2010/12/31 by Karin Erdmann, Erdmann, Karin, Kai Meng Tan +1
Mathematics · #20C30 #20G43 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:20C30 #msc:20G43
paper · pdf · doi:10.48550/arxiv.1101.0254
arxiv created 2010/12/31 · openalex publication_date 2010/12/31 · arxiv updated 2011/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Lie module of the group algebra FSn of the symmetric group is known to be not projective if and only if the characteristic p of F divides n. We show that in this case its non-projective summands belong to the principal block of FSn. Let V be a vector space of dimension m over F, and let Ln(V) be the n-th homogeneous part of the free Lie algebra on V; this is a polynomial representation of GLm(F) of degree n, or equivalently, a module of the Schur algebra S(m,n). Our result implies that, when m ≥ n, every summand of Ln(V) which is not a tilting module belongs to the principal block of S(m,n), by which we mean the block containing the n-th symmetric power of V.