I realised after finishing the last post that it’s simpler than I had thought to compute the total Chern classes of the tautological bundles appearing in the short exact sequence
on the Grassmannian . I’m keeping it in a separate post, though, both because the last post is already very long and because I haven’t yet defined the Schubert varieties that appear in the answer.
In particular, it’s easy to compute , as follows. First, is globally generated, and all its global sections come from the trivial bundle (including global sections). So consider choosing general global sections, that is, choosing , a “general, constant choice” of vectors from our ambient vector space. To compute the Chern class, we need to know: for which subspaces do our sections become linearly dependent in ? (Recall that is the fiber of over .)
Well, they are linearly dependent in if and only if intersects the subspace nontrivially. But the set of intersecting a given -dimensional subspace nontrivially is precisely the Schubert variety , or, more prosaically, the Schubert variety corresponding to a one-row partition of the appropriate length.
So, in fact, the total Chern class of is the sum , the sum of all the one-row partitions fitting in a box.
Repeating this logic for (thinking of it as the tautological quotient bundle on the dual Grassmannian) shows that is the sum , the sum of all the single-vertical-column partitions. To turn this into the total Chern class of , just change the plus signs to alternating signs.