On blocks stably equivalent to a quantum complete intersection of dimension 9 in characteristic 3 and a case of the Abelian defect group conjecture. (Q2880399)

Let \(k\) be an algebraically closed field of characteristic \(3\), and let \(A\) be a nonnilpotent block of the group algebra \(kG\) with defect group \(P\cong C_3\times C_3\) and with a unique simple module, up to isomorphism. The author proves that \(A\) is Morita equivalent to its Brauer correspondent in \(kN_G(P)\) and thus to the \(k\)-algebra \(k\langle X,Y\rangle/(X^3,Y^3,XY+YX)\). This proves a special case of Broué's Abelian Defect Group Conjecture.