Editing Equivariant map

{{short description|Maps whose domain and codomain are acted on by the same group, and the map commutes}}
In [[mathematics]], '''equivariance''' is a form of [[symmetry]] for [[function (mathematics)|function]]s from one space with symmetry to another (such as [[symmetric space]]s). A function is said to be an '''equivariant map''' when its domain and codomain are [[Group action (mathematics)|acted on]] by the same [[symmetry group]], and when the function [[commutative property|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 [[Invariant (mathematics)|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]], 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===
[[File:Triangle.Centroid.svg|thumb|The centroid of a triangle (where the three red segments meet) is equivariant under [[affine transformation]]s: the centroid of a transformed triangle is the same point as the transformation of the centroid of the triangle.]]
In the geometry of [[triangle]]s, the [[area]] and [[perimeter]] of a triangle are invariants under [[Euclidean transformation]]s: translating, rotating, or reflecting a triangle does not change its area or perimeter. However, [[triangle center]]s 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, reflection, and scaling),<ref>{{citation
 | last = Kimberling | first = Clark | authorlink = Clark Kimberling
 | issue = 3
 | journal = [[Mathematics Magazine]]
 | jstor = 2690608
 | mr = 1573021
 | pages = 163–187
 | title = Central Points and Central Lines in the Plane of a Triangle
 | volume = 67
 | year = 1994
 | doi=10.2307/2690608}}. "Similar triangles have similarly situated centers", p.&nbsp;164.</ref>
and the centroid is equivariant under [[affine transformation]]s.<ref>The centroid is the only affine equivariant center of a triangle, but more general convex bodies can have other affine equivariant centers; see e.g. {{citation
 | last = Neumann | first = B. H.
 | journal = Journal of the London Mathematical Society
 | mr = 0000978
 | pages = 262–272
 | series = Second Series
 | title = On some affine invariants of closed convex regions
 | volume = 14
 | year = 1939
 | issue = 4
 | doi = 10.1112/jlms/s1-14.4.262 }}.</ref>

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 {{mvar|s}}, the perimeter also scales by {{mvar|s}} and the area scales by {{math|''s''<sup>2</sup>}}. 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 transformation]]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]] of a sample is equivariant for a much larger group of transformations, the (strictly) [[monotonic function]]s 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]].<ref>{{citation|title=Measurement theory: Frequently asked questions (Version 3)|date=September 14, 1997|publisher=SAS Institute Inc.|url=http://www.medicine.mcgill.ca/epidemiology/courses/EPIB654/Summer2010/EF/measurement%20scales.pdf|first=Warren S.|last=Sarle}}. Revision of a chapter in ''Disseminations of the International Statistical Applications Institute'' (4th ed.), vol.&nbsp;1, 1995, Wichita: ACG Press, pp. 61–66.</ref>

The concepts of an [[invariant estimator]] and equivariant estimator have been used to formalize this style of analysis.

===Representation theory===
{{See also|Representation theory#Equivariant maps and isomorphisms}}

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 {{mvar|G}} over a [[field (mathematics)|field]] {{mvar|K}} is the same thing as a [[module (mathematics)|module homomorphism]] of {{math|''K''[''G'']}}-[[module (mathematics)|modules]], where {{math|''K''[''G'']}} is the [[group ring]] of ''G''.<ref>{{citation
 | last1 = Fuchs | first1 = Jürgen
 | last2 = Schweigert | first2 = Christoph
 | isbn = 0-521-56001-2
 | mr = 1473220
 | page = 70
 | publisher = Cambridge University Press, Cambridge
 | series = Cambridge Monographs on Mathematical Physics
 | title = Symmetries, Lie algebras and representations: A graduate course for physicists
 | url = https://books.google.com/books?id=B_JQryjNYyAC&pg=PA70
 | year = 1997}}.</ref>

Under some conditions, if ''X'' and ''Y'' are both [[irreducible representation]]s, 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 {{mvar|K}}). These properties hold when the image of {{math|''K''[''G'']}} is a simple algebra, with centre {{mvar|K}} (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.<ref>{{citation
 | last1 = Sexl | first1 = Roman U.
 | last2 = Urbantke | first2 = Helmuth K.
 | doi = 10.1007/978-3-7091-6234-7
 | isbn = 3-211-83443-5
 | location = Vienna
 | mr = 1798479
 | page = 165
 | publisher = Springer-Verlag
 | series = Springer Physics
 | title = Relativity, groups, particles: Special relativity and relativistic symmetry in field and particle physics
 | url = https://books.google.com/books?id=iyj0CAAAQBAJ&pg=PA165
 | year = 2001}}.</ref>

