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

## TL;DR

This paper compares the cohomology ring of the flag variety with the Chow cohomology ring of the Gelfand-Zetlin toric variety, revealing structural differences and computing specific cases.

## Contribution

It demonstrates that the flag variety's cohomology ring is a Gorenstein quotient of a subalgebra of the Gelfand-Zetlin toric variety's Chow ring, providing explicit computations for small cases.

## Key findings

- The cohomology ring of the flag variety is a Gorenstein quotient.
- The subalgebra generated by degree 1 elements often lacks Poincare duality.
- Explicit calculations for n=3 illustrate the general structure.

## Abstract

We compare the cohomology ring of the flag variety $FL_n$ and the Chow cohomology ring of the Gelfand-Zetlin toric variety $X_{GZ}$. We show that $H^*(FL_n, \mathbb{Q})$ is the Gorenstein quotient of the subalgebra $L$ of $A^*(X_{GZ}, \mathbb{Q})$ 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.