Pieri's formula for flag manifolds and Schubert polynomials
Annales de l'Institut Fourier, Volume 46 (1996) no. 1, pp. 89-110.

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.

Nous établissons la formule pour la multiplication par la classe d’une variété de Schubert spéciale dans l’anneau de cohomologie de la variété de drapeaux. Cette formule décrit aussi la multiplication d’un polynôme de Schubert soit par un polynôme symétrique élémentaire soit par un polynôme symétrique homogène. Ainsi nous généralisons la formule classique de Pieri sur les polynômes de Schur (associés aux grassmaniennes) au cas des polynômes de Schubert (associés aux variétés de drapeaux). Notre technique principale est une description géométrique explicite de certaines intersections des variétés de Schubert. Cette méthode nous permet de calculer des constantes de structure additionnelles pour l’anneau de cohomologie, et d’exprimer certaines de ces constantes en termes de chaînes dans l’ordre de Bruhat du groupe symétrique. Cette description induit à son tour un résultat sur l’ordre de Bruhat.

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     title = {Pieri's formula for flag manifolds and {Schubert} polynomials},
     journal = {Annales de l'Institut Fourier},
     pages = {89--110},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Sottile, Frank. Pieri's formula for flag manifolds and Schubert polynomials. Annales de l'Institut Fourier, Volume 46 (1996) no. 1, pp. 89-110. doi : 10.5802/aif.1508. https://aif.centre-mersenne.org/articles/10.5802/aif.1508/

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