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Published online by Cambridge University Press:  15 June 2015

Institute for Pure and Applied Mathematics, UCLA, 460 Portola Plaza, Box 957121, Los Angeles, CA 90095-7121, USA;
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA;


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.

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