Schubert duality for SL(n,R)-flag domains

TL;DR

This paper explores the structure of cycles in SL(n,R)-flag domains, identifying special Schubert varieties that intersect these cycles transversally, and provides explicit descriptions and counts of these intersections.

Contribution

It offers a detailed description of Schubert varieties related to SL(n,R)-flag domains and computes their intersection points with cycles, advancing understanding of their geometric and homological properties.

Findings

Identified optimal Schubert varieties intersecting cycles transversally.

Computed the total number of such Schubert varieties.

Provided explicit descriptions of intersection points in terms of flags.

Abstract

This paper is concerned with the study of spaces of naturally defined cycles associated to SL(n,R)-flag domains. These are compact complex submanifolds in open orbits of real semisimple Lie groups in flag domains of their complexification. It is known that there are optimal Schubert varieties which intersect the cycles transversally in finitely many points and in particular determine them in homology. Here we give a precise description of these Schubert varieties in terms of certain subsets of the Weyl group and compute their total number. Furthermore, we give an explicit description of the points of intersection in terms of flags and their number.