Difference between revisions of "Open problems"

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(Basic algebra dimension 9)
 
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== General problems ==
 
== General problems ==
  
* Is the isomorphism type of the defect group a Morita invariant?
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* [[(Non-)Morita invariance of the isomorphism type of a defect group|Is the isomorphism type of the defect group a Morita invariant?]] - no (see [[References#G|[GMdelR21]]])
* Is <math>{\rm Pic}_\mathcal{O}(B)</math> always finite?
 
 
* Is every Morita equivalence between <math>\mathcal{O}</math>-blocks endopermutation source?
 
* Is every Morita equivalence between <math>\mathcal{O}</math>-blocks endopermutation source?
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* Does there exist a pair of blocks Morita equivalent with respect to <math>k</math> but not with respect to <math>\mathcal{O}</math> - yes (see [[References#G|[GMdelR21]]])
  
 
== Open cases for classifications of Morita equivalence classes for a given <math>p</math>-group ==
 
== Open cases for classifications of Morita equivalence classes for a given <math>p</math>-group ==
  
 
* Which Brauer trees give rise to blocks with defect group <math>C_7</math>? (This is the smallest cyclic group for which the classification is not known).
 
* Which Brauer trees give rise to blocks with defect group <math>C_7</math>? (This is the smallest cyclic group for which the classification is not known).
 
== Picard groups ==
 
 
* What is <math>{\rm Pic}_\mathcal{O}(B_0(\mathcal{O} J_1))</math>?
 
  
 
== Basic algebras of dimension 9 ==
 
== Basic algebras of dimension 9 ==
  
Does the 9-dimensional algebra described in [[Blocks with basic algebras of low dimension]] occur as the basic algebra of a block of a finite group?
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Does the 9-dimensional algebra described in [[Blocks with basic algebras of low dimension]] occur as the basic algebra of a block of a finite group? - no (see [[References#L|[LM20]]] and [[References#S|[Sa20]]])

Latest revision as of 11:30, 21 June 2021

This page is for open problems, large and small, relating to module categories for blocks. Missing data is also flagged within tables elsewhere on this site.

General problems

Open cases for classifications of Morita equivalence classes for a given [math]p[/math]-group

  • Which Brauer trees give rise to blocks with defect group [math]C_7[/math]? (This is the smallest cyclic group for which the classification is not known).

Basic algebras of dimension 9

Does the 9-dimensional algebra described in Blocks with basic algebras of low dimension occur as the basic algebra of a block of a finite group? - no (see [LM20] and [Sa20])