Comptes Rendus
Differential Geometry/Mathematical Physics
Killing tensors as irreducible representations of the general linear group
Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 621-624.

We show that the vector space of fixed valence Killing tensors on a space of constant curvature is naturally isomorphic to a certain highest weight, irreducible representation of the general linear group. The isomorphism is equivariant in the sense that the natural action of the isometry group corresponds to the restriction of the linear action to the appropriate subgroup. As an application, we deduce the Delong–Takeuchi–Thompson formula on the dimension of the vector space of Killing tensors from the classical Weyl dimension formula.

Nous démontrons que l'espace des tenseurs de Killing d'un ordre donné est naturellement isomorphe à une représentation irréductible de plus haut poids du groupe linéaire. L'isomorphisme est équivariant ; les transformations par isométries correspondent à l'inclusion du groupe des isométries comme un sous-groupe particulier du groupe linéaire. Comme application de cet isomorphisme nous obtenons la formule de Delong–Takeuchi–Thompson sur la dimension de l'espace des tenseurs de Killing à partir de la formule classique de dimension de Weyl.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.07.017
Raymond G. McLenaghan 1; Robert Milson 2; Roman G. Smirnov 2

1 Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
2 Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada
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Raymond G. McLenaghan; Robert Milson; Roman G. Smirnov. Killing tensors as irreducible representations of the general linear group. Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 621-624. doi : 10.1016/j.crma.2004.07.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.017/

[1] R. Bott Homogeneous vector bundles, Ann. Math., Volume 66 (1957), pp. 203-247

[2] R. Delong, Killing tensors and the Hamilton–Jacobi equation, Ph.D. thesis, University of Minnesota, 1982

[3] W. Fulton; J. Harris Representation Theory, Springer, 1991

[4] S. Helgason Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, Providence, RI, 2001

[5] R.G. McLenaghan; R.G. Smirnov; D. The An extension of the classical theory of algebraic invariants to pseudo-Riemannian geometry and Hamiltonian mechanics, J. Math. Phys., Volume 45 (2004), pp. 1079-1120

[6] M. Takeuchi Killing tensor fields on spaces of constant curvature, Tsukuba J. Math., Volume 7 (1983), pp. 233-255

[7] G. Thompson Killing tensors in spaces of constant curvature, J. Math. Phys., Volume 27 (1986), pp. 2693-2699

[8] J. Wolf Spaces of Constant Curvature, Publish or Perish, Houston, TX, 1984

Cited by Sources:

* R.G.M, R.M, and R.G.S. were supported in part by the National Sciences and Engineering Research Council of Canada.

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