Specht modules with abelian vertices

TL;DR

This paper investigates Specht modules with abelian vertices, establishing their connection to $p^2$-cores and $p$-weights, and classifies such modules under certain conditions, extending previous results and computing specific vertices.

Contribution

It characterizes Specht modules with abelian vertices as $p^2$-cores and classifies them for $p eq 2$, also extending known results on module vertices.

Findings

Specht modules with abelian vertices are $p^2$-cores.

The complexity of certain Specht modules equals their $p$-weight.

Computed vertices of $S^{(p^p)}$ for $p eq 2$.

Abstract

In this article, we consider indecomposable Specht modules with abelian vertices. We show that the corresponding partitions are necessarily $p^2$-cores where $p$ is the characteristic of the underlying field. Furthermore, in the case of $p\geq 3$, or $p=2$ and $\mu$ is 2-regular, we show that the complexity of the Specht module $S^\mu$ is precisely the $p$-weight of the partition $\mu$. In the latter case, we classify Specht modules with abelian vertices. For some applications of the above results, we extend a result of M. Wildon and compute the vertices of the Specht module $S^{(p^p)}$ for $p\geq 3$.