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{{For|equivariance in estimation theory|Invariant estimator}}
In [[mathematics]], an '''equivariant map''' is a [[function (mathematics)|function]] between two [[Set (mathematics)|sets]] that commutes with the [[group action|action of a group]]. Specifically, let ''G'' be a [[group (mathematics)|group]] and let ''X'' and ''Y'' be two associated [[group action|''G''-sets]]. A function ''f'' : ''X'' → ''Y'' is said to be equivariant if
:''f''(''g''·''x'') = ''g''·''f''(''x'')
for all ''g'' ∈ ''G'' and all ''x'' in ''X''. Note that if one or both of the actions are right actions the equivariance condition must be suitably modified:
:''f''(''x''·''g'') = ''f''(''x'')·''g'' ; (right-right)
:''f''(''x''&middot;''g'') = ''g''<sup>&minus;1</sup>&middot;''f''(''x'') ; (right-left)
:''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''). Hence they are also known as '''''G''-maps''' or '''''G''-homomorphisms'''. [[Isomorphism]]s of ''G''-sets are simply [[bijective]] equivariant maps.


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>.
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[[Image:equivariant commutative diagram.png|center|175px]]
 
==Intertwiners==
A completely analogous definition holds for the case of [[linear representation]]s of ''G''.  
Specifically, if ''X'' and ''Y'' are the representation spaces of two linear representations of ''G'' then a [[linear map]] ''f'' : ''X'' → ''Y'' is called an '''intertwiner''' of the representations if it commutes with the action of ''G''. Thus an intertwiner is an equivariant map in the special case of two linear representations/actions.
 
Alternatively, an intertwiner for representations of  ''G'' over a [[field (mathematics)|field]] ''K'' is the same thing as a [[module (mathematics)|module homomorphism]] of ''K''[''G'']-[[module (mathematics)|modules]], where ''K''[''G''] is the [[group ring]] of ''G''.
 
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 ''K''). These properties hold when the image of ''K''[''G''] is a simple algebra, with centre 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.
 
==Categorical description==
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''.
 
{{DEFAULTSORT:Equivariant Map}}
[[Category:Group actions]]
[[Category:Representation theory]]

Latest revision as of 23:58, 29 December 2014


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