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SIMPLE GROUPS STABILIZING POLYNOMIALS
Published online by Cambridge University Press: 15 June 2015
Abstract
We study the problem of determining, for a polynomial function $f$ on a vector space
$V$, the linear transformations
$g$ of
$V$ such that
$f\circ g=f$. When
$f$ is invariant under a simple algebraic group
$G$ acting irreducibly on
$V$, we note that the subgroup of
$\text{GL}(V)$ stabilizing
$f$ often has identity component
$G$, and we give applications realizing various groups, including the largest exceptional group
$E_{8}$, as automorphism groups of polynomials and algebras. We show that, starting with a simple group
$G$ and an irreducible representation
$V$, one can almost always find an
$f$ whose stabilizer has identity component
$G$, and that no such
$f$ exists in the short list of excluded cases. This relies on our core technical result, the enumeration of inclusions
$G<H\leqslant \text{SL}(V)$ such that
$V/H$ has the same dimension as
$V/G$. The main results of this paper are new even in the special case where
$k$ is the complex numbers.
MSC classification
- Type
- Research Article
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s) 2015
References
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