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''of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o0700101.png" /> relative to a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o0700102.png" /> acting on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o0700103.png" /> (on the left)''
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 +
''of a point  $  x $
 +
relative to a [[Group|group]] $  G $
 +
acting on a set $  X $(
 +
on the left)''
  
 
The set
 
The set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o0700104.png" /></td> </tr></table>
+
$$
 +
G( x)  = \{ {g( x) } : {g \in G } \}
 +
.
 +
$$
  
 
The set
 
The set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o0700105.png" /></td> </tr></table>
+
$$
 
+
G _ {x}  = \{ {g \in G } : {g( x) = x } \}
is a subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o0700106.png" /> and is called the stabilizer or stationary subgroup of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o0700107.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o0700108.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o0700109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001010.png" />, induces a bijection between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001011.png" /> and the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001012.png" />. The orbits of any two points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001013.png" /> either do not intersect or coincide; in other words, the orbits define a partition of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001014.png" />. The quotient by the equivalence relation defined by this partition is called the orbit space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001015.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001016.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001017.png" />. By assigning to each point its orbit, one defines a canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001018.png" />. The stabilizers of the points from one orbit are conjugate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001019.png" />, or, more precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001020.png" />. If there is only one orbit in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001022.png" /> is a homogeneous space of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001024.png" /> is also said to act transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001026.png" /> is a [[Topological group|topological group]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001027.png" /> is a topological space and the action is continuous, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001028.png" /> is usually given the topology in which a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001029.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001030.png" /> if and only if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001031.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001032.png" />.
+
$$
  
 +
is a subgroup in  $  G $
 +
and is called the stabilizer or stationary subgroup of the point  $  x $
 +
relative to  $  G $.
 +
The mapping  $  g \mapsto g( x) $,
 +
$  g \in G $,
 +
induces a bijection between  $  G/G _ {x} $
 +
and the orbit  $  G( x) $.
 +
The orbits of any two points from  $  X $
 +
either do not intersect or coincide; in other words, the orbits define a partition of the set  $  X $.
 +
The quotient by the equivalence relation defined by this partition is called the orbit space of  $  X $
 +
by  $  G $
 +
and is denoted by  $  X/G $.
 +
By assigning to each point its orbit, one defines a canonical mapping  $  \pi _ {X,G} :  X \rightarrow X/G $.
 +
The stabilizers of the points from one orbit are conjugate in  $  G $,
 +
or, more precisely,  $  G _ {g(} x) = gG _ {x} g  ^ {-} 1 $.
 +
If there is only one orbit in  $  X $,
 +
then  $  X $
 +
is a homogeneous space of the group  $  G $
 +
and  $  G $
 +
is also said to act transitively on  $  X $.
 +
If  $  G $
 +
is a [[Topological group|topological group]],  $  X $
 +
is a topological space and the action is continuous, then  $  X/G $
 +
is usually given the topology in which a set  $  U \subset  X/G $
 +
is open in  $  X/G $
 +
if and only if the set  $  \pi _ {X,G}  ^ {-} 1 ( U) $
 +
is open in  $  X $.
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001033.png" /> be the group of rotations of a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001034.png" /> around a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001035.png" />. Then the orbits are all circles with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001036.png" /> (including the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001037.png" /> itself).
+
1) Let $  G $
 +
be the group of rotations of a plane $  X $
 +
around a fixed point $  a $.  
 +
Then the orbits are all circles with centre at $  a $(
 +
including the point $  a $
 +
itself).
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001038.png" /> be the group of all non-singular linear transformations of a finite-dimensional real vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001039.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001040.png" /> be the set of all symmetric bilinear forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001041.png" />, and let the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001042.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001043.png" /> be defined by
+
2) Let $  G $
 +
be the group of all non-singular linear transformations of a finite-dimensional real vector space $  V $,  
 +
let $  X $
 +
be the set of all symmetric bilinear forms on $  V $,  
 +
and let the action of $  G $
 +
on $  X $
 +
be defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001044.png" /></td> </tr></table>
+
$$
 +
( gf  )( u, v)  = f( g  ^ {-} 1 ( u), g  ^ {-} 1 ( v)) \  \textrm{ for }  \textrm{ any }
 +
u , v \in V.
 +
$$
  
