Isotypical representation

In group theory, an isotypical, primary or factor representation of a group G is a unitary representation formula_1 such that any two subrepresentations have equivalent sub-subrepresentations. This is related to the notion of a primary or factor representation of a C*-algebra, or to the factor for a von Neumann algebra: the representation formula_2 of G is isotypical iff formula_3 is a factor.
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 formula_1 on a separable Hilbert space formula_5 is a possibly infinite direct sum of finite dimensional irreducible representations. An isotypical representation is any direct sum of equivalent irreducible representations that appear (typically multiple times) in formula_5.