Template:Integrate

Definition

An isotypical or primary representation of a group G is a unitary representation $\pi :G⟶\mathcal{ℬ}(\mathcal{\mathscr{H}})$ such that any two subrepresentations have equivalent subsubrepresentations.

This is to relate to primary or factor representation of a C*-algebra, or to the notion of factor for a von Neumann algebra: the representation $\pi$ of G is isotypicall iff $\pi (G{)}^{{\text{'}}^{\prime }}$ is a factor.

This term more generally used in the context of semisimple module.

Example

Let G be a compact group. A corollary of the Peter-Weyl theorem has that any unitary representation $\pi :G⟶\mathcal{ℬ}(\mathcal{\mathscr{H}})$ on a separable Hilbert space $\mathcal{\mathscr{H}}$ is a possibly infinite direct sum of finite dimensional irreducible representations. An isotypical representation is a direct sum of the equivalent irreducible representations that appear, possibly multiple times, in $\mathcal{\mathscr{H}}$.

References

Mackey

"C* algebras", Jacques Dixmier, Chapter 5

"Lie Groups", Claudio Procesi, def. p. 156.

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