A Pieri-type formula for even orthogonal Grassmannians

Piotr Pragacz¹ ; Jan Ratajski²

¹ Institute of Mathematics Polish Academy of Sciences /Sniadeckich 8 P.O. Box 21 00-956 Warszawa 10, Poland
² ING Nationale-Nederlanden Polska S.A. Ludna 2 00-406 Warszawa, Poland

Fundamenta Mathematicae, Tome 178 (2003) no. 1, pp. 49-96.

Résumé

We study the cohomology ring of the Grassmannian $G$ of isotropic $n$-subspaces of a complex $2m$-dimensional vector space, endowed with a nondegenerate orthogonal form (here $1\le nm$). We state and prove a formula giving the Schubert class decomposition of the cohomology products in $H^{\ast }(G)$ of general Schubert classes by “special Schubert classes”, i.e. the Chern classes of the dual of the tautological vector bundle of rank $n$ on $G$. We discuss some related properties of reduced decompositions of “barred permutations” with even numbers of bars, and divided differences associated with the even orthogonal group $SO(2m)$.

Détail

DOI : 10.4064/fm178-1-2

Mots-clés : study cohomology ring grassmannian isotropic n subspaces complex m dimensional vector space endowed nondegenerate orthogonal form here state prove formula giving schubert class decomposition cohomology products * general schubert classes special schubert classes chern classes dual tautological vector bundle rank discuss related properties reduced decompositions barred permutations even numbers bars divided differences associated even orthogonal group