# The generating rank of a polar Grassmannian

## TL;DR

This paper determines the generating rank of various polar Grassmannians over division rings, including those from Hermitian and quadratic forms, providing new exact values and structural insights.

## Contribution

It computes the generating rank for k-polar Grassmannians over division rings, including new cases from Hermitian and quadratic forms with specific Witt indices.

## Key findings

- Generated sets for 2-Grassmannians over finite fields with q=4,8,9.
- Generated sets can be over the prime subfield for N > 6.
- Exact generating ranks are established for several classes of polar Grassmannians.

## Abstract

In this paper we compute the generating rank of $k$-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of $k$-Grassmannians arising from Hermitian forms of Witt index $n$ defined over vector spaces of dimension $N > 2n$. We also study generating sets for the $2$-Grassmannians arising from quadratic forms of Witt index $n$ defined over $V(N,{\mathbb F}_q)$ for $q=4,8,9$ and $2n \leq N \leq 2n+2$. We prove that for $N >6$ they can be generated over the prime subfield, thus determining their generating rank.