In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. Specifically, let G be a group and let X and Y be two associated G-sets. A function f : X → Y is said to be equivariant if Mathematics is the study of quantity, structure, space and change. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, groups are often used to describe symmetries of objects. ...

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 on the right the equivariance condition must be suitably modified:

f(x·g) = f(x)·g

f(x·g) = g⁻¹·f(x)

f(g·x) = f(x)·g⁻¹

Equivariant maps are homomorphisms in the category of G-sets (for a fixed G). Hence they are also known as G-maps or G-homomorphisms. Isomorphisms of G-sets are simply bijective equivariant maps. In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. ... The word category (plural categories; from Greek κατηγορια meaning assertion or accusation, hence categorical denial) has several meanings: it is used informally to mean a class of things, as in the category of all living things. See categorization. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

The equivariance condition can also be understood as the following commutative diagram. Note that denotes the map that takes an element z and returns . In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

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Intertwiners

A completely analogous definition holds for the case of linear representations of G. Specifically, if X and Y are 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. Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

Alternatively, an intertwiner for representations of G over a field K is the same thing as a module homomorphism of K[G]-modules, where K[G] is the group ring of G. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a module is a generalization of a vector space. ... In abstract algebra, a module is a generalization of a vector space. ... In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described...

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 modules). That intertwiner is then unique up to a multiplicative factor (a non-zero 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. In mathematics, the term irreducible is used in several ways. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In abstract algebra, a module is a generalization of a vector space. ... Look up Up to in Wiktionary, the free dictionary Modern Slang In modern slang, up to means you are either willing to engage in an act (Sally is up to going to the park), capable of an act (Im sorry, Im just not up to it) or are... The concept of a scalar is used in mathematics, physics, and computing. ... In mathematics, Schurs lemma is now a generic term applied to theorems on the commutant of a module M that is simple. ... In abstract algebra, a (left or right) module S over a ring R is called simple if it is not the zero module and if its only submodules are 0 and S. Understanding the simple modules over a ring is usually helpful because they form the building blocks of all...

Categorical description

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 (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. In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In category theory, a functor is a special type of mapping between categories. ... In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In mathematics, the category K_Vect has all vector spaces over a fixed field K as objects and linear transformations as morphisms. ...

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. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ...

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 continuously. An equivariant map is then a continuous map f : X → Y between representations which commutes with the action of G. The category Top has topological spaces as objects and continuous maps as morphisms. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...