Title:

Parabolic projection and generalized Cox configurations

Building on the work of LonguetHiggins in 1972 and Calderbank and Macpherson in 2009, we study the combinatorics of symmetric configurations of hyperplanes and points in projective space, called generalized Cox configurations. To do so, we use the formalism of morphisms between incidence systems. We notice that the combinatorics of Cox configurations are closely related to incidence systems associated to certain Coxeter groups. Furthermore, the incidence geometry of projective space P (V ), where V is a vector space, can be viewed as an incidence system of maximal parabolic subalgebras in a semisimple Lie algebra g, in the special case g = pgl (V ) the projective general linear Lie algebra of V . Using Lie theory, the Coxeter incidence system for the Coxeter group, whose Coxeter diagram is the underlying diagram of the Dynkin diagram of the g, can be embedded into the parabolic incidence system for g. This embedding gives a symmetric geometric configuration which we call a standard parabolic configuration of g. In order to construct a generalized Cox configuration, we project a standard parabolic configuration of type Dn into the parabolic incidence system of projective space using a process called parabolic projection, which maps a parabolic subalgebra of the Lie algebra to a parabolic subalgebra of a lower dimensional Lie algebra. As a consequence of this construction, we obtain Cox configurations and their analogues in higher dimensional projective spaces. We conjecture that the generalized Cox configurations we construct using parabolic projection are nondegenerate and, furthermore, any nondegenerate Cox configuration is obtained in this way. This conjecture yields a formula for the dimension of the space of nondegenerate generalized Cox configurations of a fixed type, which enables us to develop a recursive construction for them. This construction is closely related to LonguetHigginsâ€™ recursive construction of (generalized) Clifford configurations but our examples are more general and involve the extra parameters.
