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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, equivariance is a form of
symmetry Symmetry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

symmetry
for
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s from one space with symmetry to another (such as
symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, l ...
s). A function is said to be an equivariant map when its domain and codomain are acted on by the same
symmetry group In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their chang ...
, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation. Equivariant maps generalize the concept of invariants, functions whose value is unchanged by a symmetry transformation of their argument. The value of an equivariant map is often (imprecisely) called an invariant. In
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
, equivariance under statistical transformations of data is an important property of various estimation methods; see
invariant estimator In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...
for details. In pure mathematics, equivariance is a central object of study in
equivariant topology In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous function (topology), continuous maps f: X \to Y, and while equivariant topology ...
and its subtopics
equivariant cohomology In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''Group action (mathematics), group action''. It can be viewed as a common generalization of ...
and
equivariant stable homotopy theory In mathematics, equivariance is a form of symmetry for function (mathematics), functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are Group action ( ...
.


Examples


Elementary geometry

In the geometry of
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

triangle
s, the area and perimeter of a triangle are invariants: translating or rotating a triangle does not change its area or perimeter. However, triangle centers such as the centroid, circumcenter, incenter and orthocenter are not invariant, because moving a triangle will also cause its centers to move. Instead, these centers are equivariant: applying any Euclidean Congruence (geometry), congruence (a combination of a translation and rotation) to a triangle, and then constructing its center, produces the same point as constructing the center first, and then applying the same congruence to the center. More generally, all triangle centers are also equivariant under Similarity (geometry), similarity transformations (combinations of translation, rotation, and scaling), and the centroid is equivariant under affine transformations. The same function may be an invariant for one group of symmetries and equivariant for a different group of symmetries. For instance, under similarity transformations instead of congruences the area and perimeter are no longer invariant: scaling a triangle also changes its area and perimeter. However, these changes happen in a predictable way: if a triangle is scaled by a factor of , the perimeter also scales by and the area scales by . In this way, the function mapping each triangle to its area or perimeter can be seen as equivariant for a multiplicative group action of the scaling transformations on the positive real numbers.


Statistics

Another class of simple examples comes from statistical estimation. The mean of a sample (a set of real numbers) is commonly used as a central tendency of the sample. It is equivariant under Linear function (calculus), linear transformations of the real numbers, so for instance it is unaffected by the choice of units used to represent the numbers. By contrast, the mean is not equivariant with respect to nonlinear transformations such as exponentials. The median of a sample is equivariant for a much larger group of transformations, the (strictly) monotonic functions of the real numbers. This analysis indicates that the median is more robust statistics, robust against certain kinds of changes to a data set, and that (unlike the mean) it is meaningful for ordinal data. The concepts of an
invariant estimator In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...
and equivariant estimator have been used to formalize this style of analysis.


Representation theory

In the representation theory of finite groups, a vector space equipped with a group that acts by linear transformations of the space is called a linear representation of the group. A linear map that commutes with the action is called an intertwiner. That is, an intertwiner is just an equivariant linear map between two representations. Alternatively, an intertwiner for representations of a group over a field (mathematics), field is the same thing as a module (mathematics), module homomorphism of -module (mathematics), modules, where is the group ring of ''G''. Under some conditions, if ''X'' and ''Y'' are both irreducible representations, then an intertwiner (other than the zero map) only exists if the two representations are equivalent (that is, are isomorphic as module (mathematics), modules). That intertwiner is then unique up to a multiplicative factor (a non-zero scalar (mathematics), scalar from ). These properties hold when the image of is a simple algebra, with centre (by what is called Schur's Lemma: see simple module). As a consequence, in important cases the construction of an intertwiner is enough to show the representations are effectively the same.


Formalization

Equivariance can be formalized using the concept of a Group action (mathematics), -set for a group (mathematics), group . This is a mathematical object consisting of a set (mathematics), mathematical set and a Group action (mathematics), group action (on the left) of on . If and are both -sets for the same group , then a function is said to be equivariant if : for all and all . If one or both of the actions are right actions the equivariance condition may be suitably modified: :; (right-right) :; (right-left) :; (left-right) Equivariant maps are homomorphisms in the Category (mathematics), category of ''G''-sets (for a fixed ''G'').. Hence they are also known as ''G''-morphisms, ''G''-maps, or ''G''-homomorphisms.. Isomorphisms of ''G''-sets are simply bijective equivariant maps. The equivariance condition can also be understood as the following commutative diagram. Note that g\cdot denotes the map that takes an element z and returns g\cdot z.


Generalization

Equivariant maps can be generalized to arbitrary category (mathematics), categories in a straightforward manner. Every group ''G'' can be viewed as a category with a single object (morphisms in this category are just the elements of ''G''). Given an arbitrary category ''C'', a ''representation'' of ''G'' in the category ''C'' is a functor from ''G'' to ''C''. Such a functor selects an object of ''C'' and a subgroup of automorphisms of that object. For example, a ''G''-set is equivalent to a functor from ''G'' to the category of sets, Set, and a linear representation is equivalent to a functor to the category of vector spaces over a field, Vect''K''. Given two representations, ρ and σ, of ''G'' in ''C'', an equivariant map between those representations is simply a natural transformation from ρ to σ. Using natural transformations as morphisms, one can form the category of all representations of ''G'' in ''C''. This is just the functor category ''C''''G''. For another example, take ''C'' = Top, the category of topological spaces. A representation of ''G'' in Top is a topological space on which ''G'' acts continuous function, continuously. An equivariant map is then a continuous map ''f'' : ''X'' → ''Y'' between representations which commutes with the action of ''G''.


See also

*Curtis–Hedlund–Lyndon theorem, a characterization of cellular automata in terms of equivariant maps


References

{{DEFAULTSORT:Equivariant Map Group actions (mathematics) Representation theory Symmetry