Open problems

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Revision as of 13:54, 7 October 2019 by Charles Eaton (talk | contribs) (Pic finite solved. Added Morita over k implies Morita over O question.)
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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? Yes - see [Ei19].
  • 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 witj respect to [math]\mathcal{O}[/math]

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?