2021/07/22 by Mark Colarusso, Colarusso, Mark, Sam Evens +1
Mathematics · Physics and Astronomy · #05A15 #14L30 #14M15 #20G20 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2107.10819
openalex publication_date 2021/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is the sequel to ``Bn-1-bundles on the flag variety, I". We continue our study of the orbits of a Borel subgroup Bn-1 of Gn-1=GL(n-1) (resp. SO(n-1)) acting on the flag variety Bn 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 Bn-1-orbits on Bn 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 \|Bn-1\backslash Bn|\n≥ 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 Bn-1\backslash Bn using simple roots of both \mathfrakgn-1 and \mathfrakg and show that the closure ordering on Bn-1\backslash Bn is the standard ordering of Richardson and Springer.