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We determine which faithful irreducible representations V of a simple linear algebraic group G are generically free for Lie(G), i.e., which V have an open subset consisting of vectors whose stabilizer in Lie(G) is zero. This relies on bounds on dim V obtained in prior work (part I), which reduce the problem to a finite number of possibilities for G and highest weights for V , but still infinitely many characteristics. The remaining cases are handled individually, some by computer calculation. These results were previously known for fields of characteristic zero, although new phenomena appear in prime characteristic; we provide a shorter proof that gives the result with very mild hypotheses on the characteristic. (The few characteristics not treated here are settled in part III.) These results are related to questions about invariants and the existence of a stabilizer in general position.
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Generically free representations I: Large representations
This paper concerns a faithful representation V of a simple linear algebraic group G. Under mild assumptions, we show that if V is large enough, then the Lie algebra of G acts generically freely on V. That is, the stabilizer in Lie.G/ of a generic vector in V is zero. The bound on dim V grows like the square of the rank and holds with only mild hypotheses on the characteristic of the underlying field. The proof relies on results on generation of Lie algebras by conjugates of an element that may be of independent interest. We use the bound in subsequent works to determine which irreducible faithful representations are generically free, with no hypothesis on the characteristic of the field. This in turn has applications to the question of which representations have a stabilizer in general position.
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- Award ID(s):
- 1901595
- PAR ID:
- 10225222
- Date Published:
- Journal Name:
- Algebra and number theory
- Volume:
- 14
- Issue:
- 6
- ISSN:
- 1112-265X
- Page Range / eLocation ID:
- 1577--1611
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/doi.org/10.2140/ant.2020.14.1577
