Point-Line Geometries with a Generating Set that Depends on the Underlying Field

Abstract

Suppose Г is a Lie incidence geometry defined over some field F having a Lie incidence geometry Г0 of the same type but defined over a subfield F0 ≤ F as a subgeometry. We investigate the following question: how many points (if any at all) do we have to add to the point-set of Г0 in order to obtain a generating set for Г? We note that if Г is generated by the points of an apartment, then no additional points are needed. We then consider the long-root geometry of the group SL n +1(F) and the line-grassmannians of the polar geometries associated to the groups О2n+1(F), Sp2n (F) and O +2n (F). It turns out that in these cases the maximum number of points one needs to add to Г0 in order to generate Г equals the maximal number of roots one needs to adjoin to F0 in order to generate F. We prove that in the case of the long-root geometry of the group SL n +1(F) the point-set of Г0 does not generate Г. As a by-product we determine the generating rank of the line grassmannian of the polar geometry associated to Sp2n (F) (n ≥ 3), if F is a prime field of odd characteristic.KeywordsPolar SpaceDynkin DiagramChevalley GroupPrime FieldWitt IndexThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.