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?
• Is [math]{\rm Pic}_\mathcal{O}(B)[/math] always finite?
• Is every Morita equivalence between [math]\mathcal{O}[/math]-blocks endopermutation source?

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).

Picard groups

• What is [math]{\rm Pic}_\mathcal{O}(B_0(\mathcal{O} J_1))[/math]?

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?