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In [S], Joanna Scopes discovered a method for generating Morita equivalences between blocks of symmetric groups and thus for showing that Donavan’s conjecture, that there are only a finite number of Morita equivalence classes of blocks with a given defect group, holds for the blocks of the symmetric groups. This method has led in various different directions. It was generalized by Puig [P1] to demonstrate not only Morita equivalences but also the more restrictive Puig equivalences, thus establishing Puig’s conjecture, that there are only a finite number of Puig equivalence classes for a given defect group, for blocks of the symmetric group. A variant was adapted by the first author to prove Donovan’s conjecture for blocks of the Schur covers of the symmetric and alternating groups, [K]. A related technique was used in [Jo] for blocks of the general linear group, and an adaptation of the method was developed in [HK1], [HK2] to find Morita equivalences between blocks in various other algebraic groups. The method also lead Rickard to a way of demonstrating derived quivalences between blocks of symmetric groups, and this method was then taken up by Chuang and Rouquier [ChR] to show that for a given weight there is only one derived equivalence class of symmetric blocks which, along with [ChK], settled the Broue conjecture for symmetric blocks. In this paper we intend to return to [K] and show that, in fact, the results therein reflected only half of the picture. The results in [K] demonstrated the existence of Morita equivalences between blocks of the covering groups S̃n of Sn or between blocks of the covering groups Ãn of An. We will now reconsider the situation and show that we can equally well get “crossovers” between blocks of Ãn and S̃n. More specifically, the various characters are associated with strict partitions of n and the Morita eqivalences are obtained by an involution which is a variant of the Scopes involution used in Scopes’ original work. The cases treated in [K] were those in which the involution is parity-preserving, and in this paper we will be interested in cases where it is parity-reversing.

Crossover Morita equivalences for blocks of the covering groups of the symmetric and alternating groups