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 $X_{\mathrm{GZ}}$.We show that $H^{\ast }(\mathcal{F}{\ell }_n,\mathbb{Q})$ is the Poincaré duality quotient of the subalgebra of $A^{\ast }(X_{\mathrm{GZ}},\mathbb{Q})$ generated by degree $1$ elements. We compute these algebras for $n=3$ and see that, in general, this subalgebra does not have Poincaré duality.