# Invariants of Specht Modules

## TL;DR

This paper investigates Hemmer's conjecture about the structure of fixed point modules of Specht modules over symmetric groups in positive characteristic, providing counterexamples for all primes.

## Contribution

It presents the first known counterexamples to Hemmer's conjecture, showing that the fixed point modules do not always have a Specht series.

## Key findings

- Counterexamples exist for all primes p.
- The structure of fixed point modules is more complex than conjectured.
- Different behaviors are observed for p=2, 3, and ≥5.

## Abstract

In [14] Hemmer conjectures that the module of fixed points for the symmetric group $\Sigma_m$ of a Specht module for $\Sigma_n$ (with $n>m$), over a field of positive characteristic $p$, has a Specht series, when viewed as a $\Sigma_{n-m}$-module. We provide a counterexample for each prime $p$. The examples have the same form for $p\geq 5$ and we treat the cases $p=3$ and $p=2$ separately.