# Polarity

polar transformation

A correlation $\pi$ for which $\pi ^ {2} = \mathop{\rm id}$, that is, $\pi ( Y) = X$ if and only if $\pi ( X) = Y$. A polarity divides all subspaces into pairs; in particular, if a pair is formed by the subspaces $S _ {0}$ and $S _ {n-} 1$, where $S _ {0} = \pi ( S _ {n-} 1 )$ is a point and $S _ {n-} 1 = \pi ( S _ {0} )$ is a hyperplane, then $S _ {0}$ is called the pole of the hyperplane $S _ {n-} 1$ and $S _ {n-} 1$ is called the polar of the point $S _ {0}$. A projective space $\Pi _ {n} ( K)$ over the skew-field $K$ has a polarity if and only if $K$ admits an involutory anti-automorphism $\alpha$( that is, $\alpha ^ {2} = \mathop{\rm id}$). Suppose that $\pi$ is represented by a semi-bilinear form $f _ \alpha ( x , y )$. Then $\pi$ is a polarity if and only if $f _ \alpha ( x , y ) = 0$ implies $f _ \alpha ( y , x ) = 0$.

A polarity $\pi$ is either a symplectic correlation, characterized by the fact that $P \in \pi ( P)$ for every point $P$( in this case, $f ( x , y )$ is a skew-symmetric form on $A _ {n+} 1$, while $K$ is a field), or $\pi$ can be represented as an $\alpha$- symmetric form on $A _ {n+} 1$: $\alpha ( f _ \alpha ( x , y ) ) = f _ \alpha ( y , x )$( a symmetric polarity), in this case the existence of a non-strictly isotropic null subspace is equivalent to the fact that the characteristic of the skew-field equals 2 (in particular, if $\mathop{\rm char} K \neq 2$, then any null subspace is strictly isotropic).

Relative to a polarity $\pi$ one defines decomposition of a projective space into subspaces, which makes it possible to reduce the semi-bilinear form representing $\pi$ to canonical form. The most important among these are the following:

$M$— a maximal non-isotropic null subspace; its dimension is $n ( \pi ) - 1$, where $n$ is even and is called the deficiency of $\pi$, and $f$ is skew-symmetric;

$U$— a maximal strictly-isotropic subspace; its dimension is $i ( \pi ) - 1$, $i$ is called the index, $f \equiv 0$;

$J$— a component, free or null subspace, non-isotropic, where $f$ is positive or negative definite, $M \cap J = \emptyset$.

$W = M + U$— a maximal null subspace; its dimension is $i ( \pi ) + n ( \pi ) - 1$.

A projective transformation $F$ is called $\pi$- admissible (relative to the polarity $\pi$) if $\pi F = F \pi$. A semi-linear transformation $( \overline{F}\; , \phi )$ induces a $\pi$- admissible projective transformation if and only if in $K$ there is a $c$ for which $f ( \overline{F}\; x , \overline{F}\; y ) = c \phi ( f ( x , y ) )$. The $\pi$- admissible transformations form a group, $G _ \pi$( called the polarity group). If the group $G _ \pi$ is transitive, either every point of the space $\Pi _ {n}$ is null (and $G _ \pi$ is called symplectic) or there is no null point (and in this case $G _ \pi$ is called orthogonal for $\alpha = \mathop{\rm id}$, and unitary for $\alpha \neq \mathop{\rm id}$).

#### References

 [1] N.V. Efimov, "Higher geometry" , MIR (1980) (Translated from Russian)

Let $G = ( P, E)$ be a bipartite graph, and let $P = A \amalg B$ be the corresponding partition of $P$. A polarity on $G$ is an automorphism $\alpha$ of the graph $G$ such that $\alpha ^ {2} = \mathop{\rm id}$ and $\alpha ( A) = B$, $\alpha ( B) = A$.

The term polarity is mostly encountered in a geometric setting such as that of a projective space or incidence system. In this case the two sets of vertices are the lines and points of the incidence structure and there is an edge between a "point-vertex" and a "line-vertex" if and only if the point and line are incident.

The classical setting is that of a projective space $\mathbf P ^ {n}$ with a non-degenerate bilinear form $Q$ on it. The corresponding polarity between $d$- dimensional subspaces and $( n- d- 1)$- dimensional subspaces is defined by $\alpha ( V) = N ^ \perp = \{ {x \in \mathbf P ^ {n} } : {Q( x, y) = 0 \textrm{ for all } y \in V } \}$.

In the setting of a (Desarguesian or not) projective space $P$ a polarity is also viewed as a symmetric relation $\sigma \subset P \times P$ such that for all $v \in P$, $v ^ \perp = \{ {w \in P } : {( v, w) \in \sigma } \}$ is either a hyperplane or $P$ itself. If $P ^ \perp = \cap _ {v \in P } v ^ \perp = \emptyset$, the polarity is non-degenerate. A subspace $V$ is totally isotropic if $V \subset V ^ \perp = \cap _ {v \in V } v ^ \perp$.

#### References

 [a1] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French) [a2] H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953) [a3] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) [a4] R. Baer, "Linear algebra and projective geometry" , Acad. Press (1952) [a5] D. Pedoe, "Geometry. A comprehensive course" , Dover, reprint (1988) pp. Sect. 85.5 [a6] P. Dembowsky, "Finite geometries" , Springer (1968)
How to Cite This Entry:
Polarity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polarity&oldid=48230
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article