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Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

  Published:2008-04-01
 Printed: Apr 2008
  • 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. The bases for this algebra are indexed by set partitions. We show that there exists a natural inclusion of the Hopf algebra of noncommutative symmetric functions 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 and conclude two analogues of Chevalley's theorem in the noncommutative setting.
MSC Classifications: 16W30, 05A18;, 05E10 show english descriptions Coalgebras, bialgebras, Hopf algebras (See also 16S40, 57T05); rings, modules, etc. on which these act
unknown classification 05A18;
Combinatorial aspects of representation theory [See also 20C30]
16W30 - Coalgebras, bialgebras, Hopf algebras (See also 16S40, 57T05); rings, modules, etc. on which these act
05A18; - unknown classification 05A18;
05E10 - Combinatorial aspects of representation theory [See also 20C30]
 

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