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

  • J. A. J. Matthews (a1) and G. de B. Robinson (a1)

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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 S n , i.e., by means of the Young diagrams [λ], where λ 1 + λ 2 + … λ r = n. We denote such a tensor representation by 〈λ〉. Using Young's raising operator R ij we can write [1, p. 42]

1.1

where the dot denotes the inducing process. For example, [3] . [2] is that representation of S 5 induced by the identity representation of its subgroup S 3 × S 2.

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References

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1. de B. Robinson, G., Representation theory of the symmetric group, Toronto, 1961.
2. Whitworth, W. A., Choice and Chance, Cambridge, 1896.
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On the Dimension of an Irreducible Tensor Representation of the General Linear Group GL(d)

  • J. A. J. Matthews (a1) and G. de B. Robinson (a1)

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