polarity
Definition 1.

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Given finite dimensional vector spaces^{} $V$ and $W$, a duality of the projective geometry^{} $PG(V)$ to $PG(W)$ is an orderreversing bijection $f:PG(V)\to PG(W)$. If $W=V$ then we can refer to $f$ as a correlation.

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A correlation of order $2$ is called a polarity^{}.

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The set of correlations and collineations^{} $f:PG(V)\to PG(V)$ form a group denoted $P\mathrm{\Gamma}{L}^{*}(V)$ with the operation of composition^{}.
Remark 2.
Dualities are determined by where they map collinear^{} triples. Given a map define on the points of $P\mathit{}G\mathit{}\mathrm{(}V\mathrm{)}$ to the hyperplanes^{} of $P\mathit{}G\mathit{}\mathrm{(}W\mathrm{)}$ which maps collinear triples to triples of hyperplanes which intersect in a codimension 2 subspace^{}, this specifies a unique duality.
Remark 3.
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^{} (orderpreserving map) $f\mathrm{:}P\mathit{}G\mathit{}\mathrm{(}V\mathrm{)}\mathrm{\to}P\mathit{}G\mathit{}\mathrm{(}{V}^{\mathrm{*}}\mathrm{)}$.
Through the use of the fundamental theorem of projective geometry^{}, dualities and polarities can be identified with nondegenerate sesquilinear forms^{}. (See Polarities and forms (http://planetmath.org/PolaritiesAndForms).)
Title  polarity 
Canonical name  Polarity 
Date of creation  20130322 15:57:58 
Last modified on  20130322 15:57:58 
Owner  Algeboy (12884) 
Last modified by  Algeboy (12884) 
Numerical id  12 
Author  Algeboy (12884) 
Entry type  Definition 
Classification  msc 51A10 
Classification  msc 51A05 
Synonym  order reversing 
Related topic  SesquilinearFormsOverGeneralFields 
Related topic  PolaritiesAndForms 
Defines  polarity 
Defines  duality 
Defines  correlation 
Defines  pole 
Defines  polar 