Skew quasisymmetric Schur functions and noncommutative Schur functions

Abstract

Recently a new basis for the Hopf algebra of quasisymmetric functions $QSym$, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset ${\mathcal{L}}_C$ that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions $NSym$. This basis of $NSym$ is dual to the basis of quasisymmetric Schur functions and its elements are the pre-image of the Schur functions under the forgetful map $\chi :NSym\to Sym$. We prove that the multiplicative structure constants of the noncommutative Schur functions, equivalently the coefficients of the skew quasisymmetric Schur functions when expanded in the quasisymmetric Schur basis, are nonnegative integers, satisfying a Littlewood-Richardson rule analogue that reduces to the classical Littlewood-Richardson rule under $\chi$. As an application we show that the morphism of algebras from the algebra of Poirier-Reutenauer to $Sym$ factors through $NSym$. We also extend the definition of Schur functions in noncommuting variables of Rosas-Sagan in the algebra $NCSym$ to define quasisymmetric Schur functions in the algebra $NCQSym$. We prove these latter functions refine the former and their properties, and project onto quasisymmetric Schur functions under the forgetful map. Lastly, we show that by suitably labeling ${\mathcal{L}}_C$, skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets.

Final version. An omission in the statement of the Noncommutative Pieri rules in Corollary 3.8 has been amended

References in corpus (5)

• A Decomposition of Schur functions and an analogue of the Robinson-Schensted-Knuth Algorithm
• Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity
• Equality of multiplicity free skew characters
• Refinements of the Littlewood-Richardson Rule
• Skew Littlewood-Richardson rules from Hopf algebras

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• Row-strict dual immaculate functions
• 0-Hecke modules for row-strict dual immaculate functions
• Markov Chains from Descent Operators on Combinatorial Hopf Algebras
• The decomposition of 0-Hecke modules associated to quasisymmetric Schur functions
• Quasi-symmetric and non-commutative affine Schur functions
• Pieri rules for skew dual immaculate functions
• q-symmetric functions and q-quasisymmetric functions
• Rectification of Composition Tableaux
• Quasisymmetric Schur $Q$-functions and peak Young quasisymmetric Schur functions
• Multiplicity free Schur, skew Schur, and quasisymmetric Schur functions
• Backward jeu de taquin slides for composition tableaux and a noncommutative Pieri rule
• Modules of the 0-Hecke algebra and quasisymmetric Schur functions
• Comparing skew Schur functions: a quasisymmetric perspective