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An Introduction to Quasisymmetric Schur Functions is aimed at researchers and graduate students in algebraic combinatorics. The goal of this monograph is twofold. The first goal is to provide a reference text for the basic theory of Hopf algebras, in particular the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric functions and connections between them. The second goal is to give a survey of results with respect to an exciting new basis of the Hopf algebra of quasisymmetric functions, whose combinatorics is analogous to that of the renowned Schur functions.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
A brief history of the Hopf algebra of quasisymmetric functions is given, along with their appearance in discrete geometry, representation theory and algebra. A discussion on how quasisymmetric functions simplify other algebraic functions is undertaken, and their appearance in areas such as probability, topology, and graph theory is also covered. Research on the dual algebra of noncommutative symmetric functions is touched on, as is a variety of extensions to quasisymmetric functions. What is known about the basis of quasisymmetric Schur functions is also addressed.
Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

Chapter 2. Classical combinatorial concepts

Abstract
In this chapter we begin by defining partially ordered sets, linear extensions, the dual of a poset, and the disjoint union of two posets. We then define further combinatorial objects we will need including compositions, partitions, diagrams and Young tableaux, reverse tableaux, Young’s lattice and Schensted insertion.
Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

Chapter 3. Hopf algebras

Abstract
We give the basic theory of graded Hopf algebras, and then illustrate the theory in detail with three examples: the Hopf algebra of symmetric functions, Sym, the Hopf algebra of quasisymmetric functions, QSym, and the Hopf algebra of noncommutative symmetric functions, NSym. In each case we describe pertinent bases, the product, the coproduct and the antipode. Once defined we see how Sym is a subalgebra of QSym, and a quotient of NSym. We also discuss the duality of QSym and NSym and a variety of automorphisms on each. We end by defining combinatorial Hopf algebras and discussing the role QSym plays as the terminal object in the category of all combinatorial Hopf algebras.
Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

Chapter 4. Composition tableaux and further combinatorial concepts

Abstract
In order to state results in the next chapter, we extend many definitions from Chapter 2 to define composition diagrams, Young composition tableaux that correspond to Young tableaux, and the Young composition poset. We additionally define reverse composition diagrams, reverse composition tableaux that correspond to reverse tableaux, and the reverse composition poset. Finally, useful bijections between Young tableaux, Young composition tableaux, reverse tableaux and reverse composition tableaux are described.
Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

Chapter 5. Quasisymmetric Schur functions

Abstract
In this final chapter we introduce two additional bases for the Hopf algebra of quasisymmetric functions. The first is the basis of quasisymmetric Schur functions already in the literature, whose combinatorics is connected to reverse composition tableaux. The second is the new basis of Young quasisymmetric Schur functions whose combinatorics is connected to Young composition tableaux. For each of these bases we determine their expansion in terms of fundamental quasisymmetric functions, monomial quasisymmetric functions and monomials, and see how they refine Schur functions in a natural way. We then, for each basis, describe Pieri rules and define skew analogues, consequently developing a Littlewood-Richardson rule for these skew analogues and the coproduct. Finally via duality, we introduce two new bases for the Hopf algebra of noncommutative symmetric functions, each of which projects onto the basis of Schur functions under the forgetful map. Each of these new bases exhibit Pieri and Littlewood-Richardson rules, which we describe. As with their quasisymmetric counterparts, one basis involves reverse composition tableaux, while the other involves Young composition tableaux.
Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

Backmatter

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