DIII clan combinatorics for the orthogonal Grassmannian

2019/07/20 by Aram Bingham, Bingham, Aram, Özlem Uğurlu +1
Mathematics · #05A15 #05A19 #14M15 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A15 #msc:05A19 #msc:14M15

paper · pdf · doi:10.48550/arxiv.1907.08875

Second version with revisions (including title change) based on referee reports; 31 pages, 6 figures; comments are welcome

openalex publication_date 2019/07/20 · arxiv created 2020/09/20 · arxiv updated 2020/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Borel subgroup orbits of the classical symmetric space SO_2n/GL_n are parametrized by DIII (n,n)-clans. We group the clans into "sects" corresponding to Schubert cells of the orthogonal Grassmannian, thus providing a cell decomposition for SO_2n/GL_n. We also compute a recurrence for the rank polynomial of the weak order poset on DIII clans, and then describe explicit bijections between such clans, diagonally symmetric rook placements, certain pairs of minimally intersecting set partitions, and a class of weighted Delannoy paths. Clans of the largest sect are in bijection with fixed-point-free partial involutions.

Citations

• K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles splitting as direct sums

Related

• Sects and lattice paths over the Lagrangian Grassmannian
• Reduced words for clans
• Schubert cells and Whittaker functionals for GL(n,ℝ) part I: Combinatorics
• The Bruhat order on clans
• Association schemes on the Schubert cells of a Grassmannian