Vertices of Lie Modules

TL;DR

This paper investigates the structure of Lie modules over symmetric groups in positive characteristic, focusing on their indecomposable summands, vertices, and sources, with specific results for certain cases and a general reduction to prime powers.

Contribution

It reduces the problem of classifying indecomposable summands of Lie modules to the case where n is a prime power, providing explicit results for specific instances.

Findings

Explicit description of vertices and sources for n=9, p=3

Explicit description for n=8, p=2

Reduction of the problem to prime power cases

Abstract

Let Lie(n) be the Lie module of the symmetric group S_n over a field F of characteristic p>0, that is, Lie(n) is the left ideal of FS_n generated by the Dynkin-Specht-Wever element. We study the problem of parametrizing non-projective indecomposable summands of Lie(n), via describing their vertices and sources. Our main result shows that this can be reduced to the case when n is a power of p. When n=9 and p=3, and when n=8 and p=2, we present a precise answer. This suggests a possible parametrization for arbitrary prime powers.