Geodesic
Irreducible Linear Group-Subgroup Pairs with the same Invariants
Journal of Lie theory, Tome 15 (2005) no. 1, pp. 105-123.

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We consider the problem of finding all linear algebraic group-subgroup pairs such that the rational invariants of the group and of the subgroup coincide. In this paper the solution will be given for the case where both the group and the subgroup are connected complex irreducible linear groups. 
@article{JLT_2005_15_1_JLT_2005_15_1_a7,
     author = {S. Solomon },
     title = {Irreducible {Linear} {Group-Subgroup} {Pairs} with the same {Invariants}},
     journal = {Journal of Lie theory},
     pages = {105--123},
     publisher = {mathdoc},
     volume = {15},
     number = {1},
     year = {2005},
     url = {http://geodesic.mathdoc.fr/item/JLT_2005_15_1_JLT_2005_15_1_a7/}
}
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S. Solomon . Irreducible Linear Group-Subgroup Pairs with the same Invariants. Journal of Lie theory, Tome 15 (2005) no. 1, pp. 105-123. http://geodesic.mathdoc.fr/item/JLT_2005_15_1_JLT_2005_15_1_a7/