Abstract
We prove an explicit closed formula, written as a sum of Pfaffians, which describes each equivariant Schubert class for the Grassmannian of isotropic subspaces in a symplectic vector space On démontre une formule close explicite, écrite comme une somme de Pfaffiens, qui décrit toute classe de Schubert équivariante pour la Grassmannienne des sous-espaces isotropes dans un espace vectoriel symplectique.
Highlights
The classical Giambelli formula expresses a general Schubert class of the Grassmannian as the determinant of a matrix whose entries are the so-called special Schubert classes
The Giambelli problem consists in finding a “closed formula” for a Schubert class in terms of those special classes, and in the torus equivariant setting it is closely related to the theory of degeneracy loci of vector bundles
For the symplectic or orthogonal Grassmannians, there is a natural notion of special Schubert classes, which takes into account the isotropic conditions arising from the symplectic or orthogonal form
Summary
The classical Giambelli formula expresses a general Schubert class of the Grassmannian as the determinant of a matrix whose entries are the so-called special Schubert classes. For the non-maximal isotropic Grassmannians, an answer to the (non-equivariant) Giambelli problem was given by Buch, Kresch, and Tamvakis [BKTb, BKTa] Their formula expresses an arbitrary Schubert class as a polynomial in the special Schubert classes, it is defined by means of Young’s raising operators. The main result of this paper provides a formula expressing each double Schubert polynomial associated to the isotropic Grassmannians as a sum of Pfaffians whose entries are Wilson’s double theta polynomials corresponding to the special Schubert classes. This immediately leads to a proof of Wilson’s conjecture, because the raising operator formula can be rewritten as a Pfaffian sum by a formal computation. The only content which is not included in [IM15] is about a combinatorial description of the Bruhat order on a certain parabolic quotient of the type C Weyl group (Theorem 2.8)
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