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 order-reversing bijection $f:PG(V)\\rightarrow PG(W)$ . 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 $f:PG(V)\\rightarrow PG(V)$ form a group denoted $P\\Gamma L^{*}(V)$ with the operation of composition.

Remark 2.

Dualities are determined by where they map collinear triples. Given a mapdefine 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.

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 (order-preserving map) $f:PG(V)\\rightarrow PG(V^{*})$ .

Through the use of the fundamental theorem of projective geometry, dualities and polarities can be identified with non-degenerate sesquilinear forms. (See Polarities and forms (http://planetmath.org/PolaritiesAndForms).)

单词	Polarity
释义	polarity Definition 1. • Given finite dimensional vector spaces $V$ and $W$ , a 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. • A correlation of order $2$ is called a polarity. • 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. Remark 2. Dualities are determined by where they map collinear triples. Given a mapdefine 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. 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 (order-preserving map) $f:PG(V)\\rightarrow PG(V^{})$ . Through the use of the fundamental theorem of projective geometry, dualities and polarities can be identified with non-degenerate sesquilinear forms. (See Polarities and forms (http://planetmath.org/PolaritiesAndForms).)
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