The varieties for some Specht modules

J. Carlson introduced the cohomological and rank variety for a module over a finite group algebra. We give a general form for the largest component of the variety for the Specht module for the partition (pp) of p2 restricted to a maximal elementary abelian p-subgroup of rank p. We determine the varieties of a large class of Specht modules corresponding to p-regular partitions. To any partition μ of np of not more than p parts with empty p-core we associate a unique partition Φ(μ) of np, where the rank variety of the restricted Specht module Sμ↓En to a maximal elementary abelian p-subgroup En of rank n is if and only if . In some cases where Φ(μ) is a 2-part partition, we show that the rank variety is . In particular, the complexity of the Specht module Sμ is n.