B_{n-1}-bundles on the flag variety, II

TL;DR

This paper advances the understanding of B_{n-1}-orbits on flag varieties by providing a combinatorial model, explicit formulas, and representatives, linking orbit structure to algebraic and combinatorial properties.

Contribution

It introduces a complete combinatorial model for B_{n-1}-orbits, explicit formulas for their counts, and a canonical set of representatives, extending previous work.

Findings

Developed a combinatorial model using partitions into lists.

Derived explicit formulas and generating functions for orbit counts.

Connected orbit closure orderings to Richardson and Springer orderings.

Abstract

This paper is the sequel to ``$B_{n-1}$-bundles on the flag variety, I". We continue our study of the orbits of a Borel subgroup $B_{n-1}$ of $G_{n-1}=GL(n-1)$ (resp. $SO(n-1)$) acting on the flag variety $\mathcal{B}_{n}$ of $G=GL(n)$ (resp. $SO(n)$). We begin by using the results of the first paper to obtain a complete combinatorial model of the $B_{n-1}$-orbits on $\mathcal{B}_{n}$ in terms of partitions into lists. The model allows us to obtain explicit formulas for the number of orbits as well as the exponential generating functions for the sequences $\{|B_{n-1}\backslash \mathcal{B}_{n}|\}_{n\geq 1}$ . We then use the combinatorial description of the orbits to construct a canonical set of representatives of the orbits in terms of flags. These representatives allow us to understand an extended monoid action on $B_{n-1}\backslash \mathcal{B}_{n}$ using simple roots of both $\mathfrak{g}_{n-1}$ and $\mathfrak{g}$ and show that the closure ordering on $B_{n-1}\backslash \mathcal{B}_{n}$ is the standard ordering of Richardson and Springer.