The study of the cohomology of line bundles on (partial) flag varieties is an important problem at the intersection of algebraic geometry, commutative algebra, and representation theory. Over fields of characteristic zero, this is well-understood thanks to the Borel-Weil-Bott theorem, but in positive characteristics, it remains largely open.
In this talk, I will focus on the incidence correspondence, the partial flag variety parameterizing pairs consisting of a point in projective space and a hyperplane containing it. I will describe joint work with C. Raicu, A. Kyomuhangi, and E. Reed, where we establish a recursive formula for the characters of the cohomology of line bundles on the incidence correspondence in positive characteristic.
Finally, I will highlight how this problem is unexpectedly connected to other open questions in positive characteristic. In particular, I will explain how our work leads to a better understanding of the Han-Monsky representation ring, the ring of isomorphism classes of finite-length graded $k[T]$ modules.
