$B_{n-1}$-orbits on the flag variety and the Bruhat graph of the symmetric group

Abstract

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 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 the flag variety of $G$. This allows us to give a new combinatorial description of the $B_{n-1}$-orbits 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 Weyl group, and the Richardson-Springer monoid action on the orbits can be understood in terms of the classical monoid action of the Weyl group on itself. This approach makes the closure relation more transparent than in earlier work of Magyar and the monoid action significantly more computable than in our earlier papers, and also allows us to obtain new information about the orbits including a simple formula for the dimension of an orbit.