Intertwiner

For thoughtful child sponsors

In mathematical representation theory, an intertwining map or intertwiner for representations of a group 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 somewhat more concrete terms, if we have two linear representations ρ and σ of a group G, then a K-linear map f:ρ->σ is an intertwiner if g(f(x))=f(g(x)) for all g in G and all x in ρ.

Under some conditions, if ρ and σ 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 as consequence, in important cases the construction of an intertwiner is enough to show the representations are effectively the same.

Such mappings are also termed equivariant. They are a special case of natural transformations, too: given two functors from a group G (interpreted as a category with only one object) to the category of vector spaces over a field K, K-Vect, called ρ and σ, then a natural transformation from ρ to σ is an intertwiner.

In general, equivariance is the comparable property of a function F relating actions of a group or monoid:

goF = Fog

where g is an element acting on the function codomain on the left side, on the function domain on the right side. This can in the group case be written

goFog^-1 = F

which can be read as F being an invariant function.