Cohomology ring of the flag variety vs Chow cohomology ring of the Gelfand–Zetlin toric variety

This is the authors' abstract. We don't add key points for this paper.

We compare the cohomology ring of the flag variety {F}_n and the Chow cohomology ring of the Gelfand–Zetlin toric variety XGZ .We show that H^*(F{}_n, Q) is the Poincaré duality quotient of the subalgebra of A^*(XGZ, Q) generated by degree 1 elements. We compute these algebras for n=3 and see that, in general, this subalgebra does not have Poincaré duality.