Difference between revisions of "Open problems"
(Pic finite solved. Added Morita over k implies Morita over O question.) |
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* [[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?]] | ||
− | * Is <math>{\rm Pic}_\mathcal{O}(B)</math> always finite? | + | * ''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> | ||
== 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 == |
Revision as of 14:54, 7 October 2019
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
- Is the isomorphism type of the defect group a Morita invariant?
- Is [math]{\rm Pic}_\mathcal{O}(B)[/math] always finite? Yes - see [Ei19].
- 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]
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?