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Divided difference operators are 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 for arbitrary partial flag varieties of arbitrary Lie type, and show how to use them in the ordinary cohomology of G/P. 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 Billeys 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-planes in a complex n-dimensional vector space.
We define convex-geometric counterparts of divided difference (or Demazure) operators from the Schubert calculus and representation theory. These operators are used to construct inductively polytopes that capture Demazure characters of representation
For a compact Lie group $G$ with maximal torus $T$, Pittie and Smith showed that the flag variety $G/T$ is always a stably framed boundary. We generalize this to the category of $p$-compact groups, where the geometric argument is replaced by a homoto
Sums of the form add((-1)^n q^(n(n-1)/2) x^n, n>=0) are called partial theta functions. In his lost notebook, Ramanujan recorded many identities for those functions. In 2003, Warnaar found an elegant formula for a sum of two partial theta functions.
We introduce a supersymmetric analog of the classical Coxeter frieze patterns. Our approach is based on the relation with linear difference operators. We define supersymmetric analogs of linear difference operators called Hills operators. The space o
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures c