On the generation of some Lie-type geometries

TL;DR

This paper investigates the generation properties of certain Lie-type geometries called Grassmannians over division rings, proving their generating rank is infinite unless the ring is finitely generated, with implications for algebraic closures of finite fields.

Contribution

It establishes that Grassmannians of Lie-type geometries cannot be generated over proper sub-division rings, revealing their infinite generating rank in most cases.

Findings

Generating rank is infinite over non-finitely generated division rings.

Grassmannians cannot be generated over proper sub-division rings.

Embedding rank differs from generating rank in certain cases.

Abstract

Let $X_n(K)$ be a building of Coxeter type $X_n = A_n$ or $X_n = D_n$ defined over a given division ring $K$ (a field when $X_n = D_n$). For a non-connected set $J$ of nodes of the diagram $X_n$, let $\Gamma(K) = Gr_J(X_n(K))$ be the $J$-Grassmannian of $X_n(K)$. We prove that $\Gamma(K)$ cannot be generated over any proper sub-division ring $K_0$ of $K$. As a consequence, the generating rank of $\Gamma(K)$ is infinite when $K$ is not finitely generated. In particular, if $K$ is the algebraic closure of a finite field of prime order then the generating rank of $Gr_{1,n}(A_n(K))$ is infinite, although its embedding rank is either $(n+1)^2-1$ or $(n+1)^2$.