Algebra and Topology Seminar

4pm Tuesday 26 October 2004
Ralph Stöhr
University of Manchester

On 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 in which all parts are equal to 1 (which only comes into consideration when n < r). We present a combinatorial proof of Klyachko's Theorem which is based on the Littlewood-Richardson rule.




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