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Sorry for the phrasing. In retrospection, I realize myself that already "I am sitting on my carpet" is not exactly a wise choice for the beginning of a question, at least if one wants people to take it seriously. And there are some more parts... sorry, will do better in the future! Thanks very much for the answers.
This is not an answer, but maybe helpful: Probably you have come across this yourself, but in case you have not, Chris Boehning's habilitation may be useful: uni-math.gwdg.de/boehning/invariant_theory_book.pdf
