On a Question of Rickard on Tensor Products of Stably Equivalent Algebras

Let r be a positive integer, let p be a prime and Fp denote an algebraic closure of the prime field Fp. After observing that the principal block B of FpPSU(3, p) is stably equivalent of Morita type to its Brauer correspondent b, we compute the radical series of the center Z(b), and, using GAP, the radical series of Z(B) in the cases p ∈ {3, 4, 5, 7, 8}. In these cases, the dimensions of the last non zero power of the radical of Z(b) and Z(B) are different, and it follows that the algebra B ⊗Fp Fp[X]/X p is not stably equivalent of Morita type to b⊗Fp Fp[X]/X . This yields a negative answer to a question of Rickard.