Abstract
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.
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Magyar, P. Bruhat Order for Two Flags and a Line. J Algebr Comb 21, 71–101 (2005). https://doi.org/10.1007/s10801-005-6281-x
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DOI: https://doi.org/10.1007/s10801-005-6281-x