Geodesic
A combinatorial proof of Klyachko's theorem on Lie representations.
Journal of Algebraic Combinatorics, Tome 23 (2006) no. 3, pp. 225-230.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $\frac n6$ -1 fracn6-1 .
Keywords: keywords free Lie algebra, general linear group, Littlewood-Richardson rule 
@article{JAC_2006__23_3_a3,
     author = {Kov\'acs, L.G. and St\"ohr, Ralph},
     title = {A combinatorial proof of {Klyachko's} theorem on {Lie} representations.},
     journal = {Journal of Algebraic Combinatorics},
     pages = {225--230},
     publisher = {mathdoc},
     volume = {23},
     number = {3},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2006__23_3_a3/}
}
TY  - JOUR
AU  - Kovács, L.G.
AU  - Stöhr, Ralph
TI  - A combinatorial proof of Klyachko's theorem on Lie representations.
JO  - Journal of Algebraic Combinatorics
PY  - 2006
SP  - 225
EP  - 230
VL  - 23
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JAC_2006__23_3_a3/
LA  - en
ID  - JAC_2006__23_3_a3
ER  - 
%0 Journal Article
%A Kovács, L.G.
%A Stöhr, Ralph
%T A combinatorial proof of Klyachko's theorem on Lie representations.
%J Journal of Algebraic Combinatorics
%D 2006
%P 225-230
%V 23
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JAC_2006__23_3_a3/
%G en
%F JAC_2006__23_3_a3
Kovács, L.G.; Stöhr, Ralph. A combinatorial proof of Klyachko's theorem on Lie representations.. Journal of Algebraic Combinatorics, Tome 23 (2006) no. 3, pp. 225-230. http://geodesic.mathdoc.fr/item/JAC_2006__23_3_a3/