On the Generating Rank and Embedding Rank of the Hexagonic Lie Incidence Geometries

Abstract

Given a (thick) irreducible spherical building \(\Omega \) , we establish a bound on the difference between the generating rank and the embedding rank of its long root geometry and the dimension of the corresponding Weyl module, by showing that this difference does not grow when taking certain residues of \(\Omega \) (in particular the residue of a vertex corresponding to a point of the long root geometry, but also other types of vertices occur). We apply this to the finite case to obtain new results on the generating rank of mainly the exceptional long root geometries, answering an open question by Cooperstein about the generating ranks of the exceptional long root subgroup geometries. We completely settle the finite case for long root geometries of type \({{\textsf{A}}}_n\) , and the case of type \(\mathsf {F_{4,4}}\) over any field with characteristic distinct from 2 (which is not a long root subgroup geometry, but a hexagonic geometry).