2025/02/15 by Mark Colarusso, Colarusso, Mark, Sam Evens +1
Mathematics · #05E14 #14L30 #14M15 #20G20 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2502.10912
openalex publication_date 2025/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G=GL(n) be the n× n complex general linear group and let Bn be its flag variety. The standard Borel subgroup B of upper triangular matrices acts on the product Bn× ℙn-1 with finitely many orbits. In this paper, we study the B-orbits on the subvarieties Bn× Oi, where Oi is the B-orbit on ℙn-1 containing the line through the origin in the direction of the i-th standard basis vector of ℂn. For each i=1,…, n, we construct a bijection between B-orbits on Bn\timesOi and certain pairs of Schubert cells in Bn\timesBn. 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(Bn\timesOi)|\n≥ 1 and \|B\backslash (Bn× ℙn-1)|\n≥ 1. In the sequel to this paper, we use the results of this paper to construct a correspondence between B-orbits on Bn×ℙn-1 and a collection of B-orbits on the flag variety Bn+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 Bn×ℙn-1 can be understood via the Bruhat order by using our results in [CE].