Bakı Dövlət Universiteti Digital Research Lab

The local-global principle in modular representation theory asserts that the invariants of blocks of finite groups are determined by the invariants (of blocks) of local subgroups. There are several outstanding conjectures revolving around this principle. Alperin's block-wise weight conjecture (1987) predicts that the number of simple modules of a block is equal to the number of its weights. The finiteness conjectures of Donovan (1980) and Puig (1982) state that there are only finitely many blocks of finite groups with a given defect group, up to Morita and splendid Morita equivalence, respectively.

In this talk, we prove a finiteness theorem in the spirit of Donovan's and Puig's conjectures, in terms of functorial equivalences. We also give a reformulation of Alperin's conjecture in terms of diagonal $p$-permutation functors. Some parts of this work are joint with Boltje and Bouc.