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''polar transformation''
 
''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 [[Correlation|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 <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).
+
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 <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:
+
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:
  
<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;
+
$  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;
  
<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" />;
+
$  U $—  
 +
a maximal strictly-isotropic subspace; its dimension is $  i ( \pi ) - 1 $,  
 +
$  i $
 +
is called the index, $  f \equiv 0 $;
  
<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" />.
+
$  J $—  
 +
a component, free or null subspace, non-isotropic, where $  f $
 +
is positive or negative definite, $  M \cap J = \emptyset $.
  
<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" />.
+
$  W = M + U $—  
 +
a maximal null subspace; its dimension is $  i ( \pi ) + n ( \pi ) - 1 $.
  
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" />).
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,  "Higher geometry" , MIR  (1980)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,  "Higher geometry" , MIR  (1980)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====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" />.
+
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|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 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" />.
+
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 <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" />.
+
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====
 
====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>
 
<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>

Latest revision as of 08:06, 6 June 2020


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)
How to Cite This Entry:
Polarity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polarity&oldid=15854
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article