Hello!

Let $n,m\geq 0$ be integers. If I understand it correctly, there is the following description of the cohomology of the complex Grassmannian $\text{Gr}(m+n;m)$: denote by $\text{Sym}(n,m)$ the subring of the polynomial ring in $n+m$ variables consisting of polynomials which are symmetric in the first $n$ and the last $m$ variables, and by $\text{Sym}(n+m)$ the ring of symmetric polynomials in $n+m$ variables. Then $\text{Sym}(n,m)$ is a free $\text{Sym}(n+m)$ module, and its graded rank (written as a $q$-polynomial) equals the Poincare polynomial of the Grassmannian $\text{Gr}(m+n;m)$.

I'd like to know if this is true for more general flag varieties: Is it true that the graded rank of $\text{Sym}(m_1,...,m_k)$ over $\text{Sym}(m_1+...+m_k)$ is the Poincare polynomial of the variety of flags $(\{0\}=U_0,U_1,...,U_k=V)$ in $V := {\mathbb C}^{m_1+...+m_k}$ such that $\text{dim}(U_{i}) - \text{dim}(U_{i-1}) = m_i$ for $i=1,...,k$? If yes: how to prove it? :-)

On the other hand, there is a presentation of the cohomology of the full flag variety of ${\mathbb C}^N$ as the quotient of ${\mathbb C}[X_1,...,X_N]$ by the ideal generated by the symmetric polynomials of positive degree. How do these two descriptions of the cohomology of flag varieties relate?

Thank you!

Hanno