Quotients of incidence geometries

We develop a theory for quotients of geometries and obtain sufficient conditions for the quotient of a geometry to be a geometry. These conditions are compared with earlier work on quotients, in particular by Pasini and Tits. We also explore geometric properties such as connectivity, firmness and transitivity conditions to determine when they are preserved under the quotienting operation. We show that the class of coset pregeometries, which contains all flag-transitive geometries, is closed under an appropriate quotienting operation.Comment: 26 pages, 5 figure

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

On transparent embeddings of point-line geometries

We introduce the class of transparent embeddings for a point-line geometry $\Gamma = ({\mathcal P},{\mathcal L})$ as the class of full projective embeddings $\varepsilon$ of $\Gamma$ such that the preimage of any projective line fully contained in $\varepsilon({\mathcal P})$ is a line of $\Gamma$. We will then investigate the transparency of Pl\"ucker embeddings of projective and polar grassmannians and spin embeddings of half-spin geometries and dual polar spaces of orthogonal type. As an application of our results on transparency, we will derive several Chow-like theorems for polar grassmannians and half-spin geometries.Comment: 28 Pages/revised version after revie

Veronesean embeddings of dual polar spaces of orthogonal type

Given a point-line geometry P and a pappian projective space S,a veronesean embedding of P in S is an injective map e from the point-set of P to the set of points of S mapping the lines of P onto non-singular conics of S and such that e(P) spans S. In this paper we study veronesean embeddings of the dual polar space \Delta_n associated to a non-singular quadratic form q of Witt index n >= 2 in V = V(2n + 1; F). Three such embeddings are considered,namely the Grassmann embedding gr_n,the composition vs_n of the spin (projective) embedding of \Delta_n in PG(2n-1; F) with the quadric veronesean map of V(2n; F) and a third embedding w_n defined algebraically in the Weyl module V (2\lambda_n),where \lambda_n is the fundamental dominant weight associated to the n-th simple root of the root system of type Bn. We shall prove that w_n and vs_n are isomorphic. If char(F) is different from 2 then V (2\lambda_n) is irreducible and w_n is isomorphic to gr_n while if char(F) = 2 then gr_n is a proper quotient of w_n. In this paper we shall study some of these submodules. Finally we turn to universality,focusing on the case of n = 2. We prove that if F is a finite field of odd order q > 3 then sv_2 is relatively universal. On the contrary,if char(F) = 2 then vs_2 is not universal. We also prove that if F is a perfect field of characteristic 2 then vs_n is not universal,for any n>=2