The complexity of the Specht modules corresponding to hook partitions

TL;DR

This paper investigates the complexity of Specht modules for hook partitions, establishing that their complexity equals the p-weight of the partition by analyzing the generic Jordan type upon restriction to certain elementary abelian p-subgroups.

Contribution

It provides a precise computation of the complexity of Specht modules for hook partitions, linking it directly to the p-weight, which was previously not explicitly established.

Findings

The complexity of Specht modules for hook partitions equals their p-weight.

The generic Jordan type is computed for modules restricted to elementary abelian p-subgroups.

The result clarifies the relationship between module complexity and partition p-weight.

Abstract

Let w be the p-weight of a hook partition \mu and E be an elementary abelian p-subgroup generated by w disjoint p-cycles. We compute the generic Jordan type of the Specht module corresponding to the partition \mu restricted to E. In particular, we show that the complexity of the Specht module is precisely the p-weight of the partition \mu.