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

Abstract

We introduce a new basis of the non-commutative symmetric functions whose commutative images are Schur functions. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur 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.

new version includes edits, references to forthcoming research, and a 'hook-length' formula for the number of standard immaculate tableaux

References in corpus (2)

• Quasisymmetric Schur functions
• Skew quasisymmetric Schur functions and noncommutative Schur functions

Cited by in corpus (18)

• Multiplicative structures of the immaculate basis of non-commutative symmetric functions
• A lift of Schur's Q-functions to the peak algebra
• Row-strict dual immaculate functions
• The Pieri rule for dual immaculate quasi-symmetric functions
• Petrie symmetric functions
• Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions
• Structure Constants for Immaculate Functions
• 0-Hecke modules for row-strict dual immaculate functions
• Poset modules of the $0$-Hecke algebras and related quasisymmetric power sum expansions
• Representation theory of 0-Hecke-Clifford algebras
• A Note on Jing and Li's Type B Quasisymetric Schur Functions
• Positivity among P-partition generating functions
• Extended Schur functions and bases related by involutions
• Pieri rules for skew dual immaculate functions
• Equivalence classes of lower and upper descent weak Bruhat intervals
• A generalization of the dual immaculate quasisymmetric functions in partially commutative variables
• A classification of nonzero skew immaculate functions
• Quasisymmetric Schur $Q$-functions and peak Young quasisymmetric Schur functions