Orbits on a product of two flags and a line and the Bruhat Order, I

Let $G=GL(n)$ be the $n\times n$ complex general linear group and let $\mathcal{B}_{n}$ be its flag variety. The standard Borel subgroup $B$ of upper triangular matrices acts on the product $\mathcal{B}_{n}\times \mathbb{P}^{n-1}$ with finitely many orbits. In this paper, we study the $B$-orbits on the subvarieties $\mathcal{B}_{n}\times \mathcal{O}_{i}$, where $\mathcal{O}_{i}$ is the $B$-orbit on $\mathbb{P}^{n-1}$ containing the line through the origin in the direction of the $i$-th standard basis vector of $\mathbb{C}^{n}$. For each $i=1,\dots, n$, we construct a bijection between $B$-orbits on $\mathcal{B}_{n}\times\mathcal{O}_{i}$ and certain pairs of Schubert cells in $\mathcal{B}_{n}\times\mathcal{B}_{n}$. We also show that this bijection can be used to understand the Richardson-Springer monoid action on such $B$-orbits in terms of the classical monoid action of the symmetric group on itself. We also develop combinatorial models of these orbits and use these models to compute exponential generating functions for the sequences $\{|B\backslash(\mathcal{B}_{n}\times\mathcal{O}_{i})|\}_{n\geq 1}$ and $\{|B\backslash (\mathcal{B}_{n}\times \mathbb{P}^{n-1})|\}_{n\geq 1}$. In the sequel to this paper, we use the results of this paper to construct a correspondence between $B$-orbits on $\mathcal{B}_{n}\times\mathbb{P}^{n-1}$ and a collection of $B$-orbits on the flag variety $\mathcal{B}_{n+1}$ of $GL(n+1)$ and show that this correspondence respects closures relations and preserves monoid actions. As a consequence both closure relations and monoid actions for all $B$-orbits on $\mathcal{B}_{n}\times\mathbb{P}^{n-1}$ can be understood via the Bruhat order by using our results in [CE].