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.

Cardinali, I., Giuzzi, L., Pasini, A. (2023). On the generation of some Lie-type geometries. JOURNAL OF COMBINATORIAL THEORY. SERIES A, 193 [10.1016/j.jcta.2022.105673].

On the generation of some Lie-type geometries

Cardinali I.;
2023

Abstract

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.
Cardinali, I., Giuzzi, L., Pasini, A. (2023). On the generation of some Lie-type geometries. JOURNAL OF COMBINATORIAL THEORY. SERIES A, 193 [10.1016/j.jcta.2022.105673].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/1216736