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

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