In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, equivariance is a form of
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
for
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
acted on by the same
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
, 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 distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properti ...
, equivariance under statistical transformations of data is an important property of various estimation methods; see
invariant estimator for details. In pure mathematics, equivariance is a central object of study in
equivariant topology and its subtopics
equivariant cohomology and
equivariant stable homotopy theory.
Examples
Elementary geometry
In the geometry of
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
s, the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
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
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
,
circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
,
incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bis ...
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 (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 transformations (combinations of translation, rotation, and scaling),
and the centroid is equivariant under
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generall ...
s.
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
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ar ...
of a sample (a set of real numbers) is commonly used as a
central tendency of the sample. It is equivariant under
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s 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
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
of a sample is equivariant for a much larger group of transformations, the (strictly)
monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
s of the real numbers. This analysis indicates that the median is more
robust
Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
against certain kinds of changes to a data set, and that (unlike the mean) it is meaningful for
ordinal data
Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. These data exist on an ordinal scale, one of four levels of measurement described b ...
.
The concepts of an
invariant estimator 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
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
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 is the same thing as a
module homomorphism of -
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
, where is the
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
of ''G''.
Under some conditions, if ''X'' and ''Y'' are both
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _ ...
s, then an intertwiner (other than the
zero map
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, u ...
) only exists if the two representations are equivalent (that is, are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
as
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
). That intertwiner is then unique
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
a multiplicative factor (a non-zero
scalar from ). These properties hold when the image of is a simple algebra, with centre (by what is called
Schur's Lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations
of a group ...
: 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
-set for a
group . This is a mathematical object consisting of a
mathematical set
A set is the mathematical model for a collection of different things; a set contains ''elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or e ...
and a
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
(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
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s in the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of ''G''-sets (for a fixed ''G'').
[.] Hence they are also known as ''G''-morphisms,
''G''-maps, or ''G''-homomorphisms.
[.] Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
s of ''G''-sets are simply
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
equivariant maps.
The equivariance condition can also be understood as the following
commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
. Note that
denotes the map that takes an element
and returns
.
Generalization
Equivariant maps can be generalized to arbitrary
categories in a straightforward manner. Every group ''G'' can be viewed as a category with a single object (
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s 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
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s of that object. For example, a ''G''-set is equivalent to a functor from ''G'' to the
category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
, 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
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
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 In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again cont ...
. A representation of ''G'' in Top is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
on which ''G'' acts
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