Journal of Algebraic Combinatorics

, Volume 21, Issue 1, pp 71–101

Bruhat Order for Two Flags and a Line



The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a linear space V under linear transformations of V; or equivalently, it describes the closure of an orbit of GL(V acting diagonally on the product of two flag varieties.

We consider the degenerations of a triple consisting of two flags and a line, or equivalently the closure of an orbit of GL(V) acting diagonally on the product of two flag varieties and a projective space. We give a simple rank criterion to decide whether one triple can degenerate to another. We also classify the minimal degenerations, which involve not only reflections (i.e., transpositions) in the Weyl group SVSn = dim(V, but also cycles of arbitrary length. Our proofs use only elementary linear algebra and combinatorics.


quiver representations multiple flags degeneration geometric order 


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Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast Lansing

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