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We introduce a new basis of the non-commutative symmetric functions whosecommutative images are Schur functions. Dually, we build a basis of thequasi-symmetric functions which expand positively in the fundamentalquasi-symmetric functions and decompose Schur functions. We then use the basisto construct a non-commutative lift of the Hall-Littlewood symmetric functionswith similar properties to their commutative counterparts.

A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions