Just when I started understanding the basics of Schubert calculus and how the cohomology ring of Grassmannians $G(k,n)$ works, I figured I needed a generalization in terms of (partial) flags.
My goal is to answer questions somewhat like this: let $k_1\leq k_2$ and consider a flag variety of the form $Fl = \{ (\Lambda_1,\Lambda_2)\in G(k_1,n)\times G(k_2,n):\text{ } \Lambda_1\subset \Lambda_2 \}$ and take projections $\pi_1:Fl \rightarrow G(k_1,n)$, $\pi_2:Fl\rightarrow G(k_2,n)$. Given a class of a subvariety $[V]^* \in H^*(G(k_2,n))$, what is the class of $\pi_1(\pi_2^{-1}(V))$ in $H^*(G(k_1,n))$?
I think the first step is to figure out the structure of $H^*(Fl)$ but I don't know any good references. Textbooks like Hatcher, Eisenbud&Harris only touch this briefly. There is a paper by Coskun "A Littlewood - Richardson rule for two step flag varieties" but it appears to focus heavily on combinatorics.
Is there something a little more "basic" to begin with?
Thank you in advance!