Open problems

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])