Pieri's formula for flag manifolds and Schubert polynomials

Frank Sottile

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 1, page 89-110
  • ISSN: 0373-0956

Abstract

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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 or a complete symmetric polynomial. Thus, we generalize the classical Pieri’s formula for Schur polynomials (associated to Grassmann varieties) to Schubert polynomials (associated to 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, some of which we express in terms of paths in the Bruhat order on thesymmetric group, which in turn yields an enumerative result about the Bruhat order.

How to cite

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Sottile, Frank. "Pieri's formula for flag manifolds and Schubert polynomials." Annales de l'institut Fourier 46.1 (1996): 89-110. <http://eudml.org/doc/75177>.

@article{Sottile1996,
abstract = {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 or a complete symmetric polynomial. Thus, we generalize the classical Pieri’s formula for Schur polynomials (associated to Grassmann varieties) to Schubert polynomials (associated to 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, some of which we express in terms of paths in the Bruhat order on thesymmetric group, which in turn yields an enumerative result about the Bruhat order.},
author = {Sottile, Frank},
journal = {Annales de l'institut Fourier},
keywords = {multiplication by the class of a special Schubert variety; integral cohomology ring of the flag manifold; Pieri formual; Bruhat order},
language = {eng},
number = {1},
pages = {89-110},
publisher = {Association des Annales de l'Institut Fourier},
title = {Pieri's formula for flag manifolds and Schubert polynomials},
url = {http://eudml.org/doc/75177},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Sottile, Frank
TI - Pieri's formula for flag manifolds and Schubert polynomials
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 1
SP - 89
EP - 110
AB - 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 or a complete symmetric polynomial. Thus, we generalize the classical Pieri’s formula for Schur polynomials (associated to Grassmann varieties) to Schubert polynomials (associated to 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, some of which we express in terms of paths in the Bruhat order on thesymmetric group, which in turn yields an enumerative result about the Bruhat order.
LA - eng
KW - multiplication by the class of a special Schubert variety; integral cohomology ring of the flag manifold; Pieri formual; Bruhat order
UR - http://eudml.org/doc/75177
ER -

References

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