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 $f:PG(V)\to PG(W)$. 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 $f:PG(V)\to PG(V)$ 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 (order-preserving 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 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