Parabolic Projection and Generalized Cox Configurations

Abstract

Building on the work of Longuet-Higgins in 1972 and Calderbank and Macphersonin 2009, we study the combinatorics of symmetric configurations of hyperplanesand points in projective space, called generalized Cox configurations.To do so, we use the formalism of morphisms between incidence systems. Wenotice that the combinatorics of Cox configurations are closely related to incidencesystems associated to certain Coxeter groups. Furthermore, the incidencegeometry of projective space P (V ), where V is a vector space, can be viewed asan incidence system of maximal parabolic subalgebras in a semisimple Lie algebrag, in the special case g = pgl (V ) the projective general linear Lie algebra ofV . Using Lie theory, the Coxeter incidence system for the Coxeter group, whoseCoxeter diagram is the underlying diagram of the Dynkin diagram of the g, canbe embedded into the parabolic incidence system for g. This embedding givesa symmetric geometric configuration which we call a standard parabolic configurationof g. In order to construct a generalized Cox configuration, we projecta standard parabolic configuration of type Dn into the parabolic incidence systemof projective space using a process called parabolic projection, which mapsa parabolic subalgebra of the Lie algebra to a parabolic subalgebra of a lowerdimensional Lie algebra.As a consequence of this construction, we obtain Cox configurations and theiranalogues in higher dimensional projective spaces. We conjecture that the generalizedCox configurations we construct using parabolic projection are nondegenerateand, furthermore, any non-degenerate Cox configuration is obtainedin this way. This conjecture yields a formula for the dimension of the space ofnon-degenerate generalized Cox configurations of a fixed type, which enables usto develop a recursive construction for them. This construction is closely relatedto Longuet-Higgins’ recursive construction of (generalized) Clifford configurationsbut our examples are more general and involve the extra parameters.

Date of Award	29 Jan 2014
Original language	English
Awarding Institution	University of Bath
Supervisor	David Calderbank (Supervisor) & Alastair King (Supervisor)