[Tenseurs de Killing comme des représentations irréductibles du groupe linéaire.]
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.
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.
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@article{CRMATH_2004__339_9_621_0,
author = {McLenaghan, Raymond G. and Milson, Robert and Smirnov, Roman G.},
title = {Killing tensors as irreducible representations of the general linear group},
journal = {Comptes Rendus. Math\'ematique},
pages = {621--624},
publisher = {Elsevier},
volume = {339},
number = {9},
year = {2004},
doi = {10.1016/j.crma.2004.07.017},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.07.017/}
}
TY - JOUR AU - McLenaghan, Raymond G. AU - Milson, Robert AU - Smirnov, Roman G. TI - Killing tensors as irreducible representations of the general linear group JO - Comptes Rendus. Mathématique PY - 2004 SP - 621 EP - 624 VL - 339 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.07.017/ DO - 10.1016/j.crma.2004.07.017 LA - en ID - CRMATH_2004__339_9_621_0 ER -
%0 Journal Article %A McLenaghan, Raymond G. %A Milson, Robert %A Smirnov, Roman G. %T Killing tensors as irreducible representations of the general linear group %J Comptes Rendus. Mathématique %D 2004 %P 621-624 %V 339 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.07.017/ %R 10.1016/j.crma.2004.07.017 %G en %F CRMATH_2004__339_9_621_0
McLenaghan, Raymond G.; Milson, Robert; Smirnov, Roman G. Killing tensors as irreducible representations of the general linear group. Comptes Rendus. Mathématique, Tome 339 (2004) no. 9, pp. 621-624. doi : 10.1016/j.crma.2004.07.017. http://www.numdam.org/articles/10.1016/j.crma.2004.07.017/
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* R.G.M, R.M, and R.G.S. were supported in part by the National Sciences and Engineering Research Council of Canada.
