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A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions. (English) Zbl 1291.05206
Summary: We introduce a new basis of the algebra of non-commutative symmetric functions whose images under the forgetful map are Schur functions when indexed by a partition. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions. We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts.

MSC:
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
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