Bruhat Order for Two Flags and a Line
Peter Magyar
Department of Mathematics Michigan State University East Lansing MI 48824
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 S VS n = dim( V, but also cycles of arbitrary length. Our proofs use only elementary linear algebra and combinatorics.
Pages: 71–101
Keywords: keywords quiver representations; multiple flags; degeneration; geometric order
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References
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2. M. Brion, “Groupe de Picard et nombres caractéristiques des variétés sphériques,” Duke Mathematical Journal 58 (1989), 397-424.
3. P.H. Edelman, “The Bruhat order of the symmetric group is lexicographically shellable,” Proceedings of the AMS 82 (1981), 355-358.
4. W. Fulton, Young Tableaux, Cambridge University Press, 1997.
5. P. Magyar, J. Weyman, and A. Zelevinsky, “Multiple flags of finite type,” Advances in Mathematics 141 (1999), 97-118.
6. P. Magyar, J. Weyman, and A. Zelevinsky, “Symplectic multiple flags of finite type,” Journal of Algebra 230 (2000), 245-265.
7. C. Riedtmann, “Degenerations for representations of quivers with relations,” Ann. Sci. Éc. Norm. Sup. 19(4) (1986), 275-301.
8. A. Skowronski and G. Zwara, “Degenerations in module varieties with finitely many orbits,” Contemporary Mathematics 229 (1998), 343-356.
9. G. Zwara, “Degenerations of finite dimensional modules are given by extensions,” Compositio Mathematica 121 (2000), 205-218.
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