-modules and representation theory of Lie groups
Using the polytopes defined in an earlier paper, we show in this paper the existence of degeneration of a large class of Schubert varieties of to toric varieties by extending the method used by Gonciulea and Lakshmibai for a miniscule to Schubert varieties in .
We propose a theory of double Schubert polynomials for the Lie types , , which naturally extends the family of Lascoux and Schützenberger in type . These polynomials satisfy positivity, orthogonality and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When is a maximal Grassmannian element of the Weyl group, can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type formula of Kempf and Laksov....