Sch\"utzenberger's factorization on the (completed) Hopf algebra of q-stuffle product
Abstract
In order to extend the Sch\"utzenberger's factorization, the combinatorial Hopf algebra of the q-stuffles product is developed systematically in a parallel way with that of the shuffle product and and in emphasizing the Lie elements as studied by Ree. In particular, we will give here an effective construction of pair of bases in duality.
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