==Formalization==
Equivariance can be formalized using the concept of a [[Group action (mathematics)|{{mvar|G}}-set]] for a [[group (mathematics)|group]] {{mvar|G}}. This is a mathematical object consisting of a [[set (mathematics)|mathematical set]] {{mvar|S}} and a [[Group action (mathematics)|group action]] (on the left) of {{mvar|G}} on {{mvar|S}}.
If {{mvar|X}} and {{mvar|Y}} are both {{mvar|G}}-sets for the same group {{mvar|G}}, then a function {{math|''f'' : ''X'' → ''Y''}} is said to be equivariant if
:{{math|1=''f''(''g''&middot;''x'') = ''g''&middot;''f''(''x'')}}
for all {{math|''g'' ∈ ''G''}} and all {{math|''x'' in ''X''}}.<ref>{{citation|title=Nominal Sets: Names and Symmetry in Computer Science|volume=57|series=Cambridge Tracts in Theoretical Computer Science|first=Andrew M.|last=Pitts|publisher=Cambridge University Press|year=2013|isbn=9781107244689|url=https://books.google.com/books?id=VVehscCSPh8C&pg=PA14|at=Definition 1.2, p.&nbsp;14}}.</ref>

If one or both of the actions are right actions the equivariance condition may be suitably modified:
:{{math|1=''f''(''x''&middot;''g'') = ''f''(''x'')&middot;''g''}}; (right-right)
:{{math|1=''f''(''x''&middot;''g'') = ''g''<sup>&minus;1</sup>&middot;''f''(''x'')}}; (right-left)
:{{math|1=''f''(''g''&middot;''x'') = ''f''(''x'')&middot;''g''<sup>&minus;1</sup>}}; (left-right)

Equivariant maps are [[homomorphism]]s in the [[Category (mathematics)|category]] of ''G''-sets (for a fixed ''G'').<ref name="grm">{{citation|title=Groups, Rings, Modules|series=Dover Books on Mathematics|first1=Maurice|last1=Auslander|first2=David|last2=Buchsbaum|publisher=Dover Publications|year=2014|isbn=9780486490823|url=https://books.google.com/books?id=VW2TAwAAQBAJ&pg=PA86|pages=86–87}}.</ref> Hence they are also known as '''''G''-morphisms''',<ref name="grm"/> '''''G''-maps''',<ref>{{citation
 | last = Segal | first = G. B.
 | contribution = Equivariant stable homotopy theory
 | mr = 0423340
 | pages = 59–63
 | publisher = Gauthier-Villars, Paris
 | title = Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2
 | year = 1971}}.</ref> or '''''G''-homomorphisms'''.<ref>{{citation
 | last1 = Adhikari | first1 = Mahima Ranjan
 | last2 = Adhikari | first2 = Avishek
 | doi = 10.1007/978-81-322-1599-8
 | isbn = 978-81-322-1598-1
 | location = New Delhi
 | mr = 3155599
 | page = 142
 | publisher = Springer
 | title = Basic modern algebra with applications
 | url = https://books.google.com/books?id=lBO7BAAAQBAJ&pg=PA142
 | year = 2014}}.</ref> [[Isomorphism]]s of ''G''-sets are simply [[bijective]] equivariant maps.<ref name="grm"/>

The equivariance condition can also be understood as the following [[commutative diagram]].  Note that <math>g\cdot</math> denotes the map that takes an element <math>z</math> and returns <math>g\cdot z</math>.

[[Image:equivariant commutative diagram.png|center|175px]]

==Generalization==
{{See also|Representation theory#Generalizations|Category of representations#Category-theoretic definition}}
{{unreferenced section|date=April 2016}}
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 ([[morphism]]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]] of [[automorphism]]s 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'''<sub>''K''</sub>.

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''<sup>''G''</sup>.

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==
{{reflist}}

{{DEFAULTSORT:Equivariant Map}}
[[Category:Group actions]]
[[Category:Representation theory]]
[[Category:Symmetry]]