Double Schubert polynomials and degeneracy loci for the classical groups

2000/11/30 by Andrew Kresch, Harry Tamvakis
Mathematics · #math.AG #math.CO #msc:14M15 #msc:14C17 #msc:05E15

paper · pdf

published as Annales de l'Institut Fourier 52 (2002), 1681-1727 · 34 pages, LaTeX; final version

arxiv created 2003/04/22 · arxiv updated 2009/11/30

Abstract

We propose a theory of double Schubert polynomials P_w(X,Y) for the Lie types B, C, D which naturally extends the family of Lascoux of Schutzenberger in type A. These polynomials satisfy positivity, orthogonality, and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When w is a maximal Grassmannian element of the Weyl group, P_w(X,Y) can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type A formula of Kempf and Laksov. An example, motivated by quantum cohomology, shows that there are no Chern class formulas for degeneracy loci of ``isotropic morphisms'' of bundles.

Related

• Schubert polynomials and degeneracy locus formulas
• Universal Schubert polynomials
• Schubert Polynomials in Types A and C
• Degeneracy Loci, Pfaffians, and Vexillary Signed Permutations in Types B, C, and D
• Schubert polynomials, pipe dreams, equivariant classes, and a co-transition formula