Symmetric functions in superspace form a cocommutative self-dual Hopf algebra; quasisymmetric functions in superspace form a Hopf algebra whose monomial basis has an explicit dual multiplicative basis in the noncommutative dual.
Blondeau-Fournier, P
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Hopf algebra structure of symmetric and quasisymmetric functions in superspace
Symmetric functions in superspace form a cocommutative self-dual Hopf algebra; quasisymmetric functions in superspace form a Hopf algebra whose monomial basis has an explicit dual multiplicative basis in the noncommutative dual.