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structure of subrepresentations of (infinite) sums of irr. representations

Plancherel formula for compact groups from Peter-Weyl TheoremCanonical field of Hilbert spaces in Dixmier; Plancherel TheoremIrreducible Representations and Direct SumsTwo questions about orthogonal projections on Hilbert spaceRepresentations of the symmetric group $S_3$Every Irreducible Representation of a Compact Group is Finite-DimensionalBeginner question about representations and Jordan forms$(V otimes W)^H = V^H otimes W^H$?Decomposing vector spaces into Direct Sums of spans of basis vectorsProof or counterexample for isomorphism of group representations

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$begingroup$

Let $G$ be a (locally compact) group and $ ( pi_1 , V_pi_1 ) , ( pi_2 , V_pi_2 ) , ldots$ irreducible, unitary representations. I think, I can show, that for the (finite) direct sum representation $$big( pi_1 oplus ldots oplus pi_n , V_pi_1 oplus ldots oplus V_pi_n big) $$
every subrepresentation $(eta , U)$ is $G$-isomorphic to a direct sum of the representations showing up in the direct sum, so $$(eta , U) cong left( bigoplus_k=1^l pi_j_k , bigoplus_k=1^mathcall V_pi_j_k right) ,$$where $1 leq j_k leq n$ and $l leq n$.

Now the question is, when we start with the $textitinfinite$ direct sum of representations, do we have a similar statement? Namely, is every subrepresentation of $$left( bigoplus_k=1^infty pi_j_k , widehatbigoplus_k=1^infty V_pi_j_k right)$$a direct sum of the $pi_i$'s? I strongly suspect this to be false, but I can't find a counterexample.

functional-analysis representation-theory

share|cite|improve this question

edited Mar 15 at 17:40

Targon

asked Mar 15 at 13:13

TargonTargon

657

$endgroup$

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  $begingroup$
  You shouldn't put an $l$ on top of your sums if they're infinite.
  $endgroup$
  – Max
  Mar 15 at 16:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  thanks, my bad. Edited it.
  $endgroup$
  – Targon
  Mar 15 at 17:41
  

  
  
  
  

add a comment | 

0

$begingroup$

Let $G$ be a (locally compact) group and $ ( pi_1 , V_pi_1 ) , ( pi_2 , V_pi_2 ) , ldots$ irreducible, unitary representations. I think, I can show, that for the (finite) direct sum representation $$big( pi_1 oplus ldots oplus pi_n , V_pi_1 oplus ldots oplus V_pi_n big) $$
every subrepresentation $(eta , U)$ is $G$-isomorphic to a direct sum of the representations showing up in the direct sum, so $$(eta , U) cong left( bigoplus_k=1^l pi_j_k , bigoplus_k=1^mathcall V_pi_j_k right) ,$$where $1 leq j_k leq n$ and $l leq n$.

Now the question is, when we start with the $textitinfinite$ direct sum of representations, do we have a similar statement? Namely, is every subrepresentation of $$left( bigoplus_k=1^infty pi_j_k , widehatbigoplus_k=1^infty V_pi_j_k right)$$a direct sum of the $pi_i$'s? I strongly suspect this to be false, but I can't find a counterexample.

functional-analysis representation-theory

share|cite|improve this question

edited Mar 15 at 17:40

Targon

asked Mar 15 at 13:13

TargonTargon

657

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You shouldn't put an $l$ on top of your sums if they're infinite.
  $endgroup$
  – Max
  Mar 15 at 16:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  thanks, my bad. Edited it.
  $endgroup$
  – Targon
  Mar 15 at 17:41
  

  
  
  
  

add a comment | 

$begingroup$

Let $G$ be a (locally compact) group and $ ( pi_1 , V_pi_1 ) , ( pi_2 , V_pi_2 ) , ldots$ irreducible, unitary representations. I think, I can show, that for the (finite) direct sum representation $$big( pi_1 oplus ldots oplus pi_n , V_pi_1 oplus ldots oplus V_pi_n big) $$
every subrepresentation $(eta , U)$ is $G$-isomorphic to a direct sum of the representations showing up in the direct sum, so $$(eta , U) cong left( bigoplus_k=1^l pi_j_k , bigoplus_k=1^mathcall V_pi_j_k right) ,$$where $1 leq j_k leq n$ and $l leq n$.

Now the question is, when we start with the $textitinfinite$ direct sum of representations, do we have a similar statement? Namely, is every subrepresentation of $$left( bigoplus_k=1^infty pi_j_k , widehatbigoplus_k=1^infty V_pi_j_k right)$$a direct sum of the $pi_i$'s? I strongly suspect this to be false, but I can't find a counterexample.

functional-analysis representation-theory

share|cite|improve this question

edited Mar 15 at 17:40

Targon

asked Mar 15 at 13:13

TargonTargon

657

$endgroup$

share|cite|improve this question

edited Mar 15 at 17:40

Targon

asked Mar 15 at 13:13

TargonTargon

657

share|cite|improve this question

edited Mar 15 at 17:40

Targon

asked Mar 15 at 13:13

TargonTargon

657

asked Mar 15 at 13:13

TargonTargon

657

asked Mar 15 at 13:13

TargonTargon

657