Orbit

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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