Geodesic
The Dimensions of Irreducible Representations of Linear Groups
Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 436-448

Voir la notice de l'article provenant de la source Cambridge University Press

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]. 
King, R. C. The Dimensions of Irreducible Representations of Linear Groups. Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 436-448. doi: 10.4153/CJM-1970-052-6
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[1] 1. Jahn, H. A. and N. El, Samra, Dimension of the King-Abramsky mixed tensor representation of GLn (to appear). Google Scholar

[2] 2. King, R. C., Generalised Young tableaux and the general linear group, J. Math. Phys. 11 (1970), 280–294. Google Scholar

[3] 3. Robinson, G. de B., Representation theory of the symmetric group, pp. 60, 89 (The University Press, Edinburgh, 1961). Google Scholar

[4] 4. Weyl, H., The classical groups, their invariants and representations (Princeton Univ. Press, Princeton, N.J., 1939). Google Scholar

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