Bruhat order for two subspaces and a flag

TL;DR

This paper extends the classical Bruhat order to triples of two subspaces and a flag in a vector space, providing a rank criterion and classifying minimal degenerations using elementary linear algebra and combinatorics.

Contribution

It introduces a new Bruhat order framework for triples of subspaces and flags, with a rank criterion and minimal degeneration classification, expanding the classical theory.

Findings

Established a rank criterion for degenerations of triples

Classified minimal degenerations of subspace and flag triples

Extended Bruhat order concepts to more complex geometric configurations

Abstract

The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a finite-dimensional vector space V; or, equivalently, the closure of an orbit of the group GL(V) acting on the direct product of two full flag varieties. We obtain a similar result for triples consisting of two subspaces and a partial flag in V; this is equivalent to describing the closure of a GL(V)-orbit in the product of two Grassmannians and one flag variety. We give a rank criterion to check whether such a triple can be degenerated to another one, and we classify the minimal degenerations. Our methods involve only elementary linear algebra and combinatorics of graphs (originating in Auslander-Reiten quivers).