The complexities of some simple modules of the symmetric groups

We determine the complexities of the simple modules lying in two well-known classes of blocks of symmetric group algebras in characteristic p, namely the Rouquier blocks of p-weight w, with w < p, and the defect 2 blocks. For the former, we show that the simple modules have complexity w. For the latter, we show that D-(p+1,D-1p-1) has complexity 1, while the other simple modules have complexity 2.