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Classifying 2-blocks with an elementary abelian defect group

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GRAW01 - Introductory/instructional workshop

Donovan's conjecture predicts that given a $p$-group $D$ there are only finitely many Morita equivalence classes of blocks of group algebras of finite groups with defect group $D$. While the conjecture is still open for a generic $p$-group $D$, it has been proven in 2014 by Eaton, Kessar, Külshammer and Sambale when D is an elementary abelian 2-group, and in 2018 by Eaton, Eisele and Livesey when D is any abelian 2-group. The proof, however, does not describe these equivalence classes explicitly. A classification up to Morita equivalence over a complete discrete valuation ring $\mathcal{O}$ has been achieved when $p=2$ for abelian $D$ with rank $3$ or less, and for $D=(C_2)4$. In my PhD thesis I have done $(C_2)5$, and I have partial results on $(C_2)^6$. I will introduce the topic, give some definitions and then describe the process of classifying these blocks, with a focus on the process and the tools needed to produce a complete classification. All the obtained data is available on https://wiki.manchester.ac.uk/blocks/.

This talk is part of the Isaac Newton Institute Seminar Series series.

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