We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schub...

We link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth. Résumé. Nous ...

We give a combinatorial proof that the product of a Schubert polynomial by a Schur polynomial is a nonnegative sum of Schubert polynomials. Our proof uses Assaf’s theory of dual equivalence to show that a quasisymmetric function of Bergeron and Sottile is Schur-positive. By a geometric comparison theorem of Buch and Mihalcea, this implies the nonnegativity of Gromov-Witten invariants of the Gra...

Die Umsetzung eines Portals ist eine technisch schwierige und komplexe Aufgabe. Dennoch resultieren die schlimmsten Mängel des Endprodukts meist nicht aus technischen Fehlern, sondern von falsch oder unvollständig ermittelten oder umgesetzten Anforderungen. Dieser Beitrag beschreibt technische Maßnahmen, mit deren Hilfe die Anforderungsbestimmung beim Portalbau so mit der technischen Umsetzung ...

Consider a flag variety Fl over an algebraically closed field, and a subvariety V whose cycle class is a multiplicity–free sum of Schubert cycles. We show that V is arithmetically normal and Cohen–Macaulay, in the projective embedding given by any ample invertible sheaf on Fl.

In these notes, we present some fundamental results concerning flag varieties and their Schubert varieties. By a flag variety, we mean a complex projective algebraic variety X, homogeneous under a complex linear algebraic group. The orbits of a Borel subgroup form a stratification of X into Schubert cells. These are isomorphic to affine spaces; their closures in X are the Schubert varieties, ge...