Eine Symmetrieeigenschaft von Solomons Algebra und der höheren Lie-Charaktere

Abstract

We prove here three results in chain: the result of Section 2 is a symmetry property of the higher Lie characters ofS 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 ϕ,ωof the Solomon descent algebra ofS n , one hasc( ϕ)(ω) =c(ω ϕ), wherec is the Solomon mapping from this algebra to the space of central functions onS 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.