View source for Orbit

''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)''

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>

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>

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" />.

======


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).

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

<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>

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.

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

<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>

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>

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.

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" />.

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]]]).

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

<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>

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

<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>

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

<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>

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]]]).

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]].

====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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Harary,   "Graph theory" , Addison-Wesley  (1969)  pp. Chapt. 9</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Luna,   "Slices étales"  ''Bull. Soc. Math. France.'' , '''33'''  (1973)  pp. 81–105</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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Borel,   "Linear algebraic groups" , Benjamin  (1969)</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)</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</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</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</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  D. Mumford,   J. Fogarty,   "Geometric invariant theory" , Springer  (1982)</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</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  J.E. Humphreys,   "Linear algebraic groups" , Springer  (1975)</TD></TR></table>



====Comments====


====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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Kraft,   "Geometrische Methoden in der Invariantentheorie" , Vieweg  (1984)</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)</TD></TR></table>