B₍-₁-orbits on the Flag Variety and the Bruhat Graph of the Symmetric Group

Abstract Let G=G₍=GL (n) be the n n complex general linear group and embed G₍-₁=GL (n-1) in the top left hand corner of G. The standard Borel subgroup of upper triangular matrices B₍-₁ of G₍-₁ acts on the flag variety B₍ of G with finitely many orbits. In this paper, we show that each B₍-₁ -orbit is the intersection of orbits of two Borel subgroups of G acting on B₍. This allows us to give a new combinatorial description of the B₍-₁ -orbits on B₍ by associating to each orbit a pair of Weyl group elements. The closure relations for the B₍-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.