Then an orbit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001046.png" /> is the set of forms which have a fixed rank and signature.
+
Then an orbit of $  G $
 +
on $  X $
 +
is the set of forms which have a fixed rank and signature.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001047.png" /> be a real [[Lie group|Lie group]] acting smoothly on a [[Differentiable manifold|differentiable manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001048.png" /> (see [[Lie transformation group|Lie transformation group]]). For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001049.png" />, the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001050.png" /> is an immersed submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001051.png" />, diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001052.png" /> (the diffeomorphism is induced by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001054.png" />). This submanifold is not necessarily closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001055.png" /> (i.e., not necessarily imbedded). A classical example is the "winding of a toruswinding of a torus" , i.e. an orbit of the action of the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001056.png" /> on the torus
+
Let $  G $
 +
be a real [[Lie group|Lie group]] acting smoothly on a [[Differentiable manifold|differentiable manifold]] $  X $(
 +
see [[Lie transformation group|Lie transformation group]]). For any point $  x \in X $,  
 +
the orbit $  G( x) $
 +
is an immersed submanifold in $  X $,  
 +
diffeomorphic to $  G/G _ {x} $(
 +
the diffeomorphism is induced by the mapping $  g \mapsto g( x) $,  
 +
$  g \in G $).  
 +
This submanifold is not necessarily closed in $  X $(
 +
i.e., not necessarily imbedded). A classical example is the "winding of a toruswinding of a torus" , i.e. an orbit of the action of the additive group $  \mathbf R $
 +
on the torus
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001057.png" /></td> </tr></table>
+
$$
 +
T  ^ {2}  = \{ {( z _ {1} , z _ {2} ) } : {z _ {i} \in \mathbf C ,\
 +
| z _ {i} | = 1 , i = 1 , 2 } \}
 +
$$
  
 
defined by the formula
 
defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001058.png" /></td> </tr></table>
+
$$
 +
t( z _ {1} , z _ {2} )  = ( e  ^ {it} z _ {1} , e ^ {i \alpha t } z _ {2} ),\ \
 +
t \in \mathbf R ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001059.png" /> is an irrational real number; the closure of its orbit coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001060.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001061.png" /> is compact, then all orbits are imbedded submanifolds.
+
where $  \alpha $
 +
is an irrational real number; the closure of its orbit coincides with $  T  ^ {2} $.  
 +
If $  G $
 +
is compact, then all orbits are imbedded submanifolds.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001062.png" /> is an [[Algebraic group|algebraic group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001063.png" /> is an [[Algebraic variety|algebraic variety]] over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001064.png" />, with regular action (see [[Algebraic group of transformations|Algebraic group of transformations]]), then any orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001065.png" /> is a smooth algebraic variety, open in its closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001066.png" /> (in the Zariski topology), while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001067.png" /> always contains a closed orbit of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001068.png" /> (see [[#References|[5]]]). In this case the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001070.png" />, induces an isomorphism of the algebraic varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001072.png" /> if and only if it is separable (this condition is always fulfilled if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001073.png" /> is a field of characteristic zero, cf. [[Separable mapping|Separable mapping]]). The orbits of maximal dimension form an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001074.png" />.
+
If $  G $
 +
is an [[Algebraic group|algebraic group]] and $  X $
 +
is an [[Algebraic variety|algebraic variety]] over an algebraically closed field $  k $,  
 +
with regular action (see [[Algebraic group of transformations|Algebraic group of transformations]]), then any orbit $  G( x) $
 +
is a smooth algebraic variety, open in its closure $  \overline{ {G( x) }}\; $(
 +
in the Zariski topology), while $  \overline{ {G( x) }}\; $
 +
always contains a closed orbit of the group $  G $(
 +
see [[#References|[5]]]). In this case the morphism $  G \rightarrow G( x) $,  
 +
$  g \mapsto g( x) $,  
 +
induces an isomorphism of the algebraic varieties $  G/G _ {x} $
 +
and $  G( x) $
 +
if and only if it is separable (this condition is always fulfilled if $  k $
 +
is a field of characteristic zero, cf. [[Separable mapping|Separable mapping]]). The orbits of maximal dimension form an open set in $  X $.
  
