Self-Duality of Generalized Eagon-Northcott Complexes and Cohomology of Line Bundles on Flag Varieties

In characteristic 0, we prove self-duality of particular generalized Eagon-Northcott complexes. The failure to extend to positive characteristic results from cohomology of line bundles on flag varieties being characteristic dependent and moreover not full described in positive characteristic. We further discuss joint work with Luca Fiorindo, Shahriyar Roshan Zamir, and Hongmiao Yu on a uniform identification of stable cohomology groups of particular classes of these line bundles in arbitrary characteristic. Joint with Annet Kyomuhangi, Emanuela Marangone, and Claudiu Raicu, we give a recursive formula for the cohomology of line bundles on the incidence correspondence again in arbitrary characteristic. This calculation is linked to some additional results we prove for the equivariant splitting type of bundles of principal parts on the projective line, the graded Han-Monsky representation ring, and Weak Lefschetz properties for monomial complete intersections.