On the cohomology of line bundles over certain flag schemes II

Abstract

Over a field $K$ of characteristic $p$, let $Z$ be the incidence variety in ${\mathbb{P}}^d\times ({\mathbb{P}}^d{)}^{\ast }$ and let $\mathcal{L}$ be the restriction to $Z$ of the line bundle $\mathcal{O}(-n-d)⊠\mathcal{O}(n)$, where $n=p+f$ with $0\le f\le p-2$. We prove that $H^d(Z,\mathcal{L})$ is the simple ${\mathrm{GL}}_{d+1}$-module corresponding to the partition ${\lambda }_0=(p-1+f,p-1,f+1)$. When $f=0$, using the first author's description of $H^d(Z,\mathcal{L})$ and Jantzen's sum formula, we obtain as a by-product that the sum of the monomial symmetric functions $m_{\lambda }$, for all partitions $\lambda$ of $2p-1$ less than $(p-1,p-1,1)$ in the dominance order, is the alternating sum of the Schur functions $S_{p-1,p-1-i,1^{i+1}}$ for $i=0,\dots ,p-2$.

One footnote added: Conjecture 1.4.5 now proved by Darij Grinberg