# $DIII$ clan combinatorics for the orthogonal Grassmannian

## TL;DR

This paper introduces a combinatorial framework for understanding the structure of the orthogonal Grassmannian using $DIII$ clans, providing new parametrizations, cell decompositions, and bijections with various combinatorial objects.

## Contribution

It develops a novel parametrization of Borel orbit classes via $DIII$ clans, establishes a cell decomposition, and links clans to rook placements, set partitions, and Delannoy paths.

## Key findings

- Provided a cell decomposition for $SO_{2n}/GL_n$
- Derived a recurrence for the rank polynomial of the weak order
- Established bijections between clans and combinatorial objects

## 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.