Difference between revisions of "Open problems"
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== General problems == | == General problems == | ||
| − | * Is the isomorphism type of the defect group a Morita invariant? | + | * [[(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]]]) |
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* 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 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). | ||
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== 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]]]) |
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
- Is the isomorphism type of the defect group a Morita invariant? - no (see [GMdelR21])
- 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 with respect to [math]\mathcal{O}[/math] - yes (see [GMdelR21])
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])
