Simple module branching rules for K(Sp×Sp) and KS2p

We study the branching rules between K(Sp×Sp) and KS2p: that is we determine, in characteristic p for p odd, the Loewy structure of the principal block simple K(Sp×Sp)-modules induced to KS2p and the Loewy structure of the principal block simple KS2p-modules restricted to K(Sp×Sp). In characteristic zero the answer follows easily from the Littlewood–Richardson rule, but in other characteristics the structure of the induced and restricted modules is not known. Kleshchev has obtained the simple module branching rules between KSn and KSn+1 for any n and any field characteristic. We provide similar branching rules for our case.