Difference between revisions of "Miscallaneous results"

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(Added [Sa20] results)
(Morita invariance of the isomorphism type of a defect group: Section rewritten following counterexample to modular isomorphism problem)
 
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In [[References#L|[Li18b]]] Markus Linckelmann calculated the <math>k</math>-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra is identified. This final case was ruled out by Linckelmann and Murphy in [[References#L|[LM20]]]. These results do not use the classification of finite simple groups. In [[References#S|[Sa20]]] Benjamin Sambale applied the classification of finite simple groups to extend the classification to dimensions 13 and 14. See [[Blocks with basic algebras of low dimension]] for a description of these results.
 
In [[References#L|[Li18b]]] Markus Linckelmann calculated the <math>k</math>-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra is identified. This final case was ruled out by Linckelmann and Murphy in [[References#L|[LM20]]]. These results do not use the classification of finite simple groups. In [[References#S|[Sa20]]] Benjamin Sambale applied the classification of finite simple groups to extend the classification to dimensions 13 and 14. See [[Blocks with basic algebras of low dimension]] for a description of these results.
  
== Morita invariance of the isomorphism type of a defect group ==
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== Morita (non-)invariance of the isomorphism type of a defect group ==
 
[[Morita invariance of the isomorphism type of a defect group|Main article: Morita invariance of the isomorphism type of a defect group]]
 
[[Morita invariance of the isomorphism type of a defect group|Main article: Morita invariance of the isomorphism type of a defect group]]
  
It is not known whether there exist Morita equivalent blocks with non-isomorphic defect groups. In general this is a difficult problem, subsuming the modular isomorphism problem for <math>p</math>-groups. [[Glossary#Basic Morita/stable equivalence|Basic Morita equivalences]] do preserve the isomorphism type of a defect group, and part of the difficulty in resolving the question is the lack of examples of Morita equivalent blocks which are not also basic Morita equivalent (this is not to say that every known Morita equivalence is basic). A survey may be found in [[References#N|[NS18]]].
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In [[References#G|[GMdelR21]]] examples are given of non-isomorphic <math>2</math>-groups whose group algebras over a field of characteristic <math>2</math> are isomorphic, thus giving a counterexample to the modular isomorphism problem for fields of prime characteristic. This gives examples of blocks with non-isomorphic defect groups that are Morita equivalent. Note that these do no yield examples of Morita equivalent blocks defined over a local ring, so the question is still open as to whether the defect group is an invariant under Morita equivalence of such blocks.  
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Note that the examples in [[References#G|[GMdelR21]]] also yield blocks that are Morita equivalent but not via a [[Glossary#Basic Morita/stable equivalence|basic Morita equivalence]].

Latest revision as of 13:51, 4 August 2022

This page will contain results which do not fit in elsewhere on this site.

Blocks with basic algebras of low dimension

Main article: Blocks with basic algebras of low dimension

In [Li18b] Markus Linckelmann calculated the [math]k[/math]-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra is identified. This final case was ruled out by Linckelmann and Murphy in [LM20]. These results do not use the classification of finite simple groups. In [Sa20] Benjamin Sambale applied the classification of finite simple groups to extend the classification to dimensions 13 and 14. See Blocks with basic algebras of low dimension for a description of these results.

Morita (non-)invariance of the isomorphism type of a defect group

Main article: Morita invariance of the isomorphism type of a defect group

In [GMdelR21] examples are given of non-isomorphic [math]2[/math]-groups whose group algebras over a field of characteristic [math]2[/math] are isomorphic, thus giving a counterexample to the modular isomorphism problem for fields of prime characteristic. This gives examples of blocks with non-isomorphic defect groups that are Morita equivalent. Note that these do no yield examples of Morita equivalent blocks defined over a local ring, so the question is still open as to whether the defect group is an invariant under Morita equivalence of such blocks.

Note that the examples in [GMdelR21] also yield blocks that are Morita equivalent but not via a basic Morita equivalence.