Vertices of Lie Modules

2013/09/09 by R. M. Bryant, Roger M. Bryant, Susanne Danz +6
Mathematics · #20C20 #20C30 #20G43 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #math.RT #msc:20C20 #msc:20C30 #msc:20G43

paper · pdf · doi:10.48550/arxiv.1309.2085

26 pages

arxiv created 2013/09/09 · openalex publication_date 2013/09/09 · arxiv updated 2013/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• The non-projective part of the Lie module for the symmetric group
• A new lower bound on the vertices of Specht modules for the symmetric groups
• On a curious variant of the S_n-module Lie_n
• The Lie module of the symmetric group
• Variations on the S_n-module Lie_n