A Pieri-type formula for even orthogonal Grassmannians

Abstract

We study the cohomology ring of the Grassmannian G of isotropic n- subspaces of a complex 2m-dimensional vector space, endowed with a nondegenerate or- thogonal form (here 1 n < m). We state and prove a formula giving the Schubert class decomposition of the cohomology products in H (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 dierences associated with the even orthogonal group SO(2m). Contents