Cohomology on the incidence correspondence and related questions

We study a variety of questions centered around the computation of cohomology of line bundles on the incidence correspondence (the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane containing it). Over a field of characteristic zero, this problem is resolved by the Borel-Weil-Bott theorem. In positive characteristic, we give recursive formulas for cohomology, generalizing work of Donkin and Liu in the case of the 3-dimensional flag variety. In characteristic 2, we provide non-recursive formulas describing the cohomology characters in terms of truncated Schur polynomials and Nim symmetric polynomials. The main technical ingredient in our work is the recursive description of the splitting type of vector bundles of principal parts on the projective line. We also discuss properties of the structure constants in the graded Han-Monsky representation ring, and explain how our cohomology calculation characterizes the Weak Lefschetz Property for Artinian monomial complete intersections.