5. Quasisymmetric Schur functions

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.