On the generation of some Lie-type geometries

Ilaria Cardinali; Luca Giuzzi; Antonio Pasini

doi:10.1016/j.jcta.2022.105673

paper

On the generation of some Lie-type geometries

arXiv:1912.03484 · doi:10.1016/j.jcta.2022.105673

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 $Γ (K) = G r_{J} (X_{n} (K))$ be the $J$ -Grassmannian of $X_{n} (K)$ . We prove that $Γ (K)$ cannot be generated over any proper sub-division ring $K_{0}$ of $K$ . As a consequence, the generating rank of $Γ (K)$ 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 $G r_{1, n} (A_{n} (K))$ is infinite, although its embedding rank is either $(n + 1)^{2} - 1$ or $(n + 1)^{2}$ .

20 pages; revised version