Home > Equivariant

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.

The equivariance condition can also be understood diagrammatically as follows. Let ρ and σ be actions of G on the sets X and Y. We write g·x as ρ_g(x) and likewise for σ. (ρ and σ can be considered as group homomorphisms from G to the symmetric groups on X and Y, respectively). Then a map f : X → Y is equivariant iff the following diagram commutes:

1 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.

Alternatively, an intertwiner for representations of G over a field K is the same thing as a module homomorphismAlgebra In abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties of of K[G]- modulesAlgebra In abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties of, where K[G] is the group ringIn 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 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 modulesAlgebra In abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties of). 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 moduleIn 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" o). As as consequence, in important cases the construction of an intertwiner is enough to show the representations are effectively the same.