Pieri's rule for flag manifolds and Schubert polynomials

Details Pieri's rule for flag manifolds and Schubert polynomials Pieri's rule for flag manifolds and Schubert polynomials

1995-05-02

arxiv.org/abs/alg-geom/9505001v1

Ann. de l'Inst. Four., 46 (1996) 89-110

Mathematics

Algebraic Geometry

21 pages with 1 figure. AMSLaTeX v 1.1

Scientific paper

We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary symmetric polynomial or a complete homogeneous symmetric polynomial. Thus, we generalize the classical Pieri's rule for symmetric polynomials/Grassmann varieties to Schubert polynomials/flag manifolds. Our primary technique is an explicit geometric description of certain intersections of Schubert varieties. This method allows us to compute additional structure constants for the cohomology ring, which we express in terms of paths in the Bruhat order on the symmetric group.

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