# 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]]]) |

− | |||

* 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 == | ||

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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 12:30, 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])