# Bruhat order for two subspaces and a flag

Evgeny Smirnov

Number | 33 |
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Author | Evgeny Smirnov |

Year | 2007 |

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).