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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 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. 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 [NS18].