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

TL;DR

This paper studies the structure and combinatorics of Borel group orbits on products of flag varieties and projective spaces, revealing new bijections, models, and generating functions that connect orbit classifications to classical symmetric group actions.

Contribution

It introduces a novel combinatorial framework and bijections for Borel orbits on certain subvarieties, linking them to Schubert cells and symmetric group actions, and computes related generating functions.

Findings

Bijection between B-orbits and pairs of Schubert cells

Combinatorial models for orbit classification

Explicit exponential generating functions for orbit counts

Abstract

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].