information:filtering, structural:expansion, stylistic:tone
Equivariant maps and isomorphisms

If V and W are vector spaces over F, equipped with representations φ and ψ of a group G, then an equivariant map from V to W is a linear map α: V → W such that

for all g in G and v in V. In terms of φ: G → GL(V) and ψ: G → GL(W), this means

for all g in G, that is, the following diagram commutes:

Equivariant maps for representations of an associative or Lie algebra are defined similarly. If α is invertible, then it is said to be an isomorphism, in which case V and W (or, more precisely, φ and ψ) are isomorphic representations, also phrased as equivalent representations. An equivariant map is often called an intertwining map of representations. Also, in the case of a group G, it is on occasion called a G -map.

Isomorphic representations are, for practical purposes, "the same"; they provide the same information about the group or algebra being represented. Representation theory therefore seeks to classify representations up to isomorphism.