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Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

Nantel Bergeron·Christophe Reutenauer·Mercedes Rosas·Mike Zabrocki
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Abstract

We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions which was recently studied as a vector space by Rosas and Sagan. The bases for this algebra are indexed by set partitions. We show that there exist a natural inclusion of the Hopf algebra of noncommutative symmetric functions indexed by compositions in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommutative polynomials.

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Nantel Bergeron, Christophe Reutenauer, Mercedes Rosas, Mike Zabrocki. 2005-02-04. Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables. https://doi.org/10.4153/cjm-2008-013-4

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