Naoki Fujita, A model of Schubert polynomials in terms of Bott-Samelson varieties,

The purpose of this expository talk is to explain the results of the paper "Schubert polynomials and Bott-Samelson varieties" (published in 1998) by Peter Magyar. Schubert polynomials are generalizations of Schur polynomials. However, it is not clear how to generalize several classical formulas such as the Weyl character formula, the Demazure character formula, and the generating series of semistandard tableaux. In the paper above, the author introduced a geometric model of Schubert polynomials in terms of Bott-Samelson varieties and obtained these formulas. Here, Bott-Samelson varieties were realized as the orbit closures of a Borel subgroup in a product of Grassmannians. These orbit closures are determined by the Rothe diagrams, and are isomorphic to the usual Bott-Samelson varieties of certain reduced words whose associated chamber families with multiplicity are identical to the Rothe diagrams. In this setting, the generalized formulas above arise naturally from combinatorial arguments about the Rothe diagrams.