## Formulas for Lagrangian and orthogonal degeneracy loci; $$\tilde Q$$-polynomial approach.(English)Zbl 0916.14026

The main goal of the present article is to give explicit formulas for the fundamental classes of Schubert subschemes in Lagrangian and orthogonal grassmannians of maximal isotropic subbundles. This is accomplished by the usual method of desingularization and pushing down classes from the desinguarization to the original space. In general the approach gives an efficient algorithm for computing formulas for Lagrangian and orthogonal Schubert classes. In some cases, like for one or two Schubert conditions, closed formulas are obtained. The classes are expressed in terms of $$\widetilde Q$$- and $$\widetilde P$$-polynomials introduced by the authors. These are variants of Schur’s $$Q$$- and $$P$$-polynomials.
The geometric situation is technically more complicated than the well known situation for Schubert cycles in grassmannians. Computations of the class of the relative diagonal and orthogonality properties of the $$\widetilde Q$$- and $$\widetilde P$$-polynomials are used to overcome the additional difficulties. Frequent references are made to the work of the first author.

### MSC:

 14M15 Grassmannians, Schubert varieties, flag manifolds 05E05 Symmetric functions and generalizations 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry

### Keywords:

Schubert subschemes; orthogonal Grassmannians; Lagrangian
Full Text: