The varieties for some Specht modules

TL;DR

This paper investigates the complexity of Specht modules in relation to the VIGRE's conjecture, providing calculations and results that confirm the conjecture in specific cases and analyze the cohomological varieties involved.

Contribution

It calculates the cohomological variety of Specht modules in abelian defect cases and confirms the VIGRE's conjecture for certain partitions, including p^p.

Findings

Specht modules have complexities equal to p-weights of partitions.

Confirmed the conjecture for partitions with at most p parts and empty p-core.

Determined the complexity of the Specht module for partition p^p as p-1.

Abstract

We concern the VIGRE's conjecture; namely the complexity of a Specht module is the p-weight of the corresponding partition if and only if the partition is not p by p. In abelian defect case, we calculate the cohomological variety of the Specht modules. In particular, we show that the Specht modules have complexities exactly given by the p-weights of the corresponding partitions. For some p-regular partitions not more than p parts with empty p-core, we show that the Specht modules fit into the conjecture. For the partition p^p, we show that the Specht module has complexity p-1 and we study the rank variety of the Specht module restricted to a maximal elementary abelian p-subgroup of rank p.