Divided difference operators for partial flag varieties

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…