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 L n denote the homogeneous component of degree n in L. Viewed as a module for the general linear group GL(r,K), L n 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...[Show more]
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