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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| Published in | Journal of combinatorial theory. Series A Vol. 193; p. 105673 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
01.01.2023
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0097-3165 |
| DOI | 10.1016/j.jcta.2022.105673 |
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| 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. |
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| AbstractList | 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. |
| ArticleNumber | 105673 |
| Author | Giuzzi, Luca Pasini, Antonio Cardinali, Ilaria |
| Author_xml | – sequence: 1 givenname: Ilaria orcidid: 0000-0002-9494-5800 surname: Cardinali fullname: Cardinali, Ilaria email: ilaria.cardinali@unisi.it organization: Department of Information Engineering and Mathematics, University of Siena, Via Roma 56, I-53100, Siena, Italy – sequence: 2 givenname: Luca orcidid: 0000-0003-3975-7281 surname: Giuzzi fullname: Giuzzi, Luca email: luca.giuzzi@unibs.it organization: DICATAM - Section of Mathematics, University of Brescia, Via Branze 53, I-25123, Brescia, Italy – sequence: 3 givenname: Antonio surname: Pasini fullname: Pasini, Antonio email: antonio.pasini@unisi.it organization: Department of Information Engineering and Mathematics, University of Siena, Via Roma 56, I-53100, Siena, Italy |
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| Snippet | 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... |
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| SubjectTerms | Coxeter building formula omitted Generation Grassmann geometry Subgeometry |
| Title | On the generation of some Lie-type geometries |
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