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arXiv · 1804.01762

A noncommutative cycle index and new bases of quasi-symmetric functions and noncommutative symmetric functions

Jean-Christophe Novelli·Jean-Yves Thibon·Frederic Toumazet
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Abstract

We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then projecting the coefficients to $QSym$. By duality, we obtain a basis of noncommutative symmetric functions, for which a product formula and a recurrence in the form of a combinatorial complex are obtained. This basis allows to identify noncommutative symmetric functions with the quotient of FQSym induced by the pattern-replacement relation $321 \equiv 231$ and $312 \equiv 132$.

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Jean-Christophe Novelli, Jean-Yves Thibon, Frederic Toumazet. 2019-03-26. A noncommutative cycle index and new bases of quasi-symmetric functions and noncommutative symmetric functions. https://arxiv.org/abs/1804.01762

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