The description of the structure of an orbit for a given action usually reduces to giving in each orbit a unique representative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001075.png" />, the description of the stabilizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001076.png" /> and the description of a suitable class of functions which are constant on the orbit (invariants) and which separate various orbits; these functions enable one to describe the location of the orbits in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001077.png" /> (orbits are intersections of their level sets). This program is usually called the problem of orbit decomposition. Many classification problems can be reduced to this problem. Thus, Example 2) is a classification problem of bilinear symmetric forms up to equivalence; the invariants in this case — the rank and signature — are "discrete" , while the stabilizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001078.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001079.png" /> is non-degenerate, is the corresponding orthogonal group. The classical theory of the Jordan form of matrices (as well as the theory of other normal forms of matrices, cf. [[Normal form|Normal form]]) can also be incorporated in this scheme: The Jordan form is a canonical representing element (defined, admittedly, up to the order of Jordan blocks) in the orbit of the [[General linear group|general linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001080.png" /> on the space of all complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001081.png" />-matrices, for the conjugation action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001082.png" />; the coefficients of the characteristic polynomial of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001083.png" /> are important invariants (which, however, do not separate any two orbits). The idea of considering equivalent objects as orbits of a group is actively used in various classification problems, for example, in algebraic [[Moduli theory|moduli theory]] (see [[#References|[10]]]).
+
The description of the structure of an orbit for a given action usually reduces to giving in each orbit a unique representative $  x $,  
 +
the description of the stabilizer $  G _ {x} $
 +
and the description of a suitable class of functions which are constant on the orbit (invariants) and which separate various orbits; these functions enable one to describe the location of the orbits in $  X $(
 +
orbits are intersections of their level sets). This program is usually called the problem of orbit decomposition. Many classification problems can be reduced to this problem. Thus, Example 2) is a classification problem of bilinear symmetric forms up to equivalence; the invariants in this case — the rank and signature — are "discrete" , while the stabilizer $  G _ {f} $,  
 +
where $  f $
 +
is non-degenerate, is the corresponding orthogonal group. The classical theory of the Jordan form of matrices (as well as the theory of other normal forms of matrices, cf. [[Normal form|Normal form]]) can also be incorporated in this scheme: The Jordan form is a canonical representing element (defined, admittedly, up to the order of Jordan blocks) in the orbit of the [[General linear group|general linear group]] $  \mathop{\rm GL} _ {n} ( \mathbf C ) $
 +
on the space of all complex $  ( n \times n) $-
 +
matrices, for the conjugation action $  Y \mapsto AYA  ^ {-} 1 $;  
 +
the coefficients of the characteristic polynomial of a matrix $  Y $
 +
are important invariants (which, however, do not separate any two orbits). The idea of considering equivalent objects as orbits of a group is actively used in various classification problems, for example, in algebraic [[Moduli theory|moduli theory]] (see [[#References|[10]]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001085.png" /> are finite, then the [[Burnside Lemma]] holds:
+
If $  G $
 +
and $  X $
 +
are finite, then the [[Burnside Lemma]] holds:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001086.png" /></td> </tr></table>
+
$$
 +
| X/G |  =
 +
\frac{1}{| G | }
 +
\sum _ {g \in G } |  \mathop{\rm Fix}  g |,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001087.png" /> is the number of elements of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001088.png" />, and
+
where $  | Y | $
 +
is the number of elements of the set $  Y $,  
 +
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001089.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Fix}  g  = \{ {x \in X } : {g( x) = x } \}
 +
.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001090.png" /> is a compact Lie group acting smoothly on a connected smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001091.png" />, then the orbit structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001092.png" /> is locally finite, i.e. for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001093.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001094.png" /> such that the number of conjugacy classes of different stabilizers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001096.png" />, is finite. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001097.png" /> is compact, then the number of different conjugacy classes of stabilizers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o07001099.png" />, is finite. For any subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010100.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010101.png" />, each of the sets
+
If $  G $
 +
is a compact Lie group acting smoothly on a connected smooth manifold $  X $,  
 +
then the orbit structure of $  X $
 +
is locally finite, i.e. for any point $  x \in X $
 +
there is a neighbourhood $  U $
 +
such that the number of conjugacy classes of different stabilizers $  G _ {y} $,  
 +
$  y \in U $,  
 +
is finite. In particular, if $  X $
 +
is compact, then the number of different conjugacy classes of stabilizers $  G _ {y} $,  
 +
$  y \in X $,  
 +
is finite. For any subgroup $  H $
 +
in $  G $,  
 +
each of the sets
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010102.png" /></td> </tr></table>
+
$$
 +
X _ {(} H)  = \{ {x \in X } : {G _ {x}  \textrm{ is }  \textrm{ conjugate }  roman ^ {  }  H
 +
  \mathop{\rm in}  G } \}
 +
$$
  
is the intersection of an open and a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010103.png" />-invariant subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010104.png" />. Investigation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010105.png" /> in this case leads to the classification of actions (see [[#References|[1]]]).
+
is the intersection of an open and a closed $  G $-
 +
invariant subset in $  X $.  
 +
Investigation of $  X _ {(} H) $
 +
in this case leads to the classification of actions (see [[#References|[1]]]).
  
Analogues of these results have been obtained in the geometric theory of invariants (cf. [[Invariants, theory of|Invariants, theory of]]) (see [[#References|[3]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010106.png" /> be a reductive algebraic group acting regularly on an affine algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010107.png" /> (the base field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010108.png" /> is algebraically closed and has characteristic zero). The closure of any orbit contains a unique closed orbit. There exists a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010109.png" /> into a finite union of locally closed invariant non-intersecting subsets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010110.png" />, such that: a) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010112.png" /> is closed, then the stabilizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010113.png" /> is conjugate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010114.png" /> to a subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010115.png" />, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010116.png" /> is also closed, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010117.png" /> is conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010118.png" />; b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010119.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010121.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010123.png" /> are closed, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010124.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010125.png" /> are not conjugate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010126.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010127.png" /> is a smooth algebraic variety (for example, in the important case of a rational linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010128.png" /> in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010129.png" />), then there is a non-empty open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010130.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010131.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010132.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010133.png" /> are conjugate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010134.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010135.png" />. The latter result is an assertion about a property of points in general position in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010136.png" />, i.e. points of a non-empty open subset; there are also a number of other assertions of this type. For example, for a rational linear representation of a semi-simple group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010137.png" /> in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010138.png" />, the orbits of the points in general position are closed if and only if their stabilizers are reductive (see [[#References|[7]]]); when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010139.png" /> is irreducible, an explicit expression of the stabilizers of the points in general position has been found (see [[#References|[8]]], [[#References|[9]]]). The study of orbit closures is important in this context. So, the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010140.png" /> the closure of whose orbits contains the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010141.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010142.png" /> coincides with the variety of the zeros of all non-constant invariant polynomials on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010143.png" />; in many cases, and especially in the applications of the theory of invariants to the theory of moduli, this variety plays a vital part (see [[#References|[10]]]). Any two different closed orbits can be separated by invariant polynomials. The orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010144.png" /> is closed if and only if the orbit of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010145.png" /> relative to the normalizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010146.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010147.png" /> is closed (see [[#References|[4]]]). The presence of non-closed orbits is connected with properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010148.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010149.png" /> is unipotent (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010150.png" /> is affine), then any orbit is closed (see [[#References|[6]]]). One aspect of the theory of invariants concerns the study of orbit decompositions of different concrete actions (especially linear representations). One of these — the adjoint representation of a reductive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010151.png" /> — has been studied in detail (see, for example, [[#References|[11]]]). This study is connected with the theory of representations of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010152.png" />; see [[Orbit method|Orbit method]].
+
Analogues of these results have been obtained in the geometric theory of invariants (cf. [[Invariants, theory of|Invariants, theory of]]) (see [[#References|[3]]]). Let $  G $
 +
be a reductive algebraic group acting regularly on an affine algebraic variety $  X $(
 +
the base field $  k $
 +
is algebraically closed and has characteristic zero). The closure of any orbit contains a unique closed orbit. There exists a partition of $  X $
 +
into a finite union of locally closed invariant non-intersecting subsets, $  X = \cup _  \alpha  X _  \alpha  $,  
 +
such that: a) if $  x, y \in X _  \alpha  $
 +
and $  G( x) $
 +
is closed, then the stabilizer $  G _ {y} $
 +
is conjugate in $  G $
 +
to a subgroup in $  G _ {x} $,  
 +
while if $  G( y) $
 +
is also closed, then $  G _ {y} $
 +
is conjugate to $  G _ {x} $;  
 +
b) if $  x \in X _  \alpha  $,  
 +
$  y \in X _  \beta  $,  
 +
$  \alpha \neq \beta $,  
 +
and $  G( x) $
 +
and $  G( y) $
 +
are closed, then $  G _ {x} $
 +
and $  G _ {y} $
 +
are not conjugate in $  G $.  
 +
If $  X $
 +
is a smooth algebraic variety (for example, in the important case of a rational linear representation of $  G $
 +
in a vector space $  V = X $),  
 +
then there is a non-empty open subset $  \Omega $
 +
in $  X $
 +
such that $  G _ {x} $
 +
and $  G _ {y} $
 +
are conjugate in $  G $
 +
for any $  x, y \in \Omega $.  
 +
The latter result is an assertion about a property of points in general position in $  X $,  
 +
i.e. points of a non-empty open subset; there are also a number of other assertions of this type. For example, for a rational linear representation of a semi-simple group $  G $
 +
in a vector space $  V $,  
 +
the orbits of the points in general position are closed if and only if their stabilizers are reductive (see [[#References|[7]]]); when $  G $
 +
is irreducible, an explicit expression of the stabilizers of the points in general position has been found (see [[#References|[8]]], [[#References|[9]]]). The study of orbit closures is important in this context. So, the set of $  x \in V $
 +
the closure of whose orbits contains the element $  O $
 +
of $  V $
 +
coincides with the variety of the zeros of all non-constant invariant polynomials on $  V $;  
 +
in many cases, and especially in the applications of the theory of invariants to the theory of moduli, this variety plays a vital part (see [[#References|[10]]]). Any two different closed orbits can be separated by invariant polynomials. The orbit $  G( x) $
 +
is closed if and only if the orbit of the point $  x $
 +
relative to the normalizer of $  G( x) $
 +
in $  G $
 +
is closed (see [[#References|[4]]]). The presence of non-closed orbits is connected with properties of $  G $;  
 +
if $  G $
 +
is unipotent (and $  X $
 +
is affine), then any orbit is closed (see [[#References|[6]]]). One aspect of the theory of invariants concerns the study of orbit decompositions of different concrete actions (especially linear representations). One of these — the adjoint representation of a reductive group $  G $—  
 +
has been studied in detail (see, for example, [[#References|[11]]]). This study is connected with the theory of representations of the group $  G $;  
 +
see [[Orbit method|Orbit method]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Palais, "The classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010153.png" />-spaces" , Amer. Math. Soc. (1960) {{MR|177401}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9 {{MR|0256912}} {{MR|0256911}} {{MR|0250930}} {{ZBL|0196.27202}} {{ZBL|0193.28103}} {{ZBL|0182.57702}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Luna, "Slices étales" ''Bull. Soc. Math. France.'' , '''33''' (1973) pp. 81–105 {{MR|0342523}} {{ZBL|0286.14014}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Luna, "Adhérence d'orbite et invariants" ''Invent. Math.'' , '''29''' : 3 (1975) pp. 231–238 {{MR|0376704}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R. Steinberg, "Conjugacy classes in algebraic groups" , ''Lect. notes in math.'' , '''366''' , Springer (1974) {{MR|0352279}} {{ZBL|0281.20037}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.L. Popov, "Stability criteria for the action of a semisimple group on a factorial manifold" ''Math. USSR Izv.'' , '''4''' (1970) pp. 527–535 ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''34''' (1970) pp. 523–531 {{MR|}} {{ZBL|0261.14011}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.M. Popov, "Irreducible semisimple linear Lie groups with finite stationary subgroups of general position" ''Funct. Anal. Appl.'' , '''12''' : 2 (1978) pp. 154–155 ''Funkts. Anal. i Prilozhen.'' , '''12''' : 2 (1978) pp. 91–92 {{MR|0498913}} {{ZBL|0404.22018}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A.G. Elashvili, "Stationary subalgebras of points of the common state for irreducible Lie groups" ''Funct. Anal. Appl.'' , '''6''' : 2 (1972) pp. 139–148 ''Funkts. Anal. i Prilozhen.'' , '''6''' : 2 (1972) pp. 65–78 {{MR|}} {{ZBL|0252.22016}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> D. Mumford, J. Fogarty, "Geometric invariant theory" , Springer (1982) {{MR|0719371}} {{ZBL|0504.14008}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> B. Kostant, "Lie group representations on polynomial rings" ''Amer. J. Math.'' , '''85''' : 3 (1963) pp. 327–404 {{MR|0158024}} {{MR|0150240}} {{ZBL|0248.20056}} {{ZBL|0124.26802}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Palais, "The classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010153.png" />-spaces" , Amer. Math. Soc. (1960) {{MR|177401}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9 {{MR|0256912}} {{MR|0256911}} {{MR|0250930}} {{ZBL|0196.27202}} {{ZBL|0193.28103}} {{ZBL|0182.57702}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Luna, "Slices étales" ''Bull. Soc. Math. France.'' , '''33''' (1973) pp. 81–105 {{MR|0342523}} {{ZBL|0286.14014}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Luna, "Adhérence d'orbite et invariants" ''Invent. Math.'' , '''29''' : 3 (1975) pp. 231–238 {{MR|0376704}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R. Steinberg, "Conjugacy classes in algebraic groups" , ''Lect. notes in math.'' , '''366''' , Springer (1974) {{MR|0352279}} {{ZBL|0281.20037}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.L. Popov, "Stability criteria for the action of a semisimple group on a factorial manifold" ''Math. USSR Izv.'' , '''4''' (1970) pp. 527–535 ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''34''' (1970) pp. 523–531 {{MR|}} {{ZBL|0261.14011}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.M. Popov, "Irreducible semisimple linear Lie groups with finite stationary subgroups of general position" ''Funct. Anal. Appl.'' , '''12''' : 2 (1978) pp. 154–155 ''Funkts. Anal. i Prilozhen.'' , '''12''' : 2 (1978) pp. 91–92 {{MR|0498913}} {{ZBL|0404.22018}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A.G. Elashvili, "Stationary subalgebras of points of the common state for irreducible Lie groups" ''Funct. Anal. Appl.'' , '''6''' : 2 (1972) pp. 139–148 ''Funkts. Anal. i Prilozhen.'' , '''6''' : 2 (1972) pp. 65–78 {{MR|}} {{ZBL|0252.22016}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> D. Mumford, J. Fogarty, "Geometric invariant theory" , Springer (1982) {{MR|0719371}} {{ZBL|0504.14008}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> B. Kostant, "Lie group representations on polynomial rings" ''Amer. J. Math.'' , '''85''' : 3 (1963) pp. 327–404 {{MR|0158024}} {{MR|0150240}} {{ZBL|0248.20056}} {{ZBL|0124.26802}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.L. Popov, "Modern developments in invariant theory" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , Amer. Math. Soc. (1987) pp. 394–406 {{MR|0934239}} {{ZBL|0679.14024}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Kraft, "Geometrische Methoden in der Invariantentheorie" , Vieweg (1984) {{MR|0768181}} {{ZBL|0569.14003}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Kraft (ed.) P. Slodowy (ed.) T.A. Springer (ed.) , ''Algebraische Transformationsgruppen und Invariantentheorie'' , ''DMV Sem.'' , '''13''' , Birkhäuser (1989) {{MR|1044582}} {{ZBL|0682.00008}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.L. Popov, "Modern developments in invariant theory" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , Amer. Math. Soc. (1987) pp. 394–406 {{MR|0934239}} {{ZBL|0679.14024}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Kraft, "Geometrische Methoden in der Invariantentheorie" , Vieweg (1984) {{MR|0768181}} {{ZBL|0569.14003}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Kraft (ed.) P. Slodowy (ed.) T.A. Springer (ed.) , ''Algebraische Transformationsgruppen und Invariantentheorie'' , ''DMV Sem.'' , '''13''' , Birkhäuser (1989) {{MR|1044582}} {{ZBL|0682.00008}} </TD></TR></table>

Revision as of 08:04, 6 June 2020


of a point $ x $ relative to a group $ G $ acting on a set $ X $( on the left)

The set

$$ G( x) = \{ {g( x) } : {g \in G } \} . $$

The set

$$ G _ {x} = \{ {g \in G } : {g( x) = x } \} $$

is a subgroup in $ G $ and is called the stabilizer or stationary subgroup of the point $ x $ relative to $ G $. The mapping $ g \mapsto g( x) $, $ g \in G $, induces a bijection between $ G/G _ {x} $ and the orbit $ G( x) $. The orbits of any two points from $ X $ either do not intersect or coincide; in other words, the orbits define a partition of the set $ X $. The quotient by the equivalence relation defined by this partition is called the orbit space of $ X $ by $ G $ and is denoted by $ X/G $. By assigning to each point its orbit, one defines a canonical mapping $ \pi _ {X,G} : X \rightarrow X/G $. The stabilizers of the points from one orbit are conjugate in $ G $, or, more precisely, $ G _ {g(} x) = gG _ {x} g ^ {-} 1 $. If there is only one orbit in $ X $, then $ X $ is a homogeneous space of the group $ G $ and $ G $ is also said to act transitively on $ X $. If $ G $ is a topological group, $ X $ is a topological space and the action is continuous, then $ X/G $ is usually given the topology in which a set $ U \subset X/G $ is open in $ X/G $ if and only if the set $ \pi _ {X,G} ^ {-} 1 ( U) $ is open in $ X $.

1) Let $ G $ be the group of rotations of a plane $ X $ around a fixed point $ a $. Then the orbits are all circles with centre at $ a $( including the point $ a $ itself).

2) Let $ G $ be the group of all non-singular linear transformations of a finite-dimensional real vector space $ V $, let $ X $ be the set of all symmetric bilinear forms on $ V $, and let the action of $ G $ on $ X $ be defined by

$$ ( gf )( u, v) = f( g ^ {-} 1 ( u), g ^ {-} 1 ( v)) \ \textrm{ for } \textrm{ any } u , v \in V. $$

Then an orbit of $ G $ on $ X $ is the set of forms which have a fixed rank and signature.

Let $ G $ be a real Lie group acting smoothly on a differentiable manifold $ X $( see Lie transformation group). For any point $ x \in X $, the orbit $ G( x) $ is an immersed submanifold in $ X $, diffeomorphic to $ G/G _ {x} $( the diffeomorphism is induced by the mapping $ g \mapsto g( x) $, $ g \in G $). This submanifold is not necessarily closed in $ X $( i.e., not necessarily imbedded). A classical example is the "winding of a toruswinding of a torus" , i.e. an orbit of the action of the additive group $ \mathbf R $ on the torus

$$ T ^ {2} = \{ {( z _ {1} , z _ {2} ) } : {z _ {i} \in \mathbf C ,\ | z _ {i} | = 1 , i = 1 , 2 } \} $$

defined by the formula

$$ t( z _ {1} , z _ {2} ) = ( e ^ {it} z _ {1} , e ^ {i \alpha t } z _ {2} ),\ \ t \in \mathbf R , $$

where $ \alpha $ is an irrational real number; the closure of its orbit coincides with $ T ^ {2} $. If $ G $ is compact, then all orbits are imbedded submanifolds.

If $ G $ is an algebraic group and $ X $ is an algebraic variety over an algebraically closed field $ k $, with regular action (see Algebraic group of transformations), then any orbit $ G( x) $ is a smooth algebraic variety, open in its closure $ \overline{ {G( x) }}\; $( in the Zariski topology), while $ \overline{ {G( x) }}\; $ always contains a closed orbit of the group $ G $( see [5]). In this case the morphism $ G \rightarrow G( x) $, $ g \mapsto g( x) $, induces an isomorphism of the algebraic varieties $ G/G _ {x} $ and $ G( x) $ if and only if it is separable (this condition is always fulfilled if $ k $ is a field of characteristic zero, cf. Separable mapping). The orbits of maximal dimension form an open set in $ X $.

The description of the structure of an orbit for a given action usually reduces to giving in each orbit a unique representative $ x $, the description of the stabilizer $ G _ {x} $ and the description of a suitable class of functions which are constant on the orbit (invariants) and which separate various orbits; these functions enable one to describe the location of the orbits in $ X $( orbits are intersections of their level sets). This program is usually called the problem of orbit decomposition. Many classification problems can be reduced to this problem. Thus, Example 2) is a classification problem of bilinear symmetric forms up to equivalence; the invariants in this case — the rank and signature — are "discrete" , while the stabilizer $ G _ {f} $, where $ f $ is non-degenerate, is the corresponding orthogonal group. The classical theory of the Jordan form of matrices (as well as the theory of other normal forms of matrices, cf. Normal form) can also be incorporated in this scheme: The Jordan form is a canonical representing element (defined, admittedly, up to the order of Jordan blocks) in the orbit of the general linear group $ \mathop{\rm GL} _ {n} ( \mathbf C ) $ on the space of all complex $ ( n \times n) $- matrices, for the conjugation action $ Y \mapsto AYA ^ {-} 1 $; the coefficients of the characteristic polynomial of a matrix $ Y $ are important invariants (which, however, do not separate any two orbits). The idea of considering equivalent objects as orbits of a group is actively used in various classification problems, for example, in algebraic moduli theory (see [10]).

If $ G $ and $ X $ are finite, then the Burnside Lemma holds:

$$ | X/G | = \frac{1}{| G | } \sum _ {g \in G } | \mathop{\rm Fix} g |, $$

where $ | Y | $ is the number of elements of the set $ Y $, and

$$ \mathop{\rm Fix} g = \{ {x \in X } : {g( x) = x } \} . $$

If $ G $ is a compact Lie group acting smoothly on a connected smooth manifold $ X $, then the orbit structure of $ X $ is locally finite, i.e. for any point $ x \in X $ there is a neighbourhood $ U $ such that the number of conjugacy classes of different stabilizers $ G _ {y} $, $ y \in U $, is finite. In particular, if $ X $ is compact, then the number of different conjugacy classes of stabilizers $ G _ {y} $, $ y \in X $, is finite. For any subgroup $ H $ in $ G $, each of the sets

$$ X _ {(} H) = \{ {x \in X } : {G _ {x} \textrm{ is } \textrm{ conjugate } roman ^ { } H \mathop{\rm in} G } \} $$

is the intersection of an open and a closed $ G $- invariant subset in $ X $. Investigation of $ X _ {(} H) $ in this case leads to the classification of actions (see [1]).

Analogues of these results have been obtained in the geometric theory of invariants (cf. Invariants, theory of) (see [3]). Let $ G $ be a reductive algebraic group acting regularly on an affine algebraic variety $ X $( the base field $ k $ is algebraically closed and has characteristic zero). The closure of any orbit contains a unique closed orbit. There exists a partition of $ X $ into a finite union of locally closed invariant non-intersecting subsets, $ X = \cup _ \alpha X _ \alpha $, such that: a) if $ x, y \in X _ \alpha $ and $ G( x) $ is closed, then the stabilizer $ G _ {y} $ is conjugate in $ G $ to a subgroup in $ G _ {x} $, while if $ G( y) $ is also closed, then $ G _ {y} $ is conjugate to $ G _ {x} $; b) if $ x \in X _ \alpha $, $ y \in X _ \beta $, $ \alpha \neq \beta $, and $ G( x) $ and $ G( y) $ are closed, then $ G _ {x} $ and $ G _ {y} $ are not conjugate in $ G $. If $ X $ is a smooth algebraic variety (for example, in the important case of a rational linear representation of $ G $ in a vector space $ V = X $), then there is a non-empty open subset $ \Omega $ in $ X $ such that $ G _ {x} $ and $ G _ {y} $ are conjugate in $ G $ for any $ x, y \in \Omega $. The latter result is an assertion about a property of points in general position in $ X $, i.e. points of a non-empty open subset; there are also a number of other assertions of this type. For example, for a rational linear representation of a semi-simple group $ G $ in a vector space $ V $, the orbits of the points in general position are closed if and only if their stabilizers are reductive (see [7]); when $ G $ is irreducible, an explicit expression of the stabilizers of the points in general position has been found (see [8], [9]). The study of orbit closures is important in this context. So, the set of $ x \in V $ the closure of whose orbits contains the element $ O $ of $ V $ coincides with the variety of the zeros of all non-constant invariant polynomials on $ V $; in many cases, and especially in the applications of the theory of invariants to the theory of moduli, this variety plays a vital part (see [10]). Any two different closed orbits can be separated by invariant polynomials. The orbit $ G( x) $ is closed if and only if the orbit of the point $ x $ relative to the normalizer of $ G( x) $ in $ G $ is closed (see [4]). The presence of non-closed orbits is connected with properties of $ G $; if $ G $ is unipotent (and $ X $ is affine), then any orbit is closed (see [6]). One aspect of the theory of invariants concerns the study of orbit decompositions of different concrete actions (especially linear representations). One of these — the adjoint representation of a reductive group $ G $— has been studied in detail (see, for example, [11]). This study is connected with the theory of representations of the group $ G $; see Orbit method.

References

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Comments

References

[a1] V.L. Popov, "Modern developments in invariant theory" , Proc. Internat. Congress Mathematicians (Berkeley, 1986) , Amer. Math. Soc. (1987) pp. 394–406 MR0934239 Zbl 0679.14024
[a2] H. Kraft, "Geometrische Methoden in der Invariantentheorie" , Vieweg (1984) MR0768181 Zbl 0569.14003
[a3] H. Kraft (ed.) P. Slodowy (ed.) T.A. Springer (ed.) , Algebraische Transformationsgruppen und Invariantentheorie , DMV Sem. , 13 , Birkhäuser (1989) MR1044582 Zbl 0682.00008
How to Cite This Entry:
Orbit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orbit&oldid=48062
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article