A combinatorial proof of Klyachko's Theorem on Lie representations
L.G. Kovács
and Ralph Stöhr
Mathematics, Australian National University, Canberra ACT 0200, Australia
DOI: 10.1007/s10801-006-7394-6
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 \frac n6 -1 \frac{n}{6}-1 .
Pages: 225–230
Keywords: keywords free Lie algebra; general linear group; Littlewood-Richardson rule
Full Text: PDF
References
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2. Marshall Hall, Jr., “A basis for free Lie rings and higher commutators in free groups,” Proc. Amer. Math. Soc. 1 (1950), 575-581.
3. Torsten Hannebauer and Ralph St\ddot ohr, “Homology of groups with coefficients in free metabelian Lie powers and exterior powers of relation modules and applications to group theory,” in Proc. Second Internat. Group Springer J Algebr Comb (2006) 23: 225-230 Theory Conf. (Bressanone/Brixen, June 11-17, 1989), Rend. Circ. Mat. Palermo (2) Suppl. 23 (1990), 77-113.
4. A.A. Klyachko, “Lie elements in the tensor algebra,” Sibirsk Mat. \check Z. 15 (1974), 1296-1304, 1430 (Russian). English translation: Siberian J. Math. 15 (1974), 914-921 (1975).
5. Witold Kraśkiewicz and Jerzy Weyman, “Algebra of coinvariants and the action of a Coxeter element,” Bayreuth. Math. Schr. No. 63 (2001), 265-284 (Preprint, 1987).
6. Frank Levin, “Generating groups for nilpotent varieties,” J. Austral. Math. Soc. 11 (1970), 108-114.
7. I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Oxford, 1979.
8. Cristophe Reutenauer, Free Lie algebras, Oxford University Press, Oxford, 1993 (London Math. Soc. Monographs, New Ser., Vol. 7).
9. Manfred Schocker, \ddot Uber die h\ddot oheren Lie-Darstellungen der symmetrischen Gruppen, Dissertation, Kiel, 2000; Bayreuth. Math. Schr. No. 63 (2001), 103-263.
10. Manfred Schocker, “Embeddings of higher Lie modules,” J. Pure Appl. Algebra 185 (2003), 279-288.
11. Sheila Sundaram, “Decompositions of Sn-submodules in the free Lie algebra,” J. Algebra 154 (1993), 507-558.
12. R.M. Thrall, “On symmetrized Kronecker powers and the structure of the free Lie ring,” Amer. J. Math. 64 (1942), 371-388.
13. M.R. Vaughan-Lee, “Generating groups of nilpotent varieties,” Bull. Austral. Math. Soc. 3 (1970), 145- 154.
14. F. Wever, “ \ddot Uber Invarianten von Lieschen Ringen,” Math. Ann. 120 (1949), 563-580.
15. V.M. Zhuravlev, “The free Lie algebra as a module over the general linear group,” Mat. Sb. 187(2) (1996), 59-80 (Russian). English translation: Sb. Math. 187(2) (1996), 215-236.