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Noncommutative symmetric functions. II: Transformations of alphabets. (English) Zbl 0907.05055
In a previous paper these three authors, joined by three more, had developed a theory of noncommutative symmetric functions using the quasi-determinant introduced by Gelfand and Retakh [see I. M. Gelfand, D. Krob, A. Lascoux, B. Leclarc, V. S. Retakh, and J.-Y. Thibon, Adv. Math. 112, No. 2, 218-348 (1995; Zbl 0831.05063)]. They also reinterpreted many elements of the descent algebras of Solomon as noncommutative symmetric functions. In the present work they apply this theory to the combinatorics of free Lie algebras. The idea is that the descent algebra is the noncommutative analog of the character ring, so they extend to the noncommutative case the operations encountered in character computations, such as, for example, the transformation $$A \rightarrow (1-q)A$$ used by Ram in the computation of some character tables. They also study idempotents and nilpotents in descent algebras.

##### MSC:
 500000 Symmetric functions and generalizations
##### Keywords:
noncommutative symmetric functions; descent algebras
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