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)

Comments

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)