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

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

E. Cartan's attempt at bridge-building between Einstein and the Cosserats -- or how translational curvature became to be known as {\em torsion}

\'Elie Cartan's "g\'en\'eralisation de la notion de courbure" (1922) arose from a creative evaluation of the geometrical structures underlying both, Einstein's theory of gravity and the Cosserat brothers generalized theory of elasticity. In both theories groups operating in the infinitesimal played a crucial role. To judge from his publications in 1922--24, Cartan developed his concept of generalized spaces with the dual context of general relativity and non-standard elasticity in mind. In this context it seemed natural to express the translational curvature of his new spaces by a rotational quantity (via a kind of Grassmann dualization). So Cartan called his translational curvature "torsion" and coupled it to a hypothetical rotational momentum of matter several years before spin was encountered in quantum mechanics.Comment: 36 p

Conformally coupled supermultiplets in four and five dimensions

We obtain by superfield methods the exceptional representations of the OSp(2N/4,R) and SU(2,2/1) superalgebras which extend to supersingletons of SU(2,2/2N) and F(4), respectively. These representations describe superconformally coupled multiplets and appear in three- and four-dimensional superconformal field theories which are holographic descriptions of certain anti-de Sitter supergravities

Isometric embeddings of Johnson graphs in Grassmann graphs

Let $V$ be an $n$-dimensional vector space ($4\le n <\infty$) and let ${\mathcal G}_{k}(V)$ be the Grassmannian formed by all $k$-dimensional subspaces of $V$. The corresponding Grassmann graph will be denoted by $\Gamma_{k}(V)$. We describe all isometric embeddings of Johnson graphs $J(l,m)$, $1<m<l-1$ in $\Gamma_{k}(V)$, $1<k<n-1$ (Theorem 4). As a consequence, we get the following: the image of every isometric embedding of $J(n,k)$ in $\Gamma_{k}(V)$ is an apartment of ${\mathcal G}_{k}(V)$ if and only if $n=2k$. Our second result (Theorem 5) is a classification of rigid isometric embeddings of Johnson graphs in $\Gamma_{k}(V)$, $1<k<n-1$.Comment: New version -- 14 pages accepted to Journal of Algebraic Combinatoric