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Linear preservers and representations with a 1-dimensional ring of invariants


Authors: H. Bermudez, S. Garibaldi and V. Larsen
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
Published electronically: April 16, 2014
MathSciNet review: 3217699
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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)$.


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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

V. Larsen
Affiliation: Department of Mathematics and Computer Science, MSC W401, Emory University, 400 Dowman Drive, Atlanta, Georgia 30322

DOI: https://doi.org/10.1090/S0002-9947-2014-06081-9
Received by editor(s): January 24, 2012
Received by editor(s) in revised form: October 20, 2012
Published electronically: April 16, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.