PlanetMath: polarity

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polarity

(Definition)

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.)

"polarity" is owned by Algeboy.

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Other names:

order reversing

Also defines:

polarity, duality, correlation, pole, polar

Keywords:

projective geometry, projective point, hyperplane, order preserving, order reversing

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Cross-references: sesquilinear forms, non-degenerate, fundamental theorem of projective geometry, projectivity, subspace, codimension, intersect, hyperplanes, points, collinear, map, composition, operation, group, collineations, order, bijection, order-reversing, projective geometry, vector spaces, finite dimensional
There are 60 references to this entry.

This is version 9 of polarity, born on 2006-06-09, modified 2006-06-21.
Object id is 7981, canonical name is Polarity2.
Accessed 9545 times total.

Classification:

AMS MSC:	51A05 (Geometry :: Linear incidence geometry :: General theory and projective geometries)
	51A10 (Geometry :: Linear incidence geometry :: Homomorphism, automorphism and dualities)

Pending Errata and Addenda

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