Simple Groups Stabilizing Polynomials

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 ◦ g = f . When f is invariant under a simple algebraic group G acting irreducibly on V , we note that the subgroup of GL(V ) stabilizing f often has identity component G, and we give applications realizing various groups, including the largest exceptional group E8, 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 6 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. 2010 Mathematics Subject Classification: 20G15 (primary); 15A72, 20G41 (secondary)