In group theory, an isotypical, primary or factor representation of a group G is a unitary representation
Contents
This term more generally used in the context of semisimple modules.
Property
One of the interesting property of this notion lies in the fact that two isotypical representations are either quasi-equivalent or disjoint (in analogy with the fact that irreducible representations are either unitarily equivalent or disjoint).
This can be understood through the correspondence between factor representations and minimal central projection (in a von Neumann algebra),. Two minimal central projections are then either equal or orthogonal.
Example
Let G be a compact group. A corollary of the Peter–Weyl theorem has that any unitary representation