Vol. 3, No. 1, 2006

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The universal representation group of Huybrechts's dimensional dual hyperoval

Alberto Del Fra and Antonio Pasini

Vol. 3 (2006), No. 1, 121–148

A d-dimensional dual hyperoval can be regarded as the image S = ρ(Σ) of a full d-dimensional projective embedding ρ of a dual circular space Σ. The affine expansion Exp(ρ) of ρ is a semibiplane and its universal cover is the expansion of the abstract hull ρ˜ of ρ.

In this paper we consider Huybrechts’s dual hyperoval, namely ρ(Σ) where Σ is the dual of the affine space AG(n,2) PG(n,2) and ρ 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(ρ) is a truncation of a Coxeter complex of type D2n and that, if U˜ is the codomain of the abstract hull ρ˜ of ρ, then U˜ is a subgroup of the Coxeter group D of type D2n, |U˜| = 22n1 but U˜ is non-commutative. This information does not explain what the structure of U˜ is and how U˜ is placed inside D. These questions will be answered in this paper.

dimensional dual hyperovals, semibiplanes, embeddings, Coxeter groups, exterior algebras
Mathematical Subject Classification 2000
Primary: 20F55
Secondary: 15A75, 51E20
Received: 15 March 2006
Accepted: 22 May 2006
Alberto Del Fra
Antonio Pasini
Department of Information Engineering and Mathematics
University of Siena
Via Roma 56
53100 Siena