# 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