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== Definition ==<br />
An '''isotypical''' or '''primary representation''' of a group G is a unitary representation <math>\pi : G \longrightarrow \mathcal{B}(\mathcal{H}) </math> such that any two subrepresentations have equivalent subsubrepresentations.<br />
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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 <math>\pi </math> of G is isotypicall iff <math>\pi(G)^{''} </math> is a factor. <br />
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This term more generally used in the context of [[semisimple module]].<br />
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== Example ==<br />
Let G be a compact group. A corollary of the [[Peter-Weyl theorem]] has that any unitary representation <math>\pi : G \longrightarrow \mathcal{B}(\mathcal{H}) </math> on a separable Hilbert space <math>\mathcal{H}</math> 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 <math>\mathcal{H}</math>.<br />
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==References==<br />
Mackey<br />
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"C* algebras", Jacques Dixmier, Chapter 5<br />
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"Lie Groups", Claudio Procesi, def. p. 156.<br />
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[[Category:Mathematics]]<br />
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