View source for Polarity

''polar transformation''

A [[Correlation|correlation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p0734601.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p0734602.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p0734603.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p0734604.png" />. A polarity divides all subspaces into pairs; in particular, if a pair is formed by the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p0734605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p0734606.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p0734607.png" /> is a point and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p0734608.png" /> is a hyperplane, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p0734609.png" /> is called the pole of the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346011.png" /> is called the polar of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346012.png" />. A projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346013.png" /> over the skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346014.png" /> has a polarity if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346015.png" /> admits an involutory [[anti-automorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346016.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346017.png" />). Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346018.png" /> is represented by a semi-bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346019.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346020.png" /> is a polarity if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346021.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346022.png" />.

A polarity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346023.png" /> is either a symplectic correlation, characterized by the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346024.png" /> for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346025.png" /> (in this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346026.png" /> is a skew-symmetric form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346027.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346028.png" /> is a field), or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346029.png" /> can be represented as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346030.png" />-symmetric form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346031.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346032.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346033.png" />, then any null subspace is strictly isotropic).

Relative to a polarity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346034.png" /> one defines decomposition of a projective space into subspaces, which makes it possible to reduce the semi-bilinear form representing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346035.png" /> to canonical form. The most important among these are the following:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346036.png" /> — a maximal non-isotropic null subspace; its dimension is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346038.png" /> is even and is called the deficiency of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346039.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346040.png" /> is skew-symmetric;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346041.png" /> — a maximal strictly-isotropic subspace; its dimension is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346043.png" /> is called the index, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346044.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346045.png" /> — a component, free or null subspace, non-isotropic, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346046.png" /> is positive or negative definite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346047.png" />.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346048.png" /> — a maximal null subspace; its dimension is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346049.png" />.

A projective transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346050.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346051.png" />-admissible (relative to the polarity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346052.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346053.png" />. A semi-linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346054.png" /> induces a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346055.png" />-admissible projective transformation if and only if in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346056.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346057.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346058.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346059.png" />-admissible transformations form a group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346060.png" /> (called the polarity group). If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346061.png" /> is transitive, either every point of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346062.png" /> is null (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346063.png" /> is called symplectic) or there is no null point (and in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346064.png" /> is called orthogonal for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346065.png" />, and unitary for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346066.png" />).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,   "Higher geometry" , MIR  (1980)  (Translated from Russian)</TD></TR></table>



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346067.png" /> be a bipartite graph, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346068.png" /> be the corresponding partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346069.png" />. A polarity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346070.png" /> is an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346071.png" /> of the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346072.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346075.png" />.

The term polarity is mostly encountered in a geometric setting such as that of a [[Projective space|projective space]] or [[Incidence system|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346076.png" /> with a non-degenerate bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346077.png" /> on it. The corresponding polarity between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346078.png" />-dimensional subspaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346079.png" />-dimensional subspaces is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346080.png" />.

In the setting of a (Desarguesian or not) projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346081.png" /> a polarity is also viewed as a symmetric relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346082.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346084.png" /> is either a hyperplane or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346085.png" /> itself. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346086.png" />, the polarity is non-degenerate. A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346087.png" /> is totally isotropic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073460/p07346088.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Busemann,   P.J. Kelly,   "Projective geometry and projective metrics" , Acad. Press  (1953)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1963)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Baer,   "Linear algebra and projective geometry" , Acad. Press  (1952)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Pedoe,   "Geometry. A comprehensive course" , Dover, reprint  (1988)  pp. Sect. 85.5</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Dembowsky,   "Finite geometries" , Springer  (1968)</TD></TR></table>