Vector Invariants in Arbitrary Characteristic

Let k be an algebraically closed field of characteristic p ≥ 0. Let H be a subgroup of GL_n(k). We are interested in the determination of the vector invariants of H. When the characteristic of k is 0, it is known that the invariants of d vectors, d ≥ n, are obtained from those of n vectors by polarization. This result is not true when char k = p > 0 even in the case where H is a torus. However, we show that the algebra of invariants is always the p-root closure of the algebra of polarized invariants. We also give conditions for the two algebras to be equal, relating equality to good filtrations and saturated subgroups. As applications, we discuss the cases where H is finite or a classical group.