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Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahedral Groups

Abstract : 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.
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https://hal.inria.fr/hal-01185143
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Anouk Bergeron-Brlek. Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahedral Groups. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.653-664, ⟨10.46298/dmtcs.3609⟩. ⟨hal-01185143⟩

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