Journal of Algebraic Combinatorics

, Volume 21, Issue 1, pp 71–101

Bruhat Order for Two Flags and a Line

Authors

    • Department of MathematicsMichigan State University
Article

DOI: 10.1007/s10801-005-6281-x

Cite this article as:
Magyar, P. J Algebr Comb (2005) 21: 71. doi:10.1007/s10801-005-6281-x

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.

Keywords

quiver representationsmultiple flagsdegenerationgeometric order

Copyright information

© Springer Science + Business Media, Inc. 2005