A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions. (English) Zbl 1291.05206

Berg, Chris; Bergeron, Nantel; Saliola, Franco; Serrano, Luis; Zabrocki, Mike

Can. J. Math. 66, No. 3, 525-565 (2014).

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

Keywords:

Hall-Littlewood polynomial; symmetric function; quasisymmetric function; tableau