The universal representation group of Huybrechts's dimensional dual hyperoval

A d-dimensional dual hyperoval can be regarded as the image S = p(Σ) of a full d-dimensional projective embedding p of a dual circular space Σ. The affine expansion Exp(p) of p is a semibiplane and its universal cover is the expansion of the abstract hull p of p. In this paper we consider Huybrechts's dual hyperoval, namely p(Σ) where Σ is the dual of the affine space AG(n, 2) ⊂ PG(n, 2) and p is induced by the embedding of the line grassmannian of PG(n, 2) in PG ((n+1/2) — 1, 2). It is known that the universal cover of Exp(p) is a truncation of a Coxeter complex of type D2n and that, if Ũ is the codomain of the abstract hull p of p, then Ũ is a subgroup of the Coxeter group D of type D2n, | Ũ | = 22n -1 but Ũ is non-commutative. This information does not explain what the structure of Ũ is and how Ũ is placed inside D. These questions will be answered in this paper. © 2006, Mathematical Sciences Publishers. All rights reserved.