A combinatorial proof of Klyachko’s Theorem on Lie representations

Let L be a free Lie algebra of finite rank r over an arbitrary field K of characteristic 0, and let Ln denote the homogeneous component of degree n in L . Viewed as a module for the general linear group GL(r, K ), Ln is known to be semisimple with the isomorphism types of the simple summands indexed by partitions of n with at most r parts. Klyachko proved in 1974 that, for n > 6, almost all such partitions are needed here, the exceptions being the partition with just one part, and the partition in which all parts are equal to 1. This paper presents a combinatorial proof based on the Littlewood-Richardson rule. This proof also yields that if the composition multiplicity of a simple summand in Ln is greater than 1, then it is at least 6 − 1.