Schubert polynomials, the Bruhat order, and the geometry of flag manifolds

Details Schubert polynomials, the Bruhat order, and the geometry of flag manifolds Schubert polynomials, the Bruhat order, and the geometry of flag manifolds

1997-02-28

arxiv.org/abs/alg-geom/9703001v1

Duke Math. J., Vol 95 (1998) pp. 373-423.

Mathematics

Algebraic Geometry

Revised version of MSRI preprint \# 1996 - 083, 61 pages with 36 figures, where 15 of the pages and 26 of the figures are in a

Scientific paper

We illuminate the relation between the Bruhat order on the symmetric group and structure constants (Littlewood-Richardson coefficients) for the cohomology of the flag manifold in terms of its basis of Schubert classes. Equivalently, the structure constants for the ring of polynomials in variables $x_1,x_2,...$ in terms of its basis of Schubert polynomials. We use combinatorial, algebraic, and geometric methods, notably a study of intersections of Schubert varieties and maps between flag manifolds. We establish a number of new identities among these structure constants. This leads to formulas for some of these constants and new results on the enumeration of chains in the Bruhat order. A new graded partial order on the symmetric group which contains Young's lattice arises from these investigations. We also derive formulas for certain specializations of Schubert polynomials.

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