Divided difference operators for partial flag varieties
Julianna Tymoczko · Contemporary mathematics - American Mathematical Society · 2026
Divided difference operators are well-known degree-reducing operators on the cohomology of flag varieties that are used to compute algebraic invariants of the ring (for instance, structure constants). We identify divided difference operators on the equivariant cohomology of G / P G/P for arbitrary partial flag varieties of arbitrary Lie type, resolving ambiguity in the literature about these operators. We give explicit presentations of the equivariant cohomology of Grassmannians in classical Lie types and show how to use these divided difference operators in the ordinary cohomology of G / P G/P as well. We provide three applications. The first shows that all Schubert classes of partial flag varieties can be generated from a sequence of divided difference operators on the highest-degree Schubert class. The second is a generalization of Billey’s formula for the localizations of equivariant Schubert classes of flag varieties to arbitrary partial flag varieties. The third gives a choice of Schubert polynomials for partial flag varieties as well as an explicit formula for each. We focus on the example of maximal Grassmannians, including Grassmannians of k k -planes in a complex n n -dimensional vector space.