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Subgeometries in the André/Bruck-Bose representation

We consider the André/Bruck-Bose representation of the projective plane $\mathrm{PG}(2,q^n)$ in $\mathrm{PG}(2n,q)$. We investigate the representation of $\mathbb{F}_{q^k}$-sublines and $\mathbb{F}_{q^k}$-subplanes of $\mathrm{PG}(2,q^n)$, extending the results for $n=3$ of \cite{BarJack2} and correcting the general result of \cite{BarJack1}. We characterise the representation of $\mathbb{F}_{q^k}$-sublines tangent to or contained in the line at infinity, $\mathbb{F}_q$-sublines external to the line at infinity, $\mathbb{F}_q$-subplanes tangent to and $\mathbb{F}_{q^k}$-subplanes secant to the line at infinity.