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 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.

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).)

 Title polarity Canonical name Polarity Date of creation 2013-03-22 15:57:58 Last modified on 2013-03-22 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