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 $F{L}_n$ and the Chow cohomology ring of the Gelfand-Zetlin toric variety $X_{GZ}$. We show that $H^{\ast }(F{L}_n,\mathbb{Q})$ is the Gorenstein quotient of the subalgebra $L$ of $A^{\ast }(X_{GZ},\mathbb{Q})$ generated by degree $1$ elements. We compute these algebras for $n=3$ to see that, in general, the subalgebra $L$ does not have Poincare duality.

16 pages, 3 figures. To appear in Journal of Combinatorial Algebra