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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 12:04:48 |
| Classification: |
msc:14N05 |
| Keywords: |
projective geometry, projective point, hyperplane, order preserving, order reversing |
| Defines: |
polarity, duality |
| 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 map $f:PG(V)\rightarrow PG(V)$.
\item A duality of order $2$ is called a \emph{polarity}.
\item The set of all order preserving and reversing maps $f:PG(V)\rightarrow PG(V)$ form a group denoted $P\Gamma L^*(V)$.
\end{itemize}
\end{defn}
From the fundamental theorem of projective geometry it follows if $\dim V\neq 2$ then every order preserving map is induced by a semi-linear transformation of $V$. In similar fashion we have
\begin{prop}
$P\Gamma L^*(V)=P\Gamma L(V)\ltimes \mathbb{Z}_2$, meaning that every order
reversing map $f:PG(V)\rightarrow PG(V)$ can be decomposed as a $f=sr$ where
$s$ is induced from a semi-linear transformation and $r$ is a polarity.
\end{prop}
\begin{remark}
The group $P\Gamma L^*(V)$ is the full auotmorphism group of $PSL(V)$.
In particular, the polarities account for the graph automorphisms of the
Dynkin diagram of $A_{d-1}$, $d=\dim V$. When $\dim V=2$ there is no
graph automorphism, just as there are no dualities (points are hyperplanes
when $\dim V=2$.
\end{remark}
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