Linear preservers and representations with a 1-dimensional ring of invariants
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Abstract:
We determine the group of linear transformations preserving a polynomial function $f$ on a vector space $V$ for several interesting pairs $(V,f)$ by introducing a subgroup $G$ of $\mathrm {GL}(V)$ and applying the theory of semisimple algebraic groups. Along the way, we give an explicit description of the normalizer $N_{\mathrm {GL}(V)}(G)$ and prove that, under a mild technical assumption, the normalizer agrees with the stabilizer in $\mathrm {GL}(V)$ of the orbit of the highest weight vector in $\mathbb {P}(V)$.References
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Additional Information
- H. Bermudez
- Affiliation: Department of Mathematics and Computer Science, MSC W401, Emory University, 400 Dowman Drive, Atlanta, Georgia 30322
- S. Garibaldi
- Affiliation: Department of Mathematics and Computer Science, MSC W401, Emory University, 400 Dowman Drive, Atlanta, Georgia 30322
- MR Author ID: 622970
- ORCID: 0000-0001-8924-5933
- V. Larsen
- Affiliation: Department of Mathematics and Computer Science, MSC W401, Emory University, 400 Dowman Drive, Atlanta, Georgia 30322
- Received by editor(s): January 24, 2012
- Received by editor(s) in revised form: October 20, 2012
- Published electronically: April 16, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 4755-4780
- MSC (2010): Primary 47B49; Secondary 15A04, 15A72, 20G15
- DOI: https://doi.org/10.1090/S0002-9947-2014-06081-9
- MathSciNet review: 3217699