Armin Jöllenbeck, Christophe Reutenauer:

Eine Symmetrieeigenschaft von Solomons Algebra und der höheren Lie-Charaktere

We prove here three results in chain: the result of Section 2 is a symmetry property of the higher Lie characters of S_n (which are indexed by partitions): their character table is essentially symmetric, up to well-known factors. This is established using plethystic methods in the algebra of symmetric functions. In Section 3, we show that for any elements varphi, psi of the Solomon descent algebra of S_n one has c(varphi)(psi)=c(psi)(varphi), where c is the Solomon mapping from this algebra to the space of central functions on S_n (implicitly extended to its group algebra). We address also the question whether this is true for each finite Coxeter group. Then in the last section, we deduce a new proof of a result of Gessel and the second author that gives the number of permutations with given cycle type and descent set as scalar product of two special characters.

Mathematics Subject Classification (1991): 05E10, 20C30, 17B01, 17B35

Bibliographical note: Abh. Math. Sem. Univ. Hamburg 71 (2001), 105-111

Keywords: freie Lie-Algebra, Poincare-Birkhoff-Witt-Basis, Darstellung, symmetrische Gruppe, Solomons Algebra

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[Thu Feb 19 18:56:35 2009]

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