SIMPLE GROUPS STABILIZING POLYNOMIALS

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.