linear Groups

Published online by Cambridge University Press:  20 November 2018

R. C. King
R. C. King*
Affiliation:
The University, Southampton, England
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The theory of the relationship between the symmetric group on a symbols, Σa, and the general linear group in n-dimensions, GL(n), was greatly developed by Weyl [4] who, in this connection, made use of tensor representations of GL(n). The set of mixed tensors

forms the basis of a representation of GL(n) if all the indices may take the values 1, 2, …, n, and if the linear transformation

is associated with every non-singular n × n matrix A. The representation is irreducible if the tensors are traceless and if the sets of covariant indices (α)a and contra variant indices (β)b themselves form the bases of irreducible representations (IRs) of Σa and Σb, respectively. These IRs of Σa and Σb may be specified by Young tableaux [μ]a and [v]b in the usual way [4].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 436 - 448
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Jahn, H. A. and N. El, Samra, Dimension of the King-Abramsky mixed tensor representation of GLn (to appear).Google Scholar
2. King, R. C., Generalised Young tableaux and the general linear group, J. Math. Phys. 11 (1970), 280–294.Google Scholar
3. Robinson, G. de B., Representation theory of the symmetric group, pp. 60, 89 (The University Press, Edinburgh, 1961).Google Scholar
4. Weyl, H., The classical groups, their invariants and representations (Princeton Univ. Press, Princeton, N.J., 1939).Google Scholar