Definition 1.
  • Given finite dimensional vector spacesMathworldPlanetmath V and W, a duality of the projective geometryMathworldPlanetmath PG(V) to PG(W) is an order-reversing bijection f:PG(V)PG(W). If W=V then we can refer to f as a correlation.

  • A correlation of order 2 is called a polarityMathworldPlanetmathPlanetmath.

  • The set of correlations and collineationsMathworldPlanetmath f:PG(V)PG(V) form a group denoted PΓL*(V) with the operation of compositionMathworldPlanetmath.

Remark 2.

Dualities are determined by where they map collinearMathworldPlanetmath triples. Given a map define on the points of PG(V) to the hyperplanesMathworldPlanetmathPlanetmath of PG(W) which maps collinear triples to triples of hyperplanes which intersect in a codimension 2 subspacePlanetmathPlanetmath, 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 projectivityMathworldPlanetmath (order-preserving map) f:PG(V)PG(V*).

Through the use of the fundamental theorem of projective geometryMathworldPlanetmath, dualities and polarities can be identified with non-degenerate sesquilinear formsPlanetmathPlanetmath. (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