Bₙ₋₁-orbits on the Flag Variety and the Bruhat Graph of the Symmetric Group

Let G=Gₙ=GL(n) be the n× n complex general linear group and embed Gₙ₋₁=GL(n-1) in the top left hand corner of G . The standard Borel subgroup of upper triangular matrices Bₙ₋₁ of Gₙ₋₁ acts on the flag variety Bₙ of G with finitely many orbits. In this paper, we show that each Bₙ₋₁ -orbit is the intersection of orbits of two Borel subgroups of G acting on Bₙ . This allows us to give a new combinatorial description of the Bₙ₋₁ -orbits on Bₙ by associating to each orbit a pair of Weyl group elements. The closure relations for the Bₙ₋₁ -orbits can then be understood in terms of the Bruhat order on the symmetric group, and the Richardson-Springer monoid action on the orbits can be understood in terms of a well-understood monoid action on the symmetric group. This approach makes the closure relation more transparent than in Magyar (J. Algebraic Combin 21:71–101, 2005) and the monoid action significantly more computable than in our papers (Colarusso and Evens, J. Algebra 596:128–154, 2022) and (Colarusso and Evens, J. Algebra 619:249–297, 2023), and also allows us to obtain new information about the orbits including a simple formula for the dimension of an orbit.