Isotypical representation: Difference between revisions

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In group theory, an isotypical, primary or factor representation^[1] of a group G is a unitary representation $\pi :G\longrightarrow {\mathcal {B}}({\mathcal {H}})$ such that any two subrepresentations have equivalent sub-subrepresentations.^[2] 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 $\pi$ of G is isotypical iff $\pi (G)^{''}$ 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),.^[3] 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 $\pi :G\longrightarrow {\mathcal {B}}({\mathcal {H}})$ on a separable Hilbert space ${\mathcal {H}}$ 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 ${\mathcal {H}}$ .

References

^ Deitmar & Echterhoff 2014, § 8.3 p.162
^ Higson, Nigel; Roe, John. "Operator Algebras" (PDF). psu.edu. Retrieved 11 March 2016.
^ Dixmier 1982, Prop. 5.2.7 p.117

Bibliography

Deitmar, A.; Echterhoff, S. (2014). Principles of Harmonic Analysis. Universitext. Springer International Publishing. ISBN 978-3-319-05792-7.
Dixmier, Jacques, (1982). C*-algebras. North-Holland Publ. Co. ISBN 0-444-86391-5. OCLC 832825844.{{cite book}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)

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-In [[group theory]], an '''isotypical''', '''primary''' or '''factor representation'''<ref>Deitmar {{harvnb| Principles of Harmonic analysis| date=2009|loc= § 8.3 p.162}}</ref> of a group G is a [[unitary representation]] <math>\pi : G \longrightarrow \mathcal{B}(\mathcal{H}) </math> such that any two [[subrepresentation]]s have equivalent sub-subrepresentations.<ref name="Higson">{{cite web|last1=Higson|first1=Nigel|last2=Roe|first2=John|title=Operator Algebras|url=http://www.personal.psu.edu/users/n/d/ndh2/math/Papers_files/Higson,%20Roe%20-%202006%20-%20Operator%20algebras.pdf|website=psu.edu|accessdate=11 March 2016}}</ref>  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 <math>\pi </math> of G is isotypical iff <math>\pi(G)^{''} </math> is a factor.
+In [[group theory]], an '''isotypical''', '''primary''' or '''factor representation'''<ref>{{harvnb|Deitmar|Echterhoff|2014|loc= § 8.3 p.162}}</ref> of a group G is a [[unitary representation]] <math>\pi : G \longrightarrow \mathcal{B}(\mathcal{H}) </math> such that any two [[subrepresentation]]s have equivalent sub-subrepresentations.<ref name="Higson">{{cite web |last1=Higson |first1=Nigel |last2=Roe |first2=John |title=Operator Algebras |url=http://www.personal.psu.edu/users/n/d/ndh2/math/Papers_files/Higson,%20Roe%20-%202006%20-%20Operator%20algebras.pdf |website=psu.edu |accessdate=11 March 2016 }}</ref>  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 <math>\pi </math> of G is isotypical iff <math>\pi(G)^{''} </math> is a factor.
 This term more generally used in the context of [[semisimple module]]s.
@@ Line 11: / Line 11: @@
 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 representation]]s 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),.<ref>Dixmier {{harvnb|C*-algebras|date= 1977|loc= Prop. 5.2.7 p.117}}</ref> Two minimal central projections are then either equal or orthogonal.
+This can be understood through the correspondence between factor representations and minimal [[central projection]] (in a von Neumann algebra),.<ref>{{harvnb|Dixmier|1982|loc= Prop. 5.2.7 p.117}}</ref> Two minimal central projections are then either equal or orthogonal.
 == Example ==
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 ==References==
 {{reflist}}
+==Bibliography==
+* {{cite book | last=Deitmar | first=A. | last2=Echterhoff | first2=S. | title=Principles of Harmonic Analysis | publisher=Springer International Publishing | series=Universitext | year=2014 | isbn=978-3-319-05792-7 | url=https://books.google.com/books?id=BMcpBAAAQBAJ }}
+* {{cite book | last=Dixmier | first=Jacques, | title=C*-algebras | publisher=North-Holland Publ. Co. | year=1982 | isbn=0-444-86391-5 | oclc=832825844}}
+==Further reading==
 *Mackey
-*"C* algebras", Jacques Dixmier, Chapter 5
 *"Lie Groups", Claudio Procesi, def. p.&nbsp;156.
 * "Group and symmetries", [[Yvette Kosmann-Schwarzbach]]