Difference between revisions of "Open problems"

From Block library
Jump to: navigation, search
(Pic finite solved. Added Morita over k implies Morita over O question.)
(Updated solved problems)
Line 3: Line 3:
 
== General problems ==
 
== General problems ==
  
* [[Morita invariance of the isomorphism type of a defect group|Is the isomorphism type of the defect group a Morita invariant?]]
+
* [[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?'' Yes - see [[References#E|[Ei19]]].
 
 
* Is every Morita equivalence between <math>\mathcal{O}</math>-blocks endopermutation source?
 
* Is every Morita equivalence between <math>\mathcal{O}</math>-blocks endopermutation source?
* Does there exist a pair of blocks Morita equivalent with respect to <math>k</math> but not witj respect to <math>\mathcal{O}</math>
+
* 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 ==
Line 14: Line 13:
 
== 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?
+
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]]])

Revision as of 10:47, 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])