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            <title xml:lang="en">Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahedral Groups</title>
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            <idno type="halRefHtml">&lt;i&gt;20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)&lt;/i&gt;, 2008, Viña del Mar, Chile. pp.653-664, &lt;a target="_blank" href="https://dx.doi.org/10.46298/dmtcs.3609"&gt;&amp;#x27E8;10.46298/dmtcs.3609&amp;#x27E9;&lt;/a&gt;</idno>
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                <title xml:lang="en">Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahedral Groups</title>
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                  <title>20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)</title>
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                  <settlement>Viña del Mar</settlement>
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                <editor>Krattenthaler</editor>
                <editor>Christian and Sagan</editor>
                <editor>Bruce</editor>
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                <term xml:lang="en">Hopf algebra</term>
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              <p>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.</p>
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              <p>Nous considérons l'algèbre de Hopf graduée $NCSym$ des fonctions symétriques en variables non-commutatives, qui est analogue à l'algèbre $Sym$ des fonctions symétriques en variables commutatives. Nous donnons des formules pour le produit et coproduit sur certaines des bases analogues à celles de $Sym$, ainsi qu'une expression pour le produit $\textit{shuffle}$ sur $NCSym$. Nous considérons aussi les invariants du groupe hyperoctaédral dans le cas non-commutatif et énonçons quelques résultats.</p>
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