## Innovations in Incidence Geometry

### The universal representation group of Huybrechts's dimensional dual hyperoval

#### Abstract

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

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

Dates
Accepted: 22 May 2006
First available in Project Euclid: 28 February 2019

https://projecteuclid.org/euclid.iig/1551323244

Digital Object Identifier
doi:10.2140/iig.2006.3.121

Mathematical Reviews number (MathSciNet)
MR2267610

Zentralblatt MATH identifier
1229.51008

#### Citation

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. https://projecteuclid.org/euclid.iig/1551323244

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