Orbits on a product of two flags and a line and the Bruhat order, II

Let $G=GL(n)$ be the $n\times n$ complex general linear group and let $\B_{n}$ be its flag variety. A Borel subgroup $B$ of $G$ acts on $\B_{n}\times \mathbb{P}^{n-1}$ diagonally with finitely many orbits. In this paper, we give an embedding of the $B$-orbits on $\B_{n}\times \mathbb{P}^{n-1}$ into the $B$-orbits on the flag variety $\B_{n+1}$ of $GL(n+1)$ and show that this correspondence respects closure relations and preserves monoid actions. As a consequence both closure relations and monoid actions on the set of all $B$-orbits on $\B_{n}\times\mathbb{P}^{n-1}$ can be understood via the Bruhat order on the symmetric group on $n+1$ letters by using our results in \cite{Shpairs}. This amplifies work of Magyar \cite{Magyar} by making the closure relation more transparent and allows us to compute the monoid action using Demazure products. If $S_i$ is the stabilizer in $B$ of the line through the ith standard basis vector, we give an embedding of the $S_i$-orbits on $\B_n$ into the $B$-orbits in a single $G$-orbit in $\B_{n+1},$ and this embedding plays an essential role in the above results. We extend results from our papers \cite{CE21I}, \cite{CE21II}, and \cite{Shpairs}, and in particular show that for $S_i$-orbits on $\B_n,$ the closure ordering is given by the Richardson-Springer standard order.