COHOMOLOGY OF SCHUBERT SUBVARIETIES OF GLN/P

Date

1992-05-01

Author

AKYILDIZ, E
LASCOUX, A
PRAGACZ, P

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Let GL(n) be the group of n x n invertible complex matrices, and P a parabolic subgroup of GL(n). In this paper we give a geometric description of the cohomology ring of a Schubert subvariety Y of GL(n)/P. Our main result (Theorem 3.1) states that the coordinate ring A(Y intersect Z) of the scheme-theoretic intersection of Y and the zero scheme Z of the vector field V associated to a principal regular nilpotent element n of gl(n) is isomorphic to the cohomology algebra H*(Y ; C) of Y. This theorem was conjectured for any reductive algebraic group G in [4], and it was proved for the Grassmannian manifolds in [2]. We were recently informed that Professor D. H. Peterson has just proved that GL(n) is exactly the algebraic group G where the cohomology ring of any Schubert subvariety Y of the space G/B is isomorphic to A(Y intersect Z). Here B stands for a Borel subgroup of G. It is also interesting to note that the cohomology ring of the union of two Schubert subvarieties in GL(n)/P may not admit such a description. This result is due to Professor J. B. Carrell.

Subject Keywords

Geometry and Topology, Algebra and Number Theory, Analysis

URI

https://hdl.handle.net/11511/66478

Journal

JOURNAL OF DIFFERENTIAL GEOMETRY

DOI

https://doi.org/10.4310/jdg/1214448268

Collections

Department of Mathematics, Article

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Citation Formats

E. AKYILDIZ, A. LASCOUX, and P. PRAGACZ, “COHOMOLOGY OF SCHUBERT SUBVARIETIES OF GLN/P,” JOURNAL OF DIFFERENTIAL GEOMETRY, pp. 511–519, 1992, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/66478.