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 o 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 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 6 <= (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.