Quasisymmetric Schur functions

Abstract

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the $t$ parameter from Hall-Littlewood theory.

30 pages; references added; new subsections on transition matrices, how to include the $t$ parameter from Hall-Littlewood theory and further avenues; new survey of combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions

References in corpus (6)

• Fractional Quantum Hall States and Jack Polynomials
• Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
• Non-commutative Pieri operators on posets
• Skew quasisymmetric Schur functions and noncommutative Schur functions
• A Decomposition of Schur functions and an analogue of the Robinson-Schensted-Knuth Algorithm
• A matroid-friendly basis for the quasisymmetric functions

Cited by in corpus (35)

• Skew quasisymmetric Schur functions and noncommutative Schur functions
• A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions
• Kohnert tableaux and a lifting of quasi-Schur functions
• Set-Valued Skyline Fillings
• Non-symmetric Macdonald polynomials and Demazure-Lusztig operators
• Multiplicative structures of the immaculate basis of non-commutative symmetric functions
• Row-strict Quasisymmetric Schur Functions, Characterizations of Demazure Atoms, and Permuted Basement Nonsymmetric Macdonald Polynomials
• Row-strict dual immaculate functions
• Newton Polytopes in Algebraic Combinatorics
• The Pieri rule for dual immaculate quasi-symmetric functions
• $P$-partitions and $p$-positivity
• Polynomials from combinatorial $K$-theory
• Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$
• Asymmetric function theory
• The genomic Schur function is fundamental-positive
• 0-Hecke modules for row-strict dual immaculate functions
• The projective cover of tableau-cyclic indecomposable $H_n(0)$-modules
• Refinements of the Littlewood-Richardson Rule
• Poset modules of the $0$-Hecke algebras and related quasisymmetric power sum expansions
• A Note on Jing and Li's Type B Quasisymetric Schur Functions
• Rigidity for the Hopf algebra of quasi-symmetric functions
• Quasi-symmetric and non-commutative affine Schur functions
• Representation stability for sequences of 0-Hecke modules
• Extended Schur functions and bases related by involutions
• Positivity among P-partition generating functions
• Pieri rules for skew dual immaculate functions
• A generalization of the dual immaculate quasisymmetric functions in partially commutative variables
• QSym over Sym has a stable basis
• Comparing skew Schur functions: a quasisymmetric perspective
• Quasisymmetric Schur $Q$-functions and peak Young quasisymmetric Schur functions
• Skew row-strict quasisymmetric Schur functions
• Renormalization of quasisymmetric functions
• Rectification of Composition Tableaux
• The "Young" and "reverse" dichotomy of polynomials
• Composition Tableaux basis for Schur functors and the Plücker algebra