Journal of Lie Theory

no. 1
On Orbit Dimensions under a Simultaneous Lie Group Action on n Copies of a Manifold
Journal of Lie Theory, Volume 12 (2002) no. 1, pp. 191-203
We show that the maximal orbit dimension of a simultaneous Lie group action on n copies of a manifold does not pseudo-stabilize when n increases. We also show that if a Lie group action is (locally) effective on subsets of a manifold, then the induced Cartesian action is locally free on an open and dense subset of a sufficiently big (but finite) number of copies of the manifold. The latter is the analogue for the Cartesian action to Olver-Ovsiannikov's theorem on jet bundles and is an important fact relative to the moving frame method and the computation of joint invariants. Some interesting corollaries are presented.
DOI: 10.5802/jolt.257
Classification: 57S25
Keywords: isotropy group, stabilization order, stabilization dimension 
@article{JOLT_2002_12_1_a8,
     author = {M. Boutin},
     title = {On {Orbit} {Dimensions} under a {Simultaneous} {Lie} {Group} {Action} on n {Copies} of a {Manifold}},
     journal = {Journal of Lie Theory},
     pages = {191--203},
     year = {2002},
     volume = {12},
     number = {1},
     doi = {10.5802/jolt.257},
     zbl = {1006.57012},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.257/}
}
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M. Boutin. On Orbit Dimensions under a Simultaneous Lie Group Action on n Copies of a Manifold. Journal of Lie Theory, Volume 12 (2002) no. 1, pp. 191-203. doi: 10.5802/jolt.257

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