The Bruhat order on clans

TL;DR

This paper provides an explicit combinatorial description of the Bruhat order on clans, which parametrize orbits of GL_p x GL_q on the flag variety, clarifying orbit closures via incidence conditions.

Contribution

It introduces a detailed combinatorial characterization of the Bruhat order on clans, linking orbit closures to incidence conditions on flags.

Findings

Explicit description of the Bruhat order on clans

Characterization of orbit closures via incidence conditions

Enhanced understanding of orbit structure on flag varieties

Abstract

We give an explicit description of the closure containment order (or "Bruhat order") on the set of orbits of GL_p \times GL_q on the flag variety GL_{p+q}/B, relative to the parametrization of the orbits by combinatorial objects called "clans". This leads to a corresponding description of the closures of such orbits as sets of flags satisfying certain incidence conditions encoded by the parametrizing clans.