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