Manfred Schocker:

On the Root Symmetry of Higher Lie Characters

The higher Lie characters $\lambda_p$ of the symmetric group $S_n$ arise naturally from the Poincaré-Birckhoff-Witt basis of the free associative algebra. They are indexed by the partitions $p$ of $n$ and sum up to the regular character of $S_n$. It is shown that the sum of the values of $\lambda_p$ on the $m$-th roots of the elements $\pi\in S_n$ with cycle partition $q$ is symmetric in $p$ and $q$, for all partitions $p$ and $q$ of $n$. This root symmetry of the Lie character table reduces to the well-known symmetry property due to Gessel e.a. in the special case of $m=1$. For $q=1.1\ldots 1$, it implies that the $m$-th root number function of $S_n$ is a sum of higher Lie characters and hence is itself a character - a result due to Scharf. More generally, it follows that the image of any higher Lie character under the adjoint Adams operator for the inner plethysm is again a character of $S_n$. Further applications are obtained concerning symmetry properties of Solomon's algebra generalizing the one discovered by J\"ollenbeck and Reutenauer.

Mathematics Subject Classification (1991): 20C30*, 05E05, 17B01

Bibliographical note: Archive Math. 80, No. 4 (2003), 337--346

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[Thu Feb 19 18:56:35 2009]

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