Highest weight modules and polarized embeddings of shadow spaces

Let Gamma be the K-shadow space of a spherical building Delta. An embedding V of Gamma is called polarized if it affords all "singular" hyperplanes of Gamma. Suppose that Delta is associated to a Chevalley group G. Then Gamma can be embedded into what we call the Weyl module for G of highest weight lambda_K. It is proved that this module is polarized and that the associated minimal polarized embedding is precisely the irreducible G-module of highest weight lambda_K. In addition a number of general results on polarized embeddings of shadow spaces are proved. The last few sections are devoted to the study of specific shadow spaces, notably minuscule weight geometries, polar grassmannians, and projective flag-grassmannians. The paper is in part expository in nature so as to make this material accessible to a wide audience.Comment: Improvement in exposition of Sections 1-3 and . Notation improved. References added. Main results unchange

The generating rank of the unitary and symplectic Grassmannians

We prove that the Grassmannian of totally isotropic $k$-spaces of the polar space associated to the unitary group $\mathsf{SU}_{2n}(\mathbb{F})$ ($n\in \mathbb{N}$) has generating rank ${2n\choose k}$ when $\mathbb{F}\ne \mathbb{F}_4$. We also reprove the main result of Blok [Blok2007], namely that the Grassmannian of totally isotropic $k$-spaces associated to the symplectic group $\mathsf{Sp}_{2n}(\mathbb{F})$ has generating rank ${2n\choose k}-{2n\choose k-2}$, when $\rm{Char}(\mathbb{F})\ne 2$

A Quasi Curtis-Tits-Phan theorem for the symplectic group

We obtain the symplectic group \SP(V) as the universal completion of an amalgam of low rank subgroups akin to Levi components. We let \SP(V) act flag-transitively on the geometry of maximal rank subspaces of $V$. We show that this geometry and its rank $\ge 3$ residues are simply connected with few exceptions. The main exceptional residue is described in some detail. The amalgamation result is then obtained by applying Tits' lemma. This provides a new way of recognizing the symplectic groups from a small collection of small subgroups

The generating rank of the unitary and symplectic Grassmannians

AbstractWe prove that the Grassmannian of totally isotropic k-spaces of the polar space associated to the unitary group SU2n(F) (n∈N) has generating rank (2nk) when F≠F4. We also reprove the main result of Blok (2007) [3], namely that the Grassmannian of totally isotropic k-spaces associated to the symplectic group Sp2n(F) has generating rank (2nk)−(2nk−2), when Char(F)≠2

Far from a Point in the F4(q) Geometry

We take the long-root geometry associated with the Chevalley group F4(q), q even, and consider the subgeometry induced on the set of points at maximal distance from a given point. We shall describe this geometry and in particular determine the parameters of a 12-class association scheme on its point set obtained by joining certain classes of a group scheme