Bruhat order for two flags and a line
Abstract
Description
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_n, n=dim(V), but also cycles of arbitrary length. Our proofs use only elementary linear algebra and combinatorics.
36 pages. Version 2 adds an Introduction stating results in terms of S_n; and corrects typos, including in statement of move (v). Version 3: typos
36 pages. Version 2 adds an Introduction stating results in terms of S_n; and corrects typos, including in statement of move (v). Version 3: typos