polarityPolarity22006-06-09 12:04:482006-06-21 13:46:42Definitionpolaritydualitycorrelationpolepolarprojective geometryprojective pointhyperplaneorder preservingorder reversing\usepackage{latexsym}
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\begin{defn}
\begin{itemize}
\item Given finite dimensional vector spaces $V$ and $W$, a \emph{duality} of the projective geometry $PG(V)$ to $PG(W)$ is an order-reversing bijection
$f:PG(V)\rightarrow PG(W)$. If $W=V$ then we can refer to $f$ as a correlation.
\item A correlation of order $2$ is called a \emph{polarity}.
\item The set of correlations and collineations $f:PG(V)\rightarrow PG(V)$ form a group denoted $P\Gamma L^*(V)$ with the operation of composition.
\end{itemize}
\end{defn}
\begin{remark}
Dualities are determined by where they map collinear triples. Given a map
define on the points of $PG(V)$ to the hyperplanes of $PG(W)$ which maps collinear triples to triples of hyperplanes which intersect in a codimension 2 subspace, this specifies a unique duality.
\end{remark}
\begin{remark}
A polarity/duality necessarily interchanges points with hyperplanes. In this context points are called ``poles'' and hyperplanes ``polars.''
An alternative definition of a duality is a projectivity (order-preserving map) $f:PG(V)\rightarrow PG(V^*)$.
\end{remark}
Through the use of the fundamental theorem of projective geometry, dualities and polarities can be identified with non-degenerate sesquilinear forms. (See \PMlinkname{Polarities and forms}{PolaritiesAndForms}.)