Morita equivalent blocks with an endotrivial source that are not globally isotypic

Abstract

Let $(K,\mathcal{O},k)$ be a large enough $2$-modular system. We show that there is a block algebra $B$ of $\mathcal{O}{\mathrm{SU}}_3(2)$ which is Morita equivalent to $\mathcal{O}{Q}_8$ via a bimodule with a $3$-dimensional endotrivial source. We also prove that $B$ and $\mathcal{O}{Q}_8$ are not globally isotypic. Hence they are not $p$-permutation equivalent and not splendidly Rickard equivalent.