On the generation of some Lie-type geometries

Let Xn(K) be a building of Coxeter type Xn=An or Xn=Dn defined over a given division ring K (a field when Xn=Dn). For a non-connected set J of nodes of the diagram Xn, let Γ(K)=GrJ(Xn(K)) be the J-grassmannian of Xn(K). We prove that Γ(K) cannot be generated over any proper sub-division ring K0 of K...

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Bibliographic Details
Published inJournal of combinatorial theory. Series A Vol. 193; p. 105673
Main Authors Cardinali, Ilaria, Giuzzi, Luca, Pasini, Antonio
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.01.2023
Subjects
Coxeter building
formula omitted
Generation
Grassmann geometry
Subgeometry
Coxeter building
Generation
Dn
Subgeometry
Grassmann geometry
An
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ISSN0097-3165
DOI10.1016/j.jcta.2022.105673

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Summary:Let Xn(K) be a building of Coxeter type Xn=An or Xn=Dn defined over a given division ring K (a field when Xn=Dn). For a non-connected set J of nodes of the diagram Xn, let Γ(K)=GrJ(Xn(K)) be the J-grassmannian of Xn(K). We prove that Γ(K) cannot be generated over any proper sub-division ring K0 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 Gr1,n(An(K)) is infinite, although its embedding rank is either (n+1)2−1 or (n+1)2.
ISSN:0097-3165
DOI:10.1016/j.jcta.2022.105673