Innovations in Incidence Geometry

The universal representation group of Huybrechts's dimensional dual hyperoval

Alberto Del Fra and Antonio Pasini

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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 D 2 n 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 D 2 n , | U ˜ | = 2 2 n 1 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.

Article information

Innov. Incidence Geom., Volume 3, Number 1 (2006), 121-148.

Received: 15 March 2006
Accepted: 22 May 2006
First available in Project Euclid: 28 February 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 15A75: Exterior algebra, Grassmann algebras 51E45 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx]

dimensional dual hyperovals semibiplanes embeddings Coxeter groups exterior algebras


Del Fra, Alberto; Pasini, Antonio. The universal representation group of Huybrechts's dimensional dual hyperoval. Innov. Incidence Geom. 3 (2006), no. 1, 121--148. doi:10.2140/iig.2006.3.121.

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