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 mapdefine 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).)