Compositio Mathematica



Formulas for Lagranigian and orthogonal degeneracy loci; $\tilde{Q}$-polynomial approach


P. PRAGACZ a1 and J. RATAJSKI a2
a1 Max–Planck-Institut für Mathematik, Gottfried-Claren Strasse 26, D-53225 Bonn, Germany
a2 Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, PL-00950 Warsaw, Poland

Article author query
pragacz p   [Google Scholar] 
ratajski j   [Google Scholar] 
 

Abstract

The main goal of the paper is to give explicit formulas for the fundamental classes of Schubert subschemes in Lagrangian and orthogonal Grassmannians of maximal isotropic subbundles as well as some globalizations of them. The used geometric tools overlap appropriate desingularizations of such Schubert subschemes and Gysin maps for such Grassmannian bundles. The main algebraic tools are provided by the families of $\tilde{Q}$- and $\tilde{P}$-polynomials introduced and investigated in the present paper. The key technical result of the paper is the computation of the class of the (relative) diagonal in isotropic Grassmannian bundles based on the orthogonality property of $\tilde{Q}$- and $\tilde{P}$-polynomials. Some relationships with quaternionic Schubert varieties and Schubert polynomials for classical groups are also discussed.


Key Words: Lagrangian and orthogonal Grassmannians; Schubert subschemes; Gysin maps; divided differences; $\tilde{Q}$- and $\tilde{P}$-polynomials.