Morita equivalence classes of 2-blocks of defect three. (Q2790914)

Let \(\mathcal O\) be a suitable complete discrete valuation ring whose residue field \(k\) has characteristic 2, and let \(D\) be an elementary abelian 2-group of order \(2^d\). By a result of \textit{C. W. Eaton, R. Kessar, B. Külshammer} and \textit{B. Sambale} [Adv. Math. 254, 706-735 (2014; Zbl 1341.20006)], there are only finitely many Morita equivalence classes of blocks (over \(k\)) with defect group \(D\).NEWLINENEWLINE In the present paper, the author gives explicit representatives for these Morita equivalence classes (even over \(\mathcal O\)), in the case \(d=3\). There are precisely 8 such equivalence classes, one with inertial index \(t=1\), two with \(t=3\), two with \(t=7\) and three with \(t=21\). Moreover, there are precisely 4 derived equivalence classes of such blocks, one for each \(t\in\{1,3,7,21\}\). This implies that Broué's abelian defect group conjecture holds for 2-blocks of defect at most 3.