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.

在特征0（characteristic 0）的情形下，我们证明了特定广义Eagon-Northcott复形（generalized Eagon-Northcott complex）的自对偶性。该结论无法推广至正特征的原因在于，旗簇（flag variety）上线丛（line bundle）的上同调具有特征依赖性，且在正特征下尚未得到完整刻画。我们进一步讨论了与Luca Fiorindo、Shahriyar Roshan Zamir以及于红淼（Hongmiao Yu）合作的工作：该工作针对任意特征下特定类别的此类线丛的稳定上同调群（stable cohomology group）给出了统一的刻画。此外，我们还与Annet Kyomuhangi、Emanuela Marangone及Claudiu Raicu合作，针对任意特征下关联对应（incidence correspondence）上的线丛上同调，给出了递推公式。该计算结果与我们针对射影直线（projective line）上主部丛（principal parts bundle）的等变分裂型、分次Han-Monsky表示环（graded Han-Monsky representation ring）以及单项式完全交（monomial complete intersection）的弱莱夫谢茨性质（Weak Lefschetz property）所证明的若干额外结论存在关联。