Difference between revisions of "Miscallaneous results"
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This page will contain results which do not fit in elsewhere on this site. | This page will contain results which do not fit in elsewhere on this site. | ||
| − | == Blocks with basic algebras of dimension | + | == Blocks with basic algebras of low dimension == |
| + | [[Blocks with basic algebras of low dimension|Main article: Blocks with basic algebras of low dimension]] | ||
| − | In [[References|[ | + | 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. |
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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]] | ||
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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.
