The generating rank of the unitary and symplectic Grassmannians

TL;DR

This paper determines the generating rank of certain isotropic Grassmannians associated with unitary and symplectic groups, providing explicit formulas under specific field conditions, thus advancing understanding of their algebraic and geometric properties.

Contribution

It establishes the generating rank formulas for isotropic Grassmannians related to unitary and symplectic groups, extending previous results and clarifying field characteristic dependencies.

Findings

Generating rank for unitary case: ${2nrace k}$ when $ eq ext{F}_4$.

Generating rank for symplectic case: ${2nrace k}-{2nrace k-2}$ when $ ext{Char}( extbf{F}) eq 2$.

Reproves and extends previous results on isotropic Grassmannians.

Abstract

We prove that the Grassmannian of totally isotropic $k$-spaces of the polar space associated to the unitary group $\mathsf{SU}_{2n}(\mathbb{F})$ ($n\in \mathbb{N}$) has generating rank ${2n\choose k}$ when $\mathbb{F}\ne \mathbb{F}_4$. We also reprove the main result of Blok [Blok2007], namely that the Grassmannian of totally isotropic $k$-spaces associated to the symplectic group $\mathsf{Sp}_{2n}(\mathbb{F})$ has generating rank ${2n\choose k}-{2n\choose k-2}$, when $\rm{Char}(\mathbb{F})\ne 2$.