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Morita equivalence classes of blocks with elementary abelian defect groups of order 32. (English) Zbl 07327760
In modular representation theory of finite groups one of the most important and interesting problems is to prove (or find out a counter-example) Donovan’s conjecture. It was first stated in public in Alperin’s celebrated survey [J. L. Alperin, Proc. Symp. Pure Math. 37, 369–375 (1980; Zbl 0449.20019)]. In the article he puts a conjecture as Conjecture M: Fix arbitrary prime \(p\) and finite \(p\)-group \(D\). Then up to Morita equivalences there are only a finite number of \(p\)-block algebras of finite groups with a defect group \(D\). The conjecture has not yet been solved. See information in [20]. In particular when \(D\) is an arbitrary abelian \(2\)-group, Donovan’s conjecture is proved by C. W. Eaton and M. Livesey [Proc. Am. Math. Soc. 147, No. 3, 963–970 (2019; Zbl 1430.20011)] over an algebraically closed field \(k\) of characteristic \(2\), and then even over a suitable discrete valuation ring whose residue field is \(k\) [C. W. Eaton et al., Math. Z. 295, No. 1–2, 249–264 (2020; Zbl 07203115)]. Now, the paper under review concerns and looks closely at those \(2\)-blocks \(B\) when their defect group \(D\) is elementary ablian of order \(32\). The authro gives a precise list of \(B\)s up to Morita equivalences. Actually there are \(31\) Morita equivalence classes when \(B\)s are the principal \(2\)-block algebras and those of three when \(B\) is non-principal \(2\)-block algebras. A comment on References: the page numbers of [18] is missing.

MSC:
20C20 Modular representations and characters
16D90 Module categories in associative algebras
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
Software:
GAP; Magma
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