Presenting the cohomology of a Schubert variety
نویسنده
چکیده
We extend the short presentation due to [Borel ’53] of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its Schubert varieties. Our result is stated in a root-system uniform manner by introducing the essential set of a Coxeter group element, generalizing and giving a new characterization of [Fulton ’92]’s definition for permutations. Further refinements are obtained in type A.
منابع مشابه
A Positive Monk Formula in the S1-equivariant Cohomology of Type a Peterson Varieties
Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in ...
متن کاملGlobal F -regularity of Schubert Varieties with Applications to D-modules Niels Lauritzen, Ulf Raben-pedersen and Jesper Funch Thomsen
A projective algebraic variety X over an algebraically closed field k of positive characteristic is called globally F -regular if the section ring S(L) = ⊕n≥0 H (X,Ln) of an ample line bundle L on X is strongly F -regular in the sense of Hochster and Huneke [9] (cf. Definition 1.1). This important notion was introduced by Karen Smith in [18]. In this paper we prove that Schubert varieties are g...
متن کاملSchubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Lagrangian Grassmannian
We give positive formulas for the restriction of a Schubert Class to a T -fixed point in the equivariant K-theory and equivariant cohomology of the Lagrangian Grassmannian. Our formulas rely on a result of GhorpadeRaghavan, which gives an equivariant Gröbner degeneration of a Schubert variety in the neighborhood of a T -fixed point of the Lagrangian Grassmannian.
متن کاملSchubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Grassmannian
We give positive formulas for the restriction of a Schubert Class to a T -fixed point in the equivariant K-theory and equivariant cohomology of the Grassmannian. Our formulas rely on a result of Kodiyalam-Raghavan and Kreiman-Lakshmibai, which gives an equivariant Gröbner degeneration of a Schubert variety in the neighborhood of a T -fixed point of the Grassmannian.
متن کاملSchubert Polynomials and Classes of Hessenberg Varieties
Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a “Giambelli formula” expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hess...
متن کامل