Plücker degrees of Quot schemes

We study the Plücker degree of the main component of the Quot scheme of length $l$ quotients of a locally free sheaf on a smooth projective scheme $\mathrm{S}$ of dimension $d⩾1$. This degree is determined by classes in the Chow ring of the symmetric product ${\mathrm{S}}^{(l)}$, which are given by the pushforward of the powers of $c_1({\mathcal{O}}^{[l]})$ with respect to the canonical morphism from the Quot scheme to ${\mathrm{S}}^{(l)}$. We describe a decomposition of these classes, allowing us to compute the (in a certain sense) leading term of the Plücker degree. We also obtain a higher-dimensional analogue of a classical result of Schubert.