0.1. Topological properties of intersections of pairs and, more generally, of k-tuples of Schubert cells belonging to distinct Schubert cell decompositions of a flag space are of particular importance in representation theory and have been intensively studied during the last 15 years, see e.g. [BB, KL1, KL2, De1, GS]. Intersections of certain special arrangements of Schubert cells are related d...

We study the intersections of general Schubert varieties Xw with permuted big cells, and give an inductive degeneration of each such “Schubert patch” to a StanleyReisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. Wemaintain reducedness using the results of [Knutson 2007] on automatically reduced degenerations, or throughmore standard cohomolog...

We present an analog of the Robinson-Schensted-Knuth correspondence which bijectively proves the Cauchy identity for Schubert polynomials; thus, it generalizes the classical correspondence, which proves the similar identity for Schur functions. Our correspondence is based on a new insertion procedure for certain binary tableaux of staircase shape, which are the analogs of semistandard Young tab...

We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of the orthogonal flag variety X = SON/B. We use these polynomials to describe the arithmetic Schubert calculus on X. Moreover, we give a method to compute the natural arithmetic Chern numbers on X, and show that they are all rational numbers.

We show the equivalence of the Pieri formula for ag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a nite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a uniied deenition o...