Imbrex geometries

We introduce an axiom on strong parapolar spaces of diameter 2, which arises naturally in the framework of Hjelmslev geometries. This way, we characterize the Hjelmslev-Moufang plane and its relatives (line Grassmannians, certain half-spin geometries and Segre geometries). At the same time we provide a more general framework for a Lemma of Cohen, which is widely used to study parapolar spaces. As an application, if the geometries are embedded in projective space, we provide a common characterization of (projections of) Segre varieties, line Grassmann varieties, half-spin varieties of low rank, and the exceptional variety $\mathcal{E}_{6,1}$ by means of a local condition on tangent spaces

Veronese representation of projective Hjelmslev planes over some quadratic alternative algebras

We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type Ã_2 and respective absolute type Ã_2, Ã_5 and Ẽ_6

A geometric construction of panel-regular lattices in buildings of types ~A_2 and ~C_2

Using Singer polygons, we construct locally finite affine buildings of types ~A_2 and ~C_2 which admit uniform lattices acting regularly on panels. This construction produces very explicit descriptions of these buildings as well as very short presentations of the lattices. All but one of the ~C_2-buildings are necessarily exotic. To the knowledge of the author, these are the first presentations of lattices in buildings of type ~C_2. Integral and rational group homology for the lattices is also calculated.Comment: 42 pages, small corrections and cleanup. Results are unchanged

Symplectic polarities of buildings of type E₆

A symplectic polarity of a building Delta of type E (6) is a polarity whose fixed point structure is a building of type F (4) containing residues isomorphic to symplectic polar spaces. In this paper, we present two characterizations of such polarities among all dualities. Firstly, we prove that, if a duality theta of Delta never maps a point to a neighbouring symp, and maps some element to a non-opposite element, then theta is a symplectic duality. Secondly, we show that, if a duality theta never maps a chamber to an opposite chamber, then it is a symplectic polarity. The latter completes the programme for dualities of buildings of type E (6) of determining all domestic automorphisms of spherical buildings, and it also shows that symplectic polarities are the only polarities in buildings of type E (6) for which the Phan geometry is empty

A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two more references adde

Pure Designation. Deleuze’s Reading of Hjelmslev in The Time-Image

In the second chapter of The Time-Image, Deleuze addresses the conditions of possibility of a semiology of cinema. These conditions depend on the relations between cinema and language: under what conditions can cinematic images and signs be understood as a language? In other words, (how) can cinematic images and signs be inscribed in the discursive plane of the signifiable? Discussing Christian Metz’s semiological approach of cinema, Deleuze argues that the structural conditions of linguistics and of post-Saussurian semiology cannot adequately render intelligible the specificity of cinematic semiosis. Drawing on Louis Hjelmslev’s semiotics, Deleuze redefines the specificity of a relation of designation distinct from a relation of signification (strictly linguistic), a specificity that concerns the fact that the designative relation is antecedent and heterogeneous to any signifying relation. Put differently, the very constitution of the sign is redefined: in opposition to semiology, semiotics becomes the study of images and signs as (1) being independent of language in general and (2) expressing a “non-language material”. This article explicates the importance of Hjelmslev’s semiotic theory in The Time-Image by offering a detailed account of the constitution of the sign in Hjelmslev’s Prolegomena to a Theory of Language and by tracing Deleuze’s earlier appreciation of Hjelmslev in Anti-Oedipus and its intricate relation to his appreciation of Jean-François Lyotard’s theory of designation in Discourse Figure