Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Schubert polynomials, the Bruhat order, and the geometry of flag
  manifolds   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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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph