Morita equivalences of spin blocks of symmetric and alternating groups

Abstract

We complete the demonstration of source algebra equivalences between spin blocks of families of covering groups $\{\widetilde {S}_n\}$ and $\{\widetilde {A}_n\}$ of symmetric and alternating groups, for pairs of blocks at the ends of maximal strings. These equivalences remain within the family of groups if cores of the two blocks have the same parity and cross over from one family to the other if the cores are of opposite parity. This demonstrates Kessar and Schaps' crossover conjecture for the easier case of extremal points of maximal strings. We use this result to give an improved bound for the highest degree necessary in order to get representatives of all Morita equivalence classes of spin blocks for a given weight.