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
Abstract

A $d$-dimensional dual hyperoval can be regarded as the image $\mathsc{S}=\rho \left(\Sigma \right)$ of a full $d$-dimensional projective embedding $\rho$ of a dual circular space $\Sigma$. The affine expansion $Exp\left(\rho \right)$ of $\rho$ is a semibiplane and its universal cover is the expansion of the abstract hull $\stackrel{˜}{\rho }$ of $\rho$.

In this paper we consider Huybrechts’s dual hyperoval, namely $\rho \left(\Sigma \right)$ where $\Sigma$ is the dual of the affine space $\mathsf{AG}\left(n,2\right)\subset \mathsf{PG}\left(n,2\right)$ and $\rho$ is induced by the embedding of the line grassmannian of $\mathsf{PG}\left(n,2\right)$ in $\mathsf{PG}\left(\left(\genfrac{}{}{0.0pt}{}{n+1}{2}\right)-1,2\right)$.

It is known that the universal cover of $Exp\left(\rho \right)$ is a truncation of a Coxeter complex of type ${\mathsf{D}}_{{2}^{n}}$ and that, if $\stackrel{˜}{U}$ is the codomain of the abstract hull $\stackrel{˜}{\rho }$ of $\rho$, then $\stackrel{˜}{U}$ is a subgroup of the Coxeter group $D$ of type ${\mathsf{D}}_{{2}^{n}}$, $|\stackrel{˜}{U}|={2}^{{2}^{n}-1}$ but $\stackrel{˜}{U}$ is non-commutative. This information does not explain what the structure of $\stackrel{˜}{U}$ is and how $\stackrel{˜}{U}$ is placed inside $D$. These questions will be answered in this paper.

Keywords
dimensional dual hyperovals, semibiplanes, embeddings, Coxeter groups, exterior algebras
Mathematical Subject Classification 2000
Primary: 20F55
Secondary: 15A75, 51E20