[Invariants et espaces propres des groupes de réflexion complexes]
En utilisant des variantes d'une formule de Orlik et Solomon relative aux invariants d'un groupe de réflexions complexes G, nous redémontrons de façon élémentaire deux résultats de Lehrer and Springer, en particulier le fait qu'un entier est régulier pour G si et seulement si il divise le même nombre de degrés et de codegrés. Notre preuve évite l'analyse cas par cas.
We prove some variations of formulas of Orlik and Solomon in the invariant theory of finite unitary reflection groups, and use them to give elementary and case-free proofs of some results of Lehrer and Springer, in particular that an integer is regular for a reflection group G if and only if it divides the same number of degrees and codegrees.
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@article{CRMATH_2003__336_10_795_0,
author = {Lehrer, Gustav I. and Michel, Jean},
title = {Invariant theory and eigenspaces for unitary reflection groups},
journal = {Comptes Rendus. Math\'ematique},
pages = {795--800},
publisher = {Elsevier},
volume = {336},
number = {10},
year = {2003},
doi = {10.1016/S1631-073X(03)00192-4},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00192-4/}
}
TY - JOUR AU - Lehrer, Gustav I. AU - Michel, Jean TI - Invariant theory and eigenspaces for unitary reflection groups JO - Comptes Rendus. Mathématique PY - 2003 SP - 795 EP - 800 VL - 336 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(03)00192-4/ DO - 10.1016/S1631-073X(03)00192-4 LA - en ID - CRMATH_2003__336_10_795_0 ER -
%0 Journal Article %A Lehrer, Gustav I. %A Michel, Jean %T Invariant theory and eigenspaces for unitary reflection groups %J Comptes Rendus. Mathématique %D 2003 %P 795-800 %V 336 %N 10 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(03)00192-4/ %R 10.1016/S1631-073X(03)00192-4 %G en %F CRMATH_2003__336_10_795_0
Lehrer, Gustav I.; Michel, Jean. Invariant theory and eigenspaces for unitary reflection groups. Comptes Rendus. Mathématique, Tome 336 (2003) no. 10, pp. 795-800. doi : 10.1016/S1631-073X(03)00192-4. http://www.numdam.org/articles/10.1016/S1631-073X(03)00192-4/
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