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

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 $\{|B_{n-1}\backslash {\mathcal{B}}_n|{\}}_{n\ge 1}$ . 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.

45 pages

References in corpus (3)

• Orbits of strongly solvable spherical subgroups on the flag variety
• Eigenvalue Coincidences and Multiplicity Free Spherical Pairs
• The Complex Orthogonal Gelfand-Zeitlin System