Abstract

A $d$-dimensional dual hyperoval can be regarded as the image $\mathcal{S}=\rho \left(\Sigma \right)$ 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 $\tilde{\rho }$ 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 $\tilde{U}$ is the codomain of the abstract hull $\tilde{\rho }$ of $\rho$, then $\tilde{U}$ is a subgroup of the Coxeter group $D$ of type ${\mathsf{D}}_{2^n}$, $\left|\tilde{U}\right|=2^{2^n-1}$ but $\tilde{U}$ is non-commutative. This information does not explain what the structure of $\tilde{U}$ is and how $\tilde{U}$ is placed inside $D$. These questions will be answered in this paper.

Keywords

Mathematical Subject Classification 2000

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