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'polarity'
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| Title of object: |
polarity |
| Canonical Name: |
Polarity2 |
| Type: |
Definition |
| Created on: |
2006-06-09 12:04:48 |
| Modified on: |
2006-06-09 15:59:00 |
| Classification: |
msc:51A05 |
| Keywords: |
projective geometry, projective point, hyperplane, order preserving, order reversing |
| Defines: |
polarity, duality, correlation |
| Synonyms: |
polarity=order reversing |
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Content:
\begin{defn}
\begin{itemize}
\item Given a finite dimensional vector space $V$, a \emph{duality} of the
projective geometry $PG(V)$ is an order-reversing bijection
$f:PG(V)\rightarrow PG(V)$.
\item A duality of order $2$ is called a \emph{polarity}.
\item The set of all order preserving and reversing bijections $f:PG(V)\rightarrow PG(V)$ form a group denoted $P\Gamma L^*(V)$.
\end{itemize}
\end{defn}
\begin{remark}
Dualities are also called \emph{correlations}. This is in contrast to
projectivities (collineations) which are order preserving bijections.
\end{remark}
\begin{remark}
A polarity/duality necessarily interchanges points with hyperplanes.
Thus an alternative definition of a duality is an projectivity $f:PG(V)\rightarrow PG(V^*)$.
\end{remark}
Through the use of the fundamental theorem of projective geometry, dualities and polarities can be identified with non-degenerate sesquilinear forms. |
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