# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1903.07229