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 $\chi: NSym \rightarrow Sym$. 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.Comment: Final version. An omission in the statement of the Noncommutative Pieri rules in Corollary 3.8 has been amende

Skew row-strict quasisymmetric Schur functions

“…where F S is the Gessel fundamental quasisymmetric function indexed by the set S. We provide an analogous combinatorial formula for the skew Young row-strict quasisymmetric Schur functions, similar to that given for skew quasisymmetric Schur functions in [3].…”

Dual graphs from noncommutative and quasisymmetric Schur functions

“…This operator arises in jeu de taquin slides on semistandard reverse composition tableaux and in the right Pieri rules for noncommutative Schur functions [22]. Our fourth operator is the box adding operator t, which plays the same role in the left Pieri rules for noncommutative Schur functions [3] as u does in the right Pieri rules. Each of these operators is defined on weak compositions for every integer i ≥ 0 and we note that d 0 = a 0 = u 0 = t 0 = Id namely the identity map, which fixes the weak composition it is acting on.…”

“…Example 2.5. Let us compute u 4 ((3, 1, 4, 2, 1)) = a 4 d [3] ((3, 1, 4, 2, 1)). [3] ((3, 1, 4, 2, 1)) = (2, 1, 4, 1, 0), and hence u 4 (3, 1, 4, 2, 1) = (2, 1, 4, 1, 0, 4).…”

“…Observe that the relation < q on compositions defined by β < q α if and only if β = d I (α) does not give rise to a poset structure, since transitivity is not satisfied. For example, d {1,4} ((4, 1, 4, 1)) = (4, 1, 3) and d {1,4} ((4, 1, 3)) = (3, 3), but no I exists such that d I ((4, 1, 4, 1)) =(3,3).…”

Rigidity for the Hopf Algebra of Quasisymmetric Functions

“…Let γ be a composition. According to [6,Theorem 3.5], the coproduct in terms of the quasisymmetric Schur basis is defined by…”