Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups

Abstract

We calculate the (super)decomposition matrix for a RoCK block of a double cover of the symmetric group with abelian defect, verifying a conjecture of the first author. To do this, we exploit a theorem of the second author and Livesey that a RoCK block is Morita superequivalent to a wreath superproduct of a certain quiver (super)algebra with the symmetric group . We develop the representation theory of this wreath superproduct to compute its Cartan invariants. We then directly construct projective characters for to calculate its decomposition matrix up to a triangular adjustment, and show that this adjustment is trivial by comparing Cartan invariants.