Asymptotic behaviour of Lie powers and Lie modules

2010/09/06 by Roger M. Bryant, Kay Jin Lim, Bryant, Roger M. +3
Mathematics · #17B01 #20C20 (secondary) #20C30 (primary) #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B01 #msc:20C20 #msc:20C30

paper · pdf · doi:10.48550/arxiv.1009.0974

9 pages

arxiv created 2010/12/03 · arxiv updated 2010/12/06

Abstract

Let V be a finite-dimensional FG-module, where F is a field of prime characteristic p and G is a group. We show that, when r is not a power of p, the Lie power L^r(V) has a direct summand B^r(V) which is a direct summand of the tensor power V^{⊗ r} and which satisfies dim B^r(V)/dim L^r(V) → 1 as r → ∞. Similarly, for the same values of r, we obtain a projective submodule C(r) of the Lie module \Lie(r) over F such that dim C(r)/dim \Lie(r) → 1 as r → ∞.

Related

• The Decomposition of Lie Powers
• Factorisation of Lie Resolvents
• Asymptotic Growth of Trivial Summands in Tensor Powers
• Lie powers of the natural module for GL(2,K)
• Asymptotic properties of tensor powers in symmetric tensor categories