Sects and lattice paths over the Lagrangian Grassmannian

TL;DR

This paper explores the structure of Borel subgroup orbits in a symmetric space, establishing bijections with lattice paths and involutions, and providing a cell decomposition linked to combinatorial objects.

Contribution

It introduces a novel combinatorial framework connecting clans, lattice paths, and involutions to describe orbit structures in the symmetric space.

Findings

Bijections between clans, lattice paths, and involutions

Cell decomposition of the symmetric space based on clans

Largest sect's poset isomorphic to Bruhat order on partial involutions

Abstract

We examine Borel subgroup orbits in the classical symmetric space of type CI, which are parametrized by skew symmetric (n, n)-clans. We describe bijections between such clans, certain weighted lattice paths, and pattern-avoiding signed involutions, and we give a cell decomposition of the symmetric space in terms of collections of clans called sects. The largest sect with a conjectural closure order is isomorphic (as a poset) to the Bruhat order on partial involutions.