Varieties for Specht modules

• Main Author: Lim, Kay Jin
• Published: University of Aberdeen 2009
• Subjects: 510; Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.509180

This thesis concerns the conjecture on the complexities of Specht modules made by VIGRE research group in Georgia. We show that the cohomological variety of a Specht module is the cohomological variety of a defect group of the block containing the Specht module if the defect group is abelian. In this case, the complexity of the Specht module is the p-weight of the corresponding partition. We also show that the latter statement holds if the partition is a hook partition or belongs to a large class of p-regular partitions. We give formulae for the generic Jordan types of signed permutation modules re- stricted to elementary abelian p-subgroups generated by disjoint p-cycles. This al- lows us to formulate the generic Jordan type of the Specht module corresponding to a hook partition restricted to elementary abelian p-subgroups generated by m disjoint p-cycles where m is the p-weight of the the hook. In the case where the partition is (pp), we show that the complexity of the Specht module is p-1 and we investigate the rank variety of the restricted module S(pp)↓E where E is a maximal elementary abelian p-subgroup of Sp² of rank p. In particular, we show that the degree of the projectivized rank variety is non-zero and divisible by (p-1)².