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Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahedral Groups
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008).

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We consider the graded Hopf algebra $NCSym$ of symmetric functions with non-commutative variables, which is analogous to the algebra $Sym$ of the ordinary symmetric functions in commutative variables. We give formulaes for the product and coproduct on some of the analogues of the $Sym$ bases and expressions for a shuffle product on $NCSym$. We also consider the invariants of the hyperoctahedral group in the non-commutative case and a state a few results. 
@article{DMTCS_2008_special_255_a17,
     author = {Bergeron-Brlek, Anouk},
     title = {Invariants in {Non-Commutative} {Variables} of the {Symmetric} and {Hyperoctahedral} {Groups}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)},
     year = {2008},
     doi = {10.46298/dmtcs.3609},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3609/}
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Bergeron-Brlek, Anouk. Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahedral Groups. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008). doi : 10.46298/dmtcs.3609. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3609/

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