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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 thatD    (   p  +  1  ,   1   p  ‐  1)has complexity 1, while the other simple modules have complexity 2.

The complexities of some simple modules of the symmetric groups