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

Abstract

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