Higher symmetric powers of tautological bundles over Hilbert schemes of points on a surface

We will discuss some results about higher symmetric powers of tautological bundles over Hilbert schemes of n points over a smooth complex algebraic surface X, associated to a line bundle L on X. The main point will be a structure theorem for (the push-forward of) triple and quadruple symmetric powers of tautological bundles over the symmetric variety, in terms of a filtration whose graded sheaves are invariants of exterior tensor products of L over the product X^n, with prescribed vanishing along diagonals. We believe that a similar picture should remain valid in general. Finally we will give some applications of the previous result; in particular we will present an effective vanishing theorem for (triple and quadruple) symmetric powers of tautological bundles twisted by a determinant, in presence of adequate positivity hypothesis.