A combinatorial proof of Klyachko's Theorem on Lie representations
Date
2006
Authors
Kovacs, L
Stohr, Ralph
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
Abstract
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. 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 L n is greater than 1, then it is at least n6-1.
Description
Keywords
Keywords: Algebra; Computational complexity; Mathematical models; Numerical methods; Problem solving; Theorem proving; Composition multiplicity; Free Lie algebra; General linear group; Littlewood-Richardson rule; Combinatorial switching Free Lie algebra; General linear group; Littlewood-Richardson rule
Citation
Collections
Source
Journal of Algebraic Combinatorics
Type
Journal article
Book Title
Entity type
Access Statement
License Rights
Restricted until
2037-12-31
Downloads
File
Description