The complexities of some simple modules of the symmetric groups

We show that the simple modules of the Rouquier blocks of symmetric groups, in characteristic $p$ and having $p$-weight $w$ with $w < p$, have a common complexity $w$, and that when $p$ is odd, $D^{(p+1,1^{p-1})}$ has complexity 1, while the other simple modules labelled by a partition having $p$-weight 2 have complexity 2.