The Lie module of the symmetric group

TL;DR

This paper establishes an upper bound on the size of the largest projective submodule within the Lie module of a symmetric group over a field of prime characteristic, specifically when the group order is divisible by the prime.

Contribution

It introduces a new upper bound for the dimension of the maximal projective submodule of the Lie module in prime characteristic, under specific divisibility conditions.

Findings

Derived an explicit upper bound for the dimension.

Focused on symmetric groups with order divisible by prime p.

Provides insights into the structure of Lie modules in modular representation theory.

Abstract

We provide an upper bound for the dimension of the maximal projective submodule of the Lie module of the symmetric group of $n$ letters in prime characteristic $p$, where $n = pk$ with $p \nmid k$.