On the Free Lie Algebra and Cyclic Characters of Symmetric Groups

概要

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The cyclic characters of the symmetric groups are discussed. It is known that some special cyclic characters give the character of the Lie representation of S_n, which is afforded by an S_n-modlue defined by the Klyachko idempotent. The purpose of this article is to construct such idempotents for all the cyclic characters except a single particular case. H. O. Foulkes shows that the characteristics of the cyclic characters are described by the Ramanujan sums. On the other hand, A. Garsia consider the case corresponding to the Lie representation, and show that it can be computed by enumerating the weights of Lyndon words. The Lyndon words are known to be describing the homogeneous basis of the free Lie algebra. In this article, applying Garsia's argument, we compute the characteristics of any other cyclic characters by enumerating the weights of some kind of words.
東海大学の論文
2004-03-30