TY - JOUR

T1 - Skew quasisymmetric Schur functions and noncommutative Schur functions

AU - Bessenrodt, C.

AU - Luoto, K.

AU - van Willigenburg, Stephanie

N1 - Funding Information: ✩ The second and third authors were supported in part by the National Sciences and Engineering Research Council of Canada. The third author was supported in part by the Alexander von Humboldt Foundation. * Corresponding author. E-mail addresses: bessen@math.uni-hannover.de (C. Bessenrodt), kwluoto@math.ubc.ca (K. Luoto), steph@math.ubc.ca (S. van Willigenburg).

PY - 2011/3/20

Y1 - 2011/3/20

N2 - 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 LC 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 Χ:NSym→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 Χ. 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 LC, skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets.

AB - 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 LC 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 Χ:NSym→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 Χ. 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 LC, skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets.

KW - Composition

KW - Coproduct

KW - Free Schur function

KW - Littlewood-Richardson rule

KW - NCQSym

KW - NCSym

KW - Noncommutative symmetric function

KW - Pieri operator

KW - Pieri rule

KW - Poset

KW - Primary

KW - Quasisymmetric function

KW - Secondary

KW - Skew Schur function

KW - Symmetric function

KW - Tableau

UR - http://www.scopus.com/inward/record.url?scp=79551605312&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2010.12.015

DO - 10.1016/j.aim.2010.12.015

M3 - Article

AN - SCOPUS:79551605312

VL - 226

SP - 4492

EP - 4532

JO - Advances in mathematics

JF - Advances in mathematics

SN - 0001-8708

IS - 5

